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EXPLORING GEOMETRIC FIGURES Grade 10 6-Day Lesson Plan

Tangrams Geoboards Equation Grapher Green Globs AlphaShapes Protractor Compass

Created by Sandra Metzler

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OVERALL OBJECTIVES

1- Students will increase their knowledge of geometric shapes such astriangles, polygons, circles, and quadrilaterals.

2- Students will be able to classify triangles, polygons andquadrilaterals.

3- Students will find the measures of angles within geometric shapesand also outside these shapes.

4- Students will become confident using a compass and a protractor.

5- Students will become confident using computer technology such asequation grapher.

6- Students will be able to recognize parallel and perpendicular lines bytheir slopes.

7- Students will be able to measure central angles and arcs of circles.

8- Students will be able to display and read data in a circle graph.

9- Students will make connections between geometric figures andeveryday life.

10- Students will experience math hands-on and work cooperatively.

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LEARNING STANDARDS

ELA Standard 1: Students will read, write, listen and speak for informationand understanding.

ELA Standard 4: Students will read, write, listen and speak for social interaction.

MST Standard 1: Students will use mathematical analysis and scientific inquiry to pose questions, seek answers, and develop solutions.

MST Standard 2: Students will access, generate, process, and transfer information using appropriate technologies.

MST Standard 3: Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of geometry and algebra.

MST Standard 5: Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs.

MST Standard 6: Students will understand the relationships and common themes that connect mathematics, science, and technology to address real -life problems and make informed decisions.

MST Standard 7: Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions.

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RESOURCES AND TOOLS NEEDED:

1- Textbook: Bass, Laurie E., et al. New York Math A/B an Integrated Approach Volume 2. New Jersey: Prentice-Hall, Inc., 2001 pages 66-101.

2- Class set of Geoboards and rubber bands

3- Class set of Tangrams

4- Computers

5- Equation Grapher computer program

6- Green Globs computer program

7- Class set of AlphaShapes

8- Protractors

9- Compasses

10- Scissors

11- Construction paper

12- Graph paper

13- Paper

14- Pencils or pens

15- Overhead

16- Overhead markers

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OVERVIEW

My goal for the next six days is to involve the students in hands on learning of differentgeometric shapes and their relationships with one another. I hope that after thisinstructional period, the students will walk away with an enhanced learning ofmeasurement and classification of the different geometric shapes.

Day 1:A. Triangles

1. Construction of triangles2. Measurement of angles3. Triangle Angle-Sum Theorem4. Exterior Angle Theorem

Day 2:A. Triangles continued

1. Classification of triangles2. Use of tangrams for triangle exploration

Day 3:B. Polygons

1. Properties of polygons2. Use of geoboards to explore polygons and their relationship to

triangles3. Polygon Interior Angle-Sum Theorem4. Polygon Exterior Angle-Sum Theorem5. Exploring and classifying polygons using AlphaShapes

Day 4 and Day 5:C. Quadrilaterals

1. Classifying quadrilaterals2. Identifying slopes of different lines and how it relates to the

classification of quadrilaterals3. Practice with Equation Grapher4. Fun practice with Green Globs

Day 6:D. Circles

1. Definition of a circle and its parts2. Practice constructing circles and parts using compass and

protractor3. Graphing circles from coordinates and locating the center4. Constructing and reading a circle graph

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DAILY LESSON PLANS

DAY 1

Topic: Triangles

Learner Objectives: Students will be able to find the measure of interior and exterior angles of a triangle using the theorems presented.

Students will be comfortable using a protractor.

Instructional Presentation:•The teacher will give a brief overview of the chapter to the students.•The teacher will ask leading questions about the angles of a triangle to

activate the learner’s prior knowledge. For example, one question may be “if we know we have a right triangle what is one of the angle measures of that triangle?”

•The teacher will pass out a protractor warm up worksheet to each student.(See below)

•The teacher will instruct the students to get out their rulers, scissors, protractors, and a pencil.

•The teacher will model and instruct the students on how to do the protractor sheet.

•The teacher will give the students time to complete the three problems.•After sufficient time the teacher will model the steps of how to do each

problem. •The teacher will then pass out a piece of construction paper to each student.

•The teacher will instruct the students to draw a triangle that has sides of length 9 cm and 4 cm and a 70-degree angle between these two sides.(see below)

•The teacher will also instruct the students to find the measurement of the other two angles.

•While the students are doing this activity, the teacher will assist those students who need help or if needed the teacher will model how to do this on the overhead.

•Once everyone has completed the triangle, the students will label each angle with its measurement and then cut out their triangle. •The angles will then be torn off the triangle.

•The students will be instructed to put the three angles adjacent to one another and note their findings.

•The teacher will then lead in a discussion about the sum of the angles of a triangle.

•The Triangle Angle-Sum theorem will then be introduced formally from page 68 in their math textbooks.

•The teacher will then go through example 3 on the bottom of page 68.•The teacher will then ask the students to try number 5 on page 69.(see

below)

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DAY 1cont.

•The teacher will then ask for a volunteers to do the each part of the problem on the chalkboard.

•If the students are not grasping the Triangle Angle-Sum Theorem, then more examples will be done.

•Moving on to exterior angles, the students will be instructed to perform a similar activity as we did earlier with the construction paper and constructing a triangle.

•The teacher will again pass out a piece of construction paper to each student.

•While doing so the teacher will discuss exterior angles and remote interior angles.

•For this activity, the students will draw a triangle, where one side there is an extension, so the triangle will not be cut out for this exercise. The students will use the example in the book on page 69 as a guide.

•The students will cut out the two remote interior angles and place them on the extension line which is the exterior angle of the triangle.

•The teacher will then ask the students what have we learned by doing this and a discussion will follow.

•The teacher will then formally introduce the Exterior Angle Theorem which is described on page 70.

•The students will then be asked to try problem 7 on page 70.•Volunteers will again be asked to show their work to each of the

problems on the chalkboard.•A discussion will follow about the corollary to the theorem and problems 8 and 9 on page 70.•The teacher will then assign homework that consists of problems 10-19

and problem 21 A,B,C on page 72 of their textbook. Time permitting, the students will be able to start it in class and the teacher can assist if needed.

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PROTRACTOR WARMUP

1. Using your protractor, draw a right triangle that has a sidethat measures 8.1 cm and another side that measures 5.3cm.

2. Using your protractor, construct a triangle that has sides oflength 4.3 cm and 5.2 cm and a 29-degree angle betweenthe two sides.

3. Create your own triangle. Label the angle measurementsand the lengths of the sides. Then find the area of the triangleyou created.

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12DAY 2

Topic: Triangles and Tangrams

Learner Objectives:Students will be able to classify types of triangles by using a combination

of prior knowledge and congruency of sides and angles.Students will work cooperatively and investigate triangles by using

tangrams.

Instructional Presentation:

•The teacher will review the theorems discussed last class.•The teacher will ask the students if there was any questions from the

homework assigned and will address them.•The teacher will collect the homework for evaluation later.(see below)•Discussion will then commence with classifying triangles. In order to

activate prior knowledge the students will be asked how we determine if a triangle is scalene, obtuse, acute, or equilateral.

•Discussion will continue with page 71 of their math textbooks. The Math A Prep questions and problem 11 will be discussed as a group.

•The students will then be asked to do problems 1- 9 on page 71.•After sufficient time is given to the students to complete the problems, the teacher will go through each problem on the overhead projector with

the students’ involvement.•The students will then be grouped into pairs and each given a tanagram

set. The students will perform the activity given to them to investigate more about triangles.(see below)

•Problem number 23 on page 72 is assigned for extra credit and is due by the following Monday. It must be typed, double spaced and be no longer than one page.

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Names: ____________________________________________

TANGRAM FUN

With your partner try the following exercises.

1. Make a triangle using two of the pieces from the set.What classification does this triangle belong?

_____________________________________

Are there other two-piece combinations you can use to make a different triangle? _________________ If so, are they still the same classification or different? __________________________________________

2. Make a triangle using three of the pieces form the set.What classification does this triangle belong?

Is there another three-piece combination you could use to make adifferent triangle? ______________________If so, is the new triangle in the same classification or different?________________________________________

3. Take any one of the triangles and trace the angles one by onenext to each other. What can you conclude from this about thesum of the triangles angles?

_______________________________________________

4. Make a conjecture: Is it possible to make a triangle out of all 7pieces? _______Now investigate and try doing so.Was your conjecture true or false? _______

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MORE TANGRAM FUN

SEE IF YOU CAN MAKE THESE FIGURES USING ALL 7 OF YOUR TANGRAMS.

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18Names:_____SAMPLE STUDENT WORK

TANGRAM FUN

With your partner try the following exercises.

1. Make a triangle using two of the pieces from the set.What classification does this triangle belong?

__Isosceles__________________________

Are there other two-piece combinations you can use to make a different triangle?_______yes____ If so, are they still the same classification or different? ____Isosceles__________________________

2. Make a triangle using three of the pieces form the set.What classification does this triangle belong?Isosceles

Is there another three piece combination you could use to make adifferent triangle?__none found__________If so, is the new triangle in the same classification or different?________________________________________

3. Take any one of the triangles and trace the angles one by one next to each other. What can you conclude from this about the sum of the triangles angles?

The sum of the angles is equal to 180 or a straight line.

4. Make a conjecture: Is it possible to make a triangle out of all 7pieces?__no___Now investigate and try doing so.Was your conjecture true or false?_false_

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MORE TANGRAM FUNSEE IF YOU CAN MAKE THESE FIGURES USING ALL 7 TANGRAM PIECES.

20DAY 3:

Topic: Polygons

Learner Objectives:Students will be able to classify polygons.Students will also be able to find the sum of the measures of the interior and

exterior angles of the polygons.

Instructional Presentation:•The teacher will give a brief overview of the chapter on polygons which starts on

page 76 to 81 in their math textbook.•To activate the student’s prior knowledge, the teacher will lead in a discussion

about properties of polygons and examples. The students will be asked to participate in sharing what they know with the class.

•The teacher will hand out geoboards and rubber bands to each student alongwith a worksheet on polygons.(see below) •The students will work on the worksheet in pairs and use the geoboards to construct the polygons and the diagonals they need to complete it.

•A discussion about their findings will follow and the teacher will ask about what conjectures can be made about a larger polygon, for example one with 20sides. •The teacher will then formally introduce the Polygon Interior Angle-Sum Theorem that is on page 77 of their math textbooks.

•The teacher will then do some examples.•The teacher will then discuss the sum of exterior angles of polygons and

introduce the Polygon Exterior Angle-Sum Theorem that is located on page 78 of their textbooks.

•The teacher will then go over examples 2 and 3 on pages 78 and 79.•The teacher will then hand out AlphaShapes and the AlphaShapes worksheet to

the students.(see below)•The students will complete the worksheet in groups of two or three and hand it

in at the end of the class period. The teacher will assist as needed.

•The teacher will assign page 80 #10,12,14,16,20,22,24,26.

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22Name:______________________________________________

POLYGONSUsing your geoboard and rubber bands, construct the following polygons and then add thediagonals using rubber bands to see how many triangles are formed. Compute the sum ofinterior angles.

POLYGON NUMBER OFSIDES

NUMBER OFTRIANGLESFORMED

SUM OFINTERIORANGLES

TRIANGLE

QUADRILATERAL

PENTAGON

HEXAGON

HEPTAGON

OCTAGON

NONAGON

DECAGON

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POLYGONSUsing your geoboard and rubber bands, construct the following polygons and then add thediagonals using rubber bands to see how many triangles are formed. Compute the sum ofinterior angles.

POLYGON NUMBER OFSIDES

NUMBER OFTRIANGLESFORMED

SUM OFINTERIORANGLES

TRIANGLE 3 1 180

QUADRILATERAL 4 2 360

PENTAGON 5 3 540

HEXAGON 6 4 720

HEPTAGON 7 5 900

OCTAGON 8 6 1080

NONAGON 9 7 1260

DECAGON 10 8 1440

25DAY 4 and DAY 5

Topic: Quadrilaterals

Learner Objectives:Students will be able to classify special types of quadrilaterals.Students will be able to graph quadrilaterals and explore the relevance of

slope, parallel and perpendicular lines.

Instructional Presentation:•The teacher will go over the homework (page 80) from the last class and

then collect it. (see below)•The teacher will give a brief overview of lesson on quadrilaterals, slope,

perpendicular and parallel lines.•The teacher will pass out the AlphaShapes and ask students to find each

of the special quadrilaterals that are described on page 91 of their textbooks.

•The teacher will then pass out the worksheet on classifying quadrilaterals.(see below)•A discussion will follow on concave and convex.•The teacher will hand out a worksheet on finding the slopes of lines. The

teacher will model the first example and the students will attempt the rest on their own. Aftera reasonable amount of time, the worksheet will be reviewed and discussed as a class.(see below)

•The rest of the remaining class time, the students will explore graphing lineswith different slopes using their computers and equation grapher. The teacher will

walk the students through some examples from the book, such as page 86 20-22. (see below)

•The students can then play Green Globs in the time remaining. (see below)•The teacher will assign homework and pass out the necessary graph paper for

the students to complete the assignment on. (see below)

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SAMPLE STUDENT WORKHomework page 80

10. regular pentagon interior angles (5-2) 180 = 540 540/5 = 108 degrees exterior angles= 180 – 108 = 72 degrees

12. regular 18-gon interior angles= (18-2) 180= 2880 2880/18 = 160 degreesexterior angles= 180 – 160 = 20 degrees

14. Find the number of sides when given the exterior angle of a regular polygon.

Exterior angle = 72180 – 72 = 108 108 * 5 = 540 it works so theregular polygon is a pentagon

16. Find the number of sides when given the exterior angle of a regular polygon.

Exterior angle = 18180 – 18 = 162 162 * 10 = 1620 it works so the

regular polygon is a decagon

Find the missing variables.20. 7 sides (7-2) 180 = 900 900

116 +129 + 130 + 135 + 125 + 120 = 755 -755 Y= 145

22. 5 sides (5-2) 180 = 540 540 117 + 100 + 105 + 115 = 437 -437 X= 103

24. 4 sides (4-2) 180 = 360 h + h + 2h + 2h = 360 angles are 60, 60, 120, 120

6h = 360 h = 60

2726. exterior angles are = 360 exterior angles are 121,

108,131 z + (z – 13) + (z + 10) = 360 3z-3 = 360

z = 121

x = 180 – 121 y = 180 – 131 w = 180 - 108x = 59 y = 49 w = 72

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Name:Classifying Quadrilaterals Worksheet

Using your Quadrilateral Alphashapes determine ways to sort your quadrilaterals into subsets.

SUBSET #1What is the common feature that all these quadrilaterals have in common?

How many belong? And what letters are they?

SUBSET #2What is the common feature that all these quadrilaterals have in common?

How many belong? And what letters are they?

SUBSET #3What is the common feature that all these quadrilaterals have in common?

How many belong? And what letters are they?

Now trace one of the quadrilaterals from subset #1, #2, and #3 here and draw all the diagonals.What happened?

Now trace AlphaShapes E and T and draw all the diagonals. What happened?How are these quadrilaterals different from the ones you traced above?

29Name: SAMPLE STUDENT WORKClassifying Quadrilaterals Worksheet

Using your Quadrilateral AlphaShapes determine ways to sort your quadrilaterals into subsets.SUBSET #1

What is the common feature that all these quadrilaterals have in common?TWO PAIRS OF PARALLEL SIDES

How many belong? And what letters are they?5 Q,W,K,O,U

SUBSET #2What is the common feature that all these quadrilaterals have in common?ONE PAIR OF PARALLEL SIDES

How many belong? And what letters are they?2 C,S

SUBSET #3What is the common feature that all these quadrilaterals have in common?NO PARALLEL SIDES

How many belong? And what letters are they?1 G

Now trace one of the quadrilaterals from subset #1, #2, and #3 here and draw all the diagonals.What happened?

ONLY TWO DIAGONALS WERE DRAWN IN EACH FIGURE MAKING FOURTRIANGLES.Now trace AlphaShape E and T and draw all the diagonals. What happened?How are these quadrilaterals different from the ones you traced above?

30THESE HAVE DIAGONALS THAT RUN OUTSIDE OF THE QUADRILATERALS.

31Name:Slope Worksheet

Directions: Find the slope of each of the following graphs.

1. 2.

3. 4.

5. 6.

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7. 8.

9. 10.

11. 12.

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34Slope Worksheet

Directions: Find the slope of each of the following graphs.

1. M=1 2. M=0

3. M= UNDEFINED 4.M= -2

5. M= -1/2 6. BOTH LINES UNDEFINED AND PARALLEL

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7. M= 3 FOR BOTH 8. M= 0 for blue graph; undefined for redSO THEY ARE PARALLEL. Lines are perpendicular.

8. M= -2 for blue; M = _ red 9. M= -1 for blue; M= 1 for red lines are perpendicular lines are perpendicular

11. M= 3 12. M = - 1/2

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SAMPLE STUDENT WORKPAGE 86 #20

Which of the lines is not parallel to the line y = -2/3x + 8?

a. 2x + 3y = 1 b. 4x = 3 – 6y c. x = -1.5y

38d. 24 = 2x – 3y e. 9y = -6x -2

SAMPLE STUDENT WORKPAGE 86 # 21

a. sketch a vertical line containing the point (-5,2)b. write an equation for the linec. On the same graph, sketch horizontal line containing (-5,2)d. Write an equation for the linee. What is the relationship between the two lines?

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a. Sketch line w perpendicular to y= 5, and containing (1,4)b. Write an equation for line w.c. On the same graph, sketch line r parallel to y = 5, and containing (1,4).d. Write an equation for line r.e. What is the relationship between line r and line w?

Sample student workPage 86 # 22

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SAMPLE STUDENT WORKGREEN GLOBS

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42Name:_______________________________________________Homework Assignment: Graphing

On the provided graph paper, construct the following geometric shapes on a 10 X 10grid and answer the related questions right on the graph paper next to the figureconstructed. Label each shape with the appropriate number.

1. Plot the following points on the graph paper: A(-2,1), B(0,3), C(2,1), D(0,-1).What kind of quadrilateral is this and why?

2. Plot the following points on the graph paper: A(1,2), B(3,3), C(5,2), D(3,1).What kind of quadrilateral is this and why?

3. Plot the following points on the graph paper: W(-4,-3),X(-1,3),Y(4,3),Z(-1,3).What kind of quadrilateral is this and why?

4. Plot the following points on the graph paper: J(-2,2), K(2,2), L(5,-1), M(-5,-1).What kind of quadrilateral is this and why?

43Answer Key to homework graphing assignment

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DAY 6

Topic: Circles

Learner Objectives: Students will be able to recognize and define a circle and it’s parts: radius,diameter, central angle, major and minor arcs and adjacent arcs. Students will be able to find the measures of central angles and arcs of circles. Students will be comfortable using a compass to draw circles and arcs. Students will be able to find the center of a circle and compute the radius. Students will be able to recognize the relationships that the different arcs have tothe circle and to one another.

Instructional Presentation:•The quadrilateral graphing homework will be collected for grading.•The teacher will then ask the students if they have any questions about any of

the material covered thus far.

•The teacher will give a brief overview of the next section: circles. A discussionwill begin on how to define a circle and who can come up with a good definition that has no counterexample.

•The students will then get out their compasses and protractors and begin topractice

drawing circles with them, labeling the radius, diameter, and arcs. The teacherwill

model this on the overhead with them.•The teacher will explain (while modeling) minor arcs, major arcs, adjacent arcs

and semicircles and how to find the measure of each.

•The teacher will then go through example 3 on page 98 of the textbook which involves finding the measures of arcs.

•The students will then try to do #5 and #6 of example 3 on their own. Aftersufficient

time the teacher will go through each problem with them.•The teacher will then pass out graph paper and the students will practice finding

the coordinates of the center and the radius of different circles given two points onthe circle, construct an arc, and construct a circle graph using data from a table; problems 22,23,24,37, and 50 on pages 99-101. (see below)

•The teacher will assist as needed and may model an example on the overheadto help get the students started.

•The teacher will collect the student’s work for grading at the end of the classperiod.

45•The teacher will then assign problems 1-19 odds, 25-35 odds, 41-45 odds for

homework.

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SAMPLE STUDENT WORK PAGE 99 #22-24

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STUDENT SAMPLE WORK PAGE 99, # 37 and PAGE 101 #5

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