Web viewStudent Activity Sheet. Lesson 1. ... mathematicians recognized that attempting to define every word inevitably led to circular definitions ... find Broadway

Dabbling in Definitions: Intro to Geometric IdeasAligned to the Common Core Standards

Written byJonathan Katz, Ed. D.

Joseph WalterISA Mathematics Coaches

1

Dear Math Teacher,

What is mathematics and why do we teach it? This question drives the work of the math coaches at ISA. We love mathematics and want students to have the opportunity to begin to have a similar emotion. We hope this unit will bring some new excitement to students.

This unit is the initial unit for a full-year geometry course that is aligned to the common core standards. It is a unit that revisits concepts and procedures students experienced in middle school but with an expectation that students will leave with deeper understanding. Essential to this work is an inquiry approach to teaching mathematics where students are given multiple opportunities to reason, discover and create. Problem solving is the catalyst to the inquiry process so as you look closely at this unit you will see that students are constantly placed into problem solving situations where they are asked to think for themselves and with their classmates.

The first four Common Core Standards of Practice are central to this unit. Through the constant use of problematic situations students are being asked to develop perseverance and independent thought, to reason abstractly and quantitatively, and to critique the reasoning of others.

Mathematical modeling is present throughout the unit as students are asked to describe and analyze different bare number problems and real world situations leading to geometric ideas. Students are also asked to create models including the final project, which is to create a city plan based on the ideas of lines and angles.

The other four Standards of Practice are also present in this unit. Two of them are central to the inquiry approach. You will see these two statements in the last two standards.

Mathematically proficient students look closely to discern a pattern or structure. Mathematically proficient students notice if calculations are repeated, and look both for

general methods and for shortcuts.

We believe, as do many mathematicians, that mathematics is the science of patterns. This underlying principle is present in all the work we do with teachers and studentsIn this unit you will see that students are often asked to discern a pattern within a particular situation. This leads students to making conjectures and possibly generalizations that are both conceptual and procedural.

Thank you for looking at this unit and we welcome feedback and comments.

Sincerely,

Dr. Jonathan Katz(For the ISA math coaches)

2

Unit 1 – Dabbling in Definitions: Intro to Geometric Ideas

Essential Questions: How do we come to a precise definition?How do we honor the depth and breadth of a geometric idea?

Interim Assessments/Performance TasksWhat is a Zerf? - Lesson 1 Shoe Size Problem – Lesson 6Create a City Design- Lesson 9How do I Replicate an Angle? – Lesson 10Parallel or Not – Lesson 12Can you Construct a Perpendicular Line? – Lesson 14

Final Assessment: City Design

What will students understand and be able to do at the end of the unit? Students will be able to create precise definitions through multiple experiences with

different geometric ideas. Students will be able to write a precise definition of angle, perpendicular lines, parallel

lines and line segments, based on the undefined notions of point, line and distance around an arc.

Students will prove theorems about lines and angles including vertical angles, alternate angles and corresponding angles when a transversal crosses parallel lines.

Students will understand angle and angle measurement through working with circles and protractors, angle construction and work with parallel and intersecting lines.

Students will understand parallelism both through construction and work on the coordinate plane. This includes using slope to prove two lines are parallel.

Students will understand perpendicularity both through construction and work on the coordinate plane.

What enduring understanding will students have? In order to create a precise definition, you need to know all the qualities and what makes

it unique. Precise definitions are necessary as the building blocks for other geometric ideas. Geometric ideas need to be understood in their multiple contexts, e.g. parallel lines. Through construction one can develop a deeper understanding of geometric figures.

3

Common Core Content Standards in the Unit

G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; constructing perpendicular lines; and constructing a line parallel to a given line through a point not on the line.

G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Common Core Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

4

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams,

5

and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are

6

careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

7

Dabbling in Definitions – Intro to Geometric IdeasLesson 1

Teacher GuideWhat is a Widget?

Opening Activity

1. Observe the two groups. From your observations, what are the characteristics that define a widget? How do you know? Use the pictures as evidence for your statements.

8

2. Now look at this picture. Which of the following are widgets? How do you know?

3. Create a statement that defines a widget.

(To the Teacher: The purpose of this activity is to help students to understand how to develop a definition. The big idea we want students to think about is “What things do all the widgets have in common, and what things do widgets have that others do not have?” This type of question can be used as we develop different definitions in this unit. Let your students work on questions 1 and 2 followed by a whole class discussion. Another discussion should follow question 3. This is an opportunity for students to critique each other’s statement. Is your definition precise enough? Observation is an important idea in mathematics, including geometry, While in geometry you often cannot make assumptions from a drawing, the drawing still gives you a great deal of information that we need to observe.)

9

Second Activity

Observe the following 10 geometric shapes. Select the shapes that you feel belong together. You might try to make a grouping that you think no other group will make. Create a definition for that group and be ready to show that none of the other shapes on the page can fit your definition. Be ready to share with the rest of the class.

(Note to teacher: This is a group activity. The purpose of this activity is for students to hone in on the uniqueness of objects. What are the characteristics that make each shape a member of the group? Ask students to write as precise a definition as possible. The class will then critique each others’ definitions and help each other to refine their definition.)

10

11

Third Activity:

Group Performance Task: What is a Zerf?

Now it’s your turn to create your own activity. We’re going to call these objects zerfs. You’re going to create a group of shapes that ARE zerfs, a group that are NOT zerfs and challenge the class to figure out the definition of a zerf.

12

Dabbling in Definitions – Intro to Geometric IdeasStudent Activity Sheet

Lesson 1Name_______________________Date________________________

What is a Widget?

Opening Activity

1. Observe the two groups. From your observations, what are the characteristics that define a widget? How do you know? Use the pictures as evidence for your statements

13

2. Now look at this picture. Which of the following are widgets? How do you know?

3. Create a statement that defines a widget.

14

Second Activity

Observe the following 10 geometric shapes. Select the shapes that you feel belong together. You might try to make a grouping that you think no other group will make. Create a definition for that group and be ready to show that none of the other shapes on the page can fit your definition. Be ready to share with the rest of the class.

15

Third Activity:

Group Performance Task: What is a Zerf?

16

Now it’s your turn to create your own activity. We’re going to call these objects zerfs. You’re going to create a group of shapes that ARE zerfs, a group that are NOT zerfs and challenge the class to figure out the definition of a zerf.

17


Teacher GuideWhere do these Figures Belong? - Creating Definition

Opening ActivityBased on yesterday’s lesson think about these two questions and be ready to discuss your ideas with the rest of the class

a) What are the qualities of a good definition?b) What do you need to do in order to write a good definition?

(To the Teacher: This discussion is important as it is central to the work in the whole unit. An outcome of this discussion could be a poster based on the agreed-upon answers to the two questions.)

Second Activity: A beginning step in creating definitions

(To the Teacher: Students should be encouraged to define as many groupings as they can, e.g. every figure can be in a group called points and sets of points. We want students to see those qualities that are common and what is it that makes something unique. As we know there are undefined terms in geometry. In ancient times, geometers attempted to define every term. For example, Euclid defined a point as that which has no part. In modern times, mathematicians recognized that attempting to define every word inevitably led to circular definitions, and in geometry left some words as undefined. This should be part of your discussion.)

You will be given a set of figures. In your group classify them in any way that you want. You can make as many groups as you can. You may put a figure into more than one group.Think about this question: Why did you put these figures together in this group?

Be sure to make clear your explanations as to your groupings.As each group presents their findings, other groups will ask questions and add their own findings.Through these classifications we can begin to create a definition for each of the different groups.

Geometric Figures

.p a

.b

c d

18

(To the Teacher: You should have separate charts ready to list student findings. For example, under a chart called angles, you would list all the different ideas students came up with describing the qualities of angles. These definitions will be revisited throughout the unit. We build on them and make them more precise as we dig into different ideas.)

19


Lesson 2 Where do these Figures Belong? - Creating Definition

Name_______________________Date________________________

Opening Activity

Based on yesterday’s lesson think about these two questions and be ready to discuss your ideas with the rest of the class.

What are the qualities of a good definition? What do you need to do in order to write a good definition?

Second Activity

You will be given a set of figures. In your group classify them in any way that you want. You can make as many groups as you can. You may put a figure into more than one group.Think about this question:

Why did you put these figures together in this group?

Be sure to make clear your explanations as to your groupings.As each group presents their findings, other groups will ask questions and add their own findings.

Through these classifications we can begin to create a definition for each of the different groups.

20

Geometric Figures

.p a

.b

c d

21


Teacher GuideWhat does a city’s street design tell us about that city?

(To the Teacher: This lesson is asking students to look at maps with a geometric eye. It is also about their aesthetic appreciation. Do they prefer the regularity of a city like Manhattan or the unplanned evolution of Rome or Erfurt? Another goal of the lesson is for students to take their developing geometric definitions and talk about them concretely. Students should work on questions in the opening activity followed by a discussion of their geometric sense. The second activity is quite simple and it is recommended that students work individually for two reasons: to get multiple responses and to learn about their developing understanding of the geometric definitions.)

Today we are going to revisit our definitions through looking at maps of cities around the world.

Opening Activity

You will now observe four different city maps. The maps are located at the end of this lesson.

Spend some time looking at these maps, thinking about our question on top of the page and then

answering the questions below.

1. What is similar about the streets in the maps?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

2. What is different about the streets in the maps?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

3. Which map do you feel is most interesting, a place where you might want to live? Why?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

22

4. Which city’s street design do you think is best? Why?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

5. How would you describe the geometry of the city you just defined as best?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

6. How would you describe the geometry of one of the cities that was very different?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

Second Activity

Now we’re going to focus on the particular groupings from yesterday’s lesson and look at them

on the map of Detroit.

1. Find an example of a point and a line segment. Sketch each of them and describe their

location.

2. Find an example of a curved line segment. Sketch it and describe its location.

23

3. Find an example of two roads that meet to make an acute angle. Sketch the angle that is

formed by the two roads and write the names of the streets.

4. Find an example of two roads that meet to make an obtuse angle. Sketch the angle that is formed by the two roads and write the names of the streets.

5. Find an example of two roads that meet to make a right angle. Sketch the angle that is formed by the two roads and write the names of the streets.

6. Find an example of two roads that are parallel to each other. Sketch the relationship of the two roads below and label the streets’ names.

7. Find an example of two roads that are perpendicular to each other. Sketch the relationship of the two roads below and label the streets’ names.

24


Lesson 3Name_______________________Date________________________

What does a city’s street design tell us about that city?

Today we are going to revisit our definitions through looking at maps of cities around the world.

Opening Activity

You will now observe four different city maps. The maps are located at the end of this lesson.

Spend some time looking at these maps, thinking about our question on top of the page and then

answering the questions below.

1. What is similar about the streets in the maps?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

2. What is different about the streets in the maps?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

3. Which map do you feel is most interesting, a place where you might want to live? Why?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

25

4. Which city’s street design do you think is best? Why?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

5. How would you describe the geometry of the city you just defined as best?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

6. How would you describe the geometry of one of the cities that was very different?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

26

Second Activity

Now we’re going to focus on the particular groupings from yesterday’s lesson and look at them

on the map of Detroit.

1. Find an example of a point and a line segment. Sketch each of them and describe their

location.

2. Find an example of a curved line segment. Sketch it and describe its location.

3. Find an example of two roads that meet to make an acute angle. Sketch the angle that is formed by the two roads and write the names of the streets.

4. Find an example of two roads that meet to make an obtuse angle. Sketch the angle that is formed by the two roads and write the names of the streets.

5. Find an example of two roads that meet to make a right angle. Sketch the angle that is formed by the two roads and write the names of the streets.

27

6. Find an example of two roads that are parallel to each other. Sketch the relationship of the two roads below and label the streets’ names.

7. Find an example of two roads that are perpendicular to each other. Sketch the relationship of the two roads below and label the streets’ names.

28

Detroit, Michigan

29

Erfurt, Germany

30

Manhattan, New York City

31

Rome, Italy

32


Teacher GuideWhat Is a Line Segment?

(To the Teacher: The goal of today’s lesson is to further develop the students’ definition of a line segment. After the opening activity, students will share out all their ideas. We will build this definition over the next two lessons. Please share with the students the geometric representation of a line (AB), a ray (AB), and a segment (AB, when appropriate).Opening ActivityWhat is a line segment? In your description, discuss points, end points and what distinguishes it from a line and a ray?

Second ActivityGiven a line segment AB and a straight edge (not a ruler), replicate as precisely as you can the line segment.

BA

Third Activity Given line segment CD, a straight edge and a compass, replicate the line segment. Do you think this will be more precise? If so, how?

C D

(To the Teacher: The purpose of the Second Activity is to see what students do with just a straightedge and the difficulty that presents. What might students do? Students can approach the Third Activity in two ways. First they can measure the line segment with the compass. Then they can draw a line, locate a point on it, using the same measure from the compass you swing the arc from that point to find the desired length. The second way is that a student realizes that they can measure the original line segment using the compass, and swing the arc from that point. Then they mark a point and swing the arc from that point. A question that may arise is, “Which point on the arc would I use as the endpoint to draw the line segment?” This is a big idea for kids to think about. Why won’t it matter? While we haven’t talked about circles it would be interesting if students can use the idea of radii to understand that it doesn’t matter which point you would use. The discussion you have after the Third Activity should lead students to be able to do the Fourth Activity. )

Fourth ActivityWrite in your own words, the steps one would take to copy a given line segment.

Fifth ActivityHow did these activities affect your definition of a line segment? Do you want to make any changes or additions to your definition of line segment?

(To the Teacher: Finally, you want to revisit the statements that are on the chart for a line segment, do we want to alter it (add, subtract) to have a more precise definition.)

33


Lesson 4What Is a Line Segment?

Name_______________________Date________________________

Opening Activity

What is a line segment? In your description, discuss points, end points and what distinguishes it from a line and a ray?

Second Activity

Given a line segment AB and a straight edge (not a ruler), replicate as precisely as you can the line segment.

BA

34

Third Activity

Given line segment CD, a straight edge and a compass, replicate the line segment. Do you think this will be more precise? If so, how?

C D

Fourth Activity

Write in your own words, the steps one would take to copy a given line segment.

Fifth Activity

How did these activities affect your definition of a line segment? Do you want to make any changes or additions to your definition of line segment?

35


Teacher GuideHow Do We Define a Line On the Coordinate Plane?

Opening Activity

Compare these two representations of lines. How are they similar? How are they different?

Fig.1 Fig. 2

y

x

The equation of the line in the

above figure is y = 2x + 5.

(To the Teacher: We’re expanding on our notion of a line and looking at it in different contexts. Clearly, the line in the coordinate plane has a reference and the isolated line has no reference. Some invariants include that both are defined by an infinite set of points and that they extend infinitely in both directions. That is why both of them can be called lines. On the other hand, they are in different contexts and therefore take on different meanings. For example, if a student were to say that they both have positive slopes, it’s important to validate that response, but also to point out, that if we rotate the paper clockwise, one could say that the slope of figure 2 decreases to zero and ultimately becomes negative, whereas the slope of the line in figure 1, remains the same. What is the significance of this idea?)

Second Activity

1. Why did we name this line (y = 2x + 5) and what does that tell us?

2. If the equation were y = 1x + 5 or y = 3x + 5, how would the representation of the line in Figure 1 change? Justify your answer?

36

Third Activity

Compare these two representations of lines shown below. Which has the greater slope? Justify your answer.

1) 2) A line whose equation is y = 2x + 5

Fourth Activity

Observe the table of values below. Does this table represent the graph or the equation in the third activity? Justify your answer.

x y

-3 -11

-2 -8

0 -2

4 10

7 19

(To the Teacher: In this activity, we want students to see that you can pick any two points to determine the slope and the fact that while the table contains a finite number of discrete points, one can determine

37

y = 3x - 2

all the information necessary to define the line. But you want the students to also see that you can make a table that goes on infinitely in both directions which is representative of our notion of a line in space.)

Fifth Activity

Journal Entry: Though the multiple experiences you’ve had looking at lines in space and on a coordinate plane, discuss your understanding of the idea of a line.

38


Student Activity Sheet

How Do We Define a Line On the Coordinate Plane?

Name_______________________Date________________________

Opening Activity

Compare these two representations of lines. How are they similar? How are they different?

Fig.1 Fig. 2

y

x

The equation of the line in the

above figure is y = 2x + 5

Second Activity

1. Why did we name this line (y = 2x + 5) and what does that tell us?

2. If the equation were y = 1x + 5 or y = 3x + 5, how would the representation of the line in Figure 1 change? Justify your answer?

39

Third Activity

Compare these two representations of lines shown below. Which has the greater slope? Justify your answer.

2) 2) A line whose equation is y = 2x + 5

40

y = 3x - 2

Fourth Activity

Observe the table of values below. Does this table represent the graph or the equation in the third activity? Justify your answer.

x y

-3 -11

-2 -8

0 -2

4 10

7 19

Fifth Activity

Journal Entry: Though the multiple experiences you’ve had looking at lines in space and on a coordinate plane, discuss your understanding of the idea of a line.

41


Teacher GuideHow Do We Define a Line Segment On the Coordinate Plane?

Opening Activity: Performance Task

Shoe Size Problem

Shaquille O’Neal is seven feet one inches tall and has a fifteen inch foot. What do you think is his shoe size?

How did you come up with your prediction?

Now here is some information to see if your prediction is correct:The size of a shoe a person needs varies linearly with the length of his or her foot. A study of basketball players found the following information:

Length of foot in inches Shoe Size9 911 1314 19

Use the information to figure out Shaquille O’Neal’s shoe size? (To the Teacher: We are treating this opening activity as a performance task. Let students work alone and solve this in any way they want. Many will just use the table to try to figure it out. How well do they understand ratio? If students solve it quickly have them find more than one approach to solving it. You can learn about students’ thinking from this problem. Have a beginning discussion of students’ thinking. If students solved it by graphing hold off on their ideas until the second activity.)

Second Activity

Represent the Shaquille O’Neal problem graphically. How does your graph differ from the graph of a line?

Questions to think about as you graph it:

Do you think if we graph the data we would get a line? Why?

42

Is there a largest and smallest length of foot that a human being can have? And therefore, would there be a largest and smallest shoe that is made? How would the answer to this question affect our graph? Would the solution be a line or a line segment?(To the Teacher: With this activity, we’re setting up for a conversation about a bounded domain and range versus one that is infinite in both directions. So we would expect students to come to the conclusion that this is best represented by a line segment rather than a line. This goes along with our definition of a line segment in space that is bounded at both ends. Boundedness is an invariant of all line segments. Once a performance task is finished, you are going to lead a discussion on this question: Is this a line segment or line and why.)

Third Activity

1. What is the rate of change of the effect of foot length on the shoe size? (Hint: How is rate of change connected to the slope of the graph?)

2. If we move from a line to a line segment, what changes and what stays the same? Justify your answer.

(To the teacher: It’s important that students come to the conclusion that the only difference between a line and a line segment is boundedness of the latter. In all other respects, including slope (for the coordinate plane) and the fact that they both contain an infinite number of points, they are the same.)

43


Lesson 6How Do We Define a Line Segment On the Coordinate Plane?

Name_______________________Date_______________________

Opening Activity: Performance Task

Shoe Size Problem

Shaquille O’Neal is seven feet one inches tall and has a fifteen inch foot. What do you think is his shoe size?

How did you come up with your prediction?

Now here is some information to see if your prediction is correct:The size of a shoe a person needs varies linearly with the length of his or her foot. A study of basketball players found the following information:

Length of foot in inches Shoe Size9 911 1314 19

Use the information to figure out Shaquille O’Neal’s shoe size?

44

Second Activity

Represent the Shaquille O’Neal problem graphically. How does your graph differ from the graph of a line?

Questions to think about as you graph it:

Do you think if we graph the data we would get a line? Why? Is there a largest and smallest length of foot that a human being can have? And therefore,

would there be a largest and smallest shoe that is made? How would the answer to this question affect our graph? Would the solution be a line or a line segment?

Third Activity

1. What is the rate of change of the effect of foot length on the shoe size? (Hint: How is rate of change connected to the slope of the graph?)

2. If we move from a line to a line segment, what changes and what stays the same? Justify your answer.

45


Teacher GuideHow Do We Define an Angle?

Opening Activity

In your group, read these three definitions of an angle. Come to an agreement about which definition best describes an angle and why.

1. An angle is the union of two rays with a common endpoint.

2. An angle is the region contained between two rays.

3. An angle is the turning of a ray about a point from one position to another.

(To the Teacher: We want the students to have some time to argue this through and be able to defend their choice. Then you should bring the whole class together for a discussion. It is important to know that students have difficulty defining an angle because they often think of a definition in a static way, while it is important that students understand the notion of an angle as a dynamic concept. Make certain that students leave this discussion understanding that the last definition is the one that best describes an angle. We will spend the rest of the lesson deepening what that means.)

Second Activity

1. If you’re looking at a clock with hands on it that has no numbers, how would you describe to someone, the difference between two o’clock and six o’clock?

(To the Teacher: Our purpose with this question is to get to the understanding that the definition of an angle is determined by the dynamic relationship of the position of the hands on the clock relative to each other. We chose the use of a clock because a circle is essential to defining an angle.)

2. If we put the numbers back on the clock, and it reads twelve o’clock, how would you describe the measure of the angle between the minute hand and the hour hand?

46

3. If the clock reads two o’clock, how would you describe the measure of the angle between the minute hand and hour hand?

(To the Teacher: This is an opportunity to assess students’ understanding of angular measurement as a means of helping them to define an angle.)

4. If I went one fourth of the way around the clock, how many degrees would I pass through?

5. How do these different questions help us to further understand the definition of an angle?

(To the Teacher: Students should leave this discussion with the understanding that our definition of an angle is augmented by the use of units that differentiate angles.)

Third Activity

We will now begin to use a protractor to measure angles. This is an important tool in geometry that we will be using for several months. The picture below is of a protractor:

There are three important things to remember about using the protractor:

1. Always line up the one of the rays of the angle on the line where 0 degrees is represented.2. Always measure starting from “0”, not from “180” 3. Follow the path of the first ray to the second ray. How many degrees was the path?

Do the previous directions for using a protractor follow from our definition of an angle? Explain.

47

Now that you know the most important ideas about using a protractor, use your protractor to measure the following angles. Then label it as an acute, obtuse or right angle.

48

(To the Teacher: Bring everyone together to synthesize the different experiences around angles reinforcing the definition and hopefully moving them away from any misconceptions they may have.)

49


Lesson 7How Do We Define an Angle?

Name_______________________Date_______________________

Opening Activity

In your group, read these three definitions of an angle. Come to an agreement about which definition best describes an angle and why.

1. An angle is the union of two rays with a common endpoint.

2. An angle is the region contained between two rays.

3. An angle is the turning of a ray about a point from one position to another.

50

Second Activity

1. If you’re looking at a clock with hands on it that has no numbers how would you describe to someone, the difference between two o’clock and six o’clock?

2. If we put the numbers back on the clock, and it reads twelve o’clock, how would you describe the measure of the angle between the minute hand and the hour hand?

3. If the clock reads two o’clock, how would you describe the measure of the angle between the minute hand and hour hand?

4. If I went one fourth of the way around the clock, how many degrees would I pass through?

5. How do these different questions help us to further understand the definition of an angle?

51

Third Activity

We will now begin to use a protractor to measure angles. This is an important tool in geometry that we will be using for several months. The picture below is of a protractor:

There are three important things to remember about using the protractor:

4. Always line up the one of the rays of the angle on the line where 0 degrees is represented.5. Always measure starting from “0”, not from “180” 6. Follow the path of the first ray to the second ray. How many degrees was the path?

Do the previous directions for using a protractor follow from our definition of an angle? Explain.

52

Now that you know the most important ideas about using a protractor, use your protractor to measure the following angles. Then label it as an acute, obtuse or right angle.

53

54


Teacher GuideWhat can we observe about intersecting lines?

Opening Activity:

Today you are going to make a set of discoveries about the angles formed by intersecting lines. Based on your observations of the intersecting lines below what predictions do you want to make about the relationships of the different angle measurements of the four angles?

a) Label and measure the four angles created by the intersection of the two lines below:

Angle 1:

Angle 2:

Angle 3:

Angle 4:

b) Label and measure the four angles created by the intersection of the two lines below:

Angle 1:

Angle 2:

Angle 3:

Angle 4:

55

c) Observe your results from a and b above. What do you notice about the relationships between the angles in each of the diagrams?

d) With the straight edge of your protractor, draw two lines that intersect below and test your conjecture:

e) Generalize what you have discovered. Did it go along with your predictions?

(To the Teacher: This opening activity could be a group activity where students observe special relationships of intersecting lines. During the discussion of their findings it would be important to give names to the angle relationships discovered, such as adjacent, vertical and supplementary angles. Also you can help students understand that the general statements made will be used in further work this year (e.g. The measures of vertical angles formed by intersecting lines are equal.)

Second Activity

1. Look at the map of Detroit that we used in a previous lesson yesterday. Find where Michigan Avenue crosses First Street. Do the relationships we observed about the four angles formed by two intersecting lines hold true? Justify your answer.

2. Use your knowledge of angle relationships to find the degrees of the missing angles without using your protractor:

102

52

56

3. Find the measure of each angle:

(3x)

(6x)

(2x + 5) (x)

4. If line AB intersects line CD at point E and angle AED is 30 more than angle DEB, find the measure of all four angles.

5. Use any of the maps for the following:Find two streets that intersect. Measure the four angles formed by these streets. Use your measurements to support or disprove the conjectures for intersecting lines we discovered in class today.

(To the Teacher: Once you go over students’ solutions to the problems you might ask students to write in a journal using this prompt, “What can we say will be true about the angles formed by intersecting

lines?”)

57


Lesson 8What can we observe about intersecting lines?

Name_______________________Date_______________________

Opening Activity:

Today you are going to make a set of discoveries about the angles formed by intersecting lines. Based on your observations of the intersecting lines below what predictions do you want to make about the relationships of the different angle measurements of the four angles?

a) Label and measure the four angles created by the intersection of the two lines below:

Angle 1:

Angle 2:

Angle 3:

Angle 4:

b) Label and measure the four angles created by the intersection of the two lines below:

Angle 1:

Angle 2:

Angle 3:

Angle 4:

58

c) Observe your results from a and b above. What do you notice about the relationships between the angles in each of the diagrams?

d) With the straight edge of your protractor, draw two lines that intersect below and test your conjecture:

e) Generalize what you have discovered. Did it go along with your predictions?

59

Second Activity

1. Look at the map of Detroit that we used in a previous lesson yesterday. Find where Michigan Avenue crosses First Street. Do the relationships we observed about the four angles formed by two intersecting lines hold true? Justify your answer.

2. Use your knowledge of angle relationships to find the degrees of the missing angles without using your protractor:

102

52

60

3. Find the measure of each angle:

(3x)

(6x)

(2x + 5) (x)

4. If line AB intersects line CD at point E and angle AED is 30 more than angle DEB, find the measure of all four angles.

5. Use any of the maps for the following. Find two streets that intersect. Measure the four angles formed by these streets. Use your measurements to support or disprove the rules for intersecting lines we discovered in class today.

61


Teacher GuideWhat can we observe about right angles and straight angles?

Opening ActivityNow that you know how to measure an angle can you come up with a method to draw any given angle?

Using a protractor your job is to:a) draw a 25 degree angle b) draw a 134 degree angle c) write up an explanation of how to draw any given angle.

(To the Teacher: Students were probably shown this in middle school but some will have difficulty. How will they use their understanding of measuring an angle to now draw a given angle? A class discussion can lead to a method of drawing an angle. How can we ensure precision? In the next two activities students will be making observations about complementary and supplementary angles. Once students have made the observations you can share the geometric terms with them.)

Second Activity

1. Construct a right angle.

2. Label it <ABC

3. From point B draw a ray within the interior of <ABC. Call it BD.

4. Measure <CBD and <ABD

5. How are < CBD and <ABD related to the original <ABC? Is this always true? Can you make a general statement?

6. Triangles ABC, DEF and GHI below are right triangles. Using the values of the given angles, find the unknown angles. Explain how you found them.

A F G

78 ? E 32 ? m B C

D H I

62

Third Activity

1. Construct a straight angle.

2. Label it <ABC

3. From point B draw a ray within the interior of <ABC. Call it ray BD



6. Lines AC, DF and GI below are straight lines. Using the values of the given angles, find the unknown angles. Explain how you found them.

D E F

68 ?

? 142 ? mA B C G H I

7. Look at the following figure. Find the values of each of the angles.

2x 3xx 4x

Fourth Activity

1. Using the map of the Detroit, find an example of streets that intersect to form supplementary angles. Measure these angles to confirm that this is true.

2. Repeat the same process for complementary angles.

63

Performance Task: Create a City Design

Create a design of a small town that has eight different streets that you will name. There must be an example of complementary, supplementary, vertical and adjacent angles which you will describe in words.

64


Student Activity SheetWhat can we observe about right angles and straight angles?

Name_______________________Date_______________________

Opening Activity

Now that you know how to measure an angle can you come up with a method to draw any given angle?

Using a protractor your job is to:a) draw a 25 degree angle b) draw a 134 degree angle c) write up an explanation of how to draw any given angle.

65

Second Activity1. Construct a right angle.

2. Label it <ABC




6. Triangles ABC, DEF and GHI below are right triangles. Using the values of the

given angles, find the unknown angles. Explain how you found them.

A F G

78 ? E 32 ? m B C

D H I

66

Third Activity

1. Construct a straight angle.

2. Label it <ABC




67

6. Lines AC, DF and GI below are straight lines. Using the values of the given angles, find the unknown angles. Explain how you found them.

D E F

68 ?

? 142 ? mA B C G H I

7. Look at the following figure. Find the values of each of the angles.

2x 3xx 4x

Fourth Activity

1. Using the map of the Detroit, find an example of streets that intersect to form supplementary angles. Measure these angles to confirm that this is true.

2. Repeat the same process for complementary angles.

68

Performance Task: Creating a City Design

Create a design of a small town that has eight different streets that you will name. There must be an example of complementary, supplementary, vertical and adjacent angles which you will describe in words.

69


Teacher GuideHow would you replicate an angle?

(To the Teacher: The goal of today’s lesson is to further develop the students’ definition of an angle through a construction inquiry.)

Opening Activity

We have now looked at angles over the last few days, including beginning to define an angle, measuring and constructing angles and observing angles formed by intersecting lines. Using our original definition is there anything you would want to add to the definition or change to make it more precise? Explain your reasoning.

Our original definition was:

An angle is the turning of a ray about a point from one position to another.

Second Activity-Performance Task: How do I replicate an angle?

Given an angle, compass and a straight edge (not a ruler), replicate as precisely as you can the given angle. Show all your work and explain your thinking.

(To the Teacher: It is recommended that you walk around and observe what students are doing. You can give them 10-15 minutes for the performance task. You will want to lead a discussion afterwards based on the strategies and ideas the students used to try to copy a given angle. Choose the students to share based on their ideas even if they had errors in their thinking. You should help the class come up with an approach that would make for an exact replication. )

70

Third Activity

Write in your own words, the steps one would take to copy a given angle.

Fourth Activity

How does this the method to copy an angle go along with our definition of an angle?

71


Lesson 10 How would you replicate an angle?

Name_______________________Date________________________

Opening Activity

We have now looked at angles over the last few days, including beginning to define an angle, measured angles and observed angles formed by intersecting lines. Using our original definition is there anything you would want to add to the definition or change to make it more precise? Explain your reasoning.

Our original definition was:

An angle is the turning of a ray about a point from one position to another.

72

Second Activity-Performance Task: How do I Replicate an Angle?

Given an angle, compass and a straight edge (not a ruler), replicate as precisely as you can the given angle. Show all your work and explain your thinking.

73

Third Activity

Write in your own words, the steps one would take to copy a given angle.

Fourth Activity

How does this the method to copy an angle go along with our definition of an angle?

74

Developing Definition – Intro to Geometric IdeasLesson 11

Teacher GuideHow do we define parallel lines?

Opening Activity

Here are two definitions for parallel lines. Read them closely

Lines are parallel if they lie in the same plane, and are the same distance apart over their entire length. No matter how far you extend them they will never meet.

Parallel lines are two or more coplanar lines that have no points in common or are identical (e.g., the same line)

Now with your partners you will decide if these definitions are precise. Think about the following:

1) Why use the term “on the same plane? If we left it how would it affect the definition? Why?

2) Why do we say they are “the same distance apart over their entire length?” If we left it how would it affect the definition? Why?

3) Are the definitions saying the same thing or are they saying something different? Explain.4) Do you want to make any changes? What definition does your group want to use?

(To the teacher: This activity is about helping students dissect the language of geometry. This is important because students frequently get lost in the language. It is important for them to see why these two definitions are saying the same thing. We also want them to see that if we leave parts of these definitions out we would lose the precision of the definition. Students often say the definition of parallel lines is two lines that never meet which also is true of skew lines, hence the necessity of talking about coplanar.)

Second Activity:

Using your understanding of parallel lines and our agreed upon definition, you are going to be given a line (we will call line L) and a point (we will call P) and you have to come up with a method of constructing a line parallel to Line L that passes through point P. You will be given a straightedge and a compass.

.P

L

75

(To the Teacher: This is a difficult activity. Let students struggle somewhat. Bring them together for a discussion of ideas, which will lead them to the third activity which gives directions for constructing a pair of parallel lines.)

Third Activity:

Here are directions to draw a line parallel to a given line through a given point.See if you can follow it. Why does this method work?

Constructing a Line Parallel to a Given Line through a Given Point

After doing this Your work should look like this

Start with a line segment PQ and a point R off the line.

1. Draw a transverse line through R and across the line PQ at an angle, forming the point J where it intersects the line PQ. The exact angle is not important.

2. With the compass width set to about half the distance between R and J, place the point on J, and draw an arc across both lines.

76


3. Without adjusting the compass width, move the compass to R and draw a similar arc to the one in step 2.

4. Set compass width to the distance where the lower arc crosses the two lines.

5. Move the compass to where the upper arc crosses the transverse line and draw an arc across the upper arc, forming point S.

77


6. Draw a straight line through points R and S.

Done. The line RS is parallel to the line PQ

Now that you have done this construction think about this question.

How does the construction help to further explain our definition of parallel lines?

(To the Teacher: You should have students go through the process thinking about why this would give us parallel line. It is important to think about why this works and how it goes along with the definition of parallel lines. It is related to the next lesson in which student will do a performance task based on a transversal intersecting two parallel lines. After that activity you might revisit this construction and help students see the connection.)

78


Lesson 11How do we define parallel lines?

Name_______________________Date________________________

Opening Activity:

Here are two definitions for parallel lines. Read them closely

Lines are parallel if they lie in the same plane, and are the same distance apart over their entire length. No matter how far you extend them they will never meet.

Parallel lines are two or more coplanar lines that have no points in common or are identical (e.g., the same line)

Now with your partners you will decide if these definitions are precise. Think about the following:

1. Why use the term “on the same plane? If we left it out would the definition still be precise? Why?

2. Why do we say they are “the same distance apart over their entire length?” If we left it out how would it affect the definition?

3. Are the definitions saying the same thing or are they saying something different? Explain.

4. Do you want to make any changes? What definition does your group want to use?

79

Second Activity:

Using your understanding of parallel lines and our agreed upon definition, you are going to be given a line (we will call line L) and a point (we will call P) and you have to come up with a method of constructing a line parallel to Line L that passes through point P. You will be given a straightedge and a compass.

.P

L

80

Third Activity:

Here are directions to draw a line parallel to a given line through a given point.See if you can follow it. Why does this method work?

Constructing a Line Parallel to a Given Line through a Given Point


Start with a line segment PQ and a point R off the line.

1. Draw a transverse line through R and across the line PQ at an angle, forming the point J where it intersects the line PQ. The exact angle is not important.

2. With the compass width set to about half the distance between R and J, place the point on J, and draw an arc across both lines.

81


3. Without adjusting the compass width, move the compass to R and draw a similar arc to the one in step 2.

4. Set compass width to the distance where the lower arc crosses the two lines.

5. Move the compass to where the upper arc crosses the transverse line and draw an arc across the upper arc, forming point S.

82


6. Draw a straight line through points R and S.

Done. The line RS is parallel to the line PQ

Now that you have done this construction think about this question.

How does the construction help to further explain our definition of parallel lines?

83


Teacher GuideParallel or Not?

(To the Teacher: This lesson is to be done by students independently as a performance assessment. See accompanying document called “Parallel or Not.” Students can be given the whole period for this. You should follow up this lesson with a discussion about students’ discoveries about angles formed by a transversal intersecting two parallel lines.)

84


Teacher GuideMore with Transversals

Opening Activity

1. Write down all the things you discovered yesterday regarding the angles formed by a transversal cutting through two parallel lines.

(To the Teacher: Have a discussion with the students regarding their findings. You should take this opportunity to give names to the different angles formed, e.g. alternate interior, corresponding, alternate exterior, etc.)

2. We know that there are eight angles formed by the transversal. What is the minimum number of angle measurements you would need to know to be able to figure out the rest of the angle measurements? Justify your answer.

Second Activity

1. On the Detroit, Michigan map, mark a 1 and a 2 on two alternate interior angles and a 3 and a 4 on two corresponding angles.

2. Use your protractor to measure the angles you’ve labeled. Do your results support the discoveries you made yesterday?

Third Activity

1. Two parallel lines are crossed by the transversal below then find the measure of each of the angles below.

3x + 20

4x

85

2. If the measure of angle 1 = 57, explain how you can find the measure of angle 8.

1

8

3. If angle 2 = x, write an expression for the measure of angle7 and angle 8. Give evidence for your choices.

1 2

3 4

5 6 7 8

4. On the map of Manhattan, find Broadway. Discuss why this street would be called a transversal.

Describe the angles formed by the streets that can be called:

a. Alternate interior

b. Corresponding(To the Teacher: Have students discuss their responses to the various questions. A large idea we want them to think about from the two days of this work is that with parallel lines, cut by a transversal, there is more information that we immediately know than what we know when the two lines being intersected are not parallel. No we will revisit the construction and see if students can understand that the method we used is based on the theorem discoveries the students made over the last two days.)

Fourth ActivityRevisit the construction of two parallel lines. Can you explain now more fully why the method that you were shown would always create parallel lines?

86


Lesson 13More with Transversals

Name_______________________Date________________________

Opening Activity

1. Write down all the things you discovered yesterday regarding the angles formed by a transversal cutting through two parallel lines.

2. We know that there are eight angles formed by the transversal. What is the minimum number of angle measurements you would need to know to be able to figure out the rest of the angle measurements? Justify your answer.

Second Activity

1. On the Detroit, Michigan map, mark a 1 and a 2 on two alternate interior angles and a 3 and a 4 on two corresponding angles.

2. Use your protractor to measure the angles you’ve labeled. Do your results support the discoveries you made yesterday?

87

Third Activity

1. Two parallel lines are crossed by the transversal below then find the measure of each of the angles below.

3x + 20

4x

2. If the measure of angle 1 = 57, explain how you can find the measure of angle 8.

1

8

3. If angle 2 = x, write an expression for the measure of angle7 and angle 8. Give evidence for your choices.

1 2

3 4

5 6 7 8

88

4. On the map of Manhattan, find Broadway. Discuss why this street would be called a transversal.

Describe the angles formed by the streets that can be called:

a. Alternate interior

b. Corresponding

Fourth Activity

Revisit the construction of two parallel lines. Can you explain now more fully why the method that you were shown would always create parallel lines?

89


Teacher GuideHow do we define perpendicular lines?

(To the Teacher: One of the goals of today’s lesson is to help students expand their notion of perpendicular lines. After the Opening Activity in which you will get a chance to hear students’ ideas you will give students the opportunity to play with the construction activity in the Second Activity. You will learn from what students do or don’t do. How do they use their notion of perpendicular to think about the construction? One of the other goals is to continue to build students perseverance with problems. Throughout this unit students are asked to do things that they have to think about, where the answer takes time and a willingness to try different things. A discussion after the Second Activity can lead students to try to follow the construction given to them. Why does that method make sense? What does it say about the meaning of perpendicularity? The fourth Activity will be a Performance Task. Can students transfer their understanding of the construction to do a similar construction?)

Opening Activity

Write down your definition of perpendicular lines. Use examples to support your definition.

Second Activity

You are going to be given a line j and a point r above the line. Using a compass and straight edge your job is to come up with a method of drawing a line perpendicular to line j that passes through point r.

.r

j

Third Activity:

Now let us look at the general construction. Does it go along with your idea? Why does it work? What does it tell us about perpendicular lines?

90

Start with a line and point R which is not on that line.

Step 1 Place the compass on the given external point R.

Step 2 Set the compass width to a approximately 50% more than the distance to the line. The exact width does not matter.

Step 3 Draw an arc across the line on each side of R, making sure not to adjust the compass width in between. Label these points P and Q

91

Step 4 At this point, you can adjust the compass width. Recommended: leave it as is.

From each point P,Q, draw an arc below the line so that the arcs cross.

Step 5 Place a straightedge between R and the point where the arcs intersect. Draw the perpendicular line from R to the line, or beyond if you wish.

Step 6 Done. This line is perpendicular to the first line and passes through the point R. It also bisects the segment PQ (divides it into two equal parts)

92

Fourth Activity-Performance Assessment: Can you Construct a Perpendicular Line?

Now that you are comfortable with the last construction try to solve this problem.

Construct a line perpendicular to a given line j that passes though point r. Point r is located on line j.

.r j

Closing Activity

Let us revisit our definitions of perpendicular lines. Is there anything you would like to add or subtract from the ideas presented earlier to help us be more precise with our definition?

93


Lesson 14How do we define perpendicular lines?

Name_______________________Date________________________

Opening Activity:

Write down your definition of perpendicular lines. Use examples to support your definition.

Second Activity:

You are going to be given a line j and a point r above the line. Using a compass and straight edge your job is to come up with a method of drawing a line perpendicular to line j that passes through point r.

.r

j

94

Third Activity:

Now let us look at the general construction. Does it go along with your idea? Why does it work? What does it tell us about perpendicular lines?

95

Start with a line and point R which is not on that line.

Step 1 Place the compass on the given external point R.

Step 2 Set the compass width to a approximately 50% more than the distance to the line. The exact width does not matter.

Step 3 Draw an arc across the line on each side of R, making sure not to adjust the compass width in between. Label these points P and Q

96

Step 4 At this point, you can adjust the compass width. Recommended: leave it as is.

From each point P,Q, draw an arc below the line so that the arcs cross.

Step 5 Place a straightedge between R and the point where the arcs intersect. Draw the perpendicular line from R to the line, or beyond if you wish.

Step 6 Done. This line is perpendicular to the first line and passes through the point R. It also bisects the segment PQ (divides it into two equal parts)

97

Fourth Activity-Performance Assessment: Can you Construct a Perpendicular Line?

Now that you are comfortable with the last construction try to solve this problem. Show all your work and explain your thinking.

Construct a line perpendicular to a given line j that passes though point r. Point r is located on line j.

.r j

98

Closing Activity

Let us revisit our definitions of perpendicular lines. Is there anything you would like to add or subtract from the ideas presented earlier to help us be more precise with our definition?

99


Teacher GuideWhat is true about parallel and perpendicular lines on the coordinate plane?

(To the Teacher: In today’s lesson we are expanding our notion of parallel and perpendicular lines. We have thought about their qualities in space or on a plane and now we will think about them on a coordinate plane. Why is it important for students to think about ideas in different contexts? What is invariant and what is different? Today’s lesson is an investigation in groups. Students might have learned this last year but the meaning for them should be deeper. Walk around the room and see what the groups are doing. Perhaps you will choose one or two groups who have made some good observations to share with the rest of the class. The ideas in today’s lesson will be revisited when we do coordinate geometry proofs. The question at the end of the lesson should be discussed as you did with all the definitions. How does this deeper understanding of parallel and perpendicular lines affect your definition of these two ideas? This will be the last lesson before the introduction of the final assessment.)

Today you are going to do an investigation on the coordinate plane of the uniqueness of parallel and perpendicular lines. We have come up with definitions of each of these sets of lines now we are going to further develop them.

Before we begin the experiment observe the four diagrams and make a prediction about what will be true about the slopes of parallel lines and what will be true about the slopes of perpendicular lines.

Now for each coordinate plane find the slope of the two lines.

100

101

Observe the results of your calculations.

What do you notice about the slopes of parallel and perpendicular lines?

Make a conjecture for parallel and perpendicular lines that you will test out on the following coordinate plane.

102

Make a statement about the slopes of parallel and perpendicular lines. Then explain why this will always be true.

How does this deeper understanding of parallel and perpendicular lines affect your definition of these two ideas?

103


Student Activity SheetWhat is true about parallel and perpendicular lines on the coordinate plane?

Today you are going to do an investigation on the coordinate plane of the uniqueness of parallel and perpendicular lines. We have come up with definitions of each of these sets of lines now we are going to further develop them.

Before we begin the experiment observe the four diagrams and make a prediction about what will be true about the slopes of parallel lines and what will be true about the slopes of perpendicular lines.

Now for each coordinate plane find the slope of the two lines.

104

105

Observe the results of your calculations.

What do you notice about the slopes of parallel and perpendicular lines?

Make a conjecture for parallel and perpendicular lines that you will test out on the following coordinate plane.

106

Make a statement about the slopes of parallel and perpendicular lines. Then explain why this will always be true.

How does this deeper understanding of parallel and perpendicular lines affect your definition of these two ideas?

107

Geometry – City Planning ProjectFinal Project

(To the Teacher: This project/assessment gives students the opportunity to create a design of a city while showing their understanding of the multiple ideas discussed in this unit. A rubric should be created that accentuates the mathematical understanding students showed in their written piece and drawing. Do you want students to work in pairs or groups of 3? What materials will you need to share with students? Perhaps you will want a gallery walk at the end so students can evaluate each other’s work. How much time will you give students in class to work on this and how much time will give the students to work at home?)

Challenge: The mayor of Detroit, Dave Bing, has decided that all this vacant land in Detroit is being made available to residents of Detroit who want to help rebuild the city. There are many groups that are planning to apply for right to design this space. You are part of a group of young people who want to apply for this privilege. Your job is to design how to use the space for submission to the mayor’s office.

Within the plan, you must include the following: Residential spaces Government spaces Community spaces Recreational spaces An aesthetic space (a place of beauty)

Given: The plot of land is 8000 ft by 5000 ft.

You must design your city on graph paper. You should use the scale ¼ inch = 100 feet.

Written Piece: Part 1: You must include a written discussion of the design of your city. In this part, you must explain why your design will be a place where people want to live, work and play. This written part should be persuasive and professional as it is addressed to the mayor of Detroit.

Part 2:Next, you must include a discussion of the street relationships. Included in this section must be the following geometric terms and relationships:

Intersecting streets Two streets that are line segments where one is copied from the other Parallel streets formed by a construction with mathematical evidence that they are

parallel A transversal street intersecting two parallel streets with evidence that adjacent angles,

vertical angles, corresponding angles, alternate interior, and alternate exterior angles were formed.

Perpendicular streets formed by a construction with mathematical evidence that they are perpendicular.

Two streets that intersect and form an acute angle. Then copy that angle to another location on your design. Show that the measures of the two angles are equal.

108

Two streets that intersect to form an obtuse angle. Then copy that angle to another location on your design. Show that the measures of the two angles are equal.

Part 3:Create a glossary with precise definitions of all the geometric terms discussed in this unit. Your definitions can include diagrams to help in your descriptions.

109

Documents

Web viewStudent Activity Sheet. Lesson 1. ... mathematicians recognized that attempting to define every word inevitably led to circular definitions ... find Broadway