Grade 9 Applied: Content and Reporting Targets · ... related to an experiment, using ... areas of composite plane figures (e.g., combinations of rectangles, triangles, parallelograms,

TIPS: Section 3 – Grade 9 Applied: Introductory Unit © Queen’s Printer for Ontario, 2003 Page 1

Grade 9 Applied: Content and Reporting Targets

Across the strands and the terms Problem Solving, Reasoning, Communicating, Technology, and Computing - expectations to be applied to any/all content clusters.

Introductory Unit Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Analytic Geometry* • Graphs on the

xy-plane • Hand-drawn graphs

Number Sense and Algebra* • Fractions, integer

and equation concepts

Relationships* • Numerical and

graphical models Measurement and Geometry* • Optimization • Composite 2-D

shapes

Number Sense and Algebra* • Powers Relationships* • Linear vs. non-

linear graphs

Number Sense and Algebra* • Equation solving in

context Relationships* • Linear relationships Analytic Geometry* • Rate of

change/slope • Initial conditions • Graphs of linear

relationships using rate of change and initial conditions

• Equations of linear relationships in context

Number Sense and Algebra* • Equation solving in

context Measurement and Geometry* • Plane geometry

concepts

Number Sense and Algebra* • Formalization of

algebraic concepts Relationships* • Connection and

application of algebraic models

Analytic Geometry* • Equations of lines Measurement and Geometry* • Connection and

application of algebraic models

Rationale Using the first 3 or 4 days as introduction: • Accommodate

possible change in class lists and absenteeism in the first week.

• Working on small content targets rather than beginning larger content targets during this period reduces frustration.

• It is worthwhile to take this time to establish positive classroom norms of behaviour and cooperative learning routines.

• This is an opportunity to help students develop a positive view of secondary school mathematics and themselves as mathematics learners.

Positioning Measurement first: • Measurement

activities provide the opportunity for authentic tasks that appeal to kinesthetic learners, appropriate at the beginning of the Grade 9 Applied program, when students are making the transition to secondary school and before algebraic skills are well developed.

• Teachers can observe students’ reasoning, representing, and problem-solving skills in contexts that can be illustrated with concrete materials and visual representations.

• Optimization problems provide meaningful contexts for use of numerical and graphical models and for combining fractions with integers and equations.

Progression from Unit 1 and setting the stage for Unit 3: • Students build on

inquiry skills developed and extended in Unit 1.

• Students formalize the vocabulary used to describe relationships, to make predictions and to look for trends, using given data sets.

• Contextual data illustrates linear vs. non-linear relationships. Some non-linear relationships introduce the context for work with powers.

• Contexts added to class discussions can be referenced in Units 3 and 5 to clarify questions about abstractions with specific examples.

Care in introducing abstraction: • The introduction of

the abstract concept of a line should be tied to meaningful contexts.

• A wide variety of contexts ensures depth of understanding and should appeal to the wide range of interests of students.

• Students will form equations of lines out of context in Unit 5.

• Students develop an understanding of the connections between ‘rate of change’ in context and slope of the linear graph that represents the context.

Connections: • ‘Rate’ and ‘initial

value’/‘slope’ and ‘y-intercept’

• The formula for the relationship between the two variables in the context and the line that illustrates the relationship between the variables.

Positioning Geometry here: • Teachers will have

had an opportunity to establish appropriate behaviour and care in use of technology.

• Visual and hands-on activities provide variety that appeal to students with different learning styles.

Inclusion of instructional technology options: • Depending on the

availability of the school’s technological resources, teachers may choose to use: - a full lab - a limited number

of computers - teacher

demonstration

Content of this Unit: • Contextual

problems introduce the algebraic skills of polynomials.

• Students consolidate all of their skills and concepts from the course.

• Connections across units help students develop increased understanding.

• Students revisit algebraic manipulation in Grade 10 Applied Mathematics. Spend time connecting algebraic models to context rather than on developing algebraic proficiency.

Connections: • Equations and

graphs of lines • Abstract form

y = mx + b and equations of relationships studied in Unit 3

* Strands for reporting purposes.

Appendix: Curriculum Expectation Clusters


* Expectations that require that students be given the opportunity to learn through inquiry. Learning through problem solving is also recommended for most other curriculum expectations.

Grade 9 Applied: Number Sense and Algebra Introductory Unit Unit 1 Unit 2 Unit 3 Unit 4 Unit 5

Across the strands and the terms Problem Solving, Reasoning, Communicating, Technology, and Computing – expectations to be applied to any/all content clusters NA1.01 – determine strategies for mental mathematics and estimation, and apply these strategies throughout the course; NA1.02 – demonstrate facility in operations with integers, as necessary to support other topics of the course; NA1.03 – demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course; NA1.04 – use a scientific calculator effectively for applications that arise throughout the course; NA1.05 – judge the reasonableness of answers to problems by considering likely results within the situation described in the problem; NA1.06 – judge the reasonableness of answers produced by a calculator, a computer, or pencil and paper, using mental mathematics and estimation; NA2.02 – substitute into and evaluate algebraic expressions involving exponents, to support other topics of the course; NA3.04 – calculate sides in right triangles, using the Pythagorean theorem, as required in topics throughout the course; NA4.01 – use algebraic modelling as one of several problem solving strategies in various topics of the course; NA4.02 – compare algebraic modelling with other strategies used for solving the same problem; NA4.03 – communicate solutions to problems in appropriate mathematical forms and justify the reasoning used in solving the problems.

Combine fraction concepts with integers and equations NA2.01 – evaluate numerical expressions involving natural number exponents with rational number bases; NA3.03 – solve first degree equations, excluding equations with fractional coefficients, using an algebraic method; NA3.05 – substitute into measurement formulas and solve for one variable, with and without the help of technology.

Powers NA2.01 – evaluate numerical expressions involving natural number exponents with rational number bases; NA2.03 – *determine the meaning of negative exponents and of zero as an exponent from activities involving graphing, using technology, and from activities involving patterning; NA2.04 – represent very large and very small numbers, using scientific notation; NA2.05 – enter and interpret exponential notation on a scientific calculator, as necessary in calculations involving very large and very small numbers; NA2.06 – *determine, from the examination of patterns, the exponent rules for multiplying and dividing monomials and the exponent rule for the power of a power, and apply these rules in expressions involving one variable; NA3.05 – substitute into measurement formulas and solve for one variable, with and without the help of technology.

Equation-solving in Context NA3.03 – solve first degree equations, excluding equations with fractional coefficients, using an algebraic method.

Equation-solving in Context NA3.03 – solve first degree equations, excluding equations with fractional coefficients, using an algebraic method.

Formalization of algebraic concepts during consolidation and review of course content NA3.01 – add and subtract polynomials, and multiply a polynomial by a monomial; NA3.02 – expand and simplify polynomial expressions involving one variable; NA3.03 – solve first degree equations, excluding equations with fractional coefficients, using an algebraic method.



Grade 9 Applied: Relationships

Introductory Unit Unit 1 Unit 2 Unit 3 Unit 4 Unit 5

Across the strands and the terms Problem Solving, Reasoning, Communicating, Technology, and Computing – expectations to be applied to any/all content clusters RE1.01 – pose problems, identify variables, and formulate hypotheses associated with relationships; RE1.02 – demonstrate an understanding of some principles of sampling and surveying and apply the principles in designing and carrying out experiments to investigate the relationships between variables; RE1.03 – collect data, using appropriate equipment and/or technology; RE1.04 – organize and analyse data, using appropriate techniques and technology; RE1.05 – describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain the differences between the inferences and the hypotheses; RE1.06 – communicate the findings of an experiment clearly and concisely, using appropriate mathematical forms, and justify the conclusions reached; RE1.07 – solve and/or pose problems related to an experiment, using the findings of the experiment.

Numerical and graphical models RE2.02 – *construct tables of values and scatter plots for linearly related data involving direct variation collected from experiments; RE2.04 – construct tables of values and graphs to represent non linear relations derived from descriptions of realistic situations.

Linear vs. Non-linear Graphs RE2.03 – determine the equation of a line of best fit for a scatter plot, using an informal process; RE2.05 – demonstrate an understanding that straight lines represent linear relations and curves represent non linear relations; RE3.02 – describe, in written form, a situation that would explain the events illustrated by a given graph of a relationship between two variables; RE3.03 – identify, by calculating finite differences in its table of values, whether a relation is linear or non linear.

Linear Relationships RE2.01 – construct tables of values, graphs, and formulas to represent linear relations derived from descriptions of realistic situations involving direct and partial variation; RE2.02 – construct tables of values and scatter plots for linearly related data involving direct variation collected from experiments; RE2.03 – determine the equation of a line of best fit for a scatter plot, using an informal process; RE2.05 – demonstrate an understanding that straight lines represent linear relations and curves represent non linear relations; RE3.01 – determine values of a linear relation by using the formula of the relation and by interpolating or extrapolating from the graph of the relation; RE3.02 – describe, in written form, a situation that would explain the events illustrated by a given graph of a relationship between two variables; RE3.03 – identify, by calculating finite differences in its table of values, whether a relation is linear or non linear; RE3.04 – describe the effect on the graph and the formula of a relation of varying the conditions of a situation they represent.

Connection and application of algebraic models using Relationships contexts




Grade 9 Applied: Measurement and Geometry

Introductory Unit Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Across the strands and the terms Problem Solving, Reasoning, Communicating, Technology, and Computing – expectations to be applied to any/all content clusters MG3.04 – communicate the findings of investigations, using appropriate language and mathematical forms; MG2.04 – judge the reasonableness of answers to measurement problems by considering likely results within the situation described in the problem; MG2.05 – judge the reasonableness of answers produced by a calculator, a computer, or pencil and paper, using mental mathematics and estimation.

Optimization MG1.01 – construct a variety of rectangles for a given perimeter and determine the maximum area for a given perimeter; MG1.02 – construct a variety of square based prisms for a given volume and determine the minimum surface area for a square based prism with a given volume; MG1.03 – construct a variety of cylinders for a given volume and determine the minimum surface area for a cylinder with a given volume; MG1.04 – describe applications in which it would be important to know the maximum area for a given perimeter or the minimum surface area for a given volume. Composite 2-D Shapes MG2.01 – solve problems involving the areas of composite plane figures (e.g., combinations of rectangles, triangles, parallelograms, trapezoids, and circles).

Plane Geometry MG3.01 – illustrate and explain the properties of the interior and exterior angles of triangles and quadrilaterals, and of angles related to parallel lines; MG3.02 – *determine the properties of angle bisectors, medians, and altitudes in various types of triangles through investigation; MG3.03 – determine some properties of the sides and the diagonals of quadrilaterals.

Connection and application of algebraic models using Measurement and Geometry contexts




Grade 9 Applied: Analytic Geometry

Introductory Unit Unit 1 Unit 2 Unit 3 Unit 4 Unit 5

Across the strands and the terms Problem Solving, Reasoning, Communicating, Technology, and Computing – expectations to be applied to any/all content clusters AG3.05 – communicate solutions in established mathematical form, with clear reasons given for the steps taken.

Graphs on the xy-plane AG3.01 – plot points on the xy plane and use the terminology and notation of the xy plane correctly. Hand-drawn graphs AG3.02 – graph lines by hand, using a variety of techniques.

Rate of Change/Slope AG2.01 – identify practical situations illustrating slope; AG2.02 – determine the slope of a line segment, using the formula m = rise/run; AG2.04 – *identify the properties of the slopes of line segments through investigations facilitated by graphing technology, where appropriate. Initial Conditions AG3.01 – plot points on the xy plane and use the terminology and notation of the xy plane correctly. Graphing Linear Relationships Using Rate of Change and Initial Condition AG3.02 – graph lines by hand, using a variety of techniques; AG3.03 – graph lines, using graphing calculators or graphing software. Equations for linear relationships in context AG3.04 – determine the equation of a line, given the slope and y intercept, the slope and a point on the line, and two points on the line.

Equations of Lines AG1.01 – *determine, through investigations, the characteristics that distinguish the equation of a straight line from the equations of non linear relations; AG1.02 – select the equations of straight lines from a given set of equations of linear and non linear relations; AG1.03 – identify y = mx + b as a standard form for the equation of a straight line, including the special cases x = a, y = b; AG2.01 – identify practical situations illustrating slope; AG2.02 – determine the slope of a line segment, using the formula m = rise/run; AG2.03 – identify the geometric significance of m and b in the equation y = mx + b through investigation; AG2.04 – identify the properties of the slopes of line segments through investigations facilitated by graphing technology, where appropriate; AG3.01 – plot points on the xy plane and use the terminology and notation of the xy plane correctly; AG3.02 – graph lines by hand, using a variety of techniques; AG3.03 – graph lines, using graphing calculators or graphing software; AG3.04 – determine the equation of a line, given the slope and y intercept, the slope and a point on the line, and two points on the line; AG3.05 – communicate solutions in established mathematical form, with clear reasons given for the steps taken.


Interpreting the Lesson Outline Template

TIPS: Section III – Grade 9 Applied: Introductory Unit © Queen’s Printer for Ontario, 2003 Page 6

Sequence of Units

Grade Level

Unit 4: Plane Geometry Grade 9 Applied Lesson Outline

BIG PICTURE Students will: •• investigate properties of geometric objects using dynamic geometry software and manipulatives; •• determine the properties of medians, altitudes, and angles bisectors in triangles; •• illustrate and explain the relationship between angles in parallel lines and interior and exterior angles of triangles and

quadrilaterals; •• determine some properties of sides and diagonals of quadrilaterals. Note: Students may have a very broad range of experience with using The Geometer’s Sketchpad®. Skills can be taught as they are needed for each lesson or alternatively the Unit4Tutorial.gsp file could be used at the beginning of the unit.

Day Lesson Title Description Expectations 1 Plane Geometry –

Introduction Lesson included GSP sketches provided (See Unit4Lesson1.gsp)

•• Explore geometrical concepts (angles, triangles, parallel lines)

•• Build skills required for future use of The Geometer’s Sketchpad® (GSP)

Review of Grade 7 and 8 Geometry expectations MG3.01 CGE 2a, 5a

2 What’s So Special? Part 1 Lesson included GSP sketches provided (See Unit4Lesson2.gsp)

•• Build investigation skills by dynamic exploration of geometrical concepts

•• Develop communication skills and geometric vocabulary

MG3.04 CGE 2c, 4b, 5a

3 What’s So Special? Part 2 Lesson included

•• Present observations and conclusions from dynamic geometry software explorations

MG3.04, RE3.04 CGE 2a, 2c

4 Interior and Exterior Angles of Triangles and Quadrilaterals Sample demonstration and investigation sketches are provided (see Unit4Lesson4.gsp files)

•• Through investigation and demonstration examine the sum of the interior and exterior angles of triangles and quadrilaterals using The Geometer’s Sketchpad® (assigned activity from Day 4 prepares students for further investigation in this lesson)

•• Investigate the relationship between the number of sides of a polygon and the sum of the interior angles of the polygon – this is an opportunity to connect the Relationships strand and the Measurement and Geometry Strand

MG3.01, MG3.04, RE1.03, RE1.04, RE1.06, RE2.01

NOTES a) While planning lessons, teachers must judge whether or not pre-made activities support development of big ideas and provide

opportunities for students to understand and communicate connections to the “Big Picture.” b) Catholic Graduation Expectations (CGEs) are included for use by teachers in Catholic schools. c) Consider auditory, kinesthetic, and visual learners in the planning process and create lessons that allow students to learn in

different ways. d) The number of lessons in a unit will vary. e) Grade 9 Applied lessons are based on 75-minute classes. The time/bar graph suggests the fractions of the class to spend on

the Minds On, Action!, and Consolidate/Debrief portions of the class. f) Although some assessment is suggested during each lesson, the assessment is often meant to inform adjustments the teacher

will make to later parts of the lesson or to future lessons. A variety of more formal assessments of student achievement could be added

Download the template at www.curriculum.org/occ/tips/downloads.shtml.

Lessons are planned to help students develop and demonstrate the skills and knowledge detailed in the curriculum expectations. • To help students value and remember the mathematics they learn, each lesson is connected to and focussed on important

mathematics as described in the Big Picture. • Since students need to be active to develop understanding of these larger ideas, each point begins with a verb. • Sample starter verbs: represent, relate, investigate, generate, explore, develop, design, create, connect, apply

• Two or three points to describe the content of this lesson. • Points begin with a verb. • Individual lesson plans elaborate on how objectives are met.

Catholic Graduate Expectations by code for developed lessons

Indicates that a full lesson outline is included

Indicates that this element is provided where there is not a full lesson outline.

List curriculum expectations by code More detail is provided with the

outline if lessons are not included

http://www.curriculum.org/occ/tips/downloads.shtml

Interpreting the Lesson Planning Template

TIPS: Section III – Grade 9 Applied: Introductory Unit © Queen’s Printer for Ontario, 2003 Page 7

Tips for the TeacherThese include: - instructional hints - explanations - background - references to

resources - sample responses

to questions/tasks

Unit 1: Day 4: Down By the Bay Grade 9 Applied

75 min.

Description • Apply the Inquiry Process to a problem that requires an enclosed space on only

three sides • Explore the results for a variety of “What if ?” conditions

Materials grid paper rulers BLM 1.4.1

Assessment Opportunities

Minds On ... Whole Class Discussion Pose the question: If you wanted a rectangular swimming area at the beach, how many sides of the rectangle would you rope off? Explain. Do you suppose this beach swimming area would still be a square?

Action! Small Groups Investigation Distribute BLM 1.4.1. Learning Skills (Works Habits/Initiative)/Observation/Anecdotal: Observe as pairs work through the Explore stage of the investigation. Whole Class Check for Understanding Briefly reconvene the whole class to check for understanding so that all students can proceed with the task from this point. Individual Performance Task Curriculum Expectations/Performance Task/Rubric: Students complete the task independently. (See Assessment Tool 1.4.1.)

Consolidate Debrief

Whole Class Discussion Discuss strategies. Introduce the follow-up questions and facilitate a brief exchange of ideas.

Students responses should consider the effect on the number of possible areas when enclosing an area on three sides. Encourage students to sketch the area. When returning graded work to students, consider photocopying samples of Level 3 and Level 4 responses with student names removed. Select and discuss with the class samples that illustrate a variety of strategies.

Concept Practice Exploration Reflection

Home Activity or Further Classroom Consolidation You answered the question: Which rectangle gives the largest swimming area? Consider how your recommendation would change if you were asked: Which rectangle provides the safest swimming area? or Which rectangle gives the best access to the deeper water?

Same objectives listed in the lesson outline Unit #: Day #: Lesson Title

Grade level

Materials used in the lessonTime column-coded to

the three parts of the day’s lesson.

Suggested student grouping teaching/learning strategy for the activity.

Indicates assessment opportunities - what is assessed/strategy/scoring tool

• Mentally engages students at start of class • Makes connections between different math strands, previous lessons or groups of lessons,

students’ interests, jobs, etc. • Introduces a problem or a motivating activity - orients students to an activity or materials.

Indicates suggested assessment

• “Pulls” out´ the math of the activities and investigations • Prepares students for Home/Further Classroom Consolidation

Focus for the follow-up activity Meaningful and appropriate follow-up to the lesson

• Students do mathematics: reflecting, discussing, observing, investigating, exploring, creating, listening, reasoning, making connections, demonstrating understanding, discovering, hypothesizing

• Teachers listen, observe, respond and prompt

Download the template at www.curriculum.org/occ/tips/downloads.shtml



Introductory Unit Grade 9 Applied Lesson Outline

BIG PICTURE Students will: •• extend plotting of points to all four quadrants; •• graph lines using a starting point and a series of consistent up, down, left, and/or right changes in position; •• begin to use a structure for mathematical inquiry referenced throughout the Grade 9 program.

Day Lesson Title Description Expectations1 From One Point

to Another Lesson included Presentation software file: Points and their Coordinates (p. 14) included

•• Review of plotting points by hand on the xy-plane •• Play a game to reinforce the skill of plotting points

on the xy-plane

AG3.01 CGE 5a

2 Chief Petty Officer’s Dilemma Lesson included

•• Application of plotting points on the xy-plane to solve a problem

•• Introduction of the inquiry process through a context

AG3.01, AG3.02 CGE 2b, 3c

3 Building a Rubric Lesson included

•• Introduction of setting criteria for a task and the design of a rubric

•• Discussion of goal setting

AG3.01, AG3.02 CGE 2c


Introductory Unit: Day 1: From One Point to Another Grade 9 Applied

75 min.

Description • Review of plotting points by hand on the xy-plane • Play a game to reinforce the skill of plotting points on the xy-plane

Materials • Presentation software file: Points and their Coordinates (p. 14) • data projector • BLM 1.1, 1.2


Minds On ... Whole Class Orientation Discuss classroom expectations. Show the electronic presentation: Points and their Co-ordinates to review plotting points in the xy-plane.

Action! Pairs Activate Prior Learning Students complete first three questions of BLM 1.1 to review plotting points in the xy-plane. Circulate and provide help as needed. Curriculum Expectations/Observation/Anecdotal: Assess student understanding of plotting points and movement along the xy-plane using Up/Down; Right/Left to indicate the direction. Pairs Game Students play the game Grid Walking based on the co-ordinate axes. (BLM 1.2.) Curriculum Expectations/Observation/Mental Note: Observe students’ strategies and facility with plotting points and with moving across the co-ordinate axes to determine how often to repeat the process in order to ensure that students are proficient with these skills.

Consolidate Debrief

Whole Class Discussion Discuss strategies that students developed during the game Grid Walking: • Which of the points were better starting position choices? Why? • Once you had chosen a point, what was your criterion for choosing the Grid

Walk directions?

Encourage students to consult with their partner to solve problems before they ask for assistance. The game “Grid Walking” provides an opportunity to become proficient with an essential skill for the investigation on Day 2 in Unit 1.

Application Skill Drill

Home Activity or Further Classroom Consolidation Use the co-ordinates for the picture designed by your partner in question 4 of worksheet 1.1 to plot points and complete the picture.


y

x

1

-1 1 -1

y

x

1

-1 1 -1

y

x

1

-1 1 -1

1.1: Plotting Points

1. Plot each set of points on the grid below. Join the points to form a quadrilateral. Identify the quadrilateral

Set 1: A(1, 1), B(1, 5), C(– 3, 5), D(– 3, 1) Set 2: J(1, – 3), K(5, 1), L(8, 1), M(4, – 3) Set 3: P(– 3, 0), Q(– 6, – 2), R(4, – 4), S(10, 0)

2. Plot these points. Connect the points in order. Name the polygon.

(1, – 1), (2, 1), (1, 3), (– 1, 4), (– 3, 3), (– 4, 1), (–3, –1), (–1, – 2), (1, – 1)

3. Plot these points. Connect the points in order. What picture do you see?

(2, 1), (5, 5), (1, 2), (0, 5), (– 1, 2), (– 5, 5), (– 2, 1), (– 5, 0), (– 2, – 1), (– 5, – 5), (– 1, – 2), (0, – 5), (1, – 2), (5, – 5), (2, – 1), (5, 0), (2,1)

4. Make your own picture. Record the points in order. Exchange your picture code with a classmate and construct each other’s picture.

-5

5

y

x5 -5


1.2: Grid Walking 1. You and your partner need a Grid Walking game board and a score sheet. 2. Each of you chooses one ‘Starting Position’ and one pair of Grid Walk directions. Record

these on the score sheet. 3. On the grid, mark your starting position and move across the grid following the Grid Walk

directions. 4. Keep moving following the Grid Walk directions until you get to an edge or a corner of the

grid. 5. Collect 1 point for each complete “step.” No points are given for partial steps! 6. The next player chooses a new starting position and new Grid Walk directions. 7. Use a different colour to mark each turn on the grid. Example: The starting position is (−3 ,7). The Grid Walk directions are: Down 3, Right 1. This play has 5 complete “steps” and earns 5 points.

y

x 5 -5

5

-5

(-3,7) Starting Position:


1.2: Grid Walking Game Board (continued)

Starting Position Choices Grid Walk Choices (2, −4) (−4, −1) Up 0 Down 0 Right 0 Left 0

(−2, 0) (1, 1) Up 1 Down 1 Right 1 Left 1

(−3, 7) (0, 0) Up 2 Down 2 Right 2 Left 2

(5, −4) (−6, 1) Up 3 Down 3 Right 3 Left 3

(0, 2) (4, −3) Up 4 Down 4 Right 4 Left 4

(4, 2) (−2, −3) Up 5 Down 5 Right 5 Left 5

y

x 5 -5

5

-5


1.2: Grid Walking Score Sheet (continued) Player: Player: Starting Position Grid Walk Points Starting

Position Grid Walk Points


Points and their Coordinates (Presentation software file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

x

yCartesian Plane

y axis

x axisorigin

René Descartes (1596-1650)

Points and their Coordinates

x

y

The Cartesian Plane is divided into fourquadrants.

III

III IV

x

y

Plotting Points in the Cartesian Plane(5, 6) is an example of an ordered pair.

x coordinate

y coordinate (5, 6)

5

6

x

y

Plotting Points in the Cartesian Plane(5, 6) is an example of an ordered pair.

x coordinate

y coordinate (5, 6)

x

y

(x, y)

in the door

up the elevator

It is like entering a hotel …

(– 3, 4)(– 3, 4)

left 3

up 4

x

y

Plot the following points.

A(– 4, 6)A(– 4, 6)B(2, – 3)

B(2, – 3)

C(– 6, – 4)

C(– 6, – 4)

D(7, 3)

D(7, 3)

These points all lie in different quadrants.What do you notice about their coordinates?

x

y

Plot the following points.

F(0, 6) F(0, 6)

E(5, 0)

H(0, – 3)

G(– 7, 0)

G(– 7, 0)H(0, – 3) E(5, 0)

These points all lie on the axes, not in quadrants.What do you notice about their coordinates?



Introductory Unit: Day 2: Chief Petty Officer’s Dilemma Grade 9 Applied

75 min.

Description • Application of plotting points on the xy-plane to solve a problem • Introduction of the inquiry process through a context

Materials • BLM 2.1


Minds On ... Whole Class Discussion Engage students in a discussion about contextual applications of the xy-plane, e.g., sonar, map grid lines, battleship games. Pairs Worksheet Use BLM 2.1, p. 1 to orient students to the investigation through a preview activity that familiarizes them with the context and framework of the investigation. Curriculum Expectations/Observation/Mental Note: Questions 4 and 5 (BLM 2.1) require students to think strategically about movement across the co-ordinate plane within the context of a sonar display. Provide additional examples as necessary.

Action! Think/Pair/Share Understanding the Problem Display Chief Petty Officer’s Dilemma (BLM 2.1, p. 2) on the overhead projector. Identify which pairs are to share their ideas. Allow approximately one minute for students to think individually before they pair with their partners to discuss what will be involved in solving this problem. Prompt groups who are not understanding the problem. Small Groups Game Distribute the group investigation guide (BLM 2.1, p. 3) and the four Sonar Display scenarios to each group (BLM 2.1, p. 4). Students investigate a solution to one of the Sonar Display problems together. Advise them that if they feel unsure about how to start, restarting the problem several times before they arrive at a solution is a reasonable strategy. Remind students that they should rely on the expertise of the members of their group and that there is more than one way to solve the problem and more than one solution. Students decide on the best solution through trial and error.

Consolidate Debrief

Small Groups Sharing Take advantage of having observed the progress of all groups to arrange peer tutoring, as appropriate. Some points may need consolidation with the whole class. Further consolidation of this activity occurs on Day 3 during the oral presentations and rubric design.

The investigation introduces a framework for many lessons – small groups exploring an authentic problem, selecting strategies and providing a rationale for solutions. The process students work through will be common to many of the learning tasks in this course.

Application Home Activity or Further Classroom Consolidation Complete the Memo to the Chief Petty Officer.


2.1: Reading a Sonar Display The Chief Petty Officer is using sonar to help three ships: a Tanker, a Corvair and a Merchant Ship navigate the ocean. The sonar display resembles an xy-plane. 1. The Tanker is located at (3, 4). Plot its position.

After one minute it has moved UP 1 and RIGHT 2. Plot and label the new position. 2. The Corvair is located at (−4, 7). Plot this position with a different coloured pencil or pen.

After one minute it has moved DOWN 1 and RIGHT 0. Plot and label the new position. 3. The Merchant Ship is located at (−2, −4). Plot this position with a different coloured pencil or

pen. After one minute it has moved UP 2 and RIGHT 3. Plot and label the new position.

4. A Submarine appears at (6, −3). Plot this position with a different coloured pencil or pen.

After one minute it has moved to (3, −5). Describe how it travelled to this position.

5. Plot the position of the Tanker after 2 more minutes and label the new location.

MERCHANT SHIP

TANKER CORVAIR

5

5-5

-5

X

Y


2.1: Chief Petty Officer’s Dilemma – The Problem

Large ocean going Merchant Ships refuel and restock their cargo while travelling at sea. A

Tanker carrying fuel and a Corvair carrying food and medical aid are expected to meet the

Merchant Ship in the middle of the ocean, service it with fuel and restock its cargo.

It is the Chief Petty Officer’s (CPO’s) job to ensure that these boats meet at the same time so

that the cargo can be unloaded. The CPO does this by tracking the positions of the three

ships on the sonar display and making recommendations to the captains of the Tanker and

the Corvair, suggesting when and how they should alter their course.

Help the Chief Petty Officer advise the captains of the Tanker and Corvair how they should

manoeuvre their ships.

CORVAIR

MERCHANT SHIP TANKER


2.1: Group Investigation Guide A. Explore the Problem – Whole Class Describe the problem.

What information do you know? What are you asked to find out? What can you "try out?"

B. Model – Partners Model the position and motion of the boats on the sonar display.

How will you keep track of the position of each ship? How will you decide if the ships will meet at the same time?

C. Manipulate – Partners Investigate to determine the number of ways you can alter the Tanker’s or the Corvair’s motion so that they meet the Merchant Ship at the same time. D. Conclude – Partners Choose the "best" plan that ensures all three ships meet at the same time. Write a memo to the Chief Petty Officer with your recommendation. The memo must include: - list of all the choices that you considered; - your recommendation to the chief petty officer that ensures all three ships meet at the same

time; - an explanation indicating why this is the best choice. E. Oral Report – Partners Prepare a short presentation that describes how you solved the CPO’s dilemma.

CORVAIR



2.1: Sonar Displays On the Sonar Display grid, plot the ships’ locations for each display. Use different colours for each sonar display.

Sonar Display Merchant Ship Corvair Tanker

1 The Merchant Ship starts at (9, 0). After one minute it is located 1 UP and 2 LEFT of its starting position.

The Corvair starts at (8, 9). After one minute it is located 3 DOWN and 2 LEFT of its starting position.

The Tanker starts at (−8, 1). After one minute it is located 1 UP and 3 RIGHT of its starting position.

2 The Merchant Ship starts at (9,−9). After one minute it is located 1 UP and 2 LEFT of its starting position.


The Tanker starts at (−6, 3). After one minute it is located 2 DOWN and 2 RIGHT of its starting position.

3 The Merchant Ship starts at (2,−3). After one minute it is located 2 UP and 1 LEFT of its starting position.

The Corvair starts at (−7, −6). After one minute it is located 3 UP and 2 RIGHT of its starting position.

The Tanker starts at (7, 7). After one minute it is located 1 DOWN and 2 LEFT of its starting position.

4 The Merchant Ship starts at (9, 2). After one minute it is located 1 DOWN and 3 RIGHT of its starting position.


The Tanker starts at (3, −7). After one minute it is located 2 UP and 1 LEFT of its starting position.

CORVAIR



2.1: Sonar Display Grid

5

5 -5

-5

X

Y

CORVAIR



Introductory Unit: Day 3: Building a Rubric Grade 9 Applied

75 min.

Description • Introduction of setting criteria for a task and the design of a rubric • Discussion of goal setting

Materials • solutions from Day 2 • BLM 3.1


Minds On ... Whole Class Presentation Curriculum Expectations/Exhibition/Anecdotal: Students from each group orally present solutions to the Chief Petty Officer’s Dilemma focussing on the rationale for their choices. Provide feedback on the group’s problem-solving process and prompt discussion using prompts such as, I noticed that … Can you tell us more about…

Action! Pairs Activate Prior Learning Discuss the differences and similarities between solutions as a starting point for identifying criteria for assessment. Prompt thinking about the problem-solving process by asking: • What did you have to know how to do? • Where would your thinking be most obvious? • What would be important to think about in communicating your findings? Record student responses in a blank rubric template under the heading Criteria in the appropriate Achievement Chart categories (Assessment Tool 3.1). Develop descriptors for Level 3 performance first, then the remaining levels, completing a rubric as you go. (Assessment Tool 3.1) Curriculum Expectations/Self Assessment/Rubric: Provide time for students to assess their solutions and receive a peer assessment. Meet with students informally to review their self-assessments and provide informal feedback comparing their self-assessments with teacher’s observations as well as evidence from the oral presentations.

Consolidate Debrief

Whole Class Discussion Discuss the experience of solving the Chief Petty Officer’s Dilemma and what they would do differently next time. What did you try when you were not sure what to do next? Tell how working with a partner was helpful. Suggest that rubrics are one of many assessment tools that will be used to provide feedback to them about their achievement of the curriculum expectations for the course.

See Section 2 – Mathematical Processes for possible criteria for a rubric and descriptors appropriate for the various levels.

Reflection

Home Activity or Further Consolidation Complete a journal entry reflecting on the investigation using the following prompts: One thing I did well was… Something I need to get better at is… Next time I will…


3.1 Assessment Tool: Chief Petty Officer’s Dilemma

Mathematical Process

(Category) Criteria Below

Level 1 Level 1 Level 2 Level 3 Level 4

Knowing Facts and Procedures (Knowledge/ Understanding)

Plotting points on the co-ordinate axes Using direction statements to move from point to point on the co-ordinate axes

Use an analytic marking scheme.

Reasoning and Proving (Thinking/ Inquiry/Problem Solving)

Completeness of the advice to the captain

- little or no evidence of best advice

- advice with major omissions

- advice with some omissions

- complete advice - complete advice that includes some evidence of reflection on the strategy used to generate the advice

Communicating (Communication)

Clarity of the oral presentation

- unclear/ confusing

- limited clarity - some clarity - clear - precise

Integration of mathematical forms into a narrative in the memo

- message demonstrates little or no integration

- message demonstrated beginning integration

- message demonstrated moderate integration

- message demonstrated well-developed integration

- message demonstrated sophisticated or complete integration

Making Connections (Application)

Appropriateness of strategies selected to determine the “best” advice to give the ship’s captain

- inappropriate for this situation

- limited appropriateness

- moderate appropriateness

- appropriate - highly appropriate

TIPS: Section 3 – Grade 9 Applied: Unit 1 © Queen’s Printer for Ontario, 2003 Page 23

Unit 1: Measurement Relationships Grade 9 Applied Lesson Outline

BIG PICTURE Students will: •• describe relationships between measured quantities; •• connect measurement problems with finding the optimal solution; •• apply knowledge and understanding of 3-D formulas to simple problems in context; •• develop numeric facility in a measurement context; •• work as effective members of learning teams.

Day Lesson Title Description Expectations1 What is the Largest

Rectangle? Lesson included

•• Use an inquiry process to determine the largest rectangle with integral sides that can be constructed for a given perimeter.

Note: Focus on the Explore stage of the Inquiry Process.

NA3.05, NA4.03, MG1.01, MG1.04, MG2.03, AG3.05 CGE 2a, 5a

2 What is the Largest Rectangle Revisited? Lesson included

•• Connect the sum of the side lengths and the perimeter formula.

Note: Focus on the Model and Hypothesize stages of the Inquiry Process.

RE1.01, RE1.03 - 1.07, RE2.04, NA4.03 CGE 4b

3 On Frozen Pond Lesson included

•• Extend fixed perimeter problems to ones that do not have integral solutions.

Note: Focus on the Manipulate/Transform and Infer/Conclude stages of the Inquiry Process.

MG1.01, MG1.04, MG2.03, RE1.01, RE1.03 - 1.07, RE2.04, AG3.05, NA3.05, NA4.03 CGE 5a


Day Lesson Title Description Expectations4 Down by the Bay

Lesson included

•• Apply the Inquiry Process to a problem that requires enclosing a rectangular space on only three sides.

•• Explore the results for a variety of “What if ?” conditions.

MG1.01, MG1.04, MG2.03, RE1.01, RE1.03 - 1.07, RE2.04, AG3.05, NA3.05, NA4.03 CGE 4c

5 Formative Assessment Task Presentation software file: Scatterplots on the Graphing Calculator included (p. 38-39)

•• Manipulate given data and scatterplot for a fixed perimeter context (e.g., a garden plot, fenced area, sandpit).

•• Discover the need to collect more data in the region of the graph where an optimal value occurs.

•• Make inferences and conclusions. •• Provide immediate feedback by having students

input the data in the graphing calculator and graphing a scatterplot.

Learning Skill (Individual Work)/Response Journal/ Anecdotal: •• Assess student understanding using a journal entry

in response to prompts such as: One thing I did well is…or I need… Student pairs create and solve their own area optimization problem then exchange it with another group and solve for homework and include it in a portfolio.

RE1.05 - 1.07, MG1.01, MG1.04, NA4.03, AG3.05

6

7

Optimizing Perimeter

Learning Skill (Teamwork/Initiative)/Observation/ Checkbric and Curriculum Expectations/ Interview/Marking Scheme: •• Investigate the optimal solution when a rectangular

area is fixed and the perimeter must be minimized through problems involving, for example, the cost of fencing or the number of tiles around a pool.

•• Include examples requiring the enclosure of only two or three sides of a rectangular region.

MG2.03, MG2.05, RE1.04, RE1.05 - 1.07, AG3.05, NA1.02, NA2.02


Day Lesson Title Description Expectations8

9

Composite Figures Presentation software file: Composite Figures included (p. 40)

Learning Skill (Work Habits/Individual Work/ Initiative)/Observation/Mental Note: •• Solve composite area problems (e.g., logos, business

signs, irregularly shaped gardens). •• Review formulas for the perimeter and area of a

circle in response to the need to solve problems involving rectangles with semi-circles (e.g., an arch-topped window).

•• Introduce use of the Pythagorean theorem to solve composite rectangle and triangle problems.

•• Complete Extend Your Thinking problem found on p. 16 of Developing Perimeter and Area Formulas (Section 2).

MG2.01, MG2.02 - 2.05, NA1.04, NA2.01, NA3.03, NA3.04, NA3.05

10 Camp Olympics Lesson included

•• Determine optimal dimensions based on factors that include function, volume, and surface area.

•• Use straightforward application of spreadsheets.

MG2.03, RE1.01, RE1.03 - 1.07, RE2.04, AG3.05, NA2.01, NA4.03 CGE 5a

11

12

Optimizing Surface Area – Prisms

Learning Skills (Works Independently/ Organization/Teamwork)/Observation/Anecdotal and Curriculum Expectations/Self-Assessment/ Rating Scale: •• Develop the formulas for surface area and volume of

square-based prisms. •• Investigate an optimal container size for a juice box

and rationalize the manufacturer’s choice. •• Pose an authentic packaging problem that requires

students to investigate the optimal surface area for a fixed volume.

•• Enter formulas into spreadsheets or graphing calculators to facilitate calculations and data manipulation.

MG1.02, MG1.04, NA1.04, NA2.01, NA3.05, NA4.03



14

15

Optimizing Surface Area – Cylinders

Learning Skills (Works Independently/ Organization/Teamwork)/Observation/Checklist and Curriculum Expectations/Interview/Checklist: •• Develop formulas for surface area and volume of

cylinders using concrete materials such as unfolded paper-sided juice cans and stacked metal juice can lids.

•• Investigate commercial packaging of pop and tuna cans, for example, to determine the optimal container size.

•• Pose a cylinder optimization problem involving a fixed volume and investigate the least expensive packaging.

•• Use spreadsheet technology or graphing calculators since they allow students to focus on the investigative process.

MG1.03, MG1.04, NA1.04, NA3.05, NA4.03

16

17

Surface area and volume problems Presentation software file: Volume of a Cylinder included (p. 44) Formative Journal BLM included (p. 45)

Learning Skills (Works Habits/Individual Work/ Initiative)/Checklist and Curriculum Expectations/ Question and Answer (Oral)/Rating Scale: •• Apply surface area and volume formulas to

contextual problems involving volume of prisms and cylinders and surface area of cones and spheres.

•• Complete Mathematical Processes questions found on p. 8 of Developing Perimeter and Area Formulas (Section 2).

Curriculum Expectations/Formative Journal/ Anecdotal Response: •• Provide feedback on accuracy in a and b and on

quality of responses in c and d. •• Complete Developing Proficiency test on p. 11 of

Developing Perimeter and Area Formulas (Section 2).

MG2.02 - 2.05, NA3.05

18 19

Review and Test Sample test included (p. 46-48)

Learning Skills (Work Habits/Organization/ Teamwork)/Observation/Checklist: •• Review knowledge and skills developed to date. Curriculum Expectations/Test/Marking Scheme: Note:

Students should have a formula sheet see TIP 19 and calculator.

NA3.05, MG1.01, MG1.02, MG1.04, MG2.01 - 2.05


Day Lesson Title Description Expectations20 Designing a

Cologne Bottle (Performance Task) Lesson included

Learning Skills (Individual Work)/Observation/ Rating Scale and Curriculum Expectations/ Performance Task/Rubric: •• Minimize surface area for a closed cylinder having

fixed volume. •• Focus on the Thinking Inquiry/Problem Solving and

Communication Categories.

NA1.02, NA1.03, NA1.04, NA2.02, NA3.05, NA4.03, RE1.02 - 1.06, MG1.03, MG1.04, MG2.02, MG2.03, AG3.05 CGE 2b, 2d

21 Review of Unit Test and Performance Task


Unit 1: Day 1: What is the Largest Rectangle? Grade 9 Applied

75 min.

Description • Use an inquiry process to determine the largest rectangle with integral sides

that can be constructed for a given perimeter.

Materials • string, geoboards, dot paper, grid paper • metre sticks, measuring tapes, chart paper • BLM 1.1.1


Minds On… Whole Class Discussion Present the problem: Your neighbour has asked your advice with his garden. He wants to fence the largest rectangular area he can with 60 m of fencing. What advice will you give him? Discuss and clarify the problem. Explain that the groups can use any of the materials provided to work on solving the problem. Describe the Explore process. Groups draw at least four rectangles that use all of the fencing before they predict what the best size is.

Action! Groups of 4 Investigation Describe and assign roles to the group members: materials manager (get/return the required materials), chart paper recorder, presenter (for whole class discussion), coordinator (keeps group on task). All members make their own notes and record their group’s explorations. Distribute BLM 1.1.1 to each group’s coordinator. Using manipulatives, students brainstorm a strategy to find the perimeter and largest area, e.g., counting squares, use a formula, scale drawing. Circulate and help each group decide when they are ready to record the largest garden and their strategy on chart paper, and prepare to present their solution. Whole Class Presentation Learning Skills (Teamwork)/Observation/Mental Note: Groups draw their best solution on chart paper and record how they solved the problem, including as many representations and strategies as possible. Post the solutions on chart paper and ask groups to present their findings. Encourage students to ask each other questions. Acknowledge the variety of representations and signal for the students that they should continue to find a variety of ways to represent their work.

Consolidate Debrief

Whole Class Discussion Summarize the key ideas, ensuring that the following concept is understood: The largest area for a rectangle of fixed perimeter is a square. Review the formulas for perimeter and area of a rectangle. Review substitution into perimeter and area of rectangle problems in context.

Focus on the Explore stage of the Inquiry Process Each group of students might investigate using a different length of fencing. Different groups could be given different lengths of fencing so that there will be sufficient evidence during the consolidation part of the lesson.

Concept Practice Skill Drill

Home Activity or Further Classroom Consolidation Solve the perimeter and area problems. (The teacher selects context problems from the textbook.)


1.1.1: What is the Largest Rectangle? Your neighbour has asked for your advice about building his garden. He wants to fence the largest rectangular garden possible with ____ metres of fencing. Investigate to determine the largest garden you can build with _____ metres of fencing. Hypothesize What do you think the largest rectangular garden will look like? Explore You can use chart grid-paper, markers, string, and rulers. Brainstorm strategies you could use to determine the largest area. Record your strategies. Model Choose a strategy. Try it out to determine the largest rectangle. Transform If you do not like your model, adjust it or try another strategy. Conclude Present your solution to the problem, checking that it satisfies all of the conditions and makes sense.


Unit 1: Day 2: What is the Largest Rectangle Revisited? Grade 9 Applied

75 min.

Description • Connect the sum of the side lengths and the perimeter formula.

Materials • grid paper • rulers


Minds On ... Whole Class Discussion Construct a chart with the measurement data collected on Day 1: perimeter, length, width, area. Recorders from each group put the data into the table on the board and each student copies the table into their notes. Coach students to note patterns in the chart, especially that the width plus the length is constant. Ask: How can you use this pattern to help with future problems? Review the formulas for perimeter and area, making connections between the sum of the length and the width and the perimeter formula. Students justify the largest area for the fixed perimeter from the chart (numerical analysis) and from the graph (graphical analysis). Draw a scatterplot for Area vs. Width. Use this opportunity to introduce terminology: dependent and independent variables, scale, relationship, labels, and titles that will be used throughout the course.

Action! Pair/Share Thinking Activity Learning Skill (Work Habits/Teamwork)/Observation/Checklist: Pairs create a plan to solve similar problems (fixed perimeter/maximum area) using the headings: Explore/Hypothesize, Model and Infer/Conclude. Combine two pairs to compare strategies and revise their work. Prompt students by reminding them of the steps they took in the original investigation and clarifying stages as necessary.

Consolidate Debrief

Whole Class Discussion Develop a class chart of student actions during an investigation based on input from various groups of four students. Discuss strategies that students can use to determine the largest area and discuss how to use the graph to predict the largest area. Extend their thinking to include situations where the solution is not an integral value.

Focus on the Model and Hypothesize stages of the Inquiry Process Note that the area of a rectangle of a given perimeter depends on the width of the rectangle.

Application Home Activity or Further Classroom Consolidation The teacher assigns a contextual problem involving a maximum area if the perimeter is 40 m.


Unit 1: Day 3: On Frozen Pond Grade 9 Applied

75 min.

Description • Extend fixed perimeter problems to ones that do not have integral solutions.

Materials • BLM 1.3.1 • grid paper, rulers • string, metre sticks, measuring tapes


Minds On ... Whole Class Discussion Curriculum Expectations/Response Journal/Checklist: Collect and assess the follow-up activity. Discuss the problem within the Inquiry framework, highlighting insights that students raised through the previous two days. Introduce the task: On Frozen Pond (BLM 1.3.1). Read the instructions and clarify the problem. Suggest that students use their plans from the previous day to record their process. Questions to prompt learning: Why are we highlighting this region of the graph? How is this problem different from the previous task? How can you be sure that you have found the maximum area?

Action! Pair/Share Guided Investigation Students explore possible ice rinks and share strategies for selecting rinks with larger areas. Prompt students to manipulate the data on the scatterplot, as required. For example: Circle the region on the scatterplot where they believe the maximum area will be found and prompt them to collect more information by drawing the rectangles that would be represented by that region of the scatterplot. Students investigate the dimensions of a sufficient number of these rectangles to make and justify a conclusion.

Consolidate Debrief

Whole Class Discussion Discuss the investigation addressing how this problem is different from their previous experiences and that they must consider lengths and widths with decimal precision. Think/Pair/Share Check for Understanding Learning Skill (Independence/Initiative)/Observation/Rating Scale: Students explain what they did to determine the optimal dimensions. Expect responses that describe how they highlighted the region of the graph where they determined they needed more data to plot, and the rationale they used to determine when they had sufficient data. Encourage use of the word “because” when they are justifying a solution. Shared results could be provided on a transparency prepared by one of the groups. Whole Class Discussion Discuss and practise applications in which it would be important to know the maximum area for a given perimeter. Curriculum Expectations/Self-Assessment/Rating Scale: After students have had an opportunity to self-assess, ask what support they need before a formal assessment.

Focus on the Manipulate/ Transform and Infer/Conclude stages of the Inquiry Process

Application Concept Practice

Home Activity or Further Classroom Consolidation Create an optimization problem that would require you to collect more data in the region of the optimal solution. Select a perimeter that is an even number.


Area = length × width Area = 5 × 56 Area = 280 m2

5 m

56 m

56 m

Area = 280 m2

1.3.1: On Frozen Pond The town planners have hired you to design a rectangular ice rink for the local park. They will provide 122 metres of fencing. Your design should enclose the greatest possible area for the skaters. Explore It is possible to build a long, narrow ice rink, as shown. On the back of this page, sketch three more ice rinks that have a larger area than this ice rink. Label the dimensions on the sketch and calculate the area. Hypothesize Based on your exploration, predict the length and the width of the largest rectangular ice rink. Model Complete the table with all possible combinations of width and length for the ice rinks.

Perimeter (m) Width (m) Length (m) Area (m2) l × w

122 0 61 0 122 5 56 280 122 10 122 122 122 122 122 122 122 122 122 122

Describe what happens to the area when the width of the ice rink increases.


1.3.1: On Frozen Pond (continued) Construct a scatter plot of Area vs. Width.

Manipulate Circle the region on the scatter plot where the area of the rink is the largest. On the back of this page, construct two more sketches of rinks with lengths in this region. Add these points to the scatter plot. Conclude Write a report to the town advising them of the dimensions that would be best for the new ice rink. Justify your recommendation. Include a sketch and the area of the ice rink that you are recommending.


Unit 1: Day 4: Down by the Bay Grade 9 Applied

75 min.

Description • Apply the Inquiry Process to a problem that requires an enclosed space on only

three sides. • Explore the results for a variety of “What if ?” conditions.

Materials • grid paper • rulers • BLM 1.4.1, 1.4.2


Minds On ... Whole Class Discussion Pose the question: If you wanted a rectangular swimming area at the beach, how many sides of the rectangle would you rope off? Explain. Do you suppose this beach swimming area would still be a square?

Action! Small Groups Investigation Distribute BLM 1.4.1. Learning Skills (Works Habits/Initiative)/Observation/Anecdotal: Observe as pairs work through the Explore stage of the investigation. Whole Class Check for Understanding Briefly reconvene the whole class to check for understanding so that all students can proceed with the task from this point. Individual Performance Task Curriculum Expectations/Performance Task/Rubric: Students complete the task independently. (1.4.2 Assessment Tool)

Consolidate Debrief

Whole Class Discussion Discuss strategies. Introduce the follow-up questions and facilitate a brief exchange of ideas.

Students’ responses should consider the effect on the number of possible areas when enclosing an area on three sides. Encourage students to sketch the area. When returning graded work to students, consider photocopying samples of Level 3 and Level 4 responses with student names removed. Select and discuss with the class samples that illustrate a variety of strategies.

Concept Practice Exploration Reflection

Home Activity or Further Classroom Consolidation You answered the question: Which rectangle gives the largest swimming area? Consider how your recommendation would change if you were asked: Which rectangle provides the safest swimming area? or Which rectangle gives the best access to the deeper water?


Area = length × width Area = 5 × 90 Area = 450 m2

5 m

90 m

5 m

1.4.1: Down by the Bay The city planners would also like you to design a swimming area at a local beach. There is 100 m of rope available to enclose the swimming area. The shore will be one side of the swimming area; so only three sides of the rectangle will be roped off. It is your job to design the largest rectangular swimming area. Explore It is possible to build a long, narrow swimming area.

On the back of this page, sketch three more swimming areas that have a larger area than this swimming area. Label the dimensions on the sketch and calculate the area as shown above. Hypothesize Based on your exploration, predict the dimensions of the largest rectangular swimming area. Model Complete the table with possible combinations of width and length for the swimming pools.

Perimeter (m) Width (m) Length (m) Area (m2) l × w

100 0 5

Describe what happens to the area when the width of the swimming area increases.


1.4.1: Down by the Bay (continued) Construct a scatter plot of Area vs. Width.

Manipulate Look at the scatter plot. Circle the region on the scatter plot where the area of the swimming area is the largest. On the back of this page, construct two more sketches of swimming areas with lengths and areas in this region. Add these points to the scatter plot. Conclude Write a report to the town advising them of the dimensions that would be best for the new swimming area. Justify your choice. Include a sketch and the area of the swimming area that you are recommending.


1.4.2 Assessment Tool: Down by the Bay





Accuracy

Use a marking scheme.

Reasoning and Proving (Thinking/Inquiry/Problem Solving)

Making inferences, conclusions and justifications that connect to the problem-solving process and models presented

- presents no justification or a justification with no connection to the problem-solving process and models presented

- presents justification of the answer that has a limited connection to the problem-solving process and models presented

- presents justification of the answer that has some connection to the problem-solving process and models presented

- presents justification of the answer that is well connected to the problem-solving process and models presented

- presents justification of the answer that has an insightful connection to the problem-solving process and models presented

Making Connections (Thinking/Inquiry/Problem Solving)

Creation of a model (table and scatter plot) to represent the data

- no model or a model that represents none of the data

- creates a model that represents little of the range of data

- creates a model that represents some of the range of data

- creates a model that represents most of the range of data

- creates a model that represents the full range of data


Clarity in conclusion and justification in reporting

- provides conclusion and justification that are confusing or contradictory

- provides conclusion and justification that have limited clarity

- provide conclusion and justification that have some clarity

- provides conclusion and justification that are clear

- provides conclusion and justification that are precise


Scatterplots on the Graphing Calculator (Presentation software file) Download this file at www.curriculm.org/occ/tips/downloads.shtml

1

Scatter Plotson the

Graphing Calculator

9/29/2003 2

1. Setting UpPress the Y= key.

Be sure there are no equations entered.

If there are any equations, use the CLEAR key to delete them.

9/29/2003 3

2. Setting UpBe sure there are no equations entered below the 7th entry.

Check that there are no equations entered below the visible screen.

Use the down arrow and clear them.

9/29/2003 4

3. Clearing ListsTo clear previous lists, press “second” +.

Choose #4.

Press ENTER.

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4. Clear Lists

Press ENTER again!!.

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5. Stat KeyThe Stat Key is used to input statistical data.

The Stat Key accesses lists similar to columns on a spreadsheet.

Pressing “1:Edit” brings us to the lists.



Scatterplots on the Graphing Calculator (Presentation software file) (continued)

9/29/2003 7

6. Edit ListsWe are now ready to input the data.

We have our blank lists.

9/29/2003 8

Type the data into the lists.L1 = Width of

rectangleL2 = Area of

rectangle

7. Input Lists

9/29/2003 9

8. Stats PlotThe stats plot key allows us to choose one of three graphs to plot with the data in the lists.Choose Plot1 to start.

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9. Plot MenuTurn Plot1 ONChoose the type of graph (the first choice is good).Choose the Xlist and Ylist. L1, L2

9/29/2003 11

Zoom 9Zoom 9 will fix the window to the data and plot the graph.

9/29/2003 12

10. The GraphAll of the data is now plotted in the window.

Reminder: using the trace button will reveal the coordinates of the points.

The End


Composite Figures (Presentation software file) Download this file at www.curriculm.org/occ/tips/downloads.shtml

Composite Figures

Pieces of the Whole

Bill is painting his basement floor.

The dimensions are:

3.1 m

1.5 m

2.3 m

2.5 m

One can of paint covers 3.5 m2.

Each can of paint costs $15.25.

What is his total cost including 15% GST and PST?

a) Determine the total area

A = 2.3×2.5 + 3.1×1.5A = 5.75 + 4.65

A = 10.4 m2

Atotal = A1 + A2

A2

A1

3.1 m

1.5 m

2.3 m

2.5 m

10.4 m2

b) If one can of paint can cover 3.5 m2, how many cans of paint will he need?

He needs 3 cans of paint.

3.1 m

1.5 m

2.3 m

2.5 m

3.5m/can

2

2

10.4mNumber of cans =

Number of cans = 2.97

10.4 m2

c) If one can of paint costs $15.25, what is his total cost including 15% GST and PST?

Paint cost: 3 × $15.25 = $45.75Taxes: = 0.15 × $73.40 = $6.86

Total cost: $52.61

3.1 m

1.5 m

2.3 m

2.5 m

Reflect…

• In what other way might the area have been found?

3.1 m

1.5 m

2.3 m

2.5 m• Why is this way of splitting the area less convenient than our original way?

We have to calculate side lengths that are not given, and this introduces more sources of error.

What shortcut could be used to determine the after-tax cost?

We had:

Paint cost: 3 × $15.25 = $45.75

Taxes: = 0.15 × $45.75 = $ 6.86

Total cost: $52.61

This calculation can be done all at once

The cost plus 15% of that cost is represented by:

3 × $15.25

× (1+0.15)

× 1.15

A multiplier of 1.15 yields an increase of 15%

What multiplier yields an increase of 20%? Of 35%?1.2 1.35

cost



Unit 1: Day 10: Camp Olympics Grade 9 Applied

75 min.

Description • Determine optimal dimensions based on factors that include function, volume,

and surface area. • Use straightforward application of spreadsheets.

Materials • BLM 1.10.1


Minds On ... Whole Class Discussion Review with students the steps of the inquiry process and the formula for determining the surface area and volume of a rectangular prism.

Action! Individual Journal Present Part A of the task on transparency. Students individually record their initial thoughts in their journals. Small Group Activity Learning Skill Organization/Teamwork/Initiative)/Observation/ Checkbric: Observe and record students’ efforts as they collaborate. Present Part B of the task on transparency. Group students to discuss their responses and complete revisions to journal entries. Students follow the instructions on the transparency and record responses on the worksheet, up to the spreadsheet. They label each model with its dimensions to prepare for the development of spreadsheet formulas. Whole Class Teacher Led Discussion With students, develop the formulas for insertion into the spreadsheet. Students identify strategies for data entry/sorting that allow them to see patterns easily. Small Group Activity Groups enter their own dimension choices into the spreadsheet and complete Part C of the activity.

Consolidate Debrief

Whole Class Discussion Students share their solutions with rationale. Highlight the need to consider many cases before identifying trends and the usefulness of spreadsheets to facilitate multiple, accurate calculations. Students should recognize all steps of the inquiry process and the value of reflection in order to make positive changes in their plans.

If data is entered by increasing cut-out dimensions, the volume will increase then decrease, while the surface area of the container will decrease.

Application Concept Practice Reflection

Home Activity or Further Classroom Consolidation Find two boxes that have equal volumes but different shapes. Measure and calculate the surface area and volume for each box.


1.10.1: Camp Olympics Part A Over breakfast, you learn that there will be Olympics during the afternoon camp program. The waterfront director of Camp Avagudtym is planning a competition for the Olympics that requires each team to fill an empty bucket with water using only the container that they have constructed. The number of container refills will be counted and the team using the least number of refills wins. You start thinking about the challenge and make some notes in your journal about how you would construct the container. Part B When you arrive at the site for the Olympics, you find out that the container must be a rectangular prism made from one sheet of 10 cm × 13 cm plastic. Your counsellors put you in teams of four and you meet as a team to compare notes about the design of your container. Using scissors, you cut squares out of each corner and fold up the sides, sealing the seams with hot glue. Using a different colour record your revised plan for construction. One of your teammates suggests that you fold a letter size piece of paper into quarters and make four different accurate models of the container by cutting different sizes of squares from the corners. Before constructing the models, record your responses to the following: 1. What do you think will happen to the shape of the box as the size of the square cut-out gets

bigger? 2. What do you think will happen to the volume of the box as the size of the square cut-out

gets bigger? Model Construct the four models and use the following chart to record the dimensions and results.

Cut-out Side

Length (cm)

Height (cm)

Length (cm)

Width (cm)

Volume (cm³)

Surface Area (cm2)

Total Area cut out (cm2)

Hypothesize Predict the side lengths of the cut-out squares for the container with the largest volume.


1.10.1: Camp Olympics (continued) Part C Transform/Manipulate It becomes apparent that you need to consider more cases, including integer and non-integer values. Recognizing that a spreadsheet would be helpful, you recreate the table on a computer. To identify the container with the largest volume, you develop formulas for the cells and extend your spreadsheet to include at least ten cases. Infer/Conclude Having identified the dimensions resulting in the largest volume, your team now creates the box. Identify the dimensions of your box and explain why you selected these dimensions. Discuss the trends in surface area and in volume as the dimension of the cut-out changes. Explain the appearance of negative values in your spreadsheet. When you arrive at the beach, you learn that the bucket has a volume of 4000 cm3. Estimate and then calculate the number of refills of your container that will be required to fill the bucket. Calculations:


The Volume of a Cylinder (Presentation software file) Download this file at www.curriculm.org/occ/tips/downloads.shtml

h

r

The Volume of a CylinderVolume = (Area of the base) × (height)

Therefore, Volume = πr2 × h, or

A = πr2 V = πr2h

What shape is the base?

What is the area of a circle?

1: Determine the volume of the cylinder

Volume = (Area of the base) × (height)

5 cm

3 cm

V = (3.14)(3)2 ×(5)V = (3.14)(9)(5)

V = 141.3 cm3

Volume = (Area of a circle) × (height)

Volume = (π × radius2) × (height)

2: Determine the volume of the cylinder


9.2 cm

7 cm

V = (3.14)(3.5)2(9.2)

V = (3.14)(12.25)(9.2)

V = 353.9 cm3

r = 3.5, h = 9.2 cm



What are radius and height measures?

3: Determine the height of the cylinder if V = 600 cm3 and r = 5 cm


h cm

5 cm

600 = (3.14)(52)h600 = (3.14)(25)h600 = (78.5)h

7.6 cm = h


Volume = (π× radius2) × (height)Substitute known values into the formula.

60078.5

h=

4. A piece of cardboard measures 20 cm by 8 cm is rolled into a cylindrical shape

20 cm

8 cm 8 cm

r

Problem:What is the volume of the cylinder?

What will we need to determine before we can find the volume?

20 cm

8 cm 8 cm

r



We need to determine the radius.

What information will help us determine the radius?

The 20 cm measurement will help.

How will it help? 20 cm is the circumference of the circle.

2(3.14)r = 20

6.28r = 20

r = 3.18 cm

Circumference = 2π × radius

2πr = 20

V = 3.14(3.18)2(8)V = 3.14(10.11)(8)

V = 254 cm3


20 cm

8 cm 8 cm

r

V = (π × radius2) × (height)20

6.28r =



50 cm

40 cm

30 cm

100 cm

20 cm

30 cm

1.16.1: Formative Journal Relationships in Measurement A family pet is a dog named Emily Carr. a) When Emily is cold she curls up into a ball to keep warm.

The shape that her body forms when she is curled can be modelled by the prism below. Calculate the volume and the surface area of this prism.

b) When Emily is hot she sprawls out to cool down.

The shape that her body forms can be modelled by the prism below. Calculate the volume and the surface area of this prism.

c) Using what you have learned from this unit, explain why Emily curls up to keep warm and

sprawls out to cool down. d) Find one more occurrence in everyday life that can be explained using the relationship

between surface area and volume. Explain your reasoning.


x 16

12

x

7 15

12 m

8 m 10 m

7 m

Unit 1 Test Describing Relationships in Measurement Name: ___________________________________ A communication level will be assigned based on use of mathematical conventions and clarity of explanations. K(2) 1. For each of the following triangles, show your substitution into the Pythagorean

theorem. Do not solve for x. a) b) A(3) 2. Find the area of the shaded region. A(6) 3. Consider the track below. a) Find the perimeter of the field. b) Find the area of the field. K(2) 4. Complete the substitutions and calculations for the surface area (S.A.) of this

triangular prism.

S.A. = two triangular sides + two rectangular sides + base = 2 (1/2)(b)(h) + 2 (l)(w) + (l)(b)

= 2 (1/2)(12)(8) + 2(7)(10) + ( )( )

200 m

30 m

9 cm

4 cm


A(2) 5. A cylindrical drinking glass has a diameter of 9 cm and height of 20 cm. Identify the formula that you would use to determine the surface area of the glass. Justify your choice.

Hint: The glass has a bottom, but no top. i) 2πr2 + 2πrh ii) πr2 + 2πrh iii) 2πr2 + πrh A(1) 6. Frozen ice cream treats are sold in cone-shaped containers. The containers are

12 cm high and have a 5-cm diameter. Indicate the formula you would use to determine the volume of one ice cream treat container. DO NOT complete the calculations.

A(1) 7. A snowball has a diameter of 7 cm. Indicate the formula you would use to determine

the volume of the snowball. DO NOT complete the calculations. A(5) 8. Math-O-Bits cereal is sold in a single-serving box. (See dimension on diagram.) The volume of this box is 200 cm3. The surface area of this box is 220 cm2.

The manufacturer would like to sell Math-O-Bits in a large size too. The manufacturer doubles the length, width, and height of the single-serving box.

a) Calculate the volume and the surface area of the new large box.

b) From your answers to a), explain why it is often cheaper for customers to buy products in larger sizes.


A(2) 9. Mrs. Coffin wants to fence a rectangular garden in her backyard.

She has 36 metres of fencing. Sketch and label the largest garden that Mrs. Coffin can enclose. Explain your

reasoning. A(3) 10. Sugar cubes are sold in packages of 64 cubes. Each sugar cube is 1cm3.

The sugar cubes could be sold in a package that is 1 cm wide, 1 cm deep, and 64 cm high.

a) State two more sets of possible dimensions for the package of 64 sugar cubes.

b) Which set of dimensions do you think is most reasonable? Explain.

1 cm 1 cm

64 cm

Volume = 1 × 1 × 64 = 64 cm3


Unit 1: Day 20: Performance Task: Designing a Cologne Bottle Grade 9 Applied

75 min.

Description • Minimize surface area for a closed cylinder having a fixed volume.

Materials • spreadsheet program • BLM 1.20.1


Minds On ... Whole Class Discussion Have a variety of cologne bottles available to facilitate a discussion of the design considerations. All students log onto computer and open the spreadsheet file.

Action! Individual Performance Task Curriculum Expectations/Performance Task/Rubric/Marking Scheme: Students work independently on the task (BLM 1.20.1). Circulate and provide scaffolding as required.

Consolidate Debrief

Whole Class Discussion Collect student work and discuss their findings and strategies. Present a model solution.

Each student must have access to their own spreadsheet file. Focus on the Thinking Inquiry/Problem Solving and Communication categories.

Explore Reflect

Home Activity or Further Classroom Consolidation Make 9 different statements based on the information shown in this graph. Pizza prices in Dryden.

This question is intended to activate prior knowledge before beginning Unit 2.

Number of Toppings

Cost of a

Pizza ($)

2 4 6 8

5

10

15 xxx

x

xxx

x

x


1.20.1: Designing a Cologne Bottle

The Smells So Good Company has a new line of cologne. The glass they buy to make the bottles is expensive, so the bottle design must use less than 150 cm² of glass. The company wants a cylindrical bottle that holds 100 mL of cologne. Your job is to advise the company about the dimensions of the bottle they should produce. You began your investigation by designing a spreadsheet that explores the possible height and surface area of bottles that hold 100 mL of cologne.

Radius (cm) r

Height (cm)

2rVolumeπ

Surface Area (cm²)

2πr2 + 2πrh Volume (mL)

1 31.85 206.28 100 2 7.96 125.12 100 3 3.54 123.19 100 4 1.99 150.48 100 5 1.27 197.00 100 6 0.88 259.41 100 7 0.65 336.29 100 8 0.50 426.92 100 9 0.39 530.90 100

10 0.32 648.00 100

What restrictions would you place on the radius in order to minimize the cost of the bottle?

Hypothesize Based on the exploration, predict the radius of a bottle that would meet the company's criterion (A volume of 100 mL and a surface area less that 150 cm²). Explain why your hypothesis is reasonable. Explore Again Using the spreadsheet, consider a number of possible values for the radius within the restrictions identified in question 1. Accuracy for the radius needs to be no more than one decimal place. Infer/Conclude Determine the dimensions of the bottle that minimizes surface area.

Write your report to the Smells So Good Company to advise them of the bottle that would be best for their new cologne. Your report must include: •• a sketch of the bottle, with the dimensions (radius and height) labelled; •• the surface area and volume of the bottle that you are recommending; •• a rationale explaining why this is the best bottle size for their cologne; •• a printout of the spreadsheet showing your calculations.


(Category) Criteria Below Level 1 Level 1 Level 2 Level 3 Level 4


Accuracy - use of spreadsheet - solution

Use a marking scheme.

Reasoning and Proving (Thinking/Inquiry/ Problem Solving)

Selection of values

- no evidence of reasoning

- demonstrates limited reasoning

- demonstrates some reasoning

- demonstrates considerable reasoning

- demonstrates a high degree of reasoning


Clarity - unclearly - with limited clarity

- some clarity - clearly - precisely


1.20.1: Designing a Cologne Bottle Answers

Radius Height Surface Area Volume 1 31.85 206.28 100 2 7.96 125.12 100 3 3.54 123.19 100 4 1.99 150.48 100 5 1.27 197.00 100 6 0.88 259.41 100 7 0.65 336.29 100 8 .050 426.92 100 9 0.39 530.90 100

10 0.32 648.00 100

Minimal surface area appears to be in this range, so the required radius is between 2 and 4.

Radius (Sample) Height Surface Area Volume

2 7.96 125.13 100 2.2 6.58 121.32 100 2.4 5.53 119.52 100 2.5 5.09 119.27 100 2.6 4.71 119.4 100 2.7 4.37 119.88 100 2.8 4.06 120.69 100 2.9 3.78 121.81 100 3 3.54 123.22 100

3.1 3.31 124.9 100 4 1.99 150.53 100

This is the lowest value for surface area. Therefore, radius 2.5 cm yields the smallest surface area.


Unit 2: Describing Relationships Grade 9 Applied Lesson Outline

BIG PICTURE Students will: •• describe relationships between variables graphically; •• connect graphical features to the characteristics of the relationship (no relationship, strong/weak

correlation, linear and non-linear, discrete/continuous); •• use a line of best fit to predict values and solve problems; •• use first differences to determine that relationships are linear or non-linear.

Day Lesson Title Description Expectations 1 Is There a

Relationship? (Explore) Lesson included

•• Identify a trend in a scatterplot. •• Use a trend to describe the relationship between the

variables. •• Make a prediction about a relationship in

preparation for an investigation on Day 2.

RE1.01 - 1.03, RE3.02 CGE 2c

2 Is There a Relationship? (Model) Lesson included

•• Investigate a relationship between measures by constructing a scatterplot using graphing technology.

•• Describe the trend in the plotted points.

NA4.03, RE1.04 - 1.07, RE3.02 CGE 5a

3 Investigating Relationships Lesson included

•• Establish criteria for drawing a line of best fit. •• Use a line of best fit to solve problems.

NA1.01, RE1.01 - 1.05, RE2.03 CGE 5a, 2a

4 Footprints Lesson included

•• Apply an inquiry process to solve a mystery. NA1.01, RE1.01 - 1.07, RE2.03, RE3.02 CGE 4b, 5a

5

6

Using Scatterplots to Solve Problems

Learning Skills (Organization/Individual Work)/ Observation/Checklist and Curriculum Expectations/ Exhibition/Checkbric: •• Construct and analyse scatterplots. •• Identify examples of strong/weak correlations and

increasing/decreasing relationships in order to solve a problem.

•• Investigate, for example, additional forensic mysteries (determine the height of a person from the arm span), track and field records and altitude of venue, Florida voting results (Bush/Gore election controversy).

NA1.01, RE1.01, RE1.04, RE1.05


Day Lesson Title Description Expectations 7 Talking About

Lines Learning Skills (Initiative)/Observation/Rating Scale and Curriculum Expectations/Response Journal/ Rating Scale: •• Informally introduce the equation of a line (referring

to the equation as a formula for a relationship between the variables in the context) by counting grid squares along the graph. For example, students construct a table of values, plot points and draw a line of best fit from a word description of a linear relationship such as: A 15 cm candle burns at 0.5 cm per minute. Using the initial condition (at the start the candle is 15 cm) and the rate (the candle burns 0.5 cm every minute), they move from point to point.

•• Solve problems such as, How tall will the candle be after 7 minutes? Using informal word descriptions of the line and check their solution with the graph. These informal experiences reading and describing graphs set the stage for equations of lines in Unit 3.

NA1.01, RE2.03

8 Multiple Representations of Data

Curriculum Expectations/Quiz/Marking Scheme: •• Consolidate scatterplots and lines of best fit in

authentic contexts. For example, students match a description of a relationship with a scatterplot, describe a context for a scatterplot, provide a rationale for an outlier and use a line of best fit to interpolate and extrapolate.

NA1.01, RE3.02

9 Relationship Project

Learning Skills (Initiative/Independent Work)/ Observation/Checklist and Curriculum Expectations/ Exhibition/Rubric: •• Determine if there is a relationship between

variables. •• Use relationships to solve a problem. For example

use data from secondary sources (smoking and cancer; income and mortality rate) or collect data from an experiment (the number of elastics and the distance stretched; the height from which a ball is dropped and the rebound height; the strength of paper roofs).

NA1.01, NA3.05, RE1.01, RE1.03 - 1.07, RE2.03

10 First Differences Lesson included

•• Investigate the pattern in the first differences to determine if a relationship is linear or non-linear.

RE2.01, RE2.04, RE2.05, RE3.01, RE3.03 CGE 5e


Day Lesson Title Description Expectations 11 Paper Folding

Lesson included

•• Investigate exponential growth. NA2.01, NA2.03, RE1.01, RE1.03 - 1.07 CGE 2c

12 Patterns in Powers MATHO game included (p. 80)

Learning Skills (Works Habits/Organization/ Teamwork)/Observation/Checklist and Curriculum Expectations/Exhibition/Checkbric: •• Discover the power rules for multiplication and

division. •• As an extension, explore the patterns that result from

repeated multiplication of integers: (−6) (−6) (−6) = −216.

•• Play MATHO game to practise and consolidate exponent laws.

NA2.01, NA2.06

13 Paper Unfolding Learning Skills (Individual Work/Initiative)/ Observation/Rating Scale and Curriculum Expectations/Interview/Rating Scale: •• Introduce powers with negative exponents. •• Use patterns in the table to predict the power and

number of layers for –1, –2 folds. Imagining a layer of paper sliced almost all the way through so you could open it up, is a powerful way for most students to be able to see that you have half a layer of paper and helps make connections between powers with negative exponents and fractions of paper layers. (Using 2-ply tissue can illustrate this point.)

•• Make connections between the negative exponent and repeated dividing.

NA2.01, NA2.03, RE1.01, RE1.03 - 1.07

14 Describing Big and Small Numbers Lesson included

•• Revisit the paper folding and unfolding investigations.

•• Determine the thickness of many sheets of paper and the thickness of a slice of a layer.

•• Introduce scientific notation.

NA2.01, NA2.04, NA2.05, NA2.06 CGE 5a


Day Lesson Title Description Expectations 15 All Powered Up! Curriculum Expectations/Quiz/Marking Scheme:

•• Explore the patterns in repeated multiplication. •• Generalize the power laws and apply them to

simplify and evaluate numeric expressions involving powers.

•• Use peer tutoring to prompt discussion about common misunderstandings of the power laws. For example: What is wrong with: 2³ + 4² = 65?

Note: Extensive manipulation of expressions involving powers is not required in this course but facility with simple numeric expressions such as: 2 × 2 = 2² and 3 × 3² = 3³ will facilitate work with polynomial expressions such as: ( )2+xx and ( )xxx 22 + in Unit 5.

NA2.01, NA2.05, NA2.06

16 Review •• Review tasks or game using a carousel

17 Review and Test Sample test included (p. 83-86)

Curriculum Expectations/Test/Marking Scheme: •• Focus on Knowledge and Application questions.

For example: Knowledge: Match a scatterplot to a description and label the axes. Application: Show a table of first differences for a linear relationship; however, the data table does not go up in equal increments in the independent variable. Ask, Is the relationship linear or non-linear? Explain.

NA2.01, NA2.04, RE2.03, RE2.05, RE3.02, RE3.03, RE1.01, RE1.03 - 1.07

18 Summative Assessment Sunflowers task included (p. 87)

Curriculum Expectations/Performance Task/Rubric: •• Focus on the Thinking/Inquiry/Problem Solving and

Communication Categories through an investigation. For example:

i) Students construct scatterplots using secondary or collected data to determine a relationship, solve a problem using their understanding of a line of best fit, outliers, and interpolating and extrapolating. Students communicate their solutions with a rationale for their choices and an explanation for their conclusions.

OR ii) Sunflowers task included in this package.

OR iii) Revisit the forensic mystery on Day 4 to solve

for another identifying suspect trait.

NA2.01, NA2.04, RE2.03, RE2.05, RE3.02, RE3.03, RE1.01, RE1.03 - 1.07

19 Review of Unit Test and Performance Task


Unit 2: Day 1: Is There a Relationship? (Explore) Grade 9 Applied

75 min.

Description • Identify a trend in a scatterplot. • Use a trend to describe the relationship between the variables. • Make a prediction about a relationship in preparation for an investigation on

Day 2.

Materials • BLM 2.1.1, 2.1.2 • Fathom™

software • graphing

calculators or spreadsheet software


Minds On ... Think/Pair/Share Worksheet Students respond to statements about a scatterplot using BLM 2.1.1, focusing on the pattern in the scatterplot and providing a rationale for the statements as they respond according to Stand Up facing forward – Agree/Stand up facing backward – Disagree/Sit down – Pass (BLM 2.1.1). Share responses with the whole class.

Action! Pair/Share Application Pairs predict relationships in response to statements (BLM 2.1.2). Pairs form groups of four students to share their responses and rationale (Repeat by forming new groups of four: Round Robin). Whole Class Sharing Share student responses and rationales as a whole class; consensus is not required. Pairs Investigation Students choose one relationship that they would like to investigate from the suggestions in BLM 2.1.2. Students gather data to facilitate measurement collection, consider using measurement stations complete with measuring tapes or metre sticks attached to the wall or floor for convenience and consistency during data collection. Honour student choice but suggest that a variety of relationships be chosen to investigate. (Relationships involving height, stride length and foot size are needed for the Inquiry on Day 4) Curriculum Expectations/Observation/Mental Note: Collect data for the relationships and record their own data on a chart using headings: Height, Armspan, Leg Length, Forearm, Running Speed, Walking Stride, Thumb Length. Enter all data into a master Fathom™ document.

Consolidate Debrief Whole Class Discussion

Note: The activity continues on Day 2.

The BLM shows one of four types of relationship: increasing, decreasing, constant, no relationship A less active and quicker way for students to indicate to the teacher that they agree/disagree/ are not sure would be to use a thumbs up/thumbs down/ thumb sideways gesture. This strategy allows the teacher to see students’ responses without students being aware of other students’ responses. Fathom™ software allows students to make changes easily and quickly.

Application

Home Activity or Further Classroom Consolidation Identify other linear measures of body parts that may have a relationship to each other or to something else, e.g., handspan to number of chords that can be played on a piano.


2.1.1: Plotted Points

1. The graph shows the plotted points rising upwards to the right. • Agree • Disagree • Pass

2. As the length of the tibia increases

the length of the leg increases • Agree • Disagree • Pass

3. The graph can be used to determine

the length of a person's leg if you know the length of the tibia bone • Agree • Disagree • Pass

1. The graph shows the plotted points scattered. • Agree • Disagree • Pass

2. As the age of the house increases the price of

the house is either large or small. • Agree • Disagree • Pass

3. The graph can't be used to determine the price

of the house if you know how old it is. • Agree • Disagree • Pass

1. The graph shows the plotted points falling to the right • Agree • Disagree • Pass

2. As the distance from the net increases

the number of baskets made decreases. • Agree • Disagree • Pass

3. The graph can be used to determine

the number of baskets you will make if you know the distance from the basket. • Agree • Disagree • Pass

Tibia Length (cm)

Leg Length (cm)

Distance from the Basket

Num

ber o

f Bas

kets

Age of House

Hou

se P

rice

($)


2.1.2: Relationships Complete the following statements by yourself then share your answers with your partner. Explain the reasons for your choice. Indicate if you and your partner agree or disagree.

Is there a relationship? My partner and I:

._________________ Agree As a person gets taller their armspan ٱ (gets wider, gets smaller, stays the same) ٱ Disagree

.Agree The longer a person's legs are, _______________they run ٱ (the faster, the slower, it makes no difference to how fast) ٱ Disagree

.Agree As a person's foot size increases, the _______ their walking stride ٱ (longer, shorter, foot size does not affect) ٱ Disagree

.___________ Agree As a person's forearm gets longer, their armspan ٱ (gets longer, gets shorter, is not affected by the forearm length) ٱ Disagree

.Agree The longer a person's thumb is, the __________their index finger ٱ (longer, shorter, length of the thumb will not affect the length) ٱ Disagree

.______________Agree As a person gets taller, their foot size ٱ (gets longer, gets shorter, is not affected) ٱ Disagree


Unit 2: Day 2: Is There a Relationship? Grade 9 Applied

75 min.

Description • Investigate a relationship between measures by constructing a scatterplot using

graphing technology. • Describe the trend in the plotted points.

Materials • BLM 2.2.1 • Fathom™

software


Minds On ... Whole Class Demonstration Demonstrate data entry using the data for the scatterplots from Day 1. Students describe the trend in the graphs.

Action! Pairs Worksheet Learning Skills (Teamwork)/Observation/Checklist and Curriculum Expectations/Demonstration/Anecdotal: Students graph and analyse the relationship for which they gathered data on Day 1 (BLM 2.2.1). Groups present results to whole class. Explain individual results and anomalies as required, using appropriate terminology. For example: Outliers may be the result of mismeasurement, growth spurts.

Consolidate Debrief Whole Class Discussion

Use the following prompts: • Which graphs showed the strongest relationship? • How does the graph show the relationship? • Which graphs showed no relationship? • How does the graph show that there is a relationship? • Which results surprised you?

Fathom ™ software allows students to make changes easily and quickly.

Home Activity or Further Classroom Consolidation Collect data from a friend or family member for the relationship you investigated. Plot the data. Does it fit the trend in the graph?


2.2.1: Is There A Relationship?

Prediction 1. Complete the data table and sketch the graph.

2. Which phrase describes the direction of the plotted points in the graph?

• The plotted points rise upward to the right. • The plotted points fall downward to the right. • The plotted points are scattered across the graph.

3. Describe the relationship: 4. Could you use this graph to predict additional measurements? Explain.


Unit 2: Day 3: Investigating Relationships Grade 9 Applied

75 min.

Description • Establish criteria for drawing a line of best fit. • Use a line of best fit to solve problems.

Materials • BLM 2.3.1, 2.3.2


Minds On ... Whole Class Discussion Discuss the context for the investigation (BLM 2.3.1). Groups of 3 or 4 Placemat Access prior experience: Who would be interested in this data and what would they do with it? Students respond in a Placemat activity (See TIP 9). Each group shares one response that is different from those already presented.

Action! Whole class Guided Investigation Discuss how the data might have been collected and the meaning of the words radius and humerus. Direct student thinking with the following questions: • Do you think the longer a person’s radius is, the longer their humerus will

be? • What strategy could we use to investigate the relationship? Manually construct a graph discussing and modelling choice of axes, scale, and titles. Students complete their graph individually and respond to questions. (BLM 2. 3.1) Discuss student responses to the questions as a whole class using vocabulary for graphical descriptions and for describing the relationship between the radius and humerus. Groups of 4 Numbered Heads Students draw a line on their graph that shows the direction of the plotted points then exchange their graph with the person to the right who writes a "descriptive" comment next to the graph. Repeat three times so all students have an opportunity to view all graphs. Determine the Line of Best Fit among the group. Person 1 moves to the next table to share their table’s best answer and the rationale for their choice. Students move to other tables until all graphs are shared and discussed. Whole Class Direct Instruction of Vocabulary Learning Skills (Organization/Initiative)/Observation/Rating Scale and Curriculum Expectations/Anecdotal/Rating Scale: Students vote on the Line of Best Fit and rationale and develop a class definition. Use a Line of Best Fit to predict the humerus length if the radius length is 24.8 cm long (interpolate) and the radius length if the humerus is 32 cm long (extrapolate).

Consolidate Debrief

Think/Pair/Share Practice Students graph additional data that shows an increasing relationship. They describe the direction and relationship of the plotted points and construct a Line of Best Fit.

Prompt students to consider science-fiction movies; unsolved mysteries, crime; archaeological digs, etc. Check students’ plotting skill using decimals (BLM 2.3.1). Consider using Think/Pair/Share to answer questions. Post the Line of Best Fit for reference during the unit

Home Activity or Further Classroom Consolidation Additional data has been found for the radius and humerus: R = 22 cm and H = 17 cm and R = 20.9 cm and H = 22.5cm

Plot the data on the graph. Does the new data follow the pattern in the graph? Explain the position of the new of data.

Expect responses such as: mismeasured data; abnormal growth; not a bone from the same species; etc.


2.3.1: Forensic Analysis

Anthropologists and forensic scientists use data to help them determine information about people. Often only a few bones are available or the evidence is in inconclusive. In spite of these difficulties, by accessing the information in large databases and investigating relationships between data scientists can determine information about the height, age, and sex of the person they are examining.

1. Construct a Graph 2. Circle the point on the graph that represents the data for a radius that is 21.9 cm long.

How long is the humerus? _____________. 3. Put a box around the point on the graph that represents the data for a humerus that is

27.1 cm long. How long is the radius? ______________. 4. Underline the statement that describes the direction of the plotted points in the graph?

• The plotted points rise upward to the right. • The plotted points fall downward to the right. • The plotted points are scattered across the graph. • The plotted points lie flat along the horizontal.

5. As the length of the radius gets longer, what happens to the length of the humerus? 6. Do you think that you can use the length of the radius to predict the length of the humerus?

Explain. 7. Do you think that you can use the length of the humerus to predict the length of the radius?

Explain.

Radius (cm)

Humerus (cm)

25 29.7 22 26.5

23.5 27.1 22.5 26 23 28

22.6 25.2 21.4 24 21.9 23.8 23.5 26.7 24.3 29 24 27


2.3.2 : Placemat Recording Sheet


Unit 2: Day 4: Footprints Grade 9 Applied

75 min.

Description • Apply an inquiry process to solve a mystery.

Materials • graphing

calculators or Fathom™ software

• BLM 2.4.1, 2.4.2


Minds On ... Small Groups Clarify the Problem Read the “Memo” in BLM 2.4.1 and study the footprint evidence. Use a cooperative learning strategy to clarify the problem by discussing the information in the evidence using these guiding questions: • What questions can be answered by a set of prints? • How can we answer those questions? • What do we know about the footprint that will help us solve the problem?

(length of foot and stride-length) Whole Class Sharing Make connections to earlier lessons in the unit and record for all students to view. Choose one question that the whole class will investigate together. The class data should show a relationship. If not, revise it or provide additional adult and child data to strengthen the relationship and give better dispersion.

Action! Small Groups Investigation Learning Skills (Teamwork)/Checkbric and Curriculum Expectations/Self Assessment/Checkbric: Students respond to prompts in BLM 2.4.1. Explore: Working in groups of three, students collect data from the footprint, e.g., measure the length of the footprint, the stride length, and retrieve class data from Day 1 that they determine helps solve the problem. Groups select one student to provide feedback/help with their plans. Model: Students create a model for the problem, including a table of values, scatterplot, and line of best fit. Manipulate/Transform: Students decide how they can manipulate the model to solve the problem. Responses should include: Describe the relationship, and use the line of best fit to interpolate or extrapolate values to solve the problem. Students identify/justify outliers and plot the footprint data on the scatterplot. Infer/Conclude: Students write a report to the principal that includes a statement about the solution and a justification using the evidence and the graph.

Consolidate Debrief

Small Groups Placement Students use the four corners of the placemat to list strategies that can be used to solve a problem. Students share and record their responses for future reference.

The evidence can be a set of footprints made prior to class by stepping on a sheet of paper that is large enough to capture one full stride and both feet. See BLM 2.4.1 for an example. Cooperative learning strategies include: Numbered Heads, Think/Pair/Share, Placemat. Students will not necessarily know that they need to graph the footprint evidence with the collected data. Problem-solving strategies include: collecting data, graphing, constructing a best fit line, interpolating/ extrapolating from the line.

Home Activity or Further Classroom Consolidation Determine the height of a person who has a tibia of length 35 cm.

Provide a graph of Tibia vs. Height.


2.4.1: Footprints

Urgent Memo Mr.____________________, principal at _________________________ found these footprints outside the math department office. ____ would like some information about the person who left these footprints.

40 cm

150 cm


2.4.2: Footprints (continued)

Explore Group Planning: What would your group like to find out about the suspect?

What data will you need?

One question that we would like help with is: Whole Class Sharing: Report your information to the whole class. Ask for feedback on your plans from one person who is not in your group. Record this person’s response. Record responses from two students identified by your teacher. Model Discuss each of the mathematical tools listed below. Circle the tools that you will use to solve this problem.

• table of values • scatterplot • graph • Fathom™ • graphing calculator • spreadsheet

Construct a model for your data.


2.4.2: Footprints (continued) Transform/Manipulate Describe the relationship between the data in your graph. What can you do with your model to solve the problem? Check the suggestions below that you will use:

.Plot the mystery data ٱ

.Draw a line of best fit ٱ

.Interpolate/extrapolate from the line of best fit ٱ

.Locate and explain outliers ٱ

.Gather more data ٱ

Infer/Conclude What information does your model provide about the suspect? Explain how you know. Complete a Suspect Profile Report using the results of your investigation.


2.4.2: Footprints (continued)

Suspect Profile Report

A. Evidence for the case Foot length: Stride length: B. Focus of the investigation Our team investigated C. Results of the investigation The results of our investigation determined that the suspect D. Process of the inquiry Describe what you did to solve this problem. Note: All information to support your conclusions must be submitted with this report.


Unit 2: Day 10: First Differences Grade 9 Applied

75 min.

Description • Investigate the pattern in the first differences to determine if a relationship is

linear or non-linear.

Materials • BLM 2.10.1,

2.10.2, 2.10.3


Minds On ... Whole Class Demonstration View two graph sketches; one suggesting a linear, the other a non-linear relationship (continuous data, both beginning at the origin). Students describe the graphs and match a table of values to the appropriate graph. Discuss their rationale for the matched graphs (accept all answers without judgement).

Action! Small Group Guided Discovery Learning Skills (Independent Work/Initiative)/Observation/Checklist: Demonstrate how to calculate the first differences as an example using BLM 2.10.1 Problem 1A so that students can work through the discovery task and reference the example. Students complete BLM 2.10.1 to determine the relationship between the shape of the graph and the first differences. They learn how to use first differences to determine linearity. They encounter a piecewise linear relationship in Problem 2. Curriculum Expectations/Self-Assessment/Marking Scheme: As a check for understanding at the end of the task, students should be able to complete BLM 2.10.2 correctly and provide a rationale for the process.

Consolidate Debrief

Whole Class Discussion Review the key learning in the guided discovery, referring to examples from the task and additional tables of values. Pairs Peer Tutoring Check for understanding using textbook questions.

Take care to line up the first differences half-way between rows in the table of values. Ensure that students know to subtract the upper value from the lower value in the chart when computing first differences.

Home Activity or Further Classroom Consolidation Complete the Using What You Have Discovered worksheet. Once students have had sufficient practice calculating first differences to determine whether or not a relationship is linear, administer a proficiency test. (See Section 2 – Patterning to Algebraic Modelling – Developing Proficiency p. 10.) Re-teach students who do not meet the proficiency target and provide further opportunities for these students to demonstrate that they have met the target.

Use BLM 2.10.3 as a quiz on Day 11. The Developing Mathematical Processes questions on p. 7 of the Patterning to Algebraic Modelling content-based package provides practice questions to help student develop proficiency.


2.10.1: First Differences Problem 1 A. Jody works at a factory that produces square tiles for bathrooms and kitchens. She helps determine shipping costs by calculating the perimeter of each tile. Calculate the perimeter and record your observations in column 2.

Side Length

(cm) Perimeter

(cm) First Differences

1 2 3 4 5

Describe what happens to the perimeter of each tile when the side length increases by one centimetre. Construct a graph of the perimeter of a tile vs. the

side length of the tile. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship

between the perimeter and side length of a tile.

d) Describe the shape of the graph. Calculate the first differences in column 3 of the table. What do you notice about the first differences? Summarize your observations.

a) When the side length increases by one centimetre, the perimeter increases by b) The plotted points suggest a c) The first differences are


2.10.1: First Differences (continued) B. Jody’s is paid $8.50/hour to calculate perimeters. Calculate her pay and record your observations in column 2.

Number of Hours Pay ($) First Differences

0 1 2 3 4 5

Describe what happens to her pay when the number of hours she works increases by one hour. Construct a graph of her pay vs. the number of hours she works. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship between her pay and the number of hours she works. d) Describe the shape of the graph. Calculate the first differences in column 3 of the table. What do you notice about the first differences? Summarize your observations.

a) When the number of hours worked increases by one, the pay increases by b) The plotted points suggest a c) The first differences are


2.10.1: First Differences (continued) C. Raj, another employee at the factory, also works with the tiles. He helps to determine the

shipping costs by calculating the area of each tile and recording his calculations in the table. Calculate the area and record your observations in column 2.

Length of sides (cm) Area (cm2) First Differences

1 2 3 4 5

Describe what happens to the area of each tile when the side length of a tile is increases by one centimetre. Construct a graph of area vs. the length of the sides of the tiles. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship between the area and the side length of the tile. d) Describe the shape of the graph. Calculate the first differences in column 3 of the table. What do you notice about the

first differences?

Summarize your observations.

a) When the side length increases by one centimetre, the area increases by b) The plotted points suggest a c) The first differences are


2.10.1: First Differences (continued) Problem 2 Chuck works on commission for sales. He earns $12.00 for each of the first 3 boxes he sells. He earns $24.00 each for boxes 4, 5, and 6, and $36.00 each for selling boxes 7, 8, 9, and 10.

1. Calculate Chuck’s earnings for the following numbers of boxes of files and record your answers in column 2.

Number of Boxes Earnings ($) First Differences

1 2 3 4 5 6 7 8 9

10

Describe what happens to his earnings when the number of boxes he sells increases by one box. Construct a graph of his earnings vs. the number of boxes he sells. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship

between his earnings and the number of boxes he sells.

d) Describe the shape of the graph. Calculate the first differences in column 3 of the table. What do you notice about the first differences? Summarize your observations.

a) When the number of boxes he sells increases

by one box, his earnings increase by b) The plotted points suggest a c) The first differences are


2.10.2: Using What You Have Discovered Deep Sea Divers The table shows data collected as divers descend below sea level. Calculate the first differences. Use the first differences to determine if the relationship is linear or non-linear. Check your solution by graphing.

Time (min)

Depth (m)

First differences

0 −2

1 −4

2 −6

3 −8

4 −10

The relationship is: Hot Air Ballooning The table shows data collected as a hot air balloon leaves the ground. Calculate the first differences. Use the first differences to determine if the relationship is linear or non-linear. Check your solution by graphing.

Time (sec)

Height (m)

First differences

0 0

1 2

2 6

3 12

4 20

The relationship is


2.10.3: Journal - Describing Relationships Use 1-cm cubes to help you.

K(3) Complete the following table.

Shape Number of Cubes Surface Area First Difference

1 6

2 10

3

4

5

6

K(2) Will the graph of the relationship between the number of cubes and the surface area be linear or non-linear? Use the first differences to explain your reasoning. K(2) Will the graph of the relationship between the number of cubes and the surface area

have a positive correlation or a negative correlation? Use the first differences to explain our reasoning.


2.10.3: Journal - Describing Relationships (continued) K(3) Make a scatter plot of the relationship between the number of cubes and the surface area. Construct a line of best fit.

1 2 3 4 5 6 7 8 9 10

10

20

30

40

50

K(1) Use your line of best fit to determine the surface area of the shape made from 10 cubes. TIPS What is the surface area of the shape made from 100 cubes? Explain your reasoning.




Reasoning and Proving (Thinking/ Inquiry Problem Solving)

Logical explanation

- no evidence of logic

- limited logic is evident

- explanation is somewhat logical

- explanation is logical

- explanation is highly logical


Unit 2: Day 11: Paper Folding Grade 9 Applied

75 min.

Description • Investigate exponential growth.

Materials • paper (various

sizes) • BLM 2.11.1,

2.11.2 • graph paper


Minds On ... Whole Class Demonstration Use a riddle to get students thinking about powers; for instance, this old English nursery rhyme: “As I was going to St. Ives, I met a man with 7 wives, every wife had 7 sacks, every sack had 7 cats and every cat had 7 kits. Wives, sacks, cats and kits, how many were going to St. Ives?” Pairs Problem Solving Working with a partner, students model a solution to this problem using a diagram. Whole Class Discussion Discuss how quickly the number going to St. Ives increases.

Action! Small Group Investigation Students fold a piece of paper and record their observations in the table, leaving the powers column blank until the whole class discussion (BLM 2.11.1). Note: They will be able to fold the paper only about 6 times. Whole Class Discussion Discuss the patterns in the table and then complete the table with the class. Students can suggest the number of layers for piece of paper with 10 folds by using the pattern of repeated multiplication by two. Record the power notation for the repeated multiplication, introducing terminology through the discussion. Using the patterns, working backwards to 20 with zero folds makes sense to them but may not be obvious. Small Group Investigation Students complete BLM 2.11.1. Discuss why students would expect the graphs to be discrete, non-linear and grow very quickly. Complete a difference table and discuss the pattern in the differences, noting that the pattern in the first differences is different from previous examples. Learning Skills (Organization)/Observation/Rating Scale and Curriculum Expectations/Observation/Mental Note:

Consolidate Debrief

Small Group Investigation Students investigate the speed of exponential growth in an application problem, focussing on the Infer/Conclude stage of the inquiry. They determine on which day the amount received is the same for either choice (BLM 2.11.2).

Consider giving students different sizes of paper to fold. Students may describe the patterns as: Doubles, 2 times as many, X2, doubled again. Consider other investigations such as exponential bacterial growth.

Home Activity or Further Classroom Consolidation Continue the investigation to answer this question: How many days would it take to have a billion dollars?


2.11.1: Investigating Paper Folds Hypothesize When a piece of paper is folded in half, there are two layers of paper. How many layers would be made by folding the paper in half ten times? Explore Fold the paper as many times as possible and complete the columns in the table.

Number of Folds

Number of Layers Pattern Power

0 1 1 2 2 3 4 5 6 7 8 9

10 Find a strategy that allows you to complete the table even though you may not be able fold the paper 10 times. Model On a grid, graph the relationship: Number of Layers vs. Number of Folds

Describe the shape of the graph. Manipulate/Transform Calculate the first differences.

Number of Folds

Number of Layers

First Differences

0 11 22 43 84 165 326 647 1288 2569 51210 1024

What do the first differences tell you about the relationship between the number of folds and the number of layers?


2.11.2: Which Would You Choose? The following question appeared on a popular television game show. The contestant's prize for the correct answer was the cash value of the correct choice. Which option would you choose?

Option 1: Receive $100 every day for a month (a month of 31 days).

OR

Option 2: Receive 1 cent the first day, 3 cents the second day, and 9 cents on the third day continuing in this way for 31 days.

Investigate which option you would choose using an inquiry process.

Hypothesize Predict the best option. Explore Collect some data. Model Organize the data in a table. Transform Look for patterns, generate some more values. Conclude Choose an option and a rationale stating why it is the best choice.


MATHO Game Create your own game card, using answers from the chart provided by the teacher. As a question is called, cover the answer on your card. Call out “MATHO” when you have covered all the answers in a row vertically, horizontally, diagonally, or the four corners.

M A T H O M A T H O

M A T H O M A T H O


MATHO Game Tracking Sheet (Teacher) Use this chart to track the questions called out randomly.

M A T H O

Multiplying Rule Dividing Rule Power of a Power Rule Two Rules Evaluate

27 22 × 25

27 210 ÷ 23

26 (23)2

27 26 × 23 ÷ 22

4 28 ÷ 26

38

33 × 35 38

310 ÷ 32 38

(34)2 311

310×34 ÷ 33 27

312 ÷ 39 49

4 × 48 49

415 ÷ 46 412

(43)4 49

48×42 ÷ 4 64

45 ÷ 42 25

23 × 22 25

26 ÷ 2 215

(23)5 29

210 ÷ 25 × 24 16

22 × 22 36

32 × 34 36

38 ÷ 32 312

(33)4 37

38 ÷ 32 × 3 81

32 × 32 47

44×43 47

410÷43 48

(44)2 47

45÷42×44 128 4×43

23 2 × 22

23 28 ÷ 25

214 (22)7

2 (24)2 ÷ 27

32 24×23 ÷ 22

34 32 × 32

34 38 ÷ 34

316 (34)4

318

(33)4 × 36 9

310 ÷ 39 × 3 45

42 × 43 45

417 ÷ 412 420

(42)10 410

414 ÷ (42)2 1

(42)4 ÷ 48 Reproduce this chart for students to select the answers for their MATHO card.

M A T H O

Multiplying Rule Dividing Rule Power of a Power Rule Two Rules Evaluate

27 27 26 27 4

38 38 38 311 27

49 49 412 49 64

25 25 215 29 16

36 36 312 37 81

47 47 48 47 128

23 23 214 2 32

34 34 316 318 9

45 45 420 410 1


Unit 2: Day 14: Describing Big and Small Numbers Grade 9 Applied

75 min.

Description • Revisit the paper folding and unfolding investigations. • Determine the thickness of many sheets of paper and the thickness of a slice of

a layer. • Introduce scientific notation.

Materials • scientific

calculator with overhead display

• scientific calculators


Minds On ... Whole Class Simulation Revisit the paper folding activity with the following question: If you could fold the paper 50 times, how thick would the folded layers be? Show the thickness of the folded paper and brainstorm and record suggestions for the thickness.

Action! Whole Class Demonstration Discussion Brainstorm ideas about how to determine the thickness of one layer of paper. Multiply 250 by the thickness of a layer of paper on their calculators to determine the height of the paper folded 50 times. Whole Class Discussion Students discuss the notation on the calculator display as a method for recording a very large number. Convert the number of millimetres to metres, then to kilometres and discuss the magnitude of this number. Make connections between the notation in standard form and the scientific notation on the calculator display. Revisit paper unfolding with the following question: If the paper was unfolded four times, how thick would the unfolded layer be? Demonstrate the effect of multiplying by a power with a negative exponent on the magnitude of a number. Make connections between the notation in standard form and the calculator display. Individual Response Journal Curriculum Expectations/Response Journal/Mental Note: Students respond to the following question: Why do you have to understand scientific notation in order to use a scientific calculator effectively? Include examples.

Consolidate Debrief

Small Groups Investigation Students translate between scientific and standard notation in a context.

• How thick is a paper that is folded in thirds when it is folded in thirds for the 50th time?

• If this three-way folded paper is unfolded for the fourth time, how thin is that slice of a sheet of paper?

Small Groups Worksheet Students practise translating between standard form and scientific notation but are not expected to manipulate these numbers beyond their use as a recording tool.

Introduce scientific notation. You can measure when the folded paper is 1 mm thick (8 layers created by 3 folds).

Application Exploration

Home Activity or Further Classroom Consolidation Find an object at home (tissue paper, paper towel) and repeat the folding and unfolding experiment to determine the thickness of many layers and the thinness of a slice of a layer.


Valu

e of

car

($)

Age of Car (years)

Value vs. Age of Car

Unit 2 Test Describing Relationships A(2) 1. Draw a scatter plot that shows the relationship between the value and the age of a car. Explain your reasoning. 2. Consider the scatter plot below. K(2) a) Does this scatter plot show a positive, a negative, or no correlation? Explain your reasoning. A(2) b) Label the axes of the scatter plot with two variables that could be represented

by this graph. 3. What type of relationship (positive correlation, negative correlation, or no correlation)

will most likely exist between the following sets of variables? Explain your reasoning. A(2) a) a player’s distance from a dart board and the player’s score A(2) b) the height of a student and the number of minutes they spent watching TV

last night 4. The Environmental Club collected the following data over a two-week period.

Number of Cans Sold

Number of Cans in Recycling Bin

60 55 87 80 55 52 20 18

100 92 42 38 68 60 90 12 50 45 60 56

K(3) a) Make a well-labelled scatter plot of the data.


A(2) b) Which is the dependent variable? Explain your reasoning. K(1) c) Circle any outliers on the scatter plot. A(2) d) Sketch a line of best fit on the scatter plot. K(1) e) Describe the relationship between the number of cans sold and the number of

cans recycled. A(1) f) Using your line of best fit: If 70 cans of pop are sold, how many cans would you expect to be in the recycling bin? A(1) g) Using your line of best fit: If 70 cans of pop are recycled, how many cans of pop would you expect to have

been sold?


5. The first figure was made using 5 toothpicks. The second figure is two connected “house” shapes; made using 8 toothpicks. K(2) a) Complete the following table.

Number of Connected “Houses”

Number of Toothpicks First Difference

1 5 2 8 3 4 5

K(2) b) Using the table, is this relation linear or non-linear? Explain. K(4) c) Make a well-labelled scatter plot of the data. A(1) d) Sketch a line of best fit. A(1) f) How many toothpicks would it take to make 10 connected “house” shapes? Explain your reasoning.

1 2 3


6. A ball is tossed into the air and the height of the ball is recorded at equal time intervals. The data is listed in the table below. K(2) a) Complete the following table.

Time (s) Height (m) First Difference 0.0 0 0.5 9 1.0 15 1.5 19 2.0 20 2.5 19 3.0 15 3.5 9 4.0 0

A(2) b) Which is the dependent variable? Explain your reasoning. K(2) b) Will the graph of the relationship between height and time be linear or non-linear? Explain your answer. K(2) c) Will the graph of the relationship between height and time have a positive correlation,

a negative correlation, or both? Explain your answer.


Day 18: Sunflowers

Roxanne, Jamal, Leslie, and Ada did a group project on sunflower growth for their biology class. They investigated how different growing conditions affect plant growth. Each student chose a different growing condition. The students started their experiment with plants that were 10 centimetres tall. They collected data every week for five weeks. At the end of five weeks, they were to write a group report that would include a table, a graph, and a story for each of the four growing conditions. Unfortunately, when the students put their work together, the pages were scattered, and some were lost. This page shows the tables, graphs, and written reports that were left. Find which table, graph and written report belong to each student. Create the missing tables, graphs, and reports.

C Time (weeks) 0 1 2 3 4 5

Height (centimetres) 10 10 10 10 10 10

G I treated my sunflower very well. It had sun, good soil and I even talked to it. Every week it grew more than the week before.

Jamal

D

Hei

ght

10

15

20

25

30

35

40

45

Time0 1 2 3 4 5 6

Sunflowers Scatter Plot

A Time (weeks) 0 1 2 3 4 5

Height (centimetres) 10 12.5 17.5 25 35 47.5

E I planted my sunflower in a shady place. The plant did grow, but not so fast. The length increased every week by equal amounts.

Roxanne

B

Hei

ght

10

15

20

25

30

35

40

45

Time0 1 2 3 4 5 6

Sunflowers Scatter Plot

F I put my plant in poor soil and didn’t give it much water. It did grow a bit, but every week less and less.

Leslie


Unit 3: Tell Me a Story Using Words and Equations Lesson Outline Grade 9 Applied

BIG PICTURE Students will: •• connect physical movement to resulting distance/time graphs; •• describe linearly related data using initial condition and constant rate of change; •• describe linearly related data graphically, in words, and algebraically; •• connect representations of linearly related data to determine optimal solutions.

Day Lesson Title Description Expectations1 Modelling Motion

Graphically Lesson included

•• Use Calculator Based Ranger (CBR) and graphing calculators to analyse motion graphs in terms of starting position, direction of motion, and speed.

NA1.01, NA1.05, NA4.03, RE1.03, RE1.06, RE2.02, RE2.05, RE3.02, RE3.04 CGE 3c, 5a

2 Writing Motion Stories Sample checklist provided (p. 93)

Learning Skill (Independent Work/Initiative)/ Observation/Mental Note and Curriculum Expectations/Interview/Checklist: •• Write motion stories of piecewise linear graphs,

demonstrating understanding of the connection between the position, direction and speed of motion and the shape of the graph.

•• Generate a variety of motion contexts for one motion graph. For example: a horizontal line on a distance vs. time graph could represent a situation when motion has stopped or it could represent motion in a circle around the position represented by the origin. Speed is described informally (She walked 0.5 metres in 3 seconds.) throughout the motion stories to prepare for a formal description of speed as the slope of the line on Day 3.

•• Include graphics that represent both horizontal (walking) or vertical (raising a flag) motions.

NA1.01, NA1.05, RE1.06, RE3.02, RE3.04, AG2.01

3 Stories with Slope Lesson included Presentation software file: Slopes included (p. 95-97)

•• Examine slope in a variety of contexts. •• Calculate slope using run

rise and connect to the unit rate of change.

NA1.01 - 1.03, NA1.05, NA4.03, AG2.01, AG2.02, AG2.04, AG3.01 - 3.02 CGE 2c, 5e


Day Lesson Title Description Expectations4 Quiz - Stories with

Slope Sample quiz included (p. 100-101)

Curriculum Expectations/Quiz/Marking Scheme: •• Focus on Knowledge and Understanding.

NA1.01 - 1.03, NA1.05, NA4.03, RE2.01, RE3.02, RE3.04, AG2.01, AG2.02, AG2.04, AG3.01 - 3.02

5 Walk the Line Lesson included

•• Use the graphing calculator and CBR to collect linear motion data in order to determine the equation of linear motion graphs using the starting distance and walking rate.

•• Use technology to verify the equation. “y = ⃞x + ⃞” or “y = ⃞ + ⃞x” is introduced only to be able to graph using technology.

•• Use the equation to find values that can be verified with the graph.

NA1.01 - 1.03, NA1.05, NA3.03, NA3.05, NA4.01, RE1.03, RE2.01, RE2.02, RE2.03, RE3.01, RE3.02, RE3.04, AG2.01 - 2.04, AG3.04 CGE 4b

6 Modelling Linear Relations with Equations Lesson included

•• Write equations from descriptions, tables of values and graphs.

NA1.01 - 1.06, NA3.03, NA3.05, NA4.01 - 4.02, RE2.01, RE3.01, AG2.01 - 2.04, AG3.03, AG3.04 CGE 5g

7 Graphing Linear Relations from an Equation Lesson included

•• Given an equation in context, graph the relationship using two methods.

•• Identify the meaning of point, slope and y-intercept in context.

NA1.01 - 1.06, NA3.03, NA4.02, RE2.01, AG3.01, AG3.02, AG3.03 CGE 2c, 5e


Day Lesson Title Description Expectations8 Linear Experiments

Lesson included

•• Investigate linear and non-linear relationships through investigation.

•• Examine first differences and the shape of the graph. •• Explore the effects of changing the conditions.

NA4.03, NA4.01, RE1.01 - 1.07, RE2.01 - 2.03, RE2.05, RE3.01, RE3.03 - 3.04, AG2.01 - 2.03, AG3.02 - 3.04 CGE 5a, 5b

9 Check for Understanding

Curriculum Expectation/Test/Marking Scheme: •• Ideas for questions include:

- Calculate the slope using rise/run (Knowledge). - Given a piecewise linear graph or equation –

write a linear story (Application). - Given a linear story – draw a graph and write

an equation (Application).

NA3.03, RE2.01, RE2.02, RE3.02, RE3.04, AG2.02, AG3.04

10

11

What Would You Choose?

Learning Skill (Work habits/Initiative)/ Observation/Rating Scale and Curriculum Expectations/Performance Task/Checkbric: •• Solve authentic problems by modelling linear

relations with an equation and graphing using graphing technology to investigate the best solution. For example: Choosing a cell phone provider based on fee plans for various call times; Choosing a repair company based on the length of the repair job; Choosing an employer based on pay scales, etc.

•• Include linear problems which provide two instances for a context such as: A three-topping pizza costs $11.75 and a six-topping pizza costs $14.00. What would a four-topping pizza cost? For example: Plot and join the points with a line segment. Extend the segment in order to determine the starting conditions, then read the starting conditions and slope from the graph to determine the equation for the relation. Use the equation to solve problems (substitution) and confirm the solution with the graph.

NA1.01 - 1.03, NA1.05, NA3.03, NA3.05, NA4.01, NA4.03, RE2.01, RE3.01, RE3.04, AG2.03, AG3.01, AG3.02, AG3.04 - 3.05

12 Kitty’s Kennel Costs Lesson included

•• Explore a variety of purchase options, propose a purchase plan and provide a rationale according to specific criterion.

•• Use graphing technology to investigate the solution.

NA1.01 - 1.03, NA1.05, NA4.01, NA4.03, RE2.01, RE3.04, AG3.01 - 3.05 CGE 2c, 5a



14

Review and Test Sample test included (p. 133-136)

Learning Skill (Work habits/Organization/ Teamwork)/Observation/Checklist: •• Provide a carousel of review tasks. Curriculum Expectations/Test/Marking Scheme: •• Focus on Knowledge and Application Questions.

NA3.03, NA4.01, NA4.03, RE2.01, RE2.05, RE3.02, RE3.04, AG2.02, AG3.04

15 Summative Performance Task Sample performance task – Pool Pass included (p. 137)

Curriculum Expectations/Performance Task/Rubric: •• Students model two or three linear relations with an

equation and graph. By reading and/or manipulating the graphs, students determine the best choice. This investigation could be similar to the Pool Pass problem included.

•• The Pool Pass problem may need more scaffolding depending on the learning needs of the students.

NA4.01, NA4.03, RE2.01, RE3.01, RE3.02, AG3.04

16 Return and take up Summative Assessment


Unit 3: Day 1: Modelling Motion Graphically Grade 9 Applied

75 min.

Description • Use Calculator Based Ranger (CBR) and graphing calculators to analyse

motion graphs in terms of starting position, direction of motion, and speed.

Materials • CBR • viewscreen • graphing calculators • BLM 3.1.1


Minds On ... Whole Class Demonstration Using the CBR (motion detector) and graphing calculator with a student volunteer, demonstrate connections between the shape and position of the graph and the direction, speed (including stopped), and starting position of their walk. Complete four demonstrations with the following instructions: • Start 1 m from the CBR and walk quickly away from the CBR at a constant

pace • Start 5 m from the CBR and walk slowly towards the CBR at a constant

pace. • Stand 3 m from the CBR. (Do not move.) • Stand 1 m from CBR and walk so that your speed gradually increases and

then gradually decrease your speed. Before each walk, students predict what they think the graph will look like by drawing in the air. Observe and provide feedback on their predictions.

Action! Pairs or Small Groups Peer Coaches Students investigate the connection between the shape and position of the graph and the direction, speed, and starting position by using the “MATCH” it application of the Ranger program. One student reads the graph to give walking instructions to the walker who cannot see the graph. Then they reverse roles. Once students have successfully matched the graphs, they investigate the effect on the graph when accelerating and decelerating and describe the curved or piecewise shape as it relates to the walker’s changing speed, position, and direction of motion and the concept of linear or non-linear. Learning Skill (Teamwork/Initiative)/Observation/Checkbric and Curriculum Expectations/Demonstration/Anecdotal: Check that students understand the difference between the path walked and shape of the graph by asking students to predict which alphabet letters can be walked. For example, a student could make the letter “w” but the letter “b” is not possible. Students explain why. Students use a CBR to verify/disprove predictions about the shape of distance time graphs.

Consolidate Debrief

Whole Class Note Making With the students, develop notes that match screen shots to descriptions of the motion for them to use as reference. These should include:

- initial position with respect to CBR - direction of motion - speed of motion (faster, slower, constant speed) - what STOP looks like

Depending on the availability of space or equipment and school policy, groups can do their walks in the hall while some of the students work on an independent assignment then change activity part way through the period. Students can use Walking and Writing Motion Stories (BLM 3.1.1) as reference in developing their stories.


Home Activity or Further Classroom Consolidation Draw a graph to match the following descriptions: • Stand 4 metres from the CBR and walk at a constant pace towards the CBR

for 5 seconds. Stand still for 3 seconds then run back to the starting position. • Begin 0.5 metres from the CBR, run for 3 seconds at a constant rate, then

gradually slow down until you have come to a complete stop.


3.2.1: Walking and Writing Motion Stories

Name: ______________________________________

Criteria: Does your story reference include: Yes √

The starting position of the walk?

The ending position of the walk?

The total time taken for the walk?

The direction of motion for each section of the walk?

The starting time(s) of any changes in direction or changes in speed?

The distance and time taken for each section of the walk?

An interesting story that ties all sections of the graph together?


Unit 3: Day 3: Stories with Slope Grade 9 Applied

75 min.

Description • Examine slope in a variety of contexts. • Calculate slope using run

rise and connect to the unit rate of change.

Materials • Presentation software file: Slopes (p. 95) • computer/projection unit


Minds On ... Whole Class Demonstration Show the electronic presentation Slopes p.95. Determine the slope of a line using the run

rise within a variety of contexts, and practise determining slope. Students understand how to connect the slope of the line and the “unit” rate of change described by the context.

Action! Pairs Worksheet Learning Skill (Work habits)/Observation/Anecdotal: Observe students’ work habits and make anecdotal comments. Students write stories from graphs using the slope (rate of change) and starting position. They draw graphs for stories that describe the starting position and rate of change, e.g., bath tub filling, bottle emptying, flag raising and lowering. They investigate and describe the effect on the graph of changing the rate and the starting conditions. Example: A flag is at half mast and is lowered at 85 cm/min. Graph and describe the effect on the graph of: a) lowering the flag at 50 cm/min. b) starting the flag at the top of the flag pole and lowering at the same

speed.

Consolidate Debrief

Whole Class Sharing Select students to share their results to draw out the mathematics.

If a projection unit is not available, the pages in the electronic presentation can be made into transparencies. NCTM has many activities that relate to slopes and graphs at www.nctm.org

Application

Home Activity or Further Classroom Consolidation • Complete Journal - Mapping a Story • Complete the following questions for additional practice: (Teacher inserts

text reference.)

Students generally find speed vs. time graphs more challenging than distance vs. time graphs. It may be helpful to guide students’ connection of matching distance vs. time and speed vs. time graphs before assigning Mapping a Story.


Slopes (Presentation software file) Download this file at www.curriculm.org/occ/tips/downloads.shtml

Slopes (or steepness) of lines are seen everywhere.

The steepness of the roof of a house is referred to as the pitch of the roof by home builders.

Give one reason why some homes have roofs which have a greater pitch.

There is less snow build up in the wintertime.

Engineers refer to the slope of a road as the grade.

They often refer to the slope as a percentage.

Slopes and Lines

rise

run

The slope of a line is the steepness of the line.

riseslope = run

8100

A grade of 8% would mean for every run of 100 units, there is a rise of 8 units.

8slope = 100

= 8%

The steepness of wheelchair ramps is of great importance to handicapped persons.

112

The slope of wheelchair ramps is usually about

112

If the rise is 1.5 m, what is the run?Ans: 18 m



Slopes (Presentation software file) (continued)

Determine the slope of the line.

5 cm

9 cm

59

m =

We use the letter m because in French the word for “to go up” is monter.

Because the slope is a ratio, there are no units such as cm or cm2.

3 m

5 m

Determine the slope (pitch) of the roof.

53

m =

23

m =

33

m =

1m =

Determine the slope of the staircase.

23

3

3

3

9

39

m =

13

m =

riserun

m =

2

4 6 1080 12 14

4

6

8

10

12

2

Determine the slope.

– 5

75

7m −=

riserun

m =


2

4 6 1080 12 14

4

6

8

10

12

2

707

m =

riserun

m =

0m =

Horizontal lines have a slope of zero.2

4 6 1080 12 14

4

6

8

10

12

2


6 60

m =

riserun

m =

(undefined)

Vertical lines have slopes which are undefined.2

4 6 1080 12 14

4

6

8

10

12

2


riserun

m =

positive

negative

zeroundefined

Summary: Types of Slopes


Slopes (Presentation software file) (continued)

Determine the slope of this line.

Which points will you use to determine rise and run?

5

40 riseslope = run405

m =

= 8

Determine the slope of the line segment.

60–2

260

m −=

130

= −

Draw a line which has a slope of 12

Draw a line which has a slope of 2

Draw a line which has a slope of 56−

–5

6

–5

6

Draw a line which has a slope of …

–3

1–3

1

a) – 3 b) 3

3

13

1


3.3.1: Journal – Mapping A Story Owen drives his motorbike to his home starting at the point where the motorbike is shown. Stop signs and traffic lights are shown on the map.

A graph of Owen’s drive home is shown below. • The vertical axis shows his speed. • The horizontal axis shows time.

Time

Spe

ed

On the map, highlight one possible route that Owen may have taken that matches the graph. Explain your reasoning.


3.3.1: Journal – Mapping A Story (continued) Select a different route for Owen to take home. Highlight this route on the map.

Time

Spe

ed

Draw a speed-time graph that describes Owen’s alternative drive home. Explain your reasoning.




Making Connections (Thinking/ Inquiry)

Creation of model to represent the data

- creates a model that represents none of the data or no model shown

- creates a model that represents little of the range of data

- creates a model that represents some of the range of data

- creates a model that represents most of the range of data

- creates a model that represents the full range of data


Clarity of explanation

- communicates unclearly

- communicates with limited clarity

- communicates with some clarity

- communicates with considerable clarity

- communicates concisely with a high degree of clarity

Conciseness of explanations

- no mathematics included

- rambling - somewhat concise

- concise - high degree of conciseness


3.4.1: Stories with Slope

One sunny June morning, Leslie decided to go for a jog. Use the graph below to write a creative story about the events that occurred during her jog. Include as much mathematical terminology as possible.

When Leslie returned from her jog at noon, she went to the fridge and took out a bottle of water. The graph to the right represents the volume of water in the bottle over time. Write a story about the volume in the bottle. Include as much mathematical terminology as possible.

At 8:00 p.m., Amir turned the water on to fill his cylindrical tub. • The water in the tub rose at a rate of 4 cm per minute. • When the water reached a height of 36 cm, he

pulled out the plug. • The water drained at a rate of 6 cm per

minute. Draw a graph showing the height of the water in the tub.

What time was it when the tub was drained?

5 10 15 20 25 30 35

50

100

150

200

250

300

350

time (min)

volu

me

(mL)

5 10 15 20 25 30

1

2

3

time (min)

dist

ance

(km

)


3.4.1: Stories with Slope (continued) At the beginning of September, Rick weighed 90 kilograms. He decided to start an exercise plan and lose some weight. The graph below represents Rick’s weight over the next 40 weeks. Use the graph to write a story about Rick’s weight loss. Include as much mathematical terminology as possible. The Ace Taxi Company charges a flat fee of $2.50 plus $0.50 per kilometre. Draw a graph to show the cost of a cab ride between 0 and 8 km long.

10

20

30

40

50

60

70

80

90

time (weeks)4 8 12 16 20 24 28 32 36 400

mas

s (k

g)


Unit 3: Day 5: Walk the Line Grade 9 Applied

75 min.

Description • Use the graphing calculator and CBR to collect linear motion data in order to

determine the equation of linear motion graphs using the starting distance and walking rate.

• Use technology to verify the equation. “y = ⃞x + ⃞” or “y = ⃞ + ⃞x” is introduced only to be able to graph using technology.

• Use the equation to find values that can be verified with the graph.

Materials • CBR • graphing calculator • BLM 3.5.1


Minds On ... Whole Class Demonstration With the help of a student volunteer (the walker), demonstrate walking away from a CBR to create a linear graph of a 10 second walk. Using the view screen calculator, project the graph onto blank chart paper posted on the wall. Trace the graph, axes, and scale onto the paper. Demonstrate the construction of a run

rise triangle under the graph. Mark the start and finish position using the co-ordinates (time, distance) of the points. Join the first and last point with a straight line. Discuss how to calculate the slope of this line using the run

rise formula. Discuss how to use the graph to extrapolate the distance from the CBR after 20 seconds.

Action! Small Groups Guided Exploration Learning Skill (Teamwork)/Observation/Checklist and Curriculum Expectations/Observation/Mental Note: Students work on the assignment (BLM 3.5.1). Encourage group members to support each other with the operation of the CBR experiment, e.g., running the Ranger Program, making sure the walking alley is clear. Students are not expected to write the motion equations using x for time and y for distance. “y = mx + b” will not be formally introduced until Unit 5. Explain to students that they must translate distance from the CBR = initial position + (rate of motion) × time of motion into y = b +mx so that the graphing calculator can be used. Discuss the issues that arise when collecting motion data towards the CBR: The run

rise triangle represents a negative slope. Note that data cannot be collected when the walker is behind the CBR.

Consolidate Debrief

Whole Class Connections Discuss what changes the students made to their equations to make a better match between the equation and the graph. Determine an equation for the demo graph constructed at the start of the lesson. Students verify their walking description statements with their equations by exchanging their work with a peer. Verify their understanding of ‘starting position’ and ‘walking rate’ by locating the graph and equation among the class set of work that begins the closest/farthest from the CBR; represents the fastest/slowest walk. Summarize how to model linear motions with an equation.

Emphasize the care and precision needed to copy the graph from the calculator to the handout. Use the TRACE key to move to the right along the line and read the position and time display at the bottom of the screen. Predicting the position after 20 seconds leads to a discussion about the range of the CBR and how close to the CBR students should stand.


Home Activity or Further Classroom Consolidation Record the ‘walking description statements’ of five of your classmates. Create the graph and equation for each. Use the information to determine the distance each classmate would be from the CBR after 30 seconds, if they walked at a constant rate.


3.5.1: Walk the Line: Set-Up Instructions Your group needs: • 1 CBR • 1 graphing calculator per student • 1 ruler per student Connect your calculator to the CBR with the Link cable and follow these instructions:

Setting up the RANGER Program • Press the APPS key • Select 2: CBL/CBR • Press ENTER • Select 3: RANGER • Press ENTER You are at the MAIN MENU Select 1: SETUP/SAMPLE Use the cursor → and ↓ keys and the ENTER key to set up the CBR: MAIN MENU START NOW REAL TIME: no TIME (S): 10 DISPLAY: DIST BEGIN ON: [ENTER] SMOOTHING: none UNITS: METRES Cursor up to START NOW

Press ENTER to start collecting data

1. Walk away at a steady pace. 2. Press ENTER then 5: REPEAT SAMPLE if necessary. 3. Press ENTER then 7: QUIT when you are satisfied with the graph. 4. Press GRAPH. This is the graph you will analyse.


3.5.1: Walk the Line: Part 1 (continued) Draw your Walk the Line graph. Stand about 0.5 metres from the CBR. Walk slowly away from the CBR at a steady pace. 1. Copy the scale markings on the distance and time axes from your calculator. 2. Mark your start and finish position on the graph using the co-ordinates (time, distance). 3. Connect the start and finish position with a line made with your ruler.

________________________’s Walk Calculate the slope of the graph (speed of your walk).

Draw a large runrise

triangle under the graph line and label it with the rise and run values.

Calculate the slope of your walk using the formula: runrisem =

Complete the following: The slope of my walk is ________________. The speed of my walk is ________________ m/s away from the CBR. Describe your walk. Use your starting position and slope to write a walking description statement: I started ______m from the CBR and walked away from it at speed of _____metres per second.

After 10 seconds, I was ___________________from the motion detector. At this rate, how far would you be from the motion detector after 30 seconds?

Time

Dis

tanc

e


3.5.1: Walk the Line: Part 1 (continued) Construct an equation to model your walk. A student started 0.52 metres from the CBR and walked away at speed of 0.19 metres/sec. The equation D = 0.52 + 0.19t models the student’s position from the CBR. To graph it on the graphing calculator use: Y = 0.52 + 0.19x. Write a walking statement and equation for your walk: _____________started _____from the CBR and walked away at speed of _____ metres/sec. The equation _____________________________models __________’s position from the CBR. The graphing calculator equation is _______________________________________________ . Verify your equation with your walk using the graphing calculator.

Turn off the STATPLOT

Type your equation into the Y= editor

Graph your equation (Press: GRAPH)

Turn on the STATPLOT. Press GRAPH again.

Change the numbers in your Y = equation until you get the best possible match for the graph you walked. The best equation that matches my walk is:


3.5.1: Walk the Line: Part 1 (continued) Use the equation to solve problems. The equation D = 0.52 + 0.19t models the student’s position from the CBR. The student's distance from the CBR after 30 seconds:

D = 0.52 + 19t D = 0.52 + 0.19 × 30 D = 6.22

The student will be 6.22 metres from the CBR. Calculate your position from the CBR after 30 seconds: The equation ____________________ models your position from the CBR. Calculate your distance from the CBR after 30 seconds. Check your answer with your graph.

First, turn off the STATPLOT Next, press: GRAPH Then, press: TRACE Arrow right until you reach 30 seconds.

Record the distance the CBR displays for 30 seconds _________. How does this compare with your answer using the equation? How does this answer compare with your estimate at the beginning of the activity? Use your equation to calculate how long it will take to walk to a position 1 km from the CBR.


3.5.1: Walk the Line: Part 2 (continued) Draw your Walk the Line graph. Stand about 3 metres from the CBR. Walk slowly towards the CBR at a steady pace. Copy the scale markings on the distance and time axes from your calculator. Mark your start and finish position on the graph using the co-ordinates (time, distance). Connect the start and finish position with a line made with your ruler.

________________________’s Walk Calculate the slope of the graph (speed of your walk).

Draw a large runrise

triangle under the graph and label it with the rise and run values.

Calculate the slope using the formula:runrisem = (Hint: The rise will be a negative number!)

Complete the following:

The slope of my walk is ________________ .

The speed of my walk is ________________ m/s away from the CBR. Describe your walk. Use your starting position and slope to write a walking description statement: I started ______m from the CBR and walked away from it at speed of _____metres per second. After 10 seconds, I was ____________from the motion detector. At this rate, how far would you be from the CBR after 30 seconds?

Time

Dis

tanc

e


3.5.1: Walk the Line: Part 2 (continued) Construct an equation to model your walk. Read this walking statement: A student started 4 metres from the CBR and walked toward it at a speed of 0.32 metres/sec. The equation D = 4 - 0.32t models the student’s position from the CBR at any time t. To graph it on the graphing calculator use: Y = 4 - 0.32x. Write a walking statement and equation for your walk: _____________started _____from the CBR and walked toward it at a speed of _____ metres/second. The equation ________________________ models my position from the CBR. To graph it on the graphing calculator use:

Verify your equation with your walk using the graphing calculator. You can change the numbers in your Y = equation until you get the best possible match for the graph you walked. The best equation that matches my walk is:


Unit 3: Day 6: Modelling Linear Relations with Equations Grade 9 Applied

75 min.

Description • Write equations from descriptions, tables of values and graphs.

Materials • BLM 3.6.1


Minds On ... Whole Class Discussion Briefly describe the activity and answer any questions the students may have.

Action! Pairs Peer Coaching Learning Skills (Teamwork/Initiative)/Checkbric: Students work in pairs to complete BLM 3.6.1. Students write the equation in the same manner that the line was described (initial conditions followed by rate × time). Students create and answer their own questions.

Whole Class Check Understanding Take up examples done by pairs. Using descriptions of relationships found in textbooks, have students orally form equations. Use the equations to solve for values that can be checked with the graph. For example: How high is the balloon after 2.5 minutes?

Consolidate Debrief

Whole Class Presentation Curriculum Expectations/Demonstration/Checkbric: Students present the graph or description and their equation to the class.

It may be appropriate to discuss with some or all students the fact that many of the relationships on BLM 3.6.1 are, in fact, discrete although a continuous line is shown on graphs.

Concept Practice Application

Home Activity or Further Classroom Consolidation Complete the following questions. (Teacher inserts text reference for additional practice on making equations and using the equation to solve for values.)


10 20 30 40 50 60 70 80 90 100 110 120 130 140

500

1000

1500

2000

2500

3000

3500

Number of Guests

Cost of Holding a Wedding at a Hotel

Cost ($)

Cos

t ($)

1 2 3 4 5 6 7 8 9 10 11 1

5

10

15

20

25

Number of Hotdogs

Cost of buying Hotdogs at the Baseball Game

Cos

t ($)

3.6.1: Modelling Linear Relations with Equations Partner A: ______________________ Partner B: _______________________ Write the equation for each relationship. Show your calculations.

A coaches B B coaches A

A banquet hall charges $100 for the hall and $20 per person for dinner.

The country club charges $270 for their facilities plus $29 per guest.

To rent a car for the weekend it costs $50 plus $0.16/km.

A race car had a head start of 0.5 km. It travels at a constant speed of 220 km/h. Write an equation for the total distance travelled over time.


3.6.1: Modelling Linear Relations with Equations (continued)

A coaches B B coaches A

Time (minutes)

Hei

ght (

m)

1 2 3 4 5 6 7 8 9 10

4

2

6

8

10

12

14

16

18

Height of a Balloon

Distance

(km) Cost of

Taxi Fare ($) 0 3.50 10 6.50 20 9.50 30 12.50 40 15.50

Number of People

Cost of Holding an Athletic Banquet

0 75 20 275 40 475 60 675 80 875


Unit 3: Day 7: Graphing Linear Relations from an Equation Grade 9 Applied

75 min.

Description • Given an equation in context, graph the relationship using two methods. • Identify the meaning of point, slope, and y-intercept in context.

Materials • BLM 3.7.1, 3.7.2, 3.7.3, 3.7.4 • graph paper


Minds On ... Whole Class Discussion Using BLM 3.7.1, discuss with students how to: • graph the equation by making a table of values (use 25, 50, 75,…for n) • relate the differences (“y” difference divided by “x” difference) to the

graph’s rate • determine the meaning of point, slope and y-intercept in context • answer questions related to the problem by solving the equation and then

verifying the answer using the graph. Scale on Graph: x-axis scale goes up by 25s, y-axis scale goes up by 200s

Action! Pairs Worksheet Curriculum Expectations/Demonstration/Mental Note: Students work in partners to complete BLM 3.7.2. Whole Class Discussion Discuss how to write the equation given the description (BLM 3.7.3). Graph the equation using the initial condition as the starting point then from this point using the rate as run

rise to build two more points on the line. Connect the points. For the remaining examples on BLM 3.7.3, guide a discussion about appropriate scales on the axes. Pairs Worksheet Learning Skill (Teamwork)/Observation/Checkbric: Students work in partners and coach each other as they complete BLM 3.7.4.

Consolidate Debrief

Whole Class Connections Discuss with students which method of graphing they preferred. Help students articulate strategies for determining scales for the x and y axes that will facilitate graphing. Decide whether a discussion about discrete vs. continuous functions or about step-wise functions is appropriate for the stage of concept attainment demonstrated by the class or by some individual students at this time. If so, point out that we often draw a continuous line graph as a short-cut even when a closer look would reveal that a function is discrete or piecewise.

Application Concept Practice Skill Drill

Home Activity or Further Classroom Consolidation Complete the following questions for additional practice. (Teacher inserts text reference.)

Suggested scales for the axes: BLM 3.7.3 Repair It x-scale: 1, y-scale: 5 Movie House x-scale: 1, y-scale: 5 Kite x-scale: 1, y-scale: 1 BLM 3.7.4 A Shape Fitness x-scale: 1, y-scale: 5 Repair Window x-scale: 1, y-scale: 10 Yum-Yum & Toy Sub x-scale: 1, y-scale: 0.5. BLM 3.7.4 B Taxi x-scale: 1, y-scale: 0.5. Bank Account x-scale: 1, y-scale: 10 Dino’s x-scale: 1, y-scale: 2 Katie x-scale: 1, y-scale: 0.5 While doing BLM 3.7.2, some students may recognize that many cell phone plans quote a fixed rate for the first so many minutes plus an additional cost for each additional minute. Such a relationship would result in a piecewise linear graph looking like: C t Piecewise linear functions are studied in Grade 10 Applied mathematics.


3.7.1: The Catering Problem Maxwell’s Catering Company prepares and serves food for large gatherings like weddings and company parties. They charge a base fee for renting the hall plus a cost per person.

Menu A Chef’s Salad Chicken Kiev

Broiled Potatoes Mixed Vegetables

Ice Cream Coffee or Tea

Menu B French Onion Soup

Chef’s Salad Roast Beef

Baked Potato Steamed Broccoli

Cheese Cake Coffee or Tea

Menu C Shrimp Cocktail

Chef’s Salad Steak & Lobster Baked Potato

Glazed Carrots Dessert Crepes Coffee of Tea

The following formulas are used to calculate the ordered pair (n, C). The number of people served is n. The total cost in dollars is C.

Menu A — C = 12n + 200 Menu B — C = 16n + 200 Menu C — C = 22n + 200

Complete the table of values for each relation: Menu A Menu B Menu C

Difference Difference Difference


3.7.1: The Catering Problem (continued) Graph the 3 relations on the same set of axes. Use an appropriate scale. Identify the slope and the C-intercept of the Menu C line. How do these relate to the total cost?

How it relates: Slope :

C-intercept :

Examine the differences. How do they relate to the graph and the equation? Compare the graphs. How are the graphs the same? different? For Menu B, what does the ordered pair (120, 2120) mean? Brad and Heather have invited 70 people to their 50th wedding anniversary. How much will it cost for each menu? Lui and Sami are planning their wedding. They have $3500 to spend on dinner. They would like to have Menu C. What is the greatest number of guests they can have at dinner? Pete’s Plastics employs 50 people. Each year the company plans a party for its employees. Find the cost for Menu B and write as the ordered pair (n,C). Find the cost for Menu C and write as the ordered pair (n,C). How many more dollars will Menu C cost than Menu B?


3.7.2: A Cell Phone Problem Two cellular phone companies charge a flat fee plus an added cost for each minute used. Call-a-Lot Plan C = 0.50t +20 Where C represents the total monthly cost and Talk-More Plan C = 0.25t +25 t represents the number of minutes. Create a table of values showing the total charges for a month for up to 30 minutes.

Call-a-Lot Talk-More Difference Difference

Graph the relations on the same set of axes. Use an appropriate scale.


3.7.2: A Cell Phone Problem (continued) Identify the slope and the C-intercept of the Call-A-Lot line. How do these relate to the total charges?

How it relates: Slope:

C-intercept:

Examine the differences. How do they relate to the graph and the equation? Compare the graphs. How are the graphs the same? different? For Talk-More, what does the ordered pair (8, 27) mean? Leslie used 13 minutes on the Talk-More plan. How much will it cost? Derek had a bill of $29 last month on the Call-a-Lot Plan. How many minutes did he use? Marsha thinks that she will use an average of 12 minutes each month. Find the cost for the Call-a-Lot Plan and write as the ordered pair (t, C). Find the cost for the Talk-More plan and write as the ordered pair (t, C). Which plan is better and how much will Marsha save?


3.7.3: The Speedy Way to Graph: Part 1 Partner A ___________________________ Partner B___________________________ Write the equation for the relationship and graph the relationship. A tennis club charges an annual membership fee of $25 plus $5 for a day pass to play tennis.

Equation:

Repair-It charges $60 for service call plus $25/h to repair the appliance.

Equation:

Movie House charges a membership fee of $10 plus $5 to rent each DVD. Equation:

A kite is 15 m above the ground when it gradually descends at a rate of 1.5 m/s. Equation:


3.7.4: The Speedy Way to Graph: Part 2 Partner A ___________________________ Partner B___________________________ Write the equation for the relationship and graph the relationship. Shape Fitness charges a monthly membership fee of $20 plus $5 per additional aerobics class.

Equation:

Repair window charges $20 service fee plus $10/h to fix the window pane. Equation:

Yum-Yum ice-cream shop charges $1.50 for the cone plus $1 per scoop of ice cream. Equation:

A submarine model starts 6.5 m above the bottom of the pool. It gradually descends at a rate of 0.25 m/s. Equation:


3.7.4: The Speedy Way to Graph: Part 2 (continued) Partner A ___________________________ Partner B___________________________ Write the equation for the relationship and graph the relationship.

A taxi cab company charges $3.50 plus $0.50/km. Equation:

Shelly has $250 in her bank account. She spends $10/week on snacks. Equation:

Dino’s Pizza charges $17 for a two-foot long pizza plus $2 per topping. Equation:

Katie sells programs at the Sky Dome. She is paid $10 per game plus 25 cents for every program she sells. Equation


Unit 3: Day 8: Linear Experiments Grade 9 Applied

75 min.

Description • Investigate linear and non-linear relationships through investigation. • Examine first differences and the shape of the graph. • Explore the effects of changing the conditions.

Materials • cube links • building blocks • BLM 3.8.1, 3.8.2 • graph paper • 1 cm grid paper • scale and 6 identical books


Minds On ... Small Groups Discussion Briefly explain what the students will be doing at each station. Write the following on the board to help students make good hypotheses: • Describe the relationship using the variables. (For example, as the number

of toppings increases, the price of the pizza increases.) • Is the relationship linear or non-linear? • Is the relationship continuous or discrete? Introduce the new terminology.

Action! Small Groups Carousel of Activities Learning Skill (Teamwork)/Observation/Checkbric and Curriculum Expectations/Investigation/Rubric: Observe and record students’ collaboration skills. Students complete each of the 6 experiments on BLM 3.8.2 and record their answers on BLM 3.8.1 and graph paper.

Consolidate Debrief

Whole Class Connections Make the connection between the first differences and the type of relationship. Discuss how changing the conditions of the experiments affects the graph and the equation (linear only). None of the linear experiments results in a continuous graph. Ask students to suggest linear experiments that would have a continuous graph, e.g., walking in front of a motion detector at a constant speed and graphing distance vs. time or letting out the string on a kite at a constant rate and graphing height vs. time.

To help students understand the difference between continuous and discrete data, tell them that discrete data is counted (number of tires on a truck) while continuous data is usually measured (distance travelled by the truck). Stations can be set up for students to work at as they complete the experiments. Be sensitive to food allergies if an actual chocolate bar is used for Experiment 2. See Experiments 1 – 6 Answers.


Home Activity or Further Classroom Consolidation Complete one of the following Journal entries: • Sally was not in class today. She doesn’t know how to use differences to

determine if a relationship is linear or non-linear. Use words, pictures, and symbols to explain it to her.

• Explain the difference between continuous and discrete data.


3.8.1: Linear and Non-Linear Investigations Record Sheet Group Experiment # ____: ________________________________________________ Make a hypothesis about the relationship. Make the following Mathematical Models: Complete the table of values and calculate the differences.

Differences

On graph paper, make a scatter plot and draw the line (or curve) of best fit. Make a conclusion.

If the relationship is linear, write the equation of a line of best fit.

OR

If the relationship is non-linear, describe the relationship. Refer to your hypothesis.


3.8.2: Experiment 1 – Building Crosses Purpose Find the relationship between the diagram number and the total number of cubes. Procedure Use Linear and Non-Linear Investigations record sheet. Record your hypothesis. Using linking cubes, make three more models by adding a cube to each end. Mathematical Models Record your observations in the table of values.

Calculate the differences.

Make a scatter plot and draw the line (or curve) of best fit.

Record your conclusion.

Answer the following questions on the back of the worksheet. 1. How many cubes are required to make model number 10? Show your work. 2. What model number will have 25 cubes? 3. How would adding two blocks to each end rather than one affect the graph and the

equation?

1 2 3


3.8.2: Experiment 2 – Pass the Chocolate Bar Purpose Find the relationship between the number of pieces of chocolate bar remaining and the total number of times the chocolate bar was passed around. Procedure Use Linear and Non-Linear Investigations record sheet. Record your hypothesis. Every time the chocolate bar is passed, someone “eats” half of what remains. Mathematical Models Record the number of pieces of the chocolate bar that remain after 0 passes, 1 pass, 2 passes, - up to 4 passes.

Calculate the differences.

Make a scatter plot and draw the line (or curve) of best fit. Record your conclusion. Answer the following questions on the back of the worksheet. 1. How many pieces of chocolate bar will remain after 6 passes? Show your work. 2. Using this method of “eating” the chocolate bar, when will it be fully “eaten?” Explain. 3. If the chocolate bar began with 32 pieces instead of 16, how would the graph be different?

Include a sketch of the original graph and the new graph on the same set of axes. Give reasons for your answer.


3.8.2: Experiment 3 – Area vs. Length of a Square Purpose Find the relationship between the area and the length of a side of a square. Procedure Use Linear and Non-Linear Investigations record sheet. Record your hypothesis. On grid paper, draw a square of side length 1cm. Draw and calculate the area of squares with sides measuring 2 cm, 3 cm, 4 cm, and so on. Mathematical Models Measure and record the area of the square. Calculate the differences. Make a scatter plot and draw the line (or curve) of best fit. Record your conclusion. Answer the following questions on the back of the worksheet. 1. What is the area of a square with side length 9 cm? 2. What side length does a square with an area of 100 cm2 have? 3. Do you notice a pattern in the differences? If, so describe it.

1 cm

2 cm


3.8.2: Experiment 4 – Mass vs. Number of Textbook Books Purpose Find the relationship between the mass of textbooks and the number of textbooks on a scale. Procedure Use the Linear and Non-Linear Investigations record sheet. Record your hypothesis.

Measure the mass of one copy of the textbook provided.

Add another book and find the mass.

Repeat until there are 6 books on the weighing scale.

Mathematical Models Record the number of books and total mass. Calculate the differences. Make a scatter plot and draw the line (or curve) of best fit. Record your conclusion. Answer the following questions on a separate page. 1. If smaller books were used, how would that change the equation and the appearance of the

graph? 2. If larger books were used, how would that change the equation and the appearance of the

graph? 3. Predict the mass of 15 books. Show your work.


3.8.2: Experiment 5 – Building Pyramids Purpose Find the relationship between the number of blocks and the model number. Procedure Use the Linear and Non-Linear Investigations record sheet. Record your hypothesis. Using blocks or cube links (do not connect them) build the next three pyramids. Mathematical Models Record the diagram number and the total number of blocks. Calculate the differences. Make a scatter plot and draw the line (or curve) of best fit. Record your conclusion. Answer the following questions on the back of the worksheet. 1. How does the diagram number relate to the number of blocks on the bottom layer? 2. Tammy thinks that if you know the diagram number, you can figure out the total number of

blocks by using the formula:

2

)2)(1( ++ nn where n is the diagram number.

Show that she is correct.

21


3.8.2: Experiment 6 – Burning the Candle at Both Ends Purpose Find the relationship between the total number of blocks and the model number. Procedure Use Linear and Non-Linear Investigations sheet. Record your hypothesis. Using cube links build a long chain with 20 blocks. To create the next diagram, remove one cube from each end. Record the number of blocks. Repeat this process three more times.

Mathematical Models Record the diagram number and the total number of blocks. Calculate the differences. Make a scatter plot and draw the line (or curve) of best fit. Record your conclusion. Answer the following questions on the back of the worksheet. 1. How many cubes are required to make model number 7? Show your work. 2. What model number will have 4 cubes? 3. How would removing 2 blocks to each end of the “candle” rather than 1 affect the graph and

the equation? 4. If 5 more blocks were added to the original diagram, how would that affect the graph and the

equation?


3.8.2: Experiments 1 – 6 (Answers) Experiment 1 – Building Crosses • linear and discrete • Equation: T =1 + 4(n – 1), where T is the total number of blocks and n is the diagram number, or T = 4n - 3

Experiment 2 – Number of pieces of chocolate bar vs. number of passes • non-linear and discrete • Pattern: 16, 8, 4, 2, 1, 2

1 Experiment 3 – Area vs. length of the side of a square • non-linear and continuous • Pattern: 1, 4, 9, 25, 36… Experiment 4 – Mass vs. number of textbooks • linear and discrete • Equation: M = ___n , where M is the mass and n is the number of books Experiment 5 – Building pyramids • non-linear and discrete • Pattern: 3, 6, 10, 15… Experiment 6 – Burning the candle at both ends • linear and discrete • Equation: T = 20 − 2n, where T is the total number of blocks and n is the diagram number


Unit 3: Day 12: Kitty’s Kennel Costs Grade 9 Applied

75 min.

Description • Explore a variety of purchase options, propose a purchase plan, and provide a

rationale according to specific criterion. • Use graphing technology to investigate the solution.

Materials • graphing calculators • BLM 3.12.1


Minds On ... Whole Group Understanding the Problem Read through the scenario and discuss what the final product must look like. (BLM 3.12.1) Using Clarify the Problem, discuss the various options for solving it. Point out that the Lees will be away for weeks but the Kennel information is given in terms of days. Is there any information that is not needed or cannot be used? Use questioning to ensure understanding. Make connections to a previous lesson in which a decision was determined by interpreting the graph according to the parameters of the problem.

Action! Pairs Performance Task Curriculum Expectations/Exhibition/Checkbric: Students complete a solution with their partner. The graphical model is provided below:

Consolidate Debrief

Whole Class Discussion Curriculum Expectations/Exhibition/Demonstration/Checkbric: Share and discuss equations and graphs. Reinforce connections between the parameters of the equations and features of the graph and how they were used to solve the problem. Students present their recommendations and justifications.

Answer questions about the scenario but do not provide a strategy for solving the problem. Notice that the 4th kennel is for dogs only.


Home Activity or Further Classroom Consolidation Revisit this problem with the following question: Tanya needs to put her cat in a kennel while she is away for 12 days; Toby plans to board his kitten for 3 weeks; and the Jones don’t know if they will be on holidays for 2 or 3 weeks. Which kennel should these people choose for their cats?


3.12.1: Kitty’s Kennel Costs The Lees and their children are planning to take an extended holiday. They may be away for 4 to 6 weeks. They need to board their cat, Kitty, while they are on holiday. They want to spend the least amount of money possible. The Lees have researched the costs of local kennels and have clipped the following ads from their local newspaper.

Your job is to: • make a recommendation to the Lees about where they should board their cat and support

your answer with evidence; • show the Lees how to analyse their situation and decide which kennel meets their

requirements; • explain your process to them so that if they change their minds later and want to use a

different kennel or change the length of their holiday they understand how to do the calculations themselves.

Your report must include: • reasons for excluding any of the advertisements; • your best guess (hypothesis) based on the information given in the advertisements before

examining the data thoroughly; • the procedures that you used to compare the information in the advertisements (including

how you made use of technology); • recommendations to the Lees about where they should board their cat and evidence to

support your answer.


3.12.1: Kitty’s Kennel Costs (continued) (A) Clarify the Problem Examine the advertisements and list any important information.

Is there any information you do not need?

Make a hypothesis about which kennel you feel is the best choice for the Lees. Briefly explain your reasoning. (B) Create Models Write an equation for each advertisement that is appropriate to investigate. Define the variables.

____ represents the cost _______________________of boarding

____ represents the number of _____________ days or weeks

Rewrite each of the equations in y = x + form for inputting into the graphing calculator.

Cozy Kennel Feline Holidays Pet Paradise Bow Wow

Weekend Equation

Graphing Calculator Equation

Create a graphical model for each of the kennels that is appropriate for their cat.

Show all graphs on the same grid and label them clearly.

Reproduce the graph you see on the graphing calculator below.

Indicate the window you used.

You may need to zoom in using different windows to see the information you need.


3.12.1: Kitty’s Kennel Costs (continued) (C) Manipulate the Model If the Lees decide to take a 4, 5, or 6 week holiday, list the kennel costs:

Hint: Use the TRACE feature on your calculator.

Number of Weeks Cozy Kennel Feline

Holidays Pet Paradise Bow Wow Weekend

4

5

6 Explain where Kitty should stay if the Lees are going away for: (a) 4 weeks (b) 5 weeks (c) 6 weeks You see an advertisement for a new kennel in their neighbourhood. It charges $50 for an initial flea check and treatment, plus $10/day. Should they consider this new kennel or reject it? Give reasons for your answer that demonstrates your understanding of the effects of changes in rates in initial conditions. (D) Reconsider the Original Problem Where do you think the Lees should board their cat? Explain your reasoning. (E) Communicate the Ideas Write your report.


Time (seconds)

Dis

tanc

e (m

etre

s)

Time (seconds) D

ista

nce

(met

res)

Time (seconds)

Dis

tanc

e (m

etre

s)

Unit 3 Test Tell Me A Story and Lines A communication level will be assigned based on the correct use of mathematical symbols, labels, and conventions. A(2) 1. The graph describes Rami’s walk with a

motion detector. Tell the story that describes this graph. Use distances and times in your story.

2. A story is described in each question. Sketch the graph that describes the story

in the screen provided. A(2) Begin 5 m from the wall.

Walk toward from the wall for 5 seconds. Stop for 5 seconds. Run back to your starting position. Stop.

A (2) Begin at the wall.

Walk very slowly away from the wall for 3 seconds. Increase your speed for 3 seconds. Stop for 3 seconds.

Walk very slowly toward the wall for 3 seconds.

Run back to the wall. Stop.

A(3) 3. Jen tried her new snowboard at the One Plank Only Resort. The graph shows her first run. Tell the

story that describes Jen’s first run.


4. Find the slope of each line segment. K(2) 5. If a wheelchair ramp has a slope greater than 0.1 in size, then it is considered

unsafe. Determine whether or not each of the following ramps is safe. Show your work and explain your answer on the back side of this page. K(4)

K(2) 6. Draw a line segment with a slope of 73

− .

210 cm

20 cm

120 cm

15 cm

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10


7. Arcadia charges players a $15 admission fee to their gaming centre. Arcadia also charges each player $5 per game.

K(2) Write an equation to model the cost of admission at the Arcadia. K(2) What is the rate for this relation and how does it relate to the cost of a day at

Arcadia? K(2) What are the initial conditions for this relation and how do they relate to the cost of

a day at Arcadia. K(4) Graph the relation. K(1) How many games can Jeremy play if he has saved $60 for a day at Arcadia? K(1) How much will it cost Renay to spend a day at Arcadia if she plays 30 games? A(2) How would the graph from a) change if Arcadia decreases the admission fee to

$10? Write an equation that represents the new cost of a day spent gaming at Arcadia.

A(2) How would the graph from a) change if Arcadia increases the cost per game to $7? Write an equation that represents the new cost of a day spent gaming at Arcadia.


A law office plans to do some landscaping around their building. They have two estimates: Company A: $240 for a full landscape plan plus $30 per hour to do the work Company B: $60 per hour to do the work (which includes the landscape plan) K(3) Complete the tables of values for each company.

Company A Company B Time (h) Cost ($) Time (h) Cost ($)

0 0 2 2 4 4 6 6 8 8

10 10 12 12 14 14 16 16 18 18 20 20

K(3) Display this data on the grid below. Label carefully and use a different colour for each company. A(3) Complete each of the following sentences.

I would choose Company A if…

I would choose Company B if…

2 4 6 8 10 12 14 16 18 20

100

200

300

400

500

600

700

800

900

1000


3.15.1: Pool Pass The local swimming pool is open 5 days a week for 8 weeks during the summer holidays. The admission prices are displayed at the entrance.

Splash World Swim Park Price List

Full Season Pass ……… $120 Partial Season Pass …… $60 plus $2 per day Daily swim pass ………… $5

Explore How much will it cost one person to go to the pool every day the pool is open? a) with a full season's pass? b) with a partial season's pass? c) with a daily pass? Discuss with your partner • Is the full season’s pass the best deal if you attend daily? Give reasons. • Do you think that the full season's pass is the best deal if you attend for 20 days?

Give reasons. Investigation Inga, Stu, and Malauia are hoping to spend some time cooling off at the pool this summer. Each person investigated the cost of each plan based on the number of days they plan to

go to the pool. • Inga determined that the daily pass was the cheapest for her. • Stu has chosen the full season's pass. • Malauia purchased the partial season’s pass.

Investigate to determine the number of days Inga, Stu, and Malauia plan to attend the pool. Include details to explain how you arrived at your solution.


Unit 4: Plane Geometry Grade 9 Applied Lesson Outline

BIG PICTURE Students will: •• investigate properties of geometric objects using dynamic geometry software and manipulatives; •• determine the properties of medians, altitudes, and angles bisectors in triangles; •• illustrate and explain the relationship between angles in parallel lines and interior and exterior angles of

triangles and quadrilaterals; •• determine some properties of sides and diagonals of quadrilaterals. Note: Students may have a very broad range of experience with using The Geometer’s Sketchpad®. Skills can be taught as they are needed for each lesson or alternatively Introduction to Geometer’s Sketchpad file: (p. 140-141) could be used at the beginning of the unit.

Day Lesson Title Description Expectations 1 Plane Geometry –

Introduction Lesson included GSP file: Plane Geometry provided (p. 143-145)

•• Explore geometrical concepts (angles, triangles, parallel lines).

•• Build skills required for future use of The Geometer’s Sketchpad® (GSP).

Review of Grade 7 and 8 Geometry expectations MG3.01 CGE 2a, 5a

2 What’s So Special? Part 1 Lesson included GSP file: What’s So Special provided (p. 149)

•• Build investigation skills by dynamic exploration of geometrical concepts.

•• Develop communication skills and geometric vocabulary.

MG3.04 CGE 2c, 4b, 5a

3 What’s So Special? Part 2 Lesson included

•• Present observations and conclusions from dynamic geometry software explorations.

MG3.04, RE3.04 CGE 2a, 2c

4 Interior and Exterior Angles of Triangles and Quadrilaterals GSP files: Exterior Angle Demonstration, Sum of the Interior Angles of a Quadrilateral, and Exterior Angles provided (p. 151, 152, 153)

•• Through investigation and demonstration examine the sum of the interior and exterior angles of triangles and quadrilaterals using The Geometer’s Sketchpad®.

•• Investigate the relationship between the number of sides of a polygon and the sum of the interior angles of the polygon – this is an opportunity to connect the Relationships strand and the Measurement and Geometry Strand.

MG3.01, MG3.04, RE1.03, RE1.04, RE1.06, RE2.01


Day Lesson Title Description Expectations 5 Using the

Properties •• Practise using the geometry explored in previous

lessons. Note:

Ensure that students see geometry in context (example: opposite angle theorem can be observed in the angles in the legs of an ironing board or other stand, railways and road crossings provide context for parallel lines, flight paths provide additional context). Assign paper-folding tasks in preparation for Lesson 6.

MG3.01, MG3.04

6

7

Triangle Centres GSP files: Map, Mystery Points, Triangles, and Exploring Triangles provided (p. 154,155, 156, 157)

•• Demonstrate how to construct medians, altitudes, and angle bisectors through paper folding (see Lesson 5).

•• Investigate the properties of medians of triangles through the context of finding the balance point of a triangle or dividing land to give equal areas.

•• Investigate altitudes – use altitudes to find the area of a triangle in three different ways (using three different bases).

•• Curriculum Expectations/Quiz/Marking Scheme: Paper-and-pencil quiz on determining unknown angle measurements and giving reasons.

MG3.02, MG3.04

8 Exploring Diagonals GSP files: Diagonals, Measure, and Quadrilaterals (p. 158, 159, 160)

•• Investigate the sides and diagonals of quadrilaterals. Note:

This lesson could provide an opportunity to connect the Measurement and Geometry strand to Number Sense and Algebra through the Pythagorean theorem.

MG3.03, MG3.04

9 Summative Assessment

Curriculum Expectations/Investigation/Rubric Curriculum Expectations/Test/Marking Scheme •• Create a paper-and-pencil test to assess

knowledge/understanding of geometric properties and vocabulary. EQAO release material provides examples of short answer questions for this strand.

•• Create a Geometer’s Sketchpad® investigation that involves the figure that is formed by the midpoints of the sides of a quadrilateral, e.g., a parallelogram Example: The boundary of a provincial park forms a quadrilateral. The four ranger stations in the park are located at the midpoints of the four sides of the park. Roads have been constructed so that rangers can travel directly from one station to another station. Give a complete description of the roads in the park.

MG3.01 - 3.04


Introduction to Geometer’s Sketchpad (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

When you are finished, click on the Link button to go to Page 2

Try experimenting with these tools by constructingvarious objects on the right of the screen.

The four tools for constructing objectsare at the left of the screen.1) Point Tool (dot)2) Compass Tool (circle)3) Segment T ool (line segment)4) Text Tool (letter A)

Geometer's Sketchpad allows us toconstruct, measure, and animate and movegeometric objects easily.

Introduction to Geometer's Sketchpad.

Click on the Link button to go to Page 3

By highlighting an object you can also Hide the objectfrom the Display tab. This does not delete it, but hides itfrom view.

Another way to show or hide the label is tohighlight the object, then from the Display tab,choose Show/Hide Label.

To hide a label, click on the object, choose the Displaytab then choose Hide Label.

A point and a line segment are drawn at right. To label them click on the Text Tool (letter A)then click on the point or the line segment.

Labeling Objects.

Constructions.

To construct the midpoint of CD, click on the linesegment then, from the Construct menu, chooseMidpoint.

To construct a line parallel to FG highlight point Eand segment FG T hen, from the Construct menu,choose Parallel Line.

To construct a line perpendicular to FG, highlight point E andsegment FG Then, from the Construct menu, choosePerpendicular Line.

To construct a line segment between points A and B,click on the points then, from the Construct menu,choose Segment


BA

E

GF

DC


To construct the angle bisector of ABC, click on thepoints ABC (order is important) then, from theConstruct Menu, choose Angle Bisector.

Constructions 2

To construct the interior of DEF , click on thepoints DEF. Then, from the Construct Menu,choose Triangle Interior.

You can change the color of the interior byhighlighting it then, from the Display Menu, chooseColor.

F

D

E

CB

A


To change the precision of the measurements, go tothe Edit tab and choose Preferences.

∠CDE is drawn at right. To measure the angle,click on points CDE (order is important) then go tothe Measure menu and choose Angle.

Be sure that only the line segement is highlighted. Tode-highlight an object click anywhere on the screen.

Measuring Objects (segments and angles).

A line segment is drawn at right. To measure thelength of the line segment, click on it then go to theMeasure menu above and choose Length. A B

C

D E

Highlight the interior of the triangle then, from theMeasure Menu, choose Perimeter

Practice: Determine the area andperimeter of parallelogram DEFG.

Highlight the interior of the triangle then, from theMeasure Menu, choose Area.

Construct the interior of ABC (click on pointsABC then choose Triangle Interior from theConstruct Menu).

Measuring Objects (perimeter and area)


E

B

A

C

G F

D



Introduction to Geometer’s Sketchpad (GSP file) (continued)

m∠DFE m∠DEF m∠EDF m∠DFE+m ∠DEF+m ∠EDF

63° 43° 74° 180 °

On the next page you will learnhow to make a tabulation table.

Move point F then double-click onthe tabulation table again.

What do you notice? ______

Measuring Angles (tabulating)

DEF is constructed at right.All of the angles have been measured as well asthe sum of the angles.Move point D to change the measures of theangles, then double-click on the tabulation table.

m∠DFE+m∠DEF+m∠EDF = 180°m∠EDF = 74°m∠DEF = 43°m∠DFE = 63°

F

D

E

Change the size of the triangle then click the tabulate box. What do you notice? Do this a few times.

Highlight all three measures as well as the sum.Then, from the Graph menu, choose Tabulate.

When you see the calculator, click on the first measure followedby the plus sign (+) then click on the second measure followedby the plus sign (+) then click on the third measure followed byOK.

When you have determined the three measures,click on the Measure menu and chooseCalculate.


Measuring (sum of angles)Determine the measures of the three anglesof the triangle (see page 5).

C

A

B


6- What do you notice? ______

5- Use the Calculator to divide the area of the large triangleby the area of the smaller triangle.

4- Determine the area of the large triangle.

3- Determine the area of the small triangle in the middleformed by joining the midpoints.

2- Join the midpoints of the sides.

1- Construct any triangle and determine themidpoints of the three sides.

Practice

Construct line segments and triangles andpractice constructing the following:1- parallel lines2- perpendicular lines3- angle bisectors4- lengths of line segments5- area and perimeters

Practice


Unit 4: Day 1: Plane Geometry – Introduction Grade 9 Applied

75 min.

Description • Explore geometrical concepts (angles, triangles, parallel lines). • Build skills required for future use of The Geometer’s Sketchpad® (GSP).

Materials • BLM 4.1.1 • GSP file: Plane Geometry (p. 143)


Minds On ... Small Groups Brainstorm Curriculum Expectations/Observation/Mental Note: Circulate while students are working to assess prior learning. Use a Graffiti activity. Post sheets of chart paper around the room with the following topic titles:

• triangles • quadrilaterals • polygons • angles • lines/line segments

Groups record prior learning on one of the concept sheets. Students might draw sketches, write definitions, state properties, etc. Rotate so that each group that visits adds notes to each sheet.

Action! Pairs Guided Exploration Learning Skills/Observation/Rubric: Circulate to assess teamwork while students are working. Students use the GSP file Plane Geometry and BLM 4.1.1 to review geometric concepts from Grades 7/8 and to build skills required for future activities with The Geometer’s Sketchpad®. During the exploration one student is the computer “driver” the second student is the “recorder.” Part way through the activity students change roles. Provide feedback to student responses to BLM 4.1.1 as they work. Students can check their answers by using the GSP sketch.

Consolidate Debrief

Whole Class Discussion Connect the Graffiti responses to the computer exploration by asking students if anything needs to be added/changed/deleted on the topic sheets. Ask students for illustrations of the concepts, e.g., a swinging door or turning pages in a book could be used to discuss complementary or supplementary angles. As a class, make up acronyms for the theorems, e.g., students may suggest OAT for the Opposite Angles Theorem. In later lessons, students use the acronyms when asked to give reasons for their answers. Ask students to bring an optical illusion to the next day’s class.

As an introductory activity, ask students to search the web to discover the meaning of the word geometry. The board media centre or school resource centre might contain a video that will add a visual component to the introduction of this unit. Start a word wall. (BLM 4.1.2.) The list could be used for a Charade word game as a vocabulary review. Resources use different acronyms for geometric theorems. Acronyms are great time-savers when students are asked to give reasons for their answers

Reflection Skill Drill

Home Activity or Further Classroom Consolidation Make a title page for this unit. Include information from the Graffiti exercise and the GSP exploration. Use your textbook as an additional resource. (Teacher inserts page references.) Practise using the theorems. (Teacher inserts appropriate exercises from textbook.)

Learning Skills/Activity/ Rubric: Collect the title page and assess for work habits.


4.1.1: Plane Geometry (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml Use this page to record your observations and conclusions. Determine the unknown angle in the right column. Give reasons for your answer.



4.1.1: Plane Geometry (GSP file) (continued)

Given: ABC and exterior angle ∠ABD.

Explore: Measure the non-adjacent angles∠ACB & ∠CAB.

Calculate the sum of these two angles. Measure the exterior angle ∠ABD.

Drag point A.

Conclude: What do you notice about the sum of the two non-adjacent angles, ∠ACB & ∠CAB, inside the triangle and the exterior angle, ∠ABD?

7) Exterior Angle Theorem

How do I ...?

A

C

D

B


4.1.1: Plane Geometry (GSP file) (continued)


4.1.2: Word Wall List Acute angle Obtuse angle

Acute triangle Obtuse triangle

Adjacent angles Octagon

Alternate angles Opposite angle

Bisector Parallelogram

Circle Pentagon

Co-interior angles Perpendicular

Complementary angles Perpendicular bisector

Congruent figures Point

Corresponding angles Polygon

Diagonal Quadrilateral

Diameter Radius

Equilateral triangle Ray

Exterior angle Rectangle

Hexagon Reflex angle

Interior angle Rhombus

Isosceles triangle Right angle

Kite Right angled triangle

Line Scalene triangle

Line segment Side

Midpoint Similar figures


Unit 4: Day 2: What’s So Special? – Part 1 Grade 9 Applied

75 min.

Description • Build investigation skills by dynamic exploration of geometrical concepts. • Develop communication skills and geometric vocabulary.

Materials • data projector • BLM 4.2.1, 4.2.2 • GSP file: What’s So Special? (p. 149)


Minds On ... Whole Class Demonstration Discuss BLM 4.2.1 as a reminder for the process students must use in formulating and testing hypothesis and display the GSP file What’s So Special?, using a data projector. Discuss the optical illusions on the first sketch – people see different things in an optical illusion – students also see different things when they look at geometric diagrams. Emphasize the importance of discussing and recording observations so students can learn from each other. Discuss the hidden geometry in the constructed equilateral triangle, e.g., some sketches are special because some things always remain true in any drag test. Use the circles to demonstrate how to determine if two quantities are proportional, e.g., measure the areas of both circles, explore by dragging, determine the ratio of the areas by using the Calculate command under the Measure menu. Use the first activity to demonstrate how to look for something special in a sketch. Explore the first two triangles. Show how to tabulate to collect evidence that supports hypotheses, e.g., for the second triangle collect at least three table entries that show that angle LKM has a measure of 90 degrees. Use optical illusions to reinforce the messages.

Action! Pairs Investigation Learning Skills/Observation/Rubric: Circulate to assess teamwork while students are working. Students investigate geometric relationships using some of the sketches in the GSP file What’s So Special?. Assign one investigation to each pair for reporting purposes. Pairs may do more than one of the investigations; however, each group reports on only one investigation (BLM 4.2.2).

Consolidate Debrief

Whole Class Discussion How do you know if you have enough evidence to convince you that something is always true? Can you actually prove that something is always true by collecting lots of evidence, e.g., if it rains for 20 days will it rain on the 21st day? What are the two general ways that triangles are classified? (by angles and by sides, example: an acute isosceles triangle). Why is “dynamic geometry software” a good way to describe The Geometer’s Sketchpad®?

Establish computer protocols, e.g., screens off, change drivers, keyboards up, etc. When students identify a property encourage them to ask themselves, Is it always true? and Have I provided evidence to support my answers? Encourage students to discuss their observations so they can learn from each other. See guiding questions for GSP activities (BLM 4.2.1) A large body of evidence can be very convincing but it does not constitute a mathematical proof. A “board” sketch is a snapshot of a GSP screen. Encourage students to visualize different snapshots of the same GSP screen.

Application Skill Drill Differentiated Instructions

Home Activity or Further Classroom Consolidation • Sketch two examples of each of the following types of triangles: acute

scalene, acute isosceles, acute equilateral, right scalene, right isosceles, right equilateral, obtuse scalene, obtuse isosceles, obtuse equilateral.

• Describe how you would divide a triangular piece of land into 3 equal areas. Describe how you would divide a circle into three equal areas.

• For practice using yesterday’s theorems do … (Teacher inserts exercises.)


4.2.1: What’s So Special? Explore Drag each vertex in the figure. As you drag vertices, look for some of the following: • Measurements that always seem to be equal to each other • Measurements that never seem to change • Measurements that might have a constant ratio (proportional) • Lines that always seem to be parallel or perpendicular • Line segments that always seem to be bisected • Figures that always seem to be congruent • Objects that don’t seem to be connected yet they move together when something is dragged Hypothesize Decide which measurements you need to test your hypothesis. Drag each vertex again while you pay close attention to the way the object moves and to the way the measurements change. Test Your Hypothesis How do you collect and record evidence to test your hypothesis?

What can you measure? What can you calculate? • angles • sums • lengths • ratios • areas • formulas • perimeters

• slopes

Guiding Questions

Examine the angles … • Are any of the angles equal? • Do any of the angle measures always add to give the same total? • Does the measure of any angle always stay the same? • Are any of the angles cut in half (bisected)? Examine the line segments … • Are any of the lengths equal? • Is any length proportional with any other length? • Are any of the line segments cut in half (bisected)? Examine the lines … • Are any of the lines parallel? • Are any of the lines perpendicular? • Are any of the slopes of the lines equal? Examine areas and perimeters … • Are any of the areas (or perimeters) equal? • Are any of the shapes congruent? • Are any of the shapes similar? • Are any of the areas (or perimeters) proportional?


4.2.2: What’s So Special? (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml Record your observations and conclusions.



Unit 4: Day 3: What’s So Special? – Part 2 Grade 9 Applied

75 min.

Description • Present observations and conclusions from dynamic geometry software

explorations.

Materials • data projector • BLM 4.2.2 • GSP files: Exterior Angle Demonstration, Sum of the Interior Angles of a Quadrilateral, and Exterior Angles (p. 151, 152, 153)


Minds On ... Whole Class Discussion To prepare for student presentations during Action! discuss presentation skills. Determine an activity for the “audience” during the presentations, e.g., note taking, preparing a question for the presenter, extending examples, etc.

Action! Small Groups Discussion Learning Skills/Observation/Checklist: Circulate while students are working to assess teamwork. Form small groups from the working pairs of the previous lesson. Students discuss observations and conclusions from the previous lesson and plan their presentation. Pairs Presentation Curriculum Expectations/Observation/Rubric: Assess communication, e.g., appropriate use of geometric vocabulary.

Consolidate Debrief

Whole Class Discussion Summarize the key ideas discovered and the new GSP skills. Some guiding questions:

• Is it always true that …? • What does proportional mean? • How does the sketch with the two squares connect to what we discovered

earlier? (If the length of a side of a square is doubled then the area of the new square is quadrupled.)

In preparation for further activity ensure that students understand how to use a protractor to measure interior angles and exterior angles in triangles and quadrilaterals (add these words to the Word Wall).

“In a dynamic geometry program, if a quadrilateral is drawn, only one shape is observed as would be the case on paper or on a geoboard. But now that quadrilateral can be stretched and altered in endless ways. Students actually explore not one shape but an enormous number of examples from that class of shapes. If a property does not change when the figure changes, the property is attributable to the class of shapes rather than any particular shape.” Elementary and Middle

School Mathematics, John A. Van der Walle

(p. 340)

Exploration

Home Activity or Further Classroom Consolidation Investigate the sum of the interior angles and the sum of the exterior angles in a variety of triangles and quadrilaterals. Use paper-and-pencil sketches and protractors. Summarize your observations and conclusions. Submit your work.

Curriculum Expectations/ Assignment/Rubric Collect the investigations and assess for: • gathering of data connected to the problem • reasonableness of conclusion • clarity of conclusions


Exterior Angle Demonstration (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

4

3

2

1

5

Angle Measurements

1.

2.3.

4.

5.

6.

7.

8.9.

Make a conclusion.

What do you noticeabout the sum?

Drag the vertex ofangle 2 or 3 andobserve the sum.

Exterior Angle Demonstration

Sketch Restriction: The polygon must remain convex.This sketch was based on a similar sketch from "Exploring Geometry"

Drag each vertex andobserve the anglemeasurements.

Angle 3 = 72.0°Angle 4 = 72.0°

Angle 2 = 72.0°

Angle 5 = 72.0°

Angle 1 = 72.0°

Show Sum

Reset

Another side less

Shrink polygon

Stretch back out

One less side



Sum of the Interior Angles of a Quadrilateral (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Quadrilateral WXYZ consists of 4 triangles . The 4triangles share vertex O inside the quadrilateral.The sum of the angles at point O is 360° (sincethere are 360° in one complete rotation).

The rest of the angles in the triangles form the interiorangles of the quadrilateral.

Therefore,Sum of the interior angles in WXYZ= (sum of interior angles in 4 triangles) - 360 °= 4 x 180 ° - 360°= 360°

Sum of Interior Angles = 360°m∠YXW = 49°m∠ZYX = 129°m∠WZY = 51°m∠XWZ = 131°

Sum of the Interior Angles of a Quadrilateral

ReturnX

Y

Z

W

O



Exterior Angles (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

x

y

zIf you move one of the vertices of the triangle, does the sum stay the same? _____

From what you have learned above, state one property of exterior angles of triangle : _____

Name: ______

Using the calculator, determine the sum of the exterior angles of the triangle. _____

Click on arcs x, y and z then go to the Measure tab and determine the measure of the ArcAngles.

The exterior angles of the ∆ABC are indicated by the arcs x, y and z.

Exterior angles of ∆ABC

A

BC

F

D

E

y

x

w

z

If we move segment CD or segment AB does the sum change? ______.

How does the sum of the exterior angles of a quadrilateral compare with the sum of the exterior angles of a triangle? ______.

The exterior angles of the ABCD are indicated at w, x, y and z by symbols.

Determine the measures of these exteriorangles of quadrilateral ABCD.

Determine the sum of the exterior anglesof quadrilateral ABCD.

Exterior angles of quadri lateral ABCD

A

B

CDE

F

H

G

zx

y

If you move point D, does the relationship remain the same? _____

Write a statement about the property of the exterior angle of triangle that you have just discovered. _____

What do you notice? _____

Determine the sum of the measures of ∠x and ∠y.

Determine the measures of ∠x and ∠y.

Determine the measure of ∠ z.

Side BC has been extended to point D.∠ACD (z) is called an exterior angle.∠x and ∠y are opposite interior angles.

An exterior angle of ∆ABC

B D

A

C

m∠FAB+m∠GBC+m∠HCA = 360°

m∠HCA = 123°

m∠GBC = 110°

m∠FAB = 127°

Click on the action bar below

Sum of the exterior angles of a triangle.

Exterior angles of a triangle are shown.If we decrease the size of the triangle,what do you notice about the sum ofthe exterior angles? _________

Hide Measuremen

Make the triangle small

Reset the triangl

A

B

C

F

G

H

Click on the action bar below

Sum of the exterior angles of a quadrilateral

Exterior angles of a quadrilateral are shown.If we decrease the size of the quadrilateral,what do you notice about the sum ofthe exterior angles? ___________

Reset the quadrilater

Make the quadrilateral smal

A

B

C

D

3) If ∠QP R = 70° and ∠PRS = 130°, what the measure of ∠PQR ___ (the diagram is not drawn to scale)

y°

70°

130°

2) Three exterior angles of the quadrilateral are given. Determine the measure of the fourth angle ∠EHK? ___ (the diagram is not drawn to scale)

120°

50°

110°

x°

1) If ∠CAD = 40° and ∠ACD = 80°, what the measure of ∠CDB ___ (the diagram is not drawn to scale)

Review Questions

x°

80°

40°

L

A B

C

EF

GH

P

SQ

D

I

JK

L

R

3- Use tabulation to determine many measurements

The Ray Tool is the second straightedgetool. Hold the button down and choosethe middle one

Practice1- Use the Ray Tool to construct atriangle showing the exterior angles.

2- Determine the measures of the exterior angles.



Map (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Welcome toKillarney Park!

Park planners aregoing to build a newranger station.

The planners havedesignated threelocations in the parkthat need to be thesame distance fromthe new station.

Press the continuebutton to see thethree locations.

Continue

2.9 cm

2.5 cm6.4 cm

The threelocations are atpoints A, B, and C

Hide the map

Drag the "RangerStation" until it isthe same distancefrom points A, B,and C.

You will beinvestigating what'sspecial about thelocation for the"Ranger Station".

Continue

Hide Map

A B

C

Ran ge r Statio n

Your challenge is to describehow to determine thelocation for the rangerstation for any three pointsthat the planners choose

You might find it useful toview triangle ABC.

The park plannershave decided tochange thelocations of pointsA, B, and C.

3.1 cm

3.9 cm

3.8 cm

Hint

New Location Reset

Show Triangle ABC

Hide Map

AB

C

Range r Station

Hmmm ... did you consider usingone of these special sets of lines?

Return

Show Perpendicular Bisectors

Show Angle Bisectors

Show A ltitudes

Show Medians

A

B

C



Mystery Points (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Mystery Point P

The method used to constructpoint P is a mystery.

1. Drag the vertices of the triangle around. What do you notice about the point P?

2. Drag the sides of the triangle around. What do you notice about the point P?

How would you make the pointP in your own triangle?

P

CB

A

Mystery Point Q

The method used to constructpoint Q is a mystery.

1. Drag the vertices of thetriangle around. What do you notice aboutthe point Q?

2. Drag the sides of thetriangle around. What do you notice aboutthe point Q?

How would you make thepoint Q in your own triangle?

Q

A

B C

The method used to constructpoint R is a mystery.

1. Drag the vertices of the triangle around. What do you notice about the point R?

2. Drag the sides of the triangle around. What do you notice about the point R?

How would you make the pointR in your own triangle?

Mystery Point R

R

A

B

C



Triangles (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

5) What type of triangle is ABC when point O is outside the triangle? ____

4) What type of triangle is ABC when point O is on the triangle? ____

3) What type of triangle is ABC when point O is inside the triangle? ____

By moving the vertices A, B or C the orthocentre (point O) is either inside the triangle, on the triangle or outside the triangle. There are three different types of triangles, depending on where point O is situated.

Point O is called the orthocentre. of ABC.

1) Click on points A, E and C (the order is important), go to the Measure tab and measure ∠AEC.____

AE, BG and CF are altitudes.

Altitudes are line segments drawn from a vertex perpendicular to the opposite side

Alti tudes

E

F

GO

C

A

B

Point O is called the incentre.

2) AO , BO and CO are angle bisectors. If you move points A, B or C does point O always remain inside the triangle? ____

1) What do you notice about the measures of the two angles? _____

Click on points CAO (in that order) then go to the Measure tab and determine the measure of the Angle. (Before you determine the measure of the angle you might have to de-highlight other objects by clicking anywhere in the workspace.)

Click on points BAO (in that order) then go to the Measure tab and determine the measure of the Angle.

Angle Bisectors 1

O

A

B

C

5) If you move point P, what do you notice?_________

4) What do you notice? _________

2) PF and PE are line segments drawn perpendicularthe sides of the triangle.

3) Determine the measures of P E and PF.

1) Point P is a point on the angle bisector of ∠C?

Angle Bisectors 2

F

E A

B

C

P

The name of the circle is the incircle.

What do you notice about the circle that is formed? ____

Click on point O and the point of intersection (in that order) then go to the Construct tab and choose Circle by Center+Point .

Click on the perpendicular line and line segment AC then go to the Construct tab and choose Intersection.

Click on point O and line segment AC then go to the Construct tab and choose Perpendicular Line.

Point O is called the incentre.

Angle Bisectors 3

O

A

B

C

If you move point A, do the measures remain equal? ____

Determine the measures of segments CE and CF.What do you notice? ____

Determine the measures of segments BD and BE.What do you notice? ____

Determine the measures of segments AD and AF.What do you notice? ____

Determine the measure of ∠OEC.____

Determine the measure of ∠ODA.____

The circle intersects the triangle at points D, E and F.OD, OE and OF are radii of the circle.

Angle Bisectors 4The incircle of ∆ABC is drawn.

E

D F

O

BC

A

Practice

1- Construct your own triangle anddetermine the orthocentre.

2- Construct a second triangle and determine theincentre. Construct the incircle as well.

Euler line

Press the action button to seehow the four points change.

Ce: CentroidO: OrthocentreI: IncentreCi: Circumcentre

m∠ACB = 83.2°Animate Point

O

CeCi

I

Click on one of the actionbuttons to animate the point

circumcentre

orthocentre

incentre

centroid



Exploring Triangles (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Exploring Triangles

PerpendicularbisectorsMedians Altitudes Angle bisectors

Sho w Hid eSho w Show HideHide HideShow

SPIN Æ Right Æ Isosceles Æ Equilateral Æ

CB

A



Diagonals (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Rhombus

Paralle logramT rapezoid

Square

Kite /Chevron

Rectangle

Diagonals - Eh?Instructions

Explore1. Drag each vertex.

hypothesize2. Is there anything special about any of the diagonals?

?Questions3. What evidence could you collect to support your hypotheses?

Show HInt

Click the H idebutton then clickone of the Showbuttons.

m∠BCA = 76°m∠ABC = 43°m∠CAB = 62°

...measure the length of a side?To measure side CB :1. Deselect.2. Select the two endpoints (C and B).3. Under Measure choose Distance.

...measure an angle?To measure ∠ABC:1. Deselect.2. Select point A , then B, then C. Order is important! The second point chosen is the vertex.3. Under Measure choose Angle..

How do I ...?Hide MEASURE AN ANGLE OR SIDE

Show MEASURE AREA or PERIMETER

Show CONSTRUCT A POLYGON INTERIOR

Show CREATE A TABLE

Show CALCULATE

C

AB



Measure (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Measures of Angles

Measures of S ides

Measure It - Eh?Instructions

Explore1. Measure all angles.2. Measure all sides.

hypothesize3. What type of quadrilateral is this?4. Drag each vertex and each side. Watch the measurements.

?Questions5. Do you think your hypothesis was correct or incorrect?6. What evidence can you give to confirm or deny your hypothesis?

Next Activity

Show Hint

Show Hint

C

A

B

D

Measures of Angles

Measures of Sides





Next Activity

Show Hint

Show HintG

E H

F

Measures of Angles

Measures of Sides





Next Activity

Show Hint

Show Hint

L

M

J

K





Measures of Angles

Measures of Sides

Next Activity

Show Hint

Show Hint

N Q

PR



hypothesize3. What type of quadrilateral is this?4. Drag each vertex and each side. Watch the measurements..


Measures of Angles

Measures of Sides

Next Activity

Show Hint

Show Hint

U

TS

V





Measures of Angles

Measures of Sides

Next Activity

Show Hint

Show Hint

Z

YX

W





Measures of Angles

Measures of Sides

Next Activity

Show Hint

Show Hint

D C

BA

Rhombus

Paralle logramTrapezoid

Square

Kite/Chevron

Rectangle


Explore1. Drag each vertex.

hypothesize2. Is there anything special about any of the diagonals?

?Questions3. What evidence could you collect to support your hypotheses?

Show HInt

Go to Beginning



Quadrilaterals (GSP file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Quadri laterals

Given: Parallelogram IJKL

Click on segment IO, JO, KO and LO then go to the Measure tab and determine the Lengths.

Name one pair of equal segments. ____

If we move vertices K or L do the segments remain equal? ____

Write a conclusion about the properties of the diagonals of a parallelogram. ___

O

IJ

KL

What do you notice? ____

Write a conclusion about the properties of the diagonals of a rectangle ___

If we move vertices S or P do the segments remain equal? ____

Click on segment PO, QO, RO and SO then go to the Measure tab and determine the Lengths.

Given: Rectangle PQRS

O

Q

RS

P

Make a conclusion about the properties of the diagonals of a rhombus ___

If we move vertices A or B does the angle remain equal? ____

Click on points AOB (order is important),go to the Measure tab and measure the Angle (∠AOB).

What do you notice about the four lengths? ____

Click on segment OA, OB, OC and OD then go to the Measure tab and determine the Lengths.

Click on segment AB, BC, CD and DA then go to the Measure tab and determine the Lengths.

Move vertices A and B.What do you notice about the lengths of the four sides of the rhombus? ____

Given:Rhombus ABCD

O

D

B

A

C

What do you notice about the four lengths? ____

Click on points AOB (order is important),go to the Measure tab and measure the Angle.

If we move vertices A or B, does the angle remain the same? ____

Make a conclusion about the properties of the diagonals of a square1- ___2- ___

Click on segment OA, OB, OC and OD then go to the Measure tab and determine the Lengths.

Given: Square ABCD

O

C

DA

B



Determine the lengths of WZ and XY.

Write a conclusion about the properties of the diagonals of an isosceles trapezoid._______

Determine the lengths of WO, XO, YO and ZO.

Given: Isosceles trapezoid WXYZ, WX || YZ.

O

Y

X

Z

W

What happens to the lengths of OB and OD if you move point A? ____

What happens to the lengths of AB and AD if you move point A? ____

Write a conclusion about the properties of the diagonals of a kite.____

Determine the Measure of Angle AOB.What do you notice? ___

What do you notice about the lengths of OB and OD? ____

Determine the Lengths of OA, OB, OC and OD

What do you notice about the lengths of AB and AD? ____

Determine the Lengths of AB, BC, CD and DA

Given: Kite ABCD .

O

B

D

CA

(The difference between drawing and constructing

2- Construct a square that will remaina square even after you move one ofthe vertices.

1- Construct a parallelogram thatwill remain a parallelogram even afteryou move one of the vertices.

Practice



Unit 5: Consolidation Emphasizing Algebraic Models Lesson Outline Grade 9 Applied

BIG PICTURE Students will: •• make connections among numeric, graphic and algebraic models; •• make connections between contextual situations and appropriate models; •• extend and consolidate algebraic skills; •• look for patterns and use variables to generalize/summarize; •• consolidate the big ideas in Units 1, 2, 3, and 4.

Day Lesson Title Description Expectations 1 Three Models:

Graphic, Numeric and Algebraic Lesson included

•• Develop an understanding of collecting like terms and expanding polynomials through guided exploration.

•• Use multiple representations: graphic, numeric and algebraic models.

NA1.01, NA1.02, NA1.04 NA1.06, NA3.01 - 3.02, NA4.01 - 4.03 CGE 5e

2 Use of Variables as Bases of Power Lesson included Presentation software files Connecting Algebra Tiles to Integer Tiles, Collecting Terms Using Algebra Tiles, and Expanding Using Algebra Tiles included (p. 173-174, 174, 175)

•• Use variables as the base of powers. NAV.02, NA2.06, NA3.01, NA3.02 CGE 4b

3 Algebraic Models in Measurement and Geometry Lesson included

•• Use variables to make connections between symbolic and concrete models from measurement and geometry application problems.

•• Evaluate expressions after substitution of a value for a variable.

•• Solve equations by inspection in preparation for formal solution procedures.

NA1.02, NA1.04, NA1.06, NA2.01, NA2.04, NA3.01 - 3.03, NA3.05, NA4.01 - 4.03, MG2.01 CGE 2b


Day Lesson Title Description Expectations 4 Solving First

Degree Equations Lesson included

•• Use technology to support a formalized algebraic solution of first degree equations.

•• Recognize that an equation is created when a specific target value is required for an expression.

NA4.01 - 4.03, NA1.02, NA1.04, NA1.06, NA3.01 - 3.03, NA3.05, MG3.01 CGE 3c

5 Connecting rate/initial conditions to y = mx + b Lesson included

•• Interpret descriptions of situations to identify: rate/slope; initial condition/y-intercept; special case(s)/point on line; formula or rule for relationship/equation of line.

RE2.01 - 2.03, RE2.05, RE3.01 - 3.02, AG2.01, AG2.02, AG3.01 - 3.05 CGE 5a

6 Investigating Slopes Lesson included

•• Make connections between the magnitude of slope and direction of a line.

•• Provide meaning for zero slope, and slopes of parallel and perpendicular lines.

NA3.01, NA3.02, AG1.03, AG2.03, AG2.04, AG3.01 - 3.03 CGE 4b

7 Investigating Equations of Linear and Non-Linear Relations Lesson included

•• Investigate the properties of linear and non-linear relations noting information to be gained from graphs, equations, or calculation of differences.

AG1.01, AG1.02, R3.03 CGE 3c

8 Review and Test •• Review the algebraic skills introduced in this unit using contexts from the Measurement, Geometry, and Relationships strands.

•• Activate prior learning, wherever possible, during the review.

Choose expectations based on observation of students’ learning needs during Days 1 to 7.

9 EQAO Preparation Using Release Items

•• Use the sample release items to acquaint students with the structure of the test.


Unit 5: Day 1: Three Models: Graphic, Numeric, and Algebraic Grade 9 Applied

75 min.

Description • Develop an understanding of collecting like terms and expanding polynomials

through guided exploration. • Use multiple representations: graphic, numeric and algebraic models.

Materials • graphing calculators • BLM 5.1.1, 5.1.2, 5.1.3, 5.1.4, 5.1.5, 5.1.6, 5.1.7 • grid paper


Minds On ... Jigsaw (Home Groups of 4) Discussion Introduce the final unit of the course outlining the concepts to be studied (algebra and polynomials) and the structure (review the concepts and skills from the earlier units) within which they will be learned. Emphasize the importance of consolidating learning, making connections and recognizing general patterns. This lesson overlays multiple representations for equations with new algebra skills. Organize students into home groups of four and designate them as A, B, C, or D within the group.

Action! Jigsaw (Expert Groups) Guided Exploration Divide class into four expert groups: A and B – expansion of an expression, and C and D - collecting like terms. Distribute grid paper and BLM 5.1.1 to Group A, BLM 5.1.2 to Group B, BLM 5.1.3 to Group C, and BLM 5.1.4 to Group D. Students work through the guided discovery and compare their individual conclusions with others in their work group. Jigsaw (Home Group) Sharing They meet with their home group and check conclusions with their “partner” with the group – A with B and C with D in preparation for sharing their learning with the other pair. Curriculum Expectations/Observation/Mental Note: Pairs share with each other what they have learned and check for understanding. All members of the home group record the learning from both tasks.

Consolidate Debrief

Whole Class Guided Connections Guide the class through BLM 5.1.5 to consolidate new expansion and collections skills and to connect algebraic expressions to measurement formulas. Clarify any misunderstandings observed while pairs were working together.

The questions require facility with integers. Provide calculators for students who have numeric difficulties. You may wish to predetermine strong/weak pairs.


Home Activity or Further Classroom Consolidation Complete the worksheets 1.6 and 1.7. Complete textbook problems requiring the collection of like terms and monomial times a binomial. (The teacher assigns appropriate exercises.)


5.1.1: Three Models: Graphic, Numeric, and Algebraic Using a graphing calculator or spreadsheet, create a table of values for:

a) y = 3(x − 1) b) y = 3x − 3

x y −2 −1 0 1 2

How do the tables compare? Using a graphing calculator or graphing software: •• Graph y = 3(x – 1). Record the graph, on the grid paper. •• Graph y = 3x – 3 and record the graph, in another colour, on the same grid paper. •• How do the graphs compare?

The tables of values in a) and the graphs in b) should be identical. •• What must this mean about the expressions 3(x – 1) and 3x – 3? •• What process would transform 3(x − 1) into 3x − 3? On the back of this paper, create tables of values and compare them for:

y = 2(x + 4) y = 2x + 8 •• What process would transform 2(x + 4) into 2x + 8?

Graph y = −4(x + 3) and y = −4x – 12 on the same axes and compare the graphs. •• What process would transform −4(x + 3) into −4x – 12? The process is called “expansion” or “removal of brackets.” Expand the following:

a) 2(x – 5)

b) 5(x + 1)

c) 4(3x – 1)

d) −3(2x + 4)

e) 2(4x – 5)

f) −5(x + 4)

x y −2 −1 0 1 2



a) y = 4(x − 2) b) y = 4x − 8

x y −2 −1 0 1 2

How do the tables compare? Using a graphing calculator or graphing software: •• Graph y = 4(x – 2). Record the graph on the grid paper. •• Graph y = 4x – 8 and record the graph, in another colour, on the same paper. •• How do the graphs compare?

The tables of values in a) and the graphs in b) should be identical. •• What must this mean about the expressions 4(x – 2) and 4x – 8? •• What process would transform 4(x – 2) into 4x – 8? On the back of this paper, create tables of values and compare them for:

y = 3(x + 1) y = 3x + 3 •• What process would transform 3(x + 1) into 3x + 3?

Graph y = −2(x + 5) and y = −2x – 10 on the same axes and compare the graphs. •• What process would transform −2(x + 5) into −2x – 10?

The process is called “expansion” or “removal of brackets.” Expand the following:

a) 3(x – 2)

b) −5(x + 4)

c) 6(2x − 1)

d) −3(x – 5)

e) 2(3x + 5)

f) −5(2x – 4)

x y −2 −1 0 1 2


5.1.3: Three Models: Graphic, Numeric, and Algebraic

Using a graphing calculator or spreadsheet, create a table of values for:

a) y = 1 + 2x + 3 + 4x b) y = 6x + 4

x y −2 −1 0 1 2

How do the tables compare? Using a graphing calculator or graphing software: •• Graph y = 1 + 2x + 3 + 4x. Record the graph on the grid paper. •• Graph y = 6x + 4 and record the graph, in another colour, on the same paper. •• How do the graphs compare?

The tables of values in a) and the graphs in b) should be identical. •• What must this mean about the expressions 1 + 2x + 3 + 4x and 6x + 4? •• What process would transform 1 + 2x + 3 + 4x into 6x + 4? On the back of this paper, create tables of values and compare them for:

y = 5x2 + 4 – 3x2 – 5 y = 2x2 – 1 •• What process would transform 5x2 + 4 – 3x2 −5 into 2x2 – 1?

Graph y = 3 + 6x – 5 – 3x and y = 3x – 2 on the same axes and compare the graphs. •• What process would transform 3 + 6x – 5 – 3x into 3x – 2?

The process is called “collection of like terms” or “simplifying.” Simplify the following: a) x + 4x − 8x – 9

b −2 + 5 – 4x – 2x

c) 9 – 1x – 5x – 3

d) 2 – 2x + 4 – 2x

e) 3x2 – 4x + 2 + x2 + 6x – 2

x y −2 −1 0 1 2



a) y = 1x + 2 + 3x +4 b) y = 4x + 6

x y −2 −1 0 1 2

How do the tables compare? Using a graphing calculator or graphing software •• Graph y = 1x + 2 + 3x + 4. Record the graph on the grid paper. •• Graph y = 4x + 6 and record the graph, in another colour, on the same paper. •• How do the graphs compare?

The tables of values in a) and the graphs in b) should be identical. •• What must this mean about the expressions 1x + 2 + 3x + 4 and 4x + 6? •• What process would transform 1x + 2 + 3x + 4 into 4x + 6? On the back of this paper, create tables of values and compare them for:

55x43xy 22 −+−= y = 8x2 – 9 •• What process would transform 5543 22 −+− xx into 8x2 – 9?

Graph y = 2x + 5 + x – 7 and y = 3x – 2 on the same axes and compare the graphs. •• What process would transform 2x + 5 + x – 7 into 3x – 2?

The process is called “collection of like terms” or “simplifying.” Simplify the following:

a) x + 4 + 3x + 5

b) −4x + 1 + 5x – 7

c) 5a + 2b −2a – 6b

d) −2y + 4x + y + 3x

e) 3x2 – 5x + 6 – x2 −6x + 2

x y −2 −1 0 1 2


5.1.5: Apply: Measurement and Geometry Write an expression for the following. Simplify.

Area = Perimeter = Volume = Surface Area =

Area = Perimeter = Sum of the interior angles =

Sum of the exterior angles =

(x+3) (2x-2)

5x

x

2x−3 −x+2

4x-5

(x−1)

2

(x−1)

2

(x+4)

3

(x+4)

(3x−2) (x+9)

34

(x − 5)

3 4

(x − 5)


5.1.6: Who Is Correct?

Reconciling Equivalent Algebraic Expressions

… How many toothpicks are needed for n squares? Whose solution is correct? Show how you know. Anju’s Solution “If T is the number of toothpicks and n is the number of squares, then, the number of toothpicks is equal to 1 plus three times the number of squares.” My equation is T = 1 + 3n

Erin’s Solution “If T is the number of toothpicks and n is the number of squares, then the number of toothpicks is equal to 4 plus three times one less than the number of squares.” My equation is T = 4 + 3(n –1).

Silva’s Solution “If T is the number of toothpicks and n is the number of squares, then the number of toothpicks is equal to 2 times the number of squares plus one more than the number of squares.” My equation is T = 2n + (n + 1).

Bijuan’s Solution “If T is the number of toothpicks and n is the number of squares, then the number of toothpicks is equal to 4 times the number of squares minus one less than the number of squares.” My equation is T = 4n – (n –1).


5.1.7: The Cube Sticker Problem

Problem The Acme Toy Company makes coloured rods. The cubes are joined in a row and a sticker machine puts “smiley” stickers on the rods. The machine places exactly 1 sticker on the outside face of every cube. Every exposed face of each cube has to have a sticker. This rod is 2 cubes long. It will need 10 stickers.

Procedure Sara and Sid work together to find an algebraic model for this problem. They build a model with Cube Links and count the number of stickers needed on rods up to five cubes long.

Number of Cubes (n)

Number of Stickers (S)

1 6

2 10

3 14

4 18

5 22

Sid and Sara determined different equations to represent the relationship between the number of cubes (n) and the number of stickers (S).

Sid’s equation: S = 4(n – 1) + 6

Sara’s equation: S = 10 + 4(n – 2)

Use your knowledge of algebra to determine if Sid and Sara have the same answer. Show your work.


Unit 5: Day 2: Use of Variables as Bases of Power Grade 9 Applied

75 min.

Description • Use variables as the base of powers.

Materials • graphing calculators • BLM 5.2.1 • Presentation software files: Algebra Tiles to Integer Tiles, (p. 173-174) Collecting Terms Using Algebra Tiles, (p. 174) and Expanding Using Algebra Tiles, (p. 175)


Minds On ... Whole Class Activate Prior Knowledge Orally review the use of exponents to show repeated multiplication and the exponent rule for multiplication of powers with the same numerical bases.

Action! Pairs Guided Exploration Students extend their learning from Day 1 to expansions involving the multiplication of powers with variable bases (BLM 5.2.1). In the last question students collect like terms to complete a solution.

Consolidate Debrief

Whole Class Guided Making Connections Curriculum Expectations/Response Journal/Rating Scale: Students draw diagrams to represent x, x2, and x3 (length, area, volume). Extend the task to representations of x2y, xy2,, 4xyz. Revisit expressions of BLM 5.2.1. Showing visual representations such as:

Ensure that students are comfortable with 3 x 3² = 3³ in preparation for: x(x²) = x³ later in this lesson. Model only positive terms to connect these concepts to algebra titles. This second way of demonstrating the concept of expansion and collection of terms may seem more convincing to some students than the graphical argument. It may be appropriate to spend another day helping students connect algebraic manipulations (expansions and collection of like terms) to use of manipulatives.

Skill Drill Home Activity or Further Classroom Consolidation Complete problems from the text to consolidate understanding. (The teacher provides appropriate exercises.)

Shows x (x +3) = x2 + 3x

Shows 2x (x +4) = 2x2 + 8x

Shows x (x2 +7) = x3 + 7x

Length Width Area

DepthSurface

Area

Volume


5.2.1: Powers with Variable Bases (Numeric, Graphical, and Algebraic Models) Using a graphing calculator or spreadsheet, create a table of values for:

a) y = x(x + 3) y = x2 + 3x

x y −2 −1 0 1 2

How do the tables compare? Using a graphing calculator or graphing software: •• Graph y = x(x + 3). Record the graph, in blue, on the grid paper. •• Graph y = x2 + 3x and record the graph, in red, on the same paper. •• How do the graphs compare? The tables of values in a) and the graphs in b) should be identical. •• What must this mean about the expressions x (x + 3) and x2 + 3x? •• What process would transform x (x + 3) into x2 + 3x? On the back of this paper, create tables of values and compare them for:

y = x(x2 + 7) y = x3 + 7x •• What process would transform x(x2 + 7) into x3 + 7x?

Graph y = 2x(x – 5) and y = 2x2 – 10x on the same axes and compare the graphs. •• What process would transform 2x(x – 5) into 2x2 – 10x? Explain why y = x(x)(x) and y = x(x2) and y = x3 have identical graphs.

The process is still called “expansion” or “removal of brackets” when you are using powers with variable bases. Expand the following:

1. 2x(x + 4)

2. 3x(x2 + 2x)

3. 4x2(3x2 + 2x – 5)

4. -3a(a3 – 4a)

5. 5x3(3x – 4)

Check your understanding:

Two students were asked to expand this expression: )32( 2 xxxx +− Which solution is better? Explain your choice.

x y−2 −1 0 1 2

Kevin’s answer Sal’s answer Ari’s answer

23223)322(

xxx

xxxx

+−=

+−

23

2

2

)()32(

xxxxx

xxxx

+=+=

+−

23

223

2

3232

xxxxx

xxxx

+=

+−=

+− )(


Connecting Algebra Tiles to Integer Tiles (Presentation software file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Connecting Algebra Tiles to Integer

Tiles

Comparing Algebra Tiles to Integer Tiles

What relationships do you notice between the measurements of the integer tile and the algebra tiles?

is a representation of +1 because it has an area of 1. Length = 1 Width = 1

is a representation of -1 because the size stays the same, but the sign/colour changes.

Therefore, represents x, and


The width of the rectangle is 1 and the length can be represented by x.

The length and width of the large square are the same as x.

1

1

x

1 x

x

represents x2


represents ____

is +1

is +x represents _____

is -1

represents ___

-x

+x2 - x2

x

x2

Representations of Zero

1 –1 = 0

x–x = 0

–x2x2 = 0

Representations of IntegersIllustrate each of the following terms.

+3

-2

Illustrate a different representation of each of these integers.

Add 0.

Add 0.



Connecting Algebra Tiles to Integer Tiles (Presentation software file) (continued)

Representations of TermsIllustrate each of the following terms.

+3x

-2x2

Draw a representation of 4x, -2x, 3x2, -5x2.

Representations of Algebraic Terms

4x

3x2

-2x

-5x2

Multiple Representations of 4x

Representing Polynomials

3x+1

-2x2-3

X2-3x+2

1

1 11

11

Collecting Terms Using Algebra Tiles (Presentation software file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Collecting Terms Using Algebra Tiles

Model (+3)+(+2) using integer tiles.x

x2(+3) +(+2) =+5

Model (+3x)+(+2x) using algebra tiles.

3x + 2x = 5x

Model 3x+1+2x-3 using algebra tiles.

3x +1 +2x +(-3) = 5x-2

Every term has a numerical coefficient and a literal coefficient.

Like terms have the same literal coefficients.

Collection is the process of collecting like terms together.



Expanding Using Algebra Tiles (Presentation software file) Download this file at www.curriculum.org/occ/tips/downloads.shtml

Expanding Using Algebra Tiles

What is the geometry question that has this answer ?

What is the area of the rectangle shown?

length width areax =

2 3 6x =

Connecting Integer Concepts to Algebra TilesIllustrate 2(3x + 1).

= 6x + 22(3x + 1)

3x+1

2

Think

of a rectangle.

length width areax =

Illustrate

x + 4

3

( )3 4 3 12+ = +x x

( )3 4+x

Expansion is the process of removing brackets.

We can think of the instruction ‘expand’ as meaning to determine the area, given length x width of a rectangle.

Expand x(2x + 1)

= 2x2 + x

x2 x2 x

2x + 1

x

x(2x + 1)

Expand x(3x + 2)

3x +2

x2 x2 xx x2 x

x(3x + 2)=3x2+2x



Unit 5: Day 3: Algebraic Models in Measurement and Geometry Grade 9 Applied

75 min.

Description • Use variables to make connections between symbolic and concrete models

from measurement and geometry application problems. • Evaluate expressions after substitution of a value for a variable. • Solve equations by inspection in preparation for formal solution procedures.

Materials • scientific calculators • BLM 5.3.1


Minds On ... Whole Class Introduction Check understanding with a question taken from the follow-up activity. Discuss these problems to clarify any misunderstandings. Verbally demonstrate the solution of a two-step algebraic equation by inspection: Two more than 3 times a number is 11 (or 3 times a number plus two is 11). What’s my number? Ask students to verbalize their thinking.

Action! Whole Class [Differentiated] Worksheets Learning Skill (Individual work)/Self-Assessment/Checkbric and Curriculum Expectations/Interview/Checklist: Discuss the opportunity to “work at your own pace.” At the beginning of the worksheet activity all students participate, guided by the teacher’s entries on the transparency but, as soon as an individual student gains confidence, he/she is welcome to turn to face a different direction and proceed to work on the tasks quietly without interaction with the group. In this way students who need extra support continue to work with the teacher while all students are encouraged to self-assess. Students solve measurement problems leading to equations that can be solved by inspection (BLM 5.3.1). Using the equation created, students review substitution for a variable and simplifying or solving by systematic trial. Make connections to prior learning regarding the use of scientific notation.

Consolidate Debrief

Whole Class Planning Discuss with students the power of algebraic models to:

• represent “all cases”; • allow solution for one variable when other variables have determined

(known) values. Collect work from everyone.

For suggestions regarding ways to efficiently solve by systematic trial see Extend Your Thinking in Solving Equations and Using Variables as Placeholders.

Exploration Reflection

Home Activity or Further Classroom Consolidation Respond in your journal: What jobs might use algebra to model measurements?


5.3.1: Solving Measurement Problems Shape 1 Sam makes rectangular paving stones that are 10 cm longer than they are wide.

Explain how this model shows that a paving stone is 10 cm larger than its width.

Determine a formula for the area in terms of w.

Determine a formula for the perimeter in terms of w.

Use this formula to calculate the area when the width= 17 cm.

Use this formula to calculate the perimeter when the width = 6.75 cm.

Using a graphing calculator or graphing software, graph the equation describing area and locate the (width, area) point corresponding to your calculation.

Using a graphing calculator or graphing software, graph the equation describing perimeter and locate the (width, perimeter) point corresponding to your calculation.

Use the formula to calculate the width when the area = 600 cm²

Use the formula to calculate the width when the perimeter = 60 cm.

w

)( 10+w


5.3.1: Solving Measurement Problems (continued) Shape 2

Explain how this model shows that the base is 4 cm larger than it’s height.

Determine a formula for the area of the triangle.

Determine a formula for the perimeter of the triangle.

Use this formula to calculate the area when the height is 5.2 cm.

Use this formula to calculate the perimeter when the height is 3 cm.

Use technology to graph your equation and locate the (altitude, area) point corresponding to your calculation.

Use technology to graph your equation and locate the (altitude, perimeter) point corresponding to your calculation.

Use the formula to determine the height when the area is 70 m2 .

Use the formula to calculate the height when the perimeter is 19 cm.

(x+4)

x(3x-2) (2x+5)



Explain how this model shows: - the length is 2 times the width. - the height is 3 times the width.

Determine a formula for the volume. Determine a formula for the surface area.

Use this formula to calculate the volume when the width = 225 m.

Use this formula to calculate the surface area when the width = 6.75 cm.

Write your answer using scientific notation. Write your answer using scientific notation.

x 2x

3x



Explain how this model shows: - the length is twice the width. - the height is 3 units longer than the width.

Determine a formula for the volume. Determine a formula for the surface area.

Use this formula to calculate the volume when the width = 0.004 m.

Use this formula to calculate the surface area when the width = 0.00025 km.

Write your solution in standard form and scientific notation.

Write your solution in standard form and in scientific notation.

x 2x

(x + 3)


Unit 5 Day 4: Solving First Degree Equations Grade 9 Applied

75 min.

Description • Use technology to support a formalized algebraic solution of first degree

equations. • Recognize that an equation is created when a specific target value is required

for an expression.

Materials • graphing calculators • BLM 5.4.1, 5.4.2, 5.4.3


Minds On ... Whole Class Discussion Connect expressions and equations – an equation sets a specific target value for an expression. Establish the concept that solving an equation means finding a value for the variable so that the statement is true – so that the expression on the left side of the equal sign balances the expression on the right side of the equal sign. Provide physical models or diagrams to establish the concept that solving an equation is like maintaining a balance.

Action! Whole Class Demonstration Demonstrate how to use the graphing calculator to solve equations using the prompts in the left column of BLM 5.4.1. Complete a few examples as a whole class and save for comparison with answers found using algebra. Guide the algebraic solution outlined in the right column of BLM 5.4.1. Ask students to “read what has happened to the variable” and work backwards, e.g., 3x − 2 = 4 reads “x has been multiplied by 3 and then 2 was subtracted.” The last thing to happen was the subtraction of 2 so that is undone by adding 2. The result is 3x = 6. Now the last thing to happen was multiplication by 3 so that is undone by division by 3. A verbal cue of “undo, simplify” may be helpful. Complete a few examples as a class. Small Groups Peer Coaching Students practise solving two-step equations using algebra. Individual Practice Learning Skill (Teamwork) Observation/Checklist and Curriculum Expectations/Question and Answer/Checklist: After students have demonstrated understanding of the process and ability to identify and complete steps in solving sample equations, provide a sequence of equations of increasing complexity: requiring the collection of like terms; requiring the expansion of a bracket; requiring the creation of an expression and equation based on a diagram; using examples taken from interior, exterior angle theorems, parallel line theorems, etc. (BLM 5.4.2, 5.4.3)

Consolidate Debrief

Whole Class Discussion Discuss the pros and cons of the two strategies – using technology and solving manually. Students should become proficient with both strategies through individual practice.

Model proper format when recording the algebraic process on paper. Some students may benefit from showing the undoing in coloured pen on both sides of the equation. Once students have had sufficient practice, assess proficiency on developing algebraic expressions and solving linear equations. (See p. 10 Solving Equations and Using Variables as Placeholders.)


Home Activity or Further Classroom Consolidation Complete Developing Mathematical Processes questions and Extend Your Thinking problem from Section 2 – Patterning to Algebraic Modelling.

See p. 7, 19 Solving Equations and Using Variables as Placeholders


5.4.1: Solving Equations Graphically and with Algebra

Graphical Algebra

Solve for x: 3x – 2 = 4

Solve for x: 3x – 2 = 4

Enter the Right side of the equation in Y1 Enter the Left side of the equation in Y2

Press: GRAPH You will see that the 2 lines intersect.

Press: 2nd TRACE (CALC) Press: 5: intersect Press: ENTER four times. The display will read the intersection. The x value at the intersection point is where the equations are equal.

The solution is x = 2

Show each step of your solution: I. Locate the variable in the equation

3x – 2 = 4. II. Identify the operations applied to the value

of x. Multiply by 3 and Subtract 2.

III. Undo one of the operations applied to x by balancing the equation 3x − 2 +2 = 4+2

IV. Simplify the equation 3x = 6 V. Undo the second operation applied to x,

by balancing the equation 36

33

=x

VI. Simplify the equation x = 2


5 4.2: Connecting Algebraic Expressions to Geometry

The angles in a triangle add to________

Write an equation to model the sum of the angles in this triangle.

Write the equation in a simplified form.

Use the equation to determine the value of x. Calculate the size of: < W: < Y: < Z:

The sum of the angles in a right angle add to______ Write equations to model the sums of the angles. Write the equations in simplified forms. Use these equations to determine the values of x and y. Calculate the size of: < CBP: < ABQ:

2x +4

5x-3

x+11

w

Y

Z

A

B

C

<ABC = 90°

2x 60° 3y+5

43°

P

Q

D


5.4.3: Connecting Algebraic Expressions to Geometry Write an equation and solve for the unknown.

a) b) c) d) e) f)

g) h)

3d + 10º

2c + 40º

2a

b + 20º60º

130º

120º 100º

e – 15º

75º

78º

f – 32º

112º

5g – 22º 2h + 44º126º


5.4.3: Connecting Algebraic Expressions to Geometry (continued) i) j)

k) l)

m) n)

75º

j º

kº

2kº

l º

2l º 3l º

80º

30º

2kº 120º 3nº

2nº

i º

2i º


Unit 5: Day 5: Connecting Rate/Initial Conditions to y = mx + b Grade 9 Applied

75 min.

Description • Interpret descriptions of situations to identify: rate/slope; initial condition/

y-intercept; special case(s)/point on line; formula or rule for relationship/equation of line.

Materials • BLM 5.5.1, 5.5.2, 5.5.3, 5.5.4 • graphing calculators • graph paper


Minds On ... Whole Class Discussion Present a contextual linear problem, e.g., video rental costs, park passes or cell phone charges with a word description of the relationship. Use rate of change, starting (initial) condition, specific set of values; and formula/rule for the relationship. Guide completion of each value in the 1st scenario on BLM 5.5.1. Small groups Practice Groups fill in the chart for other contextual examples. Encourage students to create realistic scenarios.

Action! Whole Class Discussion Introduce alternative terminology. For the starting (initial) condition, C-intercept (cost intercept), P-intercept (pay intercept) as indicated by the context or vertical intercept in general suggesting that these intercepts predict the point where the line intersects the vertical axis. For the rate of change, the slope of the line. For a specific set of values, a point on the line. For the formula/rule, the equation of the line. Add the more general titles to the chart headings and include a “sketch of the relation” column. Small Groups Practice Curriculum Expectations/Observation/Checkbric: Students work in groups to complete the last three rows of BLM 5.5.1. They keep track of the sequence in which they completed the tasks for each scenario.

Consolidate Debrief

Whole Class Discussion Use BLM 5.5.2 to discuss strategies students used to graph the relations, e.g., plot points and use the slope to find the initial condition; plot the starting condition and use the slope to find the next point then connect the points, plot two points and connect them with a line that extends back to the vertical axes so the starting conditions can be read from the graph. Discuss strategies students used to find the equations of relations, e.g., use slope and y-intercept as m and b; use two points to find the slope and substitute the slope and x and y values of one point into y = mx + b to determine the y-intercept. Students complete BLM 5.5.2 and summarize ways to graph and write equations. Note that slope and one point must be known (or found). Students should see that knowing the equation allows them to use the equation for prediction of “any value.”

This lesson revisits the contextual linear problems from Unit 3 to introduce equations of lines given: Two points on the line; the y-intercept and the slope and the y-intercept and a point on the line. Many contextual examples of linear relationships are discrete or piece-wise linear. Care should be taken to avoid or clarify such cases. Prompt students to recall strategies used to model linear relations. Graphing a relationship from the description should be one of their first choices.


Home Activity or Further Classroom Consolidation • Solve the problem on worksheet 5.5.3. Show your answers with a graph and

an equation. • Play the game Linear Invaders.

BLM 5.5.4 provides instructions for the game, Linear Invaders.


5.5.1: Making Connections

Scenario Rate of Change

Initial Condition

Specific or Sample Case Formula or Rule

Tom agrees to a gym membership which charges an initiation fee of $35.00 plus $2.00 per visit

Describe another linear scenario.

Describe another linear scenario.

115km/hr 40 km

(0, 150) (20, 650)

y = 4x + 5


5

Y

x

7

5

9(4,9)

5.5.2: Forming Equations

Case 1 Case 2 Case 3 Write an equation given slope and y-

intercept Write an equation given

a point and the slope Write an equation given two points

Method Place "m" and "b" into the equation.

Method - Substitute the x and y values of the point and the slope value as m into the equation - Solve for "b" - Place "m" and "b" into the equation

Method - Find the slope by graphing the points. - Substitute one point and the slope into the equation. - Solve for “b.” - Place "m" and "b" into the equation.

Example Write the equation through (160,100) with a slope of 0.50.

y = mx + b 100 = 0.50 (160) + b

Using y = mx + b, substitute in the point (160,100) and the slope.

100 = 80 + b 100 − 80 = 80 − 80+ b 20 = b

Solve for b.

Example Write the equation when: m = 0.50 , b = 20 The equation is: y = 0.50x + 20

m= 0.50 and b = 20, so the equation is: y = 0.50x + 20

Place "m" and "b" into the equation.

Example Write the equation of the line through: (3, 7) and (4, 9) First, find the slope by graphing the points and calculating the rise/run.

Next, substitute (3, 7) and m = 2 into the equation and solve for "b"

y = mx + b 7 = 2(3) + b

7 = 6 + b 1 = b The equation is y = 2x + 1

Check your understanding: Write the equations given: m = −1 and b = 8 m = 3.5 and b = −2 m = 0 and b = 1 m = 15 and b = 0

m = 31 and b = 6

Check your understanding: Write the equations given: Point (2, 10) and m = 3

Point (4, −18) and m = −5

Check your understanding: Write the equations given: Points (5, −1) and (3,3) Points (9,2) and (0,−7)

(3,7)


5.5.3: Which Would You Choose? There are 3 teen drop-in centres in your community that rent rooms for small gatherings of up to 20 people. The centres provide decorations, tables and chairs, food, beverages and a new-release video for your event. Families often rent the rooms for celebrations. Part of the charges for the room rental are donated to the teen help phone and the local food bank. You and 8 of your friends would like to celebrate your birthday at one of these centres. Each of the three centres has a different fee schedule advertised in their flyer as posted below. Which centre would you choose for your party?

Centre 1 Centre 2 Centre 3

Party Rooms

For all your party needs!

Cost for 2 guests is $20.

The total cost for 5

guests is $35

Room 4 U

For that special occasion!

We charge $15 for the

room plus $4.50

per guest.

Open Doors

For your important day!

Only $30 to book the

room. The total cost

for 6 guests is $51.

Explore Read the problem and the information in the flyers. Are you given the cost to book the room and the rate per guest for each centre? Discuss how you could model a solution to the problem. Model Construct a graphic and algebraic model for each drop-in centre. Transform Use the model to determine which drop-in centre that you will book for your birthday. Conclude Provide a rationale explaining why this is the best choice for your party. Extending the problem 1. If you could invite 19 guests would you book with a different centre? Explain. 2. How would your decision change if you were to invite 10 guests? Explain. 3. If you had a maximum of $50 to spend, with which centre would you book your party?

Explain.


5.5.4: Linear Invaders On this Linear Invaders game board, graph each of the equations generated by your rolls of the number cubes. Fill in the scorecard for each line graphed.

-3

+5

+5

-3

+1

+2

+1

+3

+1

-2

+3

+2

+4

+1

-2


5.5.4: Linear Invaders (continued) Given an Equation of the Line

Roll a number cube to fill-in the missing numbers. Notice that some negative signs are indicated and should be used. Graph each line on the grid. (Remember to label each line.) Calculate the score of each line. (Sum of values of all invaders your line touches.) Calculate your total score. Given a y-Intercept and the Slope

Roll a number cube to fill in the missing numbers in the second and third columns. Plot the y-intercept and use the slope to find another point. Determine the equation [spaces are provided to indicate signs]. Construct a line. (Remember to label each line.) Calculate the score of each line. Calculate your total score.

# Equation Score

+= xy

−= xy

+−= xy

−−= xy

xy =

xy −=

=y

Total Score

# y-intercept Slope Equation Score

( 0, −___ ) ______ ____ xy =

( 0, ___ ) −______ ____ xy =

( 0,− ___) ______ ____ xy =

( 0, ___ ) −______ ____ xy =

( 0, −___ ) ______ ____ xy =

( 0, ___ ) −______ ____ xy =

( 0, −___ ) ______ ____ xy =

Total Score


5.5.4: Linear Invaders (continued) Given a Point and the Slope Roll a number cube to fill-in the missing numbers in columns 2 and 3. Notice that some negative signs have been indicated and should be used. Plot the point and use the slope to construct the line. Determine the equation of the line. (Remember to label each line.) Calculate the score of each line. Calculate your total score. Given 2 Points on the Line Roll a number cube to fill-in the each of the 4 missing numbers in columns 2 and 3. Plot the points on the grid and construct a line. Determine the equation of the line (Remember to label each line.) Calculate the score of each line. Calculate your total score.

# Point Slope Equation Score

( −___, −___ ) ______ ____ xy =

( ___, ___ ) −______ ____ xy =

( ___, − ___) ______ ____ xy =

(− ___, ___ ) −______ ____ xy =

( ___, ___ ) ______ ____ xy =

(− ___,− ___ ) −______ ____ xy =

(− ___, ___ ) ______ ____ xy =

Total Score

# Point Point Equation Score

(−___, −___ ) (−___, ___) ____ xy =

(___, ___ ) (___, ___) ____ xy =

(___,− ___) (−___, −___ ) ____ xy =

(−___, ___ ) (___, ___ ) ____ xy =

( ___, ___ ) ( −___, ___ ) ____ xy =

( −___, ___ ) ( ___, ___ ) ____ xy =

( ___, ___ ) ( −___, −___ ) ____ xy =

Total Score


Unit 5: Day 6: Investigating Slopes Grade 9 Applied

75 min.

Description • Make connections between the magnitude of slope and direction of a line. • Provide meaning for zero slope, and slopes of parallel and perpendicular lines.

Materials • BLM 5.6.1 • The Geometer’s Sketchpad®


Minds On ... Whole Class Discussion Introduce the exploration by referring to properties of slopes already described in earlier units. For example: Faster walking speeds have larger slope numbers, a negatively sloped line represents decreasing distance on a distance/time graph, etc.

Action! Small Groups Investigation Before beginning each student draws horizontal lines to divide a sheet of paper into sections. Students work with a partner to investigate the following properties: (BLM 5.6.1). • The larger the size of the slope, the steeper the lines. • Lines with positive slope go up to the right. • Lines with negative slope go down to the right. • Parallel lines have the same slope. • Slopes of perpendicular lines multiply to give −1. Curriculum Expectations/Anecdotal/Checkbric: As each investigation is concluded, students record their learning. Circulate and check student understanding and progress.

Consolidate Debrief

Whole Class Discussion Check for understanding by discussing solutions to the questions. Discuss the case of a vertical line and why it is reasonable that the slope is undefined, i.e., as the line gets steeper and steeper, the slope number gets larger and larger. ‘Undefined,’ means an infinitely large-sized number, either positive or negative, depending on whether we think of the vertical line as having started with a positive or negative slope. Consider the case of the horizontal line by making the connection to motion graphs (no change in distance as time changes means a “0” speed, hence the slope is also zero).

Application Concept Practice Skill Drill

Home Activity or Further Classroom Consolidation Sort the equations of parallel and perpendicular lines on the basis of the slope in the following problems. (The teacher selects problems in which students must apply algebraic skills of collecting like-terms and multiplying a monomial times a binomial.)


5.6.1: Investigating Slopes

Open The Geometer’s Sketchpad. Under the GRAPH menu, choose rectangular grid under Grid Form. Drag the point along the x axis so that the grid displays a window from (−10 to 10) on both axes. To Plot Points: Under the Graph Menu, choose Plot Points Enter the x value by tabbing from field to field. Press PLOT after you enter each point.

Part A. Investigating positive slopes 1. Label the origin (0,0) with the letter P 2. Plot the following points on the axes and label the points with the appropriate letter.

X Y

1 2 8 This is point A (2,8)

2 3 9 This is point B (3,9)

3 5 5 This is point C (5,5)

4 8 4 This is point D (8,4)

5 9 3 This is point E (9,3) 3. Construct the segments PA, PB, PC, PD, PE 4. Select the line segments. Under the Measure menu, choose Slope. 5. Move the measurements close to the segments they match. 6. Discuss with your partner: What do you notice about the relationship between the size of the slope and the steepness

of the segment? 7. In your notebook: a) Write a statement about the relationship between the size of the slope and the

steepness of the segment. b) Put lines with these slopes in order from the steepest to the least steep:

0.2 7 12.5 21 3.99

8. Briefly record what you have determined about lines with positive slopes.


5.6.1: Investigating Slopes (continued) Part B: Investigating negative slopes 1. Delete the segments you constructed in Part A. 2. Plot these points. Construct all possible segments between these points. Hint: You should

have 6 different segments.

X Y

1 5 −4 This is point A (5,−4)

2 −7 2 This is point B (−7,2)

3 −4 7 This is point C (−4,7

4 8 5 This is point D (8,5)

3. Measure the slopes of the line segments and move the labels next to the segments. 4. Discuss with your partner: What is the difference between lines with a positive slope and lines with a negative slope. 7. In your notebook: a) Sketch these segments and match them with the best slope number. b) Is it possible to draw a triangle with all three sides having a positive slope? If so, draw

the triangle. If not, explain why. c) Is it possible to draw a quadrilateral with three positive slopes? If so, draw the

quadrilateral. If not, explain why. 8. Briefly record what you have determined about lines with negative slopes.

Line C

Line B

Line A

Y

Line D

X

Possible Slopes: 5, -5, 0.2, -0.2


5.6.1: Investigating Slopes (continued) Part C: Investigating Parallel Lines 1. Delete the segments you constructed in Part B. 2. Plot these points. Construct AB, CD, and EF.

X Y

1 −8 3 This is point A (−8,3)


3 −5 2 This is point C (−5,2)

4 5 8 This is point D(5,8)

4 −7 −2 This is point E (−7,−2)

5 3 4 This is point F (3,4)

3. Measure the slope of these segments. Move the labels next to the segments. 4. Discuss with your partner:

What do you notice about the slopes of these segments? Place your ruler on the monitor and slide it from one line to the other. What can you say about the relationship between the segments?

7. In your notebook:

Describe the relationship between lines that have the same slope. Include an example. 8. Briefly record what you have determined about lines with equal slopes.


5.6.1: Investigating Slopes (continued) Part D: Investigating Perpendicular Lines 1. Plot these points. Construct segments AB, CD.

X Y

1 −4 3 This is point A (−4,−3)


3 3 1 This is point C (3,1)

4 −1 9 This is point D (−1,9)

2. Measure the slopes of AB and CD. 3. Select AB and CD and construct the point of intersection (under Construct, choose

intersection) 4. Label the point of intersection: P. 5. Measure angle APC. 6. Discuss with your partner: What is the mathematical term for segments that form a 90° angle with each other? Multiply the two slopes together. (Under Measure, choose Calculate. Select slope CD, press

the multiply (*) sign, then select slope AB. Press OK). What do you notice when you multiply the slopes together?


5.6.1: Investigating Slopes (continued) Part E: Investigating Perpendicular Lines Further

Open The Geometer’s Sketchpad. Construct a line segment and label its endpoints A and B. Construct a point on AB. Label this point C. Select C and AB and construct a perpendicular line. Select the perpendicular line and construct a point on it. Call it point D. Select the perpendicular line and construct another point on it. Call this point E.

1. Measure the slope of AB and DE. a) Select segment AB. Measure its slope. b) Select segment DE. Measure its slope.

2. Calculate the product of the two slopes. Under the Measure menu, choose Calculate. Select one of the slope measurements. Press * (multiply) on the calculator. Select the other slope measurement. Press OK.

3. Select the point B. Move B around on the screen. a) What happens to the size of the angles? b) What happens to the product of the slopes?

4. With your partner: a) Describe the relationship between the slopes of perpendicular lines. b) Fill in the blanks.

Slope 1 Slope 2 Parallel, Perpendicular or neither

5 −51

−3 −3

2 21

−10 101

32 −

23

3 Parallel

3 Perpendicular

−8 Perpendicular

71 Perpendicular

−1.5 Parallel

5. On the back of this sheet, briefly record what you have determined about the slopes of perpendicular lines.


Unit 5: Day 7: Investigating Equations of Linear and Non-Linear Relations Grade 9 Applied

75 min.

Description • Investigate the properties of linear and non-linear relations noting information

to be gained from graphs, equations, or calculation of differences.

Materials • The Geometer’s Sketchpad® V4 • BLM 5.7.1


Minds On ... Whole Class Discussion Introduce the exploration by making reference to properties of linear relations that have already been identified in the course: First differences are the same number, linear relations graph as lines and the equation shows a slope and y-intercept.

Action! Small Groups Investigation Learning Skills/Initiative/Checklist: Use this opportunity to access prior learning in order to decide what further review is needed Students determine characteristics of linear relations (BLM 5.7.1). Students sort the equations into a graphic organizer that identifies the properties of linear equations. Consider the case where the line is vertical (not graphed on GSP) and include it in the graphic organizer.

Consolidate Debrief

Whole Class Discussion Check for understanding with examples for which students must decide (linear or non-linear) and justify the decision.

Some equations require algebraic manipulation for the features of the equation to be connected to the shape of the graph.

Application Reflection Skill Drill

Home Activity or Further Classroom Consolidation Sort the equations in the problems. (The teacher selects problems in a context.)

Use this as an opportunity to apply algebraic skills of collecting like-terms and multiplying a monomial times a binomial.


5.7.1: Investigating Linear and Non-linear Relations

The Geometer’s Sketchpad® 4 Set Up 1. Open a New Sketch. 2. In the GRAPH Menu, choose Square Grid under Grid Form. 3. Adjust the scale so that you can view a window of (−10 to 10) on the x and y axes. 4. To enter the equation: Press CTRL F then Click OK. 5. To view the graph: Press CTRL G.

How can you determine whether or not a relationship is linear from the equation?

Hint: Some equations require algebraic manipulation in order for you to connect the characteristics of the equation to the graph

Equation Sketch the Shape Table of Values and Calculation of Differences

Linear/ Non-Linear

32 += xy

2x)x(42xy 2 +−=

5342 +−= xy

)( 41 −−= xy

943 23 −−= xxy

70 += xy

xxy 925 +−−=

TIPS: Section 3 – Grade 9 Applied: Summative Task © Queen’s Printer for Ontario, 2003 Page 201

Paper Drinking Cups Summative Task

Grade 9 Applied Total time 225 minutes Materials •• The Geometer’s Sketchpad® Version 4 (dynamic geometry software),

GSP file: Cones (p. 209) •• graphing calculators or spreadsheet software, •• rulers, scissors, tape, straw, string, BLMs

Description Students investigate the relationship between the sector angle of a circle and the volume of the cone that is formed from the sector. They use manipulatives and technology to collect data for the investigation. Students submit a report that justifies and explains their conclusions.

Expectations Assessed* and addressed

Number Sense and Algebra NA1.03 – demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course; NA2.01 – evaluate numerical expressions involving natural-number exponents with rational-number bases; NA2.02 – *substitute into and evaluate algebraic expressions involving exponents, to support other topics of the course; NA3.03 – solve first-degree equations, excluding equations with fractional coefficients, using an algebraic method; NA4.03 – communicate solutions to problems in appropriate mathematical forms (e.g., written explanations, formulas, charts, tables, graphs) and justify the reasoning used in solving the problems. Measurement and Geometry MGV.01 – *determine the optimal values of various measurements through investigations facilitated by the use of concrete materials, diagrams, and calculators or computer software; MG2.02 – solve simple problems using the formulas for the surface area of prisms and cylinders, and for the volume of prisms, cylinders, cones and spheres; MG2.03 – solve problems involving perimeter, area, surface area, volume, and capacity in applications; MG3.04 – communicate the findings of investigations, using appropriate language and mathematical forms.


Paper Drinking Cups Summative Task

Expectations Assessed* and addressed

Relationships RE1.01 – pose problems, identify variables, and formulate hypotheses associated with relationships (Sample problem: Does the rebound height of a ball depend on the height from which it was dropped? Make a hypothesis and design an experiment to test it.); RE1.03 – *collect data, using appropriate equipment and/or technology; RE1.04 – *organize and analyse data, using appropriate techniques and technology; RE1.05 – describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain the differences between the inferences and the hypotheses (Sample problem: Describe any trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your original hypothesis? Discuss any outlying pieces of data and provide explanations for them. Suggest a formula relating the height of the visible region to the distance from the wall. How might you vary this experiment to examine other relationships?); RE1.06 – communicate the findings of an experiment clearly and concisely, using appropriate mathematical forms (e.g., written explanations, formulas, charts, tables, graphs), and justify the conclusions reached; RE1.07 – solve and/or pose problems related to an experiment, using the findings of the experiment; RE2.04 – *construct tables of values and graphs to represent non-linear relations derived from descriptions of realistic situations. Ontario Catholic School Graduate Expectations CGE 3c - thinks reflectively and creatively to evaluate situations and solve problems; CGE 5a - works effectively as an interdependent team member.

Prior Knowledge

Skills with The Geometer’s Sketchpad® Students should be able to: •• open/close files •• select/deselect one or more objects •• drag points

Skills with graphing calculators and/or spreadsheets Assessment Tools

Rubric

Extensions Explore other relationships within this context. Examples: a) sector angle vs. surface area (linear) b) sector angle vs. height (non-linear) c) surface area vs. volume (non-linear)


Pre-task Instructions Read the following script to students on the day before the investigation begins. Teacher Script 1. We will be working on an investigation over the next three mathematics classes. 2. An investigation is an extended problem designed to allow you to show your ability to undertake an

inquiry to make a hypothesis, formulate a plan, collect data, model and interpret the data, draw conclusions and communicate and reflect on what you have found.

3. For this investigation you will be using The Geometer’s Sketchpad®. As well you will need pencils,

pens, an eraser, a ruler, notepaper and a graphing calculator to complete the work. 4. As you do the investigation you will work as part of a group and also individually. (Distribute an

envelope or folder to each student.) I am giving each of you an envelope in which you can store your notes for the duration of the investigation. Write your name on the front of the envelope.

5. On the third day you will write a report giving your conclusions and summarizing the processes you

have followed to arrive at them. 6. Be sure to show your work and include as much explanation as needed. 7. Each section of the investigation has a recommended time limit that I will tell you so you can manage

your time. 8. You will be assigned to the following groups for the three days of the investigation. (You may wish to

assign students to their groups at this time- recommended group size is four students.) 9. Are there any questions you have regarding the format or the administration of the investigation? Teacher Notes • This summative task could be used for gathering summative assessment data or for providing formative

feedback to students before they complete another task for assessment purposes. • If a Home Activity or Further Classroom Consolidation task is to be used for gathering assessment data,

it may be most appropriate for students to work on it independently under teacher supervision. • Some suggested Home Activity or Further Classroom Consolidation tasks help prepare students for

later assessments.


Day 1: Introducing the Problem Grade 9 Applied

75 min.

Description • Explore various strategies for determining volume of a cone whose radius and

height are not given. • Build concrete models that represent a range of possible cones constructed

from the same circle.

Materials • BLM S1.1, S1.2 • centimetre grid paper, straw, tape, scissors • sand (optional) • preconstructed cone (see Action!)

Minds On ... Whole Class Guided Read the problem (BLM S1.1), asking clarifying questions: How is it possible to make different cones from the same size of circle? What is the same and what is different about each cone? Why is volume a consideration in the design of the cone? Why is the amount of waste (unused paper) an important consideration for a company? Students must recognize that volume is a variable that needs to be explored.

Action! Small Groups Creating a Model Distribute BLM S1.2. Have materials ready for groups to access. Assign one of the following sector angles to each group: 90o, 135o, 180o, 225o, 270o, 315o. Students complete questions 1 to 5 on BLM S1.1 and record solutions individually. Whole Class Discussion Discuss solutions to question 3 using the cone made from the 45o sector angle. Lead a discussion on question 5. Compare different methods of determining required measurement. Include the following ideas: • volume can be determined/estimated in a number of ways, e.g., filling the cone

with sand then measuring volume of sand using a formula • radius of the base of the cone and height are needed for the volume formula • radius of the base of the cone could be measured or calculated (using perimeter

of construction circle on BLM S1.2) • height of cone could be measured or calculated using the Pythagorean theorem

(with radius of base measure and slant height measure) Collect and display the constructed cones. • What does optimization mean? • Which cone do you think has the biggest volume? • Do you think there is ONE biggest volume? • How would you find the maximum volume if it exists? Small Groups Gathering Evidence Students review their procedures in question 3 and modify their solutions, if errors are identified. Students answer question 6.

Consolidate Debrief

Whole Class Guided Discussion Record solutions to question 6 on the board and examine the data. • What would a graph of your collected data look like? • What would it look like if there was no maximum volume? one maximum

volume? • Are there any other relevant factors in the selection of a design? • What feature does a graph have if there is a minimum value? Is the relationship between the sector angle and the volume linear or non-linear? Individual Gathering Evidence Students answer question 7 and add their results to earlier data.

Clarify vocabulary, e.g., slant, height Note that the 45o sector angle is not assigned to a group. (The teacher makes one for later use.)

Differentiated

Home Activity or Further Classroom Consolidation If you are not satisfied with your solution to question 6, redo the question using a different strategy. If you are satisfied with your solution to question 6, describe two other strategies that could have been used to determine the volume, and explain why you prefer your strategy.


S1.1 Introducing the Problem

Your company makes cone-shaped drinking cups from circular pieces of paper. Since several different sizes of cones can be made from the same circular piece of paper, you must decide which one your company will produce. Consider the volume of the cone and other relevant factors, e.g., waste material, then write a report that makes a recommendation for the design of the drinking cups.

Small groups – Creating a Model

1. Shade your group’s assigned sector in the diagram on the left.

2. Construct the corresponding cone using the net.

3. Determine each of the following: a) sector angle b) radius of circle c) perimeter of circle d) perimeter of base of cone

4. Which of the values in question 3 goes into the box?

(perimeter of base of cone) = 360 × (perimeter of circle)

Small Groups – Gathering Evidence 5. Discuss how you would determine the volume of your cone. 6. Determine the volume of your cone. Show or describe your work. Individual – Gathering Evidence 7. Cut a sector with an angle that is between 270o and 360o. Construct the cone. Determine the

volume of the cone. Show your work.


S1.2 Circle Net


Day 2: Paper Drinking Cups – Graphical Model Grade 9 Applied

75 min.

Description • Graph data representing the range of cones. • Identify the need to interpolate to determine maximum value.

Materials • graphing calculators and/or spreadsheet software • BLM S1.3

Minds On ... Whole Class Hypothesizing Display the cones from Day 1. Ask students to make a hypothesis: Which range of sector angles would produce a cone with the biggest value? (e.g., 270o to 315o) Ensure that students understand why they are trying to optimize the volume.

Action! Small Groups Discussion Assign questions 1 and 2 on BLM S1.3. Tell students that each group will report to the whole class. Whole Class Discussion Facilitate consolidation of the small group discussion. Ensure that students understand that the chart may not display the largest volume. Ask how students would add data to the chart. Which values would they add? (sector angles between 270o and 315o plus, perhaps, a few slightly larger than 315°) Ensure that students know which columns of data should be graphed and that they understand instructions for BLM S1.3 questions 3 to 6. Individual Creating a Graphical Model Students complete questions 3 to 6 individually. Collect the individual responses, return them the next day so students may use them for their reports.

Consolidate Debrief

Whole Class Consolidate Question students about the prior activity. • Do you think your answer was very accurate? • What would you have done if you had more time?

Students give reasons for their responses. Whole Class Prepare for Next Day Discuss previous GSP explorations. What special things did you look for? …do? What is a dynamic model? What are some of the advantages and disadvantages of using a dynamic model? Use the screen capture at the bottom of BLM S1.3 to discuss the next day’s activities. Homogeneous Groups Differentiated Activity Provide guided review of concepts and skills and practice questions for students who had difficulty finding volume of a cone or graphing the data. Challenge other students to write a distance vs. time story that would result in a graph the same shape as the graph of volume vs. sector angle. Further challenge these students to assign appropriate scales to the axes for the story they created.

If a student was unsuccessful with the question submitted from Day 1, confer with him/her during Day 2. Consider assessing learning skills during the small group activities. The relationship is non-linear. Notice the last two data points – one is near the top of the graph and the other is on the horizontal axis. The scatter plot barely shows that the graph is going to start going down – it is very easy to “miss” the horizontal intercept.

These stories could be shared when reviewing for a summative test.

Application Concept Practice Differentiated

Home Activity or Further Classroom Consolidation Complete the differentiated activity.


S1.3 Paper Drinking Cups – Graphical Model Small Groups – Exploring the numerical model 1. Mr. Henry’s class collected the following data. How would you use the data to determine

which sector angle would create a cone with the maximum volume?

Circle Radius

(cm)

Sector Angle

(degrees)

Circle Perimeter

(cm)

Base Circle Perimeter

(cm)

Base Radius

(cm)

Cone Height (cm)

Cone Volume

(cm3)

8.8 45 55.3 6.9 1.1 8.7 11.1 8.8 90 55.3 13.8 2.2 8.5 43.2 8.8 135 55.3 20.7 3.3 8.2 93.0 8.8 180 55.3 27.6 4.4 7.6 154.5 8.8 225 55.3 34.6 5.5 6.9 217.6 8.8 270 55.3 41.5 6.6 5.8 265.5 8.8 315 55.3 48.4 7.7 4.3 264.5 8.8 360 55.3 55.3 8.8 0.0 0.0

2. What further information do you need? Individual – Creating a graphical model 3. Create a graphical model (scatter plot) of the relationship between the size of the sector

angle and the volume of the cone. Use Mr. Henry’s class data. You might add some of your own data. If you are using a graphing calculator, use the given window settings. Sketch your graph (or print it if you are using a spreadsheet).


S1.3 Paper Drinking Cups – Graphical Model (continued) 4. Draw a curve of best fit on your graph. 5. Is the relationship between the measure of the sector angle and the volume of the cone,

linear or non-linear? Give reasons for your answer. 6. Use your graph to determine which sector angle will produce a cone with the maximum

volume. Give reasons for your answer. Whole Class - Instructions


Day 3: Graphical Model Grade 9 Applied

75 min.

Description • Use The Geometer’s Sketchpad® to investigate sector angles for cones.

Materials • The Geometer’s Sketchpad® V4 • GSP file: Cones (p. 209) • BLM S1.4

Minds On ... Whole Class Guided Review classroom procedures for working in a computer lab. Summarize the class discussion about exploring a dynamic model. Ensure that students understand how the sketch connects with the original problem (finding the sector angle for maximum volume).

Action! Pairs Exploring a Dynamic Model Pairs complete BLM S1.4 questions 1 and 2, using the GSP file (see Day 2). Download this file at www.curriculum.org/occ/tips/downloads.shtml Individual Concluding Students use their work from Days 1 and 2. Individual students do question 3. They write a report including the requirements outlined on BLM S1.4. Teacher Note: When students use the dynamic model to find the cone with the maximum volume they might notice that the shape of the cone is awkward for a drinking cup. A Level 4 response could include the correct sector angle but also suggest that the circle be cut into two equal parts. Students could justify this conclusion by stating that there would not be any waste material.

Consolidate Debrief

Group Discussion Revisit the discussion about the advantages/disadvantages of using a dynamic model. Identify all of the different types of models that were used in this investigation. (Students reflect on the connections between the models [physical, numerical, algebraic, graphical, dynamic.])

Prepare a set of instructions for opening the GSP file: Cones on your system. Consider making an observation checklist to assess students while they are working at the computers.

Application Concept Practice Exploration Reflection

Home Activity or Further Classroom Consolidation 1. What other relationship(s) could be explored with the dynamic model.

Sample responses: sector angle and height, sector angle and slant height, sector angle and surface area, volume and surface area, etc.

2. Do you think the relationship(s) would be linear or non-linear? 3. How could you apply the information about other relationships?

(Who would use this information?)



S1.4 Paper Drinking Cups – Dynamic Model Exploring a dynamic model – Pairs 1. Open the GSP file: Cones.

Determine the sector angle that will produce a cone with the maximum possible volume. Describe how you found this answer.

2. Use the GSP file: Cones to explore circles with different radii. (Drag the radius control). List

your observations. Individual 3. Write a report with the details about what you have done and the conclusions you have

come to about the original problem.

Your company makes cone-shaped drinking cups from circular pieces of paper. Since several different sizes of cones can be made from the same circular piece of paper, you need to decide which one your company will produce. Consider the volume of the cone and other relevant factors, e.g., waste material, then write a report that makes a recommendation on the design of the drinking cups.

• State a specific recommendation for the company. • Justify your conclusions – explain how you determined your answer. • Include references to any notes, sketches or tables you used. • Suggest any additional steps you would follow if you had more time.


Rubric for Assessment of Paper Drinking Cups

Criteria Below L1 Level 1 Level 2 Level 3 Level 4 Knowing Facts and Procedures (Knowledge/Understanding) Calculations and Computations

Assess for correctness using a marking scheme.

Reasoning and Proving (Thinking/Inquiry/Problem Solving) - no evidence of logic

- limited logic evident

- somewhat logical

- logical - highly logical Makes convincing argument, explanations and justifications

- no conclusion reached

- major omissions in arriving at conclusion

- some omissions in arriving at conclusions

- thorough - complete and extended

Collects suitable data, and analyzes it

- little or no evidence

- with major omissions or errors

- with some omissions or errors

- completely - completely and includes evidence of reflection

Communicates (Communication) Uses mathematical language

- undeveloped use of conventions

- minimal skill in use of conventions

- moderate skill in use of conventions

- considerable skill in the use of conventions

- high degree of skill in the use of conventions

Explains the processes and conclusions to investigations

- unclear/ confusing

- limited clarity - some clarity - clear - precise

Making Connections (Applications) Makes appropriate use of measurement formulas and organizes the data


- with major omissions or errors

- with some omissions or errors

- completely - completely and includes evidence of reflection or verification

Is able to create graphical model


- narrow fit to situation

- moderate fit to situation

- broad fit to situation

- very extensive fit to situation

Documents

Grade 9 Applied: Content and Reporting Targets · ... related to an experiment, using ... areas of composite plane figures (e.g., combinations of rectangles, triangles, parallelograms,