georgia performance standards framework for...

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One Stop Shop For Educators The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary. Many more GaDOE approved instructional plans are available by using the Search Standards feature located on GeorgiaStandards.Org. Georgia Performance Standards Framework for Mathematics – Grade 7 Georgia Department of Education Kathy Cox, State Superintendent of Schools Unit 2 Organizer STAYING IN SHAPE September 20, 2006 Page 1 of 30 Copyright 2006 © All Rights Reserved Unit 5 Organizer: “STAYING IN SHAPE” (6 weeks) OVERVIEW: In this unit students will: create similar shapes by enlarging or reducing a geometric figure in a coordinate plane describe similarities by listing corresponding parts find missing side lengths or areas in similar figures To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the type of learning activities that should be utilized from a variety of sources. ENDURING UNDERSTANDINGS: A dilation is a transformation that changes the size of a figure, but not the shape. The notation used to describe a dilation includes a scale factor and a center of dilation. A dilation of scale factor k with the center of dilation at the origin may be described by the notation (kx, ky). If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement. If the scale factor is less than 1, the image is a reduction. Two shapes are similar if the lengths of all the corresponding sides are proportional and all the corresponding angles are congruent. Two similar figures are related by a scale factor, which is the ratio of the lengths of the corresponding sides. The sides and perimeters of similar figures are related by a scale factor and the areas are related by the square of the scale factor. Scale factors, length ratios, and area ratios may be used to determine missing side lengths and areas in similar geometric figures.

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One Stop Shop For Educators

The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary. Many more GaDOE approved instructional plans are available by using the Search Standards feature located on GeorgiaStandards.Org.

Georgia Performance Standards Framework for Mathematics – Grade 7

Georgia Department of Education Kathy Cox, State Superintendent of Schools Unit 2 Organizer STAYING IN SHAPE

September 20, 2006 Page 1 of 30 Copyright 2006 © All Rights Reserved

Unit 5 Organizer: “STAYING IN SHAPE” (6 weeks)

OVERVIEW: In this unit students will:

• create similar shapes by enlarging or reducing a geometric figure in a coordinate plane • describe similarities by listing corresponding parts • find missing side lengths or areas in similar figures

To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the type of learning activities that should be utilized from a variety of sources. ENDURING UNDERSTANDINGS:

• A dilation is a transformation that changes the size of a figure, but not the shape. • The notation used to describe a dilation includes a scale factor and a center of dilation. A dilation of scale factor k with the

center of dilation at the origin may be described by the notation (kx, ky). • If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement. If the scale factor

is less than 1, the image is a reduction. • Two shapes are similar if the lengths of all the corresponding sides are proportional and all the corresponding angles are

congruent. • Two similar figures are related by a scale factor, which is the ratio of the lengths of the corresponding sides. • The sides and perimeters of similar figures are related by a scale factor and the areas are related by the square of the scale

factor. • Scale factors, length ratios, and area ratios may be used to determine missing side lengths and areas in similar geometric

figures.

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Georgia Performance Standards Framework for Mathematics – Grade 7

• Congruent figures have the same size and shape. If the scale factor of a dilation is equal to one, the image resulting from the dilation is congruent to the original figure.

ESSENTIAL QUESTIONS:

• What is a dilation and how does this transformation affect a figure in the coordinate plane? • How can I tell if two figures are similar? • In what ways can I represent the relationships that exist between similar figures using the scale factors, length ratios, and

area ratios? • What strategies can I use to determine missing side lengths and areas of similar figures? • Under what conditions are similar figures congruent?

STANDARDS ADDRESSED IN THIS UNIT

Mathematics standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical ideas. KEY STANDARDS: M7G2. Students will demonstrate understanding of transformations.

a. Demonstrate understanding of dilations. b.Given a figure in the coordinate plane, determine the coordinates resulting from a dilation.

M7G3. Students will use the properties of similarity and apply these concepts to geometric figures. a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing

corresponding parts. b.Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length

ratios, and area ratios to determine side lengths and areas of similar geometric figures. c. Understand congruence of geometric figures as a special case of similarity: The figures have the same size and shape.

M7P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b.Solve problems that arise in mathematics and in other contexts.

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c. Apply and adapt a variety of appropriate strategies to solve problems. d.Monitor and reflect on the process of mathematical problem solving.

M7P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b.Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d.Select and use various types of reasoning and methods of proof.

M7P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b.Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely.

M6P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b.Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics.

M6P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b.Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena.

RELATED STANDARDS:M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.

c. Add, subtract, multiply, and divide positive and negative rational numbers. d. Solve problems using rational numbers.

M7G2. Students will demonstrate understanding of transformations. a. Demonstrate understanding of translations, rotations, reflections, and relate symmetry to appropriate transformations. b. Given a figure in the coordinate plane, determine the coordinates resulting from a translation, rotation, or reflection.

M7A1. Students will represent and evaluate quantities using algebraic expressions. a. Translate verbal phrases to algebraic expressions. b.Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. c. Add and subtract linear expressions.

M7A2. Students will understand and apply linear equations in one variable. Georgia Department of Education

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Georgia Performance Standards Framework for Mathematics – Grade 7

a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution. b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations.

M7A3 Students will understand relationships between two variables. a. Plot points on a coordinate plane.

CONCEPTS/SKILLS TO MAINTAIN: It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• plot points in the coordinate plane • calculate the perimeter and area of fundamental geometric plane figures • use the concepts of ratio, proportion, and scale factor to demonstrate the relationships between similar plane figures

SELECTED TERMS AND SYMBOLS: The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and to how their students are able to explain and apply them. Dilation : Transformation that changes the size of a figure, but not the shape. Proportion: An equation which states that two ratios are equal. Ratio : Comparison of two quantities by division and may be written as r/s, r:s, or r to s. Scale Factor : The ratio of any two corresponding lengths of the sides of two similar figures. Similar Figures : Figures that have the same shape but not necessarily the same size. Congruent Figures : Figures that have the same size and shape. You may visit www.intermath-uga.gatech.edu and click on dictionary to see definitions and specific examples of terms and symbols used in the seventh grade GPS.

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Georgia Performance Standards Framework for Mathematics – Grade 7

EVIDENCE OF LEARNING: By the conclusion of this unit, students should be able to demonstrate the following competencies:

• Enlarge or reduce a geometric figure using a given scale factor; • Given a figure in the coordinate plane, determine the coordinates resulting from a dilation; • Compare geometric figures for similarity and describe similarities by listing corresponding parts; • Describe relationships among scale factors, length ratios, and area ratios of similar geometric figures; and • Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures.

The following task represents the level of depth, rigor, and complexity expected of all 7th grade students. This task or a task of similar depth and rigor should be used to demonstrate evidence of learning. Culminating Activity: Club Logo Students will design a new logo for a club at their school. They will use translations, reflections, or rotations in their designs for a t-shirt logo and then dilate the logo to fit on a sign. STRATEGIES FOR TEACHING AND LEARNING:

• Students should be actively engaged in developing their own understanding. • Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words. • Appropriate manipulatives and technology should be used to enhance student learning. • Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition

which includes self-assessment and reflection.

TASKS: The collection of the following tasks represents the level of depth, rigor and complexity expected of all seventh grade students to demonstrate evidence of learning.

• Playing with Dilations Georgia Department of Education

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Playing with Dilations Go to the following website for this investigation: http://www.mathsnet.net/dynamic/enlarge2.html Click on “Show Values”.

a. Change the scale by moving the red point on the segment in the top left corner. What do you observe when the scale is less than 1? Equal to 1? Greater than 1? As you are changing the scale, observe what is happening to the area of the red triangle and the ratio of the areas of the triangles. Describe what you observe. Why do you think this happens?

b. Move the point X to different locations outside, inside, and on the triangle. What changes in the values do you notice as you move X? Explain why you think this happens.

c. As you moved X in part b, other than the values, describe all the changes you noticed. Why do you think these changes occurred?

d. What are some real-world situations in which this might be used? ********************************************************************************************* Playing with Dilations Discussion, Suggestions, Possible Solutions: Another web site where students can play with dilations is the National Library of Virtual Manipulatives (NLVM) site. Below is a direct link to the activities involving dilations. http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?open=activities At the NLVM site, students can explore dilations with different shapes. Suggested questions are provided on the NLVM web site.

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In this investigation, students will observe the effects of a dilation with a given center and scale factor. Students should observe that the three lines passing through corresponding vertices of the two triangles intersect at a common point. This point is called the center of dilation. The distance from the center of dilation to the original triangle is reduced or enlarged

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according to the scale factor.

a. The shape of the triangle remains the same. When the scale is less than 1, the red triangle is a reduction of ΔABC and is closer to the center of dilation. When the scale is equal to 1, the red triangle is congruent to ΔABC. When the scale is greater than 1, the red triangle is an enlargement of ΔABC and is further away from the center of dilation. The distance from X to a vertex of the red triangle depends on the scale factor (e.g., if the scale factor is 0.5, the distance from X to a vertex of the red triangle is ½ the distance from X to the corresponding vertex on the blue triangle).

As the scale increases, the area of the red triangle increases because increasing the scale makes the red triangle larger. The ratio of the area of the red triangle to the blue triangle also increases because the numerator of the fraction representing the ratio is getting larger.

[Note to teachers: Some students might observe that the ratio of the areas of the triangles is the square of the scale factor. This might not be obvious unless students look at the ratio for scale factors such as 0.5 or 2. This concept will be encountered in other activities, so it is not essential that students make this observation at this point.]

b. The values do not change because the scale and the areas of the triangles are not changing. The only thing that changes is the location of X and the red triangle in relationship to the blue triangle.

c. The lines through the corresponding vertices of the red triangle and ΔABC always intersect at a common point, X. [Teachers may want to tell students that this common point is called the center of dilation.]

d. Projecting something on a screen (e.g., using an overhead projector), reducing or enlarging pictures, scale drawings (e.g., maps, building plans). Students may remember something about scale drawings from 6th grade.

• Dilations in the Coordinate Plane

Dilations in the Coordinate Plane Plot the ordered pairs given in the table to make six different figures. Draw each figure on a separate sheet of graph paper. Connect the points with line segments as follows:

• For Set 1, connect the points in order. Connect the last point in the set to the first point in the set. • For Set 2, connect the points in order. Connect the last point in the set to the first point in the set. • For Set 3, connect the points in order. Do not connect the last point in the set to the first point in the set.

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• For Set 4, make a dot at each point (don’t connect the dots). After drawing the six figures, compare Figure 1 to each of the other figures and answer the following questions.

1. Which figures are similar? Explain your thinking. 2. Describe any similarities and/or differences between Figure 1 and each of the other figures.

• Describe how corresponding sides compare. • Describe how corresponding angles compare.

3. How do the coordinates of each figure compare to the coordinates of Figure 1? If possible, write general rules for making Figures 2-6.

4. Is having the same angle measures enough to make two figures similar? Why or why not? 5. What would be the effect of multiplying each of the coordinates in Figure 1 by ½? 6. Translate, reflect, rotate (between 0 and 90°), and dilate Figure 1 so that it lies entirely in Quadrant III on the

coordinate plane. You may perform the transformations in any order that you choose. Draw a picture of the new figure at each step and explain the procedures you followed to get the new figure. Use coordinates to describe the transformations and give the scale factor you used. Describe the similarities and differences between your new figures and Figure 1.

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6

Set 1 Set 1 Set 1 Set 1 Set 1 Set 1 (6, 4) (12, 8) (18, 4) (18, 12) (6, 12) (8, 6) (6, -4) (12, -8) (18, -4) (18, -12) (6, -12) (8, -2) (-6, -4) (-12, -8) (-18, -4) (-18, -12) (-6, -12) (-4, -2) (-6, 4) (-12, 8) (-18, 4) (-18, 12) (-6, 12) (-4, 6) Set 2 Set 2 Set 2 Set 2 Set 2 Set 2 (1, 1) (2, 2) (3, 1) (3, 3) (1, 3) (3, 3) (1, -1) (2, -2) (3, -1) (3, -3) (1, -3) (3, 1) (-1, -1) (-2, -2) (-3, -1) (-3, -3) (-1, -3) (1, 1) (-1, 1) (-2, 2) (-3, 1) (-3, 3) (-1, 3) (1, 3) Set 3 Set 3 Set 3 Set 3 Set 3 Set 3 (4, -2) (8, -4) (12, -2) (12, -6) (4, -6) (6, 0) (3, -3) (6, -6) (9, -3) (9, -9) (3, -9) (5, -1)

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(-3, -3) (-6, -6) (-9, -3) (-9, -9) (-3, -9) (-1, -1) (-4, -2) (-8, -4) (-12, -2) (-12, -6) (-4, -6) (-2, 0) Set 4 Set 4 Set 4 Set 4 Set 4 Set 4 (4, 2) (8, 4) (12, 2) (12, 6) (4, 6) (6, 4)

(-4, 2) (-8, 4) (-12, 2) (-12, 6) (-4, 6) (-2, 4) ********************************************************************************************* Dilations in the Coordinate Plane Discussion, Suggestions, Possible Solutions Source: Adapted from Stretching and Shrinking: Similarity, Connected Mathematics, Dale Seymour Publications Students will find rules to describe transformations in the coordinate plane. Rules of the form (nx, ny) transform a figure in the plane into a similar figure in the plane. This transformation is called a dilation with the center of dilation at the origin. The coefficient of x and y is the scale factor. Adding a number to x or y results in a translation of the original figure but does not affect the size. Thus, a more general rule for dilations centered at the origin is (nx + a, ny + b). Students will also observe that congruence is a special case of similarity (n=1). Congruent figures have the same size and shape. As students learned in the previous unit, transformations that preserve congruence are translations, reflections, and rotations. Possible solutions: Note: The scale used on the x- and y-axes in the figures below is 2 units. Each square is 4 square units (2 x 2).

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Figure 1:

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Figure 2:

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Figure 3:

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Figure 4:

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Figure 5:

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Figure 6:

1. Figures 1, 2, 4 and 6 are similar. Students may observe visually that these figures have the same shape but are different sizes (except for Figure 6). Figure 6 is congruent to Figure 1. Note that congruence is a special case of similarity – figures have the same size and shape. Figures 3 and 5 are longer (or taller) and skinnier. Students may also notice that corresponding angles are equal for all figures. The scale factor from Figure 1 to Figure 2 is 2. The scale factor from Figure 1 to Figure 4 is 3. The side lengths of Figure 2 are twice the side lengths of Figure 1 and the side lengths of Figure 4 are three times the side lengths of Figure 1. The scale factor from Figure 1 to Figure 6 is 1 because it is congruent to Figure 1. In Figures 3 and 5, one dimension increases by a factor of 3 and the other does not.

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2. Figure 2 is an enlargement of Figure 1. The figures have the same shape but different sizes. The ratio of the lengths of the corresponding sides is 1 to 2. The corresponding angles are equal in measure.

Figure 3 is wider or longer than Figure 1. The figures are different shapes and sizes. The ratio of the lengths of the corresponding sides is not constant. For one dimension, the ratio is 1 to 3; for the other dimension, the ratio is 1 to 1.

The corresponding angles are equal in measure.

Figure 4 is an enlargement of Figure 1. The figures have the same shape but different sizes. The ratio of the lengths of the corresponding sides is 1 to 3.

The corresponding angles are equal in measure.

Figure 5 is taller than Figure 1. The figures have different shapes and sizes. The ratio of the lengths of the corresponding sides is not constant. For one dimension, the ratio is 1 to 3; for the other dimension, the ratio is 1 to 1. The corresponding angles are equal in measure.

Figure 6 is the same shape and size as Figure 1. Figure 1 is shifted (i.e., translated) up and to the right to get Figure 6.

The ratio of the lengths of the corresponding sides is 1 to 1. The corresponding angles are equal in measure. 3. Figure 2: Both the x and y coordinates are multiplied by 2. (2x, 2y) Figure 3: The x coordinates in Figure 3 are three times the corresponding x coordinates in Figure 1; the y coordinates are the same. (3x, y) Figure 4: Both the x and y coordinates are multiplied by 3. (3x, 3y) Figure 5: The x coordinates in Figure 5 are the same as the corresponding x coordinates in Figure 1. The y coordinates are three times the corresponding y coordinates in Figure 1. (x, 3y) Figure 6: Two is added to both the x and y coordinates. (x + 2, y + 2)

4. No. All angles of the figures (except angles of the smiles) have the same angle measures, but the figures are not

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similar. Figures 3 and 5 are long (or tall) and skinny, unlike Figure 1. 5. The figure would shrink and the lengths of the sides would be half as long. [Note to teachers: Students may say

that the new figure is “½ the size” of the original figure which might imply that the area of the new figure is ½ the area of the original. In actuality, the area of the new figure is ½ ½ or ¼ the size of the original figure. Be sure that students understand that the side lengths are reduced by a factor of ½.]

6. Answers will vary depending on the transformations that students use. Students must recall what they learned in

the transformations unit about translations, reflections, and rotations. The translation, reflection, and rotation do not change the size or shape of the figure. The final figure is a reduction or enlargement of Figure 1 and it has a different orientation in the coordinate plane because of the reflection and rotation.

• Changing Shapes

Changing Shapes Later in this unit, you are going to be designing a logo for a club at your school. To prepare for this project, draw a non-rectangular shape in the coordinate plane so that portions of the shape are in each of the four quadrants. Explain what would happen to your shape if you transformed it using each of the given rules with the center of dilation at the origin.

a. (4x, 4y) b. (0.25x, 0.25y) c. (2x, y) d. (3x, 3y + 5) e. (x + 5, y - 5) f. (½ x - 1, ½ y) g. Will any of the transformed figures be similar to the original figure? Explain. h. If you make a new figure by adding 2 units to the length of each side of your shape, will the two figures be

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similar? Why or why not? i. Write a general rule for transformations in the plane that produce similar figures.

********************************************************************************************* Changing Shapes Discussion, Suggestions, Possible Solutions This task assesses students’ ability to identify the effects of transforming a figure according to a rule involving dilations and/or translations. Possible solutions

a. The figure would grow by a scale factor of 4. The distance from the origin to the object would increase by a scale factor of 4.

b. The figure would shrink by a scale factor of 0.25. The distance from the origin to the object would decrease by a scale factor of 0.25.

c. The figure would increase on one dimension by a scale factor of 2; the other dimension would stay the same. d. The figure would grow by a scale factor of 3 and move up 5 units. e. The figure would move right five units and down five units. f. The figure would shrink by a scale factor of ½ and move left 1 unit. g. Figures a, b, d, e, and f will be similar to the original figure. Both dimensions increase by the same scale factor.

Figure e will be congruent to the original figure because the side lengths and shape do not change. The ratio of the lengths of the corresponding sides will be 1:1 and the measures of the corresponding angles will be equal. Note that congruence is a special case of similarity. [Figure e is congruent to the original figure.]

h. The figures would not be similar. Adding a constant amount to each side will distort the figure. The ratio of the lengths of the corresponding sides will not be constant.

i. (nx + a, ny + b)

• Growing Logos

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Growing Logos When you design your club logo a little later in this unit, it will need to be enlarged. Before creating your logo, you need to know what happens to the perimeter and area of a figure when it is dilated. This will help you determine the scale factor to be used for the dilation.

1. On grid paper, draw a rectangle with an area of 24 square units. Label the rectangle ABCD. 2. Draw a rectangle that is three times as long and three times as wide. Label the new rectangle A'B'C'D'. Are the

rectangles similar? Why or why not? 3. What is the perimeter of A'B'C'D' ? How does it compare to the perimeter of ABCD? How does the ratio of the

perimeters compare to the ratio of the lengths of the corresponding sides? Explain why this makes sense. 4. What is the area of A'B'C'D' ? How does it compare to the area of ABCD? How does the ratio of the areas

compare to the ratio of the lengths of the corresponding sides? 5. Draw a rectangle that covers four times as much area. Label the new rectangle A"B"C"D". How are the lengths of

the corresponding sides related? Are the areas related in the same way as the corresponding sides? Use an area formula to justify your answer.

6. What do you know now about the areas of similar figures? How can you use this knowledge when determining future dilations?

********************************************************************************************* Discussion, Suggestions, Possible Solutions:

1. Answers will vary. For example, students might draw a rectangle with dimensions 1 x 24; 2 x 12; 3 x 8; or 4 x 6. Students might also be encouraged to explore non-integer dimensions such as 2.5 x 9.6.

2. The rectangles are similar because the measures of corresponding angles are equal and the ratio of corresponding sides is constant (3:1).

3. Answers will vary depending upon the dimensions selected by students. The ratio of the perimeters of A'B'C'D' and ABCD will be 3:1, which is the same as the ratio of the lengths of the corresponding sides. Let l = length of ABCD and w = width of ABCD Perimeter of A'B'C'D' = 2(length of A'B'C'D') + 2(width of A'B'C'D') = 2(3l) + 2(3w) = 3[2l+2w] = 3(Perimeter of ABCD)

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4. The area of A'B'C'D' will be 216 and the ratio of the areas of A'B'C'D' and ABCD will be 9:1. The ratio of the areas is the square of the ratio of the lengths of the corresponding sides or the scale factor.

5. The length and width should be multiplied by 2 to get a rectangle that covers four times as much area. Since the ratio of the areas is 4:1, the ratio of the sides must be 2:1. Area of A"B"C"D" = 4(Area of ABCD) = 4LW = (2L)(2W)

6. Since the ratio of the areas is the square of the scale factor, the scale factor for an enlargement can be determined based on the ratio of the area available for the enlarged shape and the area of the original shape.

• Similar Triangles

Similar Triangles The sketch below shows two triangles, ∆ABC and ∆EFG. ∆ABC has an area of 12 square units, and its base (AB) is equal to 8 units. The base of ∆EFG is equal to 24 units.

a. How do you know that the triangles are similar? b. Name the pairs of corresponding sides and the pairs of corresponding angles. How are the corresponding sides

related and how are the corresponding angles related? Why is this true? c. What is the area of ∆ EFG? Explain your reasoning. d. What is the relationship between the area of ∆ABC and the area of ∆ EFG ? What is the relationship between the

scale factor and the ratio of the areas of the two triangles? Use an area formula to justify your answer algebraically.

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Georgia Performance Standards Framework for Mathematics – Grade 7

********************************************************************************************* Similar Triangles Discussion, Suggestions, Possible Solutions This is a great opportunity for students to use what they learned in 6th grade concerning the fact that there are 180 degrees in a triangle and the standards concerning similar figures. The teacher may wish to question students to be sure they remember that if two pairs of corresponding angles in two triangles are congruent, then the triangles are similar. Students need to realize that the scale factor is 3 and that it is applied to both the base and height, so that the area increases by a factor of 9. Possible solutions: Part a: The triangles are similar because three pairs of corresponding angles are congruent. This is because when two pairs of corresponding angles are congruent in a triangle, the third pair must also be congruent. Students should be questioned to develop this understanding. Note that in triangles (unlike the rectangles in the activity, “Dilations in the

B A

C

E F

G

24 8

70° 30°70°30°

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Georgia Performance Standards Framework for Mathematics – Grade 7

Coordinate Plane”), having the same angle measures is enough to make two triangles similar. This could be demonstrated with Geometer’s Sketchpad. Part b: AB and EF; BC and FG; AC and EG are the corresponding sides

B and ے ے A and E are the corresponding anglesے ;Gے C andے ;Fے AB = BC = AC EF FG EG m Eے A = mے G; mے C = mے F; mے = B ے These relationships are true because the triangles are similar. Part c: Solution Method 1 -- The scale factor for similar triangles ABC and EFG is 3 since the ratio of the bases is 24 to 8. In similar triangles, the ratio of the areas is the square of the scale factor, so the ratio of the area of ∆ EFG to the area of ∆ABC is 9. Therefore, the area of ∆ EFG is 12 x 9 or 108 square units. Solution Method 2 -- We can find the length of the altitude of ∆ABC by using the area formula. A = ½ bh 12 = ½(8)h 12 = 4h 3 = h The altitude of ∆ EFG is 3 times the altitude of ∆ABC, so the length of the altitude of ∆ EFG is 9. [Note to teachers: Students should verify that the length of the altitude increases by the same scale factor as the corresponding sides of similar triangles. This could be done with Geometer’s Sketchpad.] Using the area formula, A = ½ bh A = ½ (24)(9) A = (12)(9) A = 108

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Georgia Performance Standards Framework for Mathematics – Grade 7

Part d: The area of ∆ EFG is nine times the area of ∆ABC, but the side lengths of ∆ EFG are three times the corresponding side lengths of ∆ABC. In the area formula, the scale factor of 3 must be applied to both the base and height. So the area of ∆ABC is multiplied by 9. Area of ∆ABC = ½ bh Area of ∆ EFG = ½ (3b)(3h) Area of ∆ EFG = 9(½ bh)

• Making Copies

Making Copies Jamal has a 5" by 7" picture that he took on his summer vacation, and he wants to use a photo copier to enlarge it to fit in an 8" by 10" picture frame. To enlarge the picture, Jamal needs to specify a percent between 50% and 200% on the copier. For example, a setting of 125% would enlarge the picture by a scale factor of 1.25. Explain two different ways Jamal can enlarge the picture to make an 8" by 10" picture.

********************************************************************************************* Making Copies Discussion, Suggestions, Possible Solutions: 8" by 10" photo paper is not similar to 5" by 7" photo paper because 8 ÷ 5 = 1.6 and 10 ÷ 7 = 1.43. The picture will need to be copied to larger paper (e.g., 8" by 11.5") and the extra length trimmed off. If the copier is set at 160% (i.e., scale factor of 1.6), both dimensions of the original photo (5" by 7") will be multiplied by 1.6, and the new photo will have dimensions 8" by 11.2". So 1.2" would have to be trimmed off the length to make an 8" by 10" photo. Another solution would be to trim the 5" by 7" picture to an appropriate size and then enlarge it.

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Georgia Performance Standards Framework for Mathematics – Grade 7

8_ = 5 10 x 8x = 50 x = 6.25 Trim the picture to 5" by 6.25" and then enlarge it by a scale factor of 1.6 (i.e., set the copier at 160%).

• Similar Pentagons

Similar Pentagons Figure 1 below is a special kind of pentagon in which ABCD is a square and E is the midpoint of the diagonal AC; then square CEBF is drawn to make the pentagon ABFCD. Figure 2 continues drawing similar figures, starting with square CEBF, then adding square CGFH, and so on.

a. Find other pentagons that are similar to ABFCD. Describe the similarities for one pair of similar pentagons by listing the corresponding parts and their relationships.

b. If CM is 2 ¼ cm long, how long is CD? Explain your reasoning. c. If AC is 6 3⁄5 cm long, how long is KC? Explain your reasoning.

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Georgia Performance Standards Framework for Mathematics – Grade 7

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M

L

K

J

H

G FE

CD

FE

CD

A B BA

********************************************************************************************* Similar Pentagons Discussion, Suggestions, Possible Solutions Source: Adapted from Grades 6-8 Mathematics Assessment Sampler, National Council of Teachers of Mathematics, p. 116. This task requires students to compare geometric figures for similarity, to describe similarities by listing corresponding parts, and to use properties of geometric figures to determine side lengths. Some students may need to trace the different pentagons on different sheets of patty paper to see the corresponding parts. To solve part b, students will need to know that the lengths of the sides of a square are equal and that the midpoint divides a segment into two equal parts.

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Georgia Performance Standards Framework for Mathematics – Grade 7

a. The similar pentagons are ABFCD ~ BFHCE ~ FHKCG ~ HKMCJ

AB = BF = FC = CD = DA BF FH HC CE EB AB = BF = FC = CD = DA FH HK KC CG GF AB = BF = FC = CD = DA HK KM MC CJ JH m ABF = m ے BFH; m ے BFC = mے FHC; m ے FCD = m ے HCE; m ے CDA = m ے CEB; m ے DAB ے= m EBF ے m ABF = m ے FHK m ے BFC = m ے HKC; m ے FCD = m ے KCG; m ے CDA = m ے CGF; m ے DAB ے= m GFH ے m ABF = m ے HKM; m ے BFC = m ے KMC; m ے FCD = m ے MCJ; m ے CDA = m ے CJH; m ے DAB ے= m JHK ے

b. If CM is 2 ¼ or 2.25 cm., then LC is also 2.25 because CLKM is a square. Since L is the midpoint of CH and LC = 2.25, then CH = 2(LC) = 2(2.25) = 4.5. Because CHFG is a square, GC = 4.5. Since G is the midpoint of BC, then BC = 2(GC) = 2(4.5) = 9. Because BCDA is a square, CD = 9.

c. If AC is 6 3⁄5 or 6.6 cm., then EC = ½ (6.6) = 3.3 because E is the midpoint of AC. Since CEBF is a square, EC =

FC = 3.3. Because J is the midpoint of FC, then JC = ½ (FC) = ½ (3.3) = 1.65. CJHK is a square, so JC = KH = KC = 1.65.

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Georgia Performance Standards Framework for Mathematics – Grade 7

• Shadow Math

Shadow Math Jeannie is practicing on the basketball goal outside her house. She thinks that the goal seems lower than the 10 ft. goal she plays on in the gym. She wonders how far the goal is from the ground. Jeannie can’t reach the goal to measure the distance to the ground, but she remembers something from math class that may help. First, she needs to estimate the distance from the bottom of the goal post to the top of the backboard. To do this, Jeannie measures the length of the shadow cast by the goal post and backboard. She then stands a yardstick on the ground so that it is perpendicular to the ground, and measures the length of the shadow cast by the yardstick. Here are Jeannie’s measurements: Length of shadow cast by goal post and backboard: 5 ft. 9 in. Length of yardstick’s shadow: 1 ft. 6 in. Draw and label a picture to illustrate Jeannie’s experiment. Using her measurements, determine the height from the bottom of the goal post to the top of the backboard. If the goal is approximately 24 inches from the top of the backboard, how does the height of the basketball goal outside Jeannie’s house compare to the one in the gym? Justify your answer. ********************************************************************************************* Discussion, Suggestions, Possible Solutions:

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Georgia Performance Standards Framework for Mathematics – Grade 7

Find height of backboard, x, by solving the following proportion: x_ = _3_ 5.75 1.5 1.5x = 17.25 x = 17.25 ÷ 1.5 x = 11.5 ft. If the goal is approximately 24 inches or 2 ft. from the top of the backboard, then the height of the goal is approximately 9.5 ft. so Jeannie’s goal is about 6 inches (or ½ ft.) lower than the goal in the gym. Solution Method 2: Students might choose to convert all the measurements to inches.

x

5.75 ft. 1.5 ft.

3 ft.

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Georgia Performance Standards Framework for Mathematics – Grade 7

x_ = 36 69 18 18x = 2484 x = 2484 ÷ 18 x = 138 in. or 11.5 ft. If the goal is approximately 24 in. or 2 ft. from the top of the backboard, then the height of the goal is approximately 114 in. or 9.5 ft., so Jeannie’s goal is about 6 inches (or ½ ft.) lower than the goal in the gym.

• Club Logo This culminating task represents the level of depth, rigor and complexity expected of all 7th grade students to demonstrate evidence of learning.

UNIT FIVE TASK: Club Logo

A club at your school wants a new logo and is sponsoring a contest to get ideas from students. The winner of the contest will receive a $100.00 cash award. You could really use that money, so you have decided to enter the contest. Your task is to design a logo for the club that meets the following requirements:

1. You must use constructions to create the logo. 2. The design must include at least one translation, reflection, or rotation. 3. Design your logo on an 8.5 inch by 11 inch coordinate plane so that it will fit on the front or back of a t-shirt. 4. Dilate your design so that it will fit on a sign (no larger than 5 feet by 5 feet).

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Georgia Performance Standards Framework for Mathematics – Grade 7

Your design will compete against other designs to be the official logo for the club. Be prepared to present your design. Your presentation should include visuals of your logo for both the t-shirt and the sign, and a written explanation of the steps and procedures you followed in designing your logo. Use coordinates to describe the transformations you used. Give the scale factor used for the dilation and explain the similarities and differences in the t-shirt logo and the sign logo.

Suggestions for Classroom Use While this task may serve as a summative assessment, it also may be used for teaching and learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them. The task can be given to the students at the beginning of the unit. As they are working on the supporting tasks, they can be thinking about this project and using instructional time appropriately.

Discussion, Suggestions and Possible Solutions

This task is designed to reinforce concepts learned in this unit and the previous unit on transformations. If students did the summative task from Unit 4, Analyzing Quilts, they should be able to use the coordinate plane to describe transformations. You may want to review that task before engaging students in the activity of designing a logo. Be sure that students are thinking about the concepts of similarity learned in this unit as they compare the logo for the t-shirt and for the sign. They should discuss relationships between corresponding sides and angles (or other parts of the logo), perimeters, and areas. They may verify scale factors by measuring or estimating (e.g., estimate area of dilated logo by observing approximately how many t-shirt logos will fit in the sign logo). One way to dilate the t-shirt design is to trace the design on a transparency and project it onto a sheet of posterboard using an overhead projector. Students could then trace the design on the posterboard. To find the scale factor for the dilation, students could measure various parts of the logos and find the ratio of corresponding parts.

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