Illustrative Math Alignment: Grade 7 Unit 2

Introducing Proportional Relationships

Lesson 12: Using Graphs to Compare Relationships

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Nodes
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 15	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 15 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar right triangles. Two right triangles are shown, with the smaller one having sides of 4 and 3, and the larger one having sides of x and 12. The problem requires finding the length of side x by setting up a proportion: 4 / 3 = x / 12. Solving this equation leads to x = 16.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 16	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 16 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar right triangles with algebraic expressions. Two right triangles are shown, with the smaller one having sides of (3x + 4) and 6, and the larger one having sides of 3x and 3x. The problem requires setting up a proportion: 8 / 6 = (3x + 4) / 3x. Solving this equation leads to x = 4, which then allows us to find the side lengths of 12 and 16.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 17	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 17 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar right triangles. Two right triangles are shown, with the left one having sides of 12 and 5, and the right one having sides of x and 15. The problem requires finding the length of side x by setting up a proportion: 12 / 5 = x / 15. Solving this equation leads to x = 36.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 18	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 18 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar right triangles with algebraic expressions. Two right triangles are shown, with the left one having sides of 12 and 5, and the right one having sides of 10x + 8 and 5x. The problem requires setting up a proportion: 12 / 5 = (10x + 8) / 5x. Solving this equation leads to x = 4, which then allows us to find the side lengths of 20 and 48.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 19	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 19 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar isosceles right triangles. Two 45-45-90 triangles are shown, with one having sides labeled y and 5√2, and the other having sides labeled x and 12. The problem requires finding the lengths of both x and y using the special properties of isosceles right triangles and proportions.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 2	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 2 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion where b is expressed as x + 1, and c and d are given constants (c = 3, d = 2). The goal is to solve for a using the proportion a / b = c / d. By substituting the known values, we set up the equation a / (x + 1) = 3 / 2 and solve for a, resulting in the expression a = (3(x + 1)) / 2.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 20	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 20 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar isosceles right triangles with algebraic expressions. Two 45-45-90 triangles are shown, with one having sides labeled 15√2 and 15, and the other having sides labeled √2 * 5x and y. The problem requires finding the length of y in terms of x using the special properties of isosceles right triangles and proportions.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 21	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 21 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar isosceles triangles. Two isosceles triangles are shown, with the smaller one having sides of 30 and 16, and the larger one having sides of x and 20. The problem requires finding the length of side x by setting up a proportion: 30 / 16 = x / 20. Solving this equation leads to x = 37.5.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 22	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 22 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar isosceles triangles with algebraic expressions. Two isosceles triangles are shown, with the smaller one having sides of 22 and 12, and the larger one having sides of 10x + 5 and 6x. The problem requires setting up a proportion: 22 / 12 = (10x + 5) / 6x. Solving this equation leads to x = 5, which then allows us to find the side lengths of the larger triangle.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 23	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 23 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar equilateral triangles. Two equilateral triangles are shown, with the smaller one having a side length of 12, and the larger one having a side length of 6x and an additional side labeled as 12 + x. The problem requires finding the lengths of the sides of the larger triangle by setting up a proportion: 12 / 12 = (12 + x) / 6x. Solving this equation leads to x = 2.4, and the length of the larger triangle's side is found to be approximately 14.4 units.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 24	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 24 Topic Ratios, Proportions, and Percents Description This example illustrates solving a complex proportion problem using similar equilateral triangles with algebraic expressions. Two equilateral triangles are shown, with the smaller one having a side length of 3x, and the larger one having side lengths labeled as (4x + 1) and (x + 1). The problem requires setting up a proportion: 3x / (4x + 1) = 3x / (3x + (x + 1)). Solving this equation leads to a quadratic equation, which when solved gives x ≈ 7/1.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 25	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 25 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar rectangles. Two rectangles are shown, with the smaller one having sides of 2 and 3, and the larger one having sides of x and 15. The problem requires finding the length of side x by setting up a proportion: 2 / 3 = x / 15. Solving this equation leads to x = 10.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 26	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 26 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar rectangles with algebraic expressions. Two rectangles are shown, with the smaller one having sides of 5 and 12, and the larger one having sides of x + 5 and 3x + 6. The problem requires setting up a proportion: 5 / 12 = (x + 5) / (3x + 6). Solving this equation leads to x = 10, which then allows us to find the side lengths of the larger rectangle as 15 and 36.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 27	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 27 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar parallelograms. Two parallelograms are shown, with the smaller one having sides of 8 and 18, and the larger one having sides of 20 and x. The problem requires finding the length of side x by setting up a proportion: 8 / 18 = 20 / x. Solving this equation leads to x = 45.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 28	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 28 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar parallelograms with algebraic expressions. Two parallelograms are shown, with the smaller one having sides of 9 and 21, and the larger one having sides of x + 12 and 4x + 3. The problem requires setting up a proportion: 9 / 21 = (x + 12) / (4x + 3). Solving this equation leads to x = 15, which then allows us to find the side lengths of the larger parallelogram as 27 and 63.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 29	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 29 Topic Ratios, Proportions, and Percents Description This example demonstrates how to determine if two triangles are similar using proportions. Two triangles are shown, both with a 70° angle. The first triangle has sides of 12 and 10, while the second has sides of 18 and 15. The problem requires setting up a proportion to check for similarity: 12 / 10 = 18 / 15. After simplifying, both ratios are equal (6 / 5 = 6 / 5), confirming that the triangles are indeed similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 3	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 3 Topic Ratios, Proportions, and Percents Description This example demonstrates solving for b in a proportion where a = 8, c = 4, and d = 3. We set up the proportion 8 / b = 4 / 3 and solve for b, resulting in b = 6. This problem shows how to find an unknown value in the denominator of a proportion.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 30	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 30 Topic Ratios, Proportions, and Percents Description This example illustrates how to determine if two triangles are not similar using proportions. Two triangles are shown, both with a 75° angle. The first triangle has sides of 15 and 9, while the second has sides of 28 and 18. The problem requires setting up a proportion to check for similarity: 15 / 9 = 28 / 18. After simplifying, the ratios are not equal (5 / 3 ≠ 14 / 9), concluding that the triangles are not similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 31	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 31 Topic Ratios, Proportions, and Percents Description This example demonstrates how to determine if two right triangles are similar using proportions. Two right triangles are shown, one with legs of length 4 and 3, and the other with legs of length 10 and 7.5. The problem requires setting up a proportion to check for similarity: 4 / 3 = 10 / 7.5. After simplifying, both ratios are equal (4 / 3 = 4 / 3), confirming that the triangles are indeed similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 32	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 32 Topic Ratios, Proportions, and Percents Description This example illustrates how to determine if two right triangles are not similar using proportions. Two right triangles are shown, one with legs of length 12 and 5, and the other with legs of length 35 and 15. The problem requires setting up a proportion to check for similarity: 12 / 5 = 35 / 15. After simplifying, the ratios are not equal (12 / 5 ≠ 7 / 3), concluding that the triangles are not similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 33	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 33 Topic Ratios, Proportions, and Percents Description This example demonstrates how to determine if two rectangles are similar using proportions. Two rectangles are shown, with the first having dimensions 3 and 8, and the second having dimensions 9 and 24. The problem requires setting up a proportion to check for similarity: 3 / 8 = 9 / 24. After simplifying, the ratios are equal, confirming that the rectangles are indeed similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 34	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 34 Topic Ratios, Proportions, and Percents Description This example illustrates how to determine if two rectangles are similar using proportions with algebraic expressions. Two rectangles are shown, with the first having dimensions 11.5 and 23, and the second having dimensions 23x and 46x. The problem requires setting up a proportion to check for similarity: 11.5 / 23 = 23x / 46x. After simplifying, the ratios are equal, confirming that the rectangles are indeed similar for any value of x.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 35	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 35 Topic Ratios, Proportions, and Percents Description This example demonstrates how to determine if two parallelograms are similar using proportions. Two parallelograms are shown, with the first having dimensions 6 and 9, and the second having dimensions 15 and 22.5. The problem requires setting up a proportion to check for similarity: 6 / 9 = 15 / 22.5. After simplifying, the ratios are equal, confirming that the parallelograms are indeed similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 36	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 36 Topic Ratios, Proportions, and Percents Description This example demonstrates how to determine if two parallelograms are not similar using proportions. Two parallelograms are shown, with the first having dimensions 9 and 28, and the second having dimensions 18 and 54. The problem requires setting up a proportion to check for similarity: 9 / 18 ≠ 28 / 54. After simplifying, the ratios are not equal, concluding that the parallelograms are not similar.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 37	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 37 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar triangles with algebraic expressions. Two triangles are shown, one with side lengths 9 and 18, and the other with expressions 6x and 10x + 6. The problem requires setting up a proportion to determine the value of x for which the triangles are similar: 9 / 18 = 6x / (10x + 6). Solving this equation leads to x = 3.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 38	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 38 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar right triangles with algebraic expressions. Two right triangles are shown, one with side lengths 4 and 3, and the other with expressions 3x and 2x + 1. The problem requires setting up a proportion to determine the value of x for which the triangles are similar: 4 / 3 = 3x / (2x + 1). Solving this equation leads to x = 4.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 39	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 39 Topic Ratios, Proportions, and Percents Description This example illustrates solving a proportion problem using similar rectangles with an unknown side length. Two rectangles are shown, one with side lengths of 4 and x, and the other with side lengths of 15 and 4. The problem requires setting up a proportion to determine the value of x for which the rectangles are similar: 4 / 15 = x / 4. Solving this equation leads to x = 16/15.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 4	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 4 Topic Ratios, Proportions, and Percents Description This example illustrates solving for b in a proportion where a is expressed as x + 2, and c and d are given constants (c = 5, d = 2). We set up the equation (x + 2) / b = 5 / 2 and solve for b, resulting in the expression b = (2(x + 2)) / 5. This problem demonstrates how to handle algebraic expressions in proportions.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 40	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 40 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar parallelograms with algebraic expressions. Two parallelograms are shown, one with side lengths of 6 and x, and the other with expressions of 3x + 2 and 22. The problem requires setting up a proportion to determine the value of x for which the parallelograms are similar: 6 / 22 = x / (3x + 2). Solving this equation leads to x = 3.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 5	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 5 Topic Ratios, Proportions, and Percents Description This example demonstrates solving for c in a proportion where a = 9, b = 4, and d = 12. We set up the proportion 9 / 4 = c / 12 and solve for c, resulting in c = 27. This problem shows how to find an unknown value in the numerator of a proportion.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 6	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 6 Topic Ratios, Proportions, and Percents Description This example illustrates solving for c in a proportion where a = 8, b = 3, and d is expressed as x + 3. We set up the equation 8 / 3 = c / (x + 3) and solve for c, resulting in the expression c = (8(x + 3)) / 3. This problem demonstrates how to handle algebraic expressions in the denominator of a proportion.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 7	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 7 Topic Ratios, Proportions, and Percents Description This example demonstrates solving for d in a proportion where a = 12, b = 5, and c = 48. We set up the proportion 12 / 5 = 48 / d and solve for d, resulting in d = 20. This problem shows how to find an unknown value in the denominator of a proportion when all other values are known.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 8	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 8 Topic Ratios, Proportions, and Percents Description This example illustrates solving for d in a proportion where a = 11, b = 5, and c is expressed as x - 4. We set up the equation 11 / 5 = (x - 4) / d and solve for d, resulting in the expression d = (5(x - 4)) / 11. This problem demonstrates how to handle algebraic expressions in the numerator of a proportion.	Proportions
Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 9	Math Example--Ratios, Proportions, and Percents--Solving Proportions: Example 9 Topic Ratios, Proportions, and Percents Description This example demonstrates solving a proportion problem using similar triangles. Two triangles are shown, with one having sides of 3 and 4, and the other having sides of 6 and x. The problem requires finding the length of side x by setting up a proportion based on the similar triangles: 3 / 4 = 6 / x. Solving this equation leads to x = 8.	Proportions
MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines	MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines This Teacher's Guide provides an overview of the 32 worked-out examples that show how to find the equation of a line parallel or perpendicular to a given line and through a given point. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of Linear Functions and Graphs of Linear Functions
MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines	MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines This Teacher's Guide provides an overview of the 32 worked-out examples that show how to find the equation of a line parallel or perpendicular to a given line and through a given point. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of Linear Functions and Graphs of Linear Functions
MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines	MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines This Teacher's Guide provides an overview of the 32 worked-out examples that show how to find the equation of a line parallel or perpendicular to a given line and through a given point. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of Linear Functions and Graphs of Linear Functions
MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines	MATH EXAMPLES--Teacher's Guide: Equations of Parallel and Perpendicular Lines This Teacher's Guide provides an overview of the 32 worked-out examples that show how to find the equation of a line parallel or perpendicular to a given line and through a given point. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of Linear Functions and Graphs of Linear Functions
MATH EXAMPLES--Teacher's Guide: Finding the Equation of a Line, Given Two Points	MATH EXAMPLES--Teacher's Guide: Finding the Equation of a Line, Given Two Points This Teacher's Guide provides an overview of the 12 worked-out examples that show how to find the equation of a line, given two points. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of Linear Functions and Graphs of Linear Functions
MATH EXAMPLES--Teacher's Guide: Finding the Equation of a Line, Given Two Points	MATH EXAMPLES--Teacher's Guide: Finding the Equation of a Line, Given Two Points This Teacher's Guide provides an overview of the 12 worked-out examples that show how to find the equation of a line, given two points. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Applications of Linear Functions and Graphs of Linear Functions

Illustrative Math Alignment: Grade 7 Unit 2

Introducing Proportional Relationships

Lesson 12: Using Graphs to Compare Relationships

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