Common Core. Mathematics Instruction

Transcription

1 014 Common Core Mathematics Instruction 7

2 Part 1: Introduction rea of Composed Figures Develop Skills and Strategies CCSS 7.G.B.6 In previous grades you learned that you can find the area of many different shapes by breaking the shapes up into rectangles and triangles. Take a look at this problem. Maia wants to cover her garden shown below with plastic mulch to warm the soil and prevent weeds. What is the area of garden? 0 cm B 0 cm 40 cm 50 cm C E D F 0 cm H 70 cm G Explore It Use the math you already know to solve the problem. Draw two vertical lines to break the figure up into 3 rectangles. The rectangle at the left is 50 cm by 0 cm. What is its area? One of the dimensions of the rectangle in the middle is 40 cm. What is the other dimension? Explain how you figured it out. What is the area of the middle rectangle? What is the area of the rectangle at the right? Explain how you figured it out. How could you find the area of the figure? 186

3 Part 1: Introduction Find Out More composite figure is a figure made up of two or more simple geometric figures such as rectangles, squares, and triangles. To find the area of a composite figure, you can separate it into simpler shapes whose area you can find. Then add the areas together. You just need to be sure that none of the simpler shapes overlap. There is often more than one way separate the figure into simpler shapes. Here is another way to break the figure from the previous page into three rectangles by drawing horizontal lines. 0 cm 0 cm 40 cm 50 cm 70 cm 0 cm Top Rectangle Middle Rectangle Bottom Rectangle 5 lw 5 (0 cm)(0 cm) sq cm 5 lw 5 (60 cm)(10 cm) sq cm dding together the areas of these three rectangles gives the same result. 5 lw 5 (70 cm)(0 cm) 5 1,400 sq cm area of the top rectangle 1 area of the middle rectangle 1 area of the bottom rectangle 400 sq cm sq cm 1 1,400 sq cm The area of the composite shape is,400 sq cm. Reflect 1 How does separating the composite figure into rectangles help you calculate the area? 187

4 Part : Modeled Instruction Read the problem below. Then explore how to calculate the area of a composite figure. Mr. Gonzalez needs to repaint one side of his house. To decide how much paint to buy, he needs to find its area. Find the area of the figure pictured below. 15 ft ft 1 ft 15 ft 4 ft Picture It You can separate the figure into simpler figures and describe those figures. 15 ft ft 1 ft C B D 15 ft 4 ft Figures and B are right triangles. Figures C and D are rectangles. Solve It You can use formulas to find the areas of the simpler figures. You can find the areas of the right triangles. Use the formula 5 1 bh. You can find the areas of the rectangles. Use the formula 5 lw. 188

5 Part : Guided Instruction Connect It Now you will use the simpler shapes to calculate the area. What are the dimensions of Triangle? Explain. What is the area of Triangle? Show your work. 3 What are the dimensions of Triangle B? Explain. What is the area of Triangle B? Show your work. 4 What are the areas of Rectangles C and D? Explain. 5 How can you find the area of the side of the house? What is that area? Show your work. 6 Explain how to find the area of a composite figure. Try It pply what you just learned about finding areas to the following problem. 7 Describe how you could find the area of the figure below. Then find its area. 6 in. 4 in. in. 4 in. 189

6 Part 3: Modeled Instruction Read the problem below. Then use what you learned about calculating area to solve the problem. The figure below shows a plan for a new town park. Park management is going to plant grass in the shaded area, at a total cost of $.50 per square foot. How much will this project cost? 80 ft 90 ft 70 ft 10 ft Picture It To find the cost, you need to know the area to be covered with grass. To find this area, you can separate the shaded figure into rectangles and triangles. 80 ft There are four right triangles that are all the same size, labeled. 90 ft C B C 70 ft There is one larger rectangle, labeled B. There are two smaller rectangles that are the same size, labeled C. 10 ft Solve It You can find the area of each part using the formulas for areas of triangles and rectangles. rea 5 1 (10)(0) sq ft rea B 5 (80)(90) 5 7,00 sq ft rea C 5 (70)(0) 5 1,400 sq ft 190

7 Part 3: Guided Instruction Connect It Now you will find the area of the shaded region and the cost of the project. 8 Look at Picture It. How can you figure out the base and height of the triangles? 9 How can you figure out the length and width of the smaller rectangles? 10 Explain how to find the area of the shaded region. What is that area? Show your work. 11 What is the total cost of the project? Show your work. You can also find area by subtracting. Think about the shaded area as the area of the outer rectangle minus the areas of the four triangles at the corners. 1 Explain how you could find the area of the shaded region by subtracting. Try It Use what you just learned to find the area of the figure below. 13 Find the area of the hexagon shown below. Find the area in two different ways. 5 cm 8 cm 1 cm 191

8 Part 4: Guided Practice Study the model below. Then solve problems Student Model The student found the area of the figure by subtracting. What is the area of this figure? 5 Look at how you could solve this problem by seeing the figure as a rectangle, square, and triangle. Subtract the area of the square and the right triangle from the 4 Pair/Share What is another way to solve this problem? area of the large rectangle. rea of the large rectangle: (4)(5) 5 0 rea of the square: ()() 5 4 rea of the triangle: 1 1 ()(3) 5 3 Solution: square units 5 4 Should I add or subtract to find the area of the figure? 14 The figure shows the floor plan of a 10 ft carpeted area in front of a stage. How much carpeting is needed to cover this area? 15 ft 15 ft 10 ft Show your work. 0 ft Pair/Share Explain how you found the area. Solution: 19

9 Part 4: Guided Practice 15 Jaime drew a snowflake on graph paper. What is its area in square units? I think I can separate this snowflake into just two different types of figures. Show your work. Solution: 16 Which of these two kites will use more paper to make? Circle your answer. Pair/Share Could you find the area by counting? The diagonals of a kite are perpendicular. 30 in. 0 in. B 30 in. 40 in. B C D kite kite B They both use the same amount of paper. There is not enough information given. Rudy chose C as the correct answer. What did he do wrong? Pair/Share Do you have enough information to calculate the area? Explain. 193

10 Part 5: Common Core Practice Solve the problems. 1 Find the area of the pentagon in the diagram below. in. in. in. 4 in. 3 in. B C D 15 sq in. 0 sq in. 5 sq in. 8 sq in. contractor needs to buy grass seed to cover a lawn. The dimensions of the lawn are shown below. One bag of grass seed covers an area of 600 square yards. Shade in the minimum number of bags of seed the contractor needs to buy to cover the entire lawn. 90 yd Lawn 8 yd 18 yd 18 yd Porch 14 yd 14 yd [not drawn to scale] 10 yd Bags of grass seed 194

11 Part 5: Common Core Practice 3 The figure shown on the right is composed of two squares. The smaller square has a side length of x. The side of the larger square measures twice that of the smaller square. Which expression accurately represents the area of the entire figure? Select all that apply. x cm x 1 x 1 x 1 x 1 x 1 x 1 x B x 1 x C (x)(3x) x D x(3x) 1 x(x) E (x )(x) 4 The flag of Seychelles is shown below. The dimensions of the flag are 3 feet by feet. The lines drawn from the bottom left corner to the top and right sides of the flag divide those sides into 3 equal parts. What fraction of the flag is painted red? blue yellow red white green nswer of the flag is painted red. Self Check Go back and see what you can check off on the Self Check on page

12 Common Core Mathematics Teacher Resource Book 7

13 Develop Skills and Strategies (Student Book pages ) rea of Composed Figures Lesson Objectives Find the areas of two-dimensional objects composed of triangles, quadrilaterals, and polygons. pply formulas to solve real-world and mathematical problems. Prerequisite Skills Compose and decompose polygons. Use formulas to find the area of triangles and rectangles. Vocabulary There is no new vocabulary. The Learning Progression In Grade 6, students found the area of right triangles, other triangles, special quadrilaterals, and polygons by composing figures into quadrilaterals or decomposing figures into triangles and other shapes. In this grade, students build on their understanding of composing and decomposing figures, using it to find the area of complex figures. In Grade 8, composing and decomposing figures continues to be important. For example, students decompose a square in two ways to explain why the Pythagorean theorem is correct. Students will also continue to reason about the relationships between two- and three-dimensional shapes. Teacher Toolbox Teacher-Toolbox.com Ready Lessons Tools for Instruction Interactive Tutorials Prerequisite Skills 7.G.B.6 CCSS Focus 7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. standrds FOR MatheMaticaL Practice: SMP 1 8 (see page 9 for full text) 00

14 t a glance Part 1: Introduction Students use decomposition to find the area of a complex figure. Part 1: introduction area of composed Figures Develop skills and strategies ccss 7.g.b.6 Step By Step Tell students that this page models finding the area of a complex figure by dividing it into smaller figures. Have students read the problem at the top of the page. Work through Explore It as a class. For the first question, sketch the figure on the board. Have volunteers extend BC and DE to HG. Label the points where they intersect as I and J, respectively. Read the next question. sk, What do you need to know before finding the area of the left figure? [the length of BI ] What is the length of BI? [50 cm; it is the same length as H.] Have students work in pairs or small groups to answer the questions about the areas of the middle and right rectangles. Have students write the dimensions and area of each rectangle using the form: l 5, w 5, 5. [l 5 50 cm, w 5 0 cm, 5 1,000 cm ; l 5 30 cm, w 5 40 cm, 5 1,00 cm ; l 5 0 cm, w 5 10 cm, 5 00 cm ] Guide students to see that the area of the complex figure is the sum of the areas of the three rectangles. SMP Tip: Demonstrate how students can use the structure of the decomposed figure to find the two missing values (SMP 7). For example, tell students you will find the value of EF. If HG 5 70 cm, then B 1 CD 1 EF 5 70 cm. Therefore, EF cm. Use the same reasoning to find the length of DE. ELL Support Explain that one meaning of the word complex is consisting of many connected parts. In geometry, complex figures can be broken down into parts. 186 in previous grades you learned that you can find the area of many different shapes by breaking the shapes up into rectangles and triangles. take a look at this problem. Maia wants to cover her garden shown below with plastic mulch to warm the soil and prevent weeds. What is the area of garden? explore it 0 cm B 0 cm 40 cm 50 cm C Mathematical Discourse H 70 cm F 0 cm use the math you already know to solve the problem. Draw two vertical lines to break the figure up into 3 rectangles. The rectangle at the left is 50 cm by 0 cm. What is its area? 1,000 sq cm One of the dimensions of the rectangle in the middle is 40 cm. What is the other dimension? Explain how you figured it out. 30 cm; it s 0 cm shorter than the tallest rectangle. What is the area of the middle rectangle? 1,00 sq cm What is the area of the rectangle at the right? Explain how you figured it out sq cm; one dimension is 0 cm and the other is 70 cm 0 cm 40 cm. How could you find the area of the figure? add the areas of the three rectangles. Why are you not able to find the area of the complex figure with one simple formula? Responses may include the idea that there is more than one rectangle within this figure, or that there is no formula for a figure like this one. Do you see another way to break the figure into rectangles? How would that help us find the area? Responses should include drawing a horizontal line from C to H and from F to H. Would it be helpful to decompose this figure into triangles or a combination of rectangles and triangles? Explain. How you decompose a complex figure depends on the information you re given and the question you re trying to answer. There is usually more than one way to decompose a complex figure although there may be a most efficient way to do so. E D G 01

15 t a glance Part 1: Introduction Students find the area of the complex figure on page 186 by decomposing it in a different way than demonstrated previously. Step By Step Read Find Out More as a class. Have students examine the figure and discuss how it is broken apart (decomposed). s a class, review how the dimensions of the three rectangles were determined. Emphasize that even though the figure was decomposed differently, the total area remains the same. Discuss which method students prefer and why. s a class, discuss the Reflect question. ELL Support Explain that the term decompose means the opposite of compose. If you compose something, you put the parts of something together. If you decompose something, you pull it into parts. Concept Extension Use subtraction to find area. sk, Is it possible to use subtraction to find the area of the complex figure? Display for students the complex figure. Using a dashed line, extend sides B and GF of the figure to meet at point K. sk, What is the area of rectangle HGK? [70 cm 3 50 cm 5 3,500 cm ] Say, Decompose shape BCDEFK into two figures. Find their areas. [rectangle: ,000 cm ; square: cm ] sk, What is the sum of their areas? [1,100 cm ] Show students that by subtracting the areas of the new shapes (1,100 cm ) from 3,500 cm, you have found the area of the complex figure. Part 1: introduction Find out More Real-World Connection composite figure is a figure made up of two or more simple geometric figures such as rectangles, squares, and triangles. To find the area of a composite figure, you can separate it into simpler shapes whose area you can find. Then add the areas together. You just need to be sure that none of the simpler shapes overlap. There is often more than one way separate the figure into simpler shapes. Here is another way to break the figure from the previous page into three rectangles by drawing horizontal lines. 0 cm 0 cm 40 cm 50 cm 70 cm 0 cm top rectangle Middle rectangle bottom rectangle 5 lw 5 (0 cm)(0 cm) sq cm 5 lw 5 (60 cm)(10 cm) sq cm 5 lw 5 (70 cm)(0 cm) 5 1,400 sq cm dding together the areas of these three rectangles gives the same result. area of the top rectangle 1 area of the middle rectangle 1 area of the bottom rectangle 400 sq cm sq cm 1 1,400 sq cm The area of the composite shape is,400 sq cm. reflect 1 How does separating the composite figure into rectangles help you calculate the area? Possible answer: because you can figure out the dimensions of these rectangles, and use formulas to find the areas of the rectangles. Discuss with students everyday situations in which people might need to decompose figures. Examples: Realtors and people in the flooring industry often need to calculate the square footage of rooms that are complex shapes

16 t a glance Part : Modeled Instruction Students find the area of a complex figure by decomposing it into rectangles and right triangles. Step By Step sk students to cover the decomposed figure in Picture It. Then, read the problem at the top of the page as a class. Direct students to look at the dashed lines that are not a part of the complex figure. sk, How can this information help you find the area of the complex figure? [You can use this information to find the dimensions of the smaller figures that make up the complex figure.] Say, Start thinking about how you might decompose this figure. Review Picture It. sk, Is the area of the triangle formed by the dashed line and dimension labeled 15 ft equal to the area of part of the complex figure? Explain. [Yes. It is equal to the area of triangle. They have the same base and height.] For Solve It, discuss how to find the measures of the smaller figures. sk, How can you find the area of the two right triangles? [Use the given measures and subtract to find the base and/or height of each triangle. Then use the formula 5 1 bh.] ELL Support Remind students that the dimensions of a rectangle are called either length and width or base and height. triangle also has a base and a height. 188 Part : Modeled instruction read the problem below. then explore how to calculate the area of a composite figure. Mr. Gonzalez needs to repaint one side of his house. To decide how much paint to buy, he needs to find its area. Find the area of the figure pictured below. Picture it ft 1 ft Mathematical Discourse Could you find the area of the complex figure by simply dividing it using one line to form a 1 ft by 4 ft rectangle? Explain. No. You still wouldn t know the area of the upper part of the figure. 15 ft What if you then decomposed the upper part of the figure into right triangle and one other figure? You could find the area of right triangle but not the figure to its right. When decomposing a complex figure to find its area, what do you need to ask yourself? Guide students to understand that they should always question whether they have enough information to find the area of each of the smaller figures. 4 ft 15 ft you can separate the figure into simpler figures and describe those figures. ft Figures and B are right triangles. Figures C and D are rectangles. solve it 1 ft 15 ft C 4 ft B D 15 ft you can use formulas to find the areas of the simpler figures. You can find the areas of the right triangles. Use the formula 5 1 bh. You can find the areas of the rectangles. Use the formula 5 lw. 03

17 t a glance Part : Guided Instruction Students find the area of the figure on page 188. Then they apply what they have learned to find the area of another complex figure. Step By Step Problems 5 walk students through the steps of finding the area of the composite figure on page 188. For problems and 3, review how to find each of the dimensions, if necessary. Lead students to see that each base and each height is either given or is found by subtracting a given measure from another. Have a volunteer share a response to problem 6. You may want to encourage them to use one method (addition or subtraction) to find the area and the other method to confirm their answer. Part : guided instruction connect it now you will use the simpler shapes to calculate the area. What are the dimensions of Triangle? Explain. the base is given; 15 ft. the height is ft 1 ft 5 10 ft. 5 1 (15)(10) 5 75 sq ft What is the area of Triangle? Show your work. 3 What are the dimensions of Triangle B? Explain. the base is 4 ft 15 ft 5 9 ft. the height is ft 15 ft 5 7 ft. 5 1 What is the area of Triangle B? Show your work. (9)(7) 5 63 or 31 1 sq ft 4 What are the areas of Rectangles C and D? Explain. rectangle c is 15 ft by 1 ft; area of c 5 (15)(1) sq ft rectangle D is 15 ft long; 4 ft 15 ft 5 9 ft wide; area of D 5 (9)(15) sq ft 5 How can you find the area of the side of the house? add the areas of the simpler figures that make it up. What is that area? Show your work sq ft 6 Explain how to find the area of a composite figure. separate the figure into simpler figures. Find the areas of the simpler figures, then add them together. try it apply what you just learned about finding areas to the following problem. 7 Describe how you could find the area of the figure below. Then find its area. separate the figure into a triangle 6 in. SMP Tip: The Try It problem provides an opportunity for students to persevere in solving a problem (SMP 1) and to reason abstractly (SMP ). They will need to use the given measures to establish the length and width of the rectangle and the base and height of the triangle. They will then need to decide how to use those measures to find the area of the figure. and a rectangle; 1 sq in. Try it Solution 4 in. 4 in. in Solution: 1 sq in.; Rectangular part:? sq in. Triangular part: 1? 4? = 4 sq in. Therefore, sq in. ERROR LERT: Students who use the subtraction method must be certain to subtract the area of two small triangles and two small rectangles. 04

18 t a glance Part 3: Modeled Instruction Students calculate the cost of planting grass in a park in the shape of an irregular octagon by decomposing the octagon into rectangles and right triangles. Step By Step sk students to cover the decomposed figure in Picture It. Then, read the problem at the top of the page as a class. Discuss some possible ways to decompose the figure. Now direct students attention to the figure in Picture It. sk, Why do you think some of the figures have the same letter? [They have the same area.] Discuss how to use the given information to find the measures needed to calculate the area of each lettered figure. Triangle : base 5 (90 70) ft height 5 (10 80) ft Rectangle B: length 5 90 ft (given) width 5 80 ft (given) Rectangle C: length 5 70 ft (given) width 5 (10 80) ft Review the Solve It to see how these measures are applied to find the area. 190 Part 3: Modeled instruction read the problem below. then use what you learned about calculating area to solve the problem. The figure below shows a plan for a new town park. Park management is going to plant grass in the shaded area, at a total cost of $.50 per square foot. How much will this project cost? Picture it 80 ft 90 ft 70 ft 10 ft Mathematical Discourse to find the cost, you need to know the area to be covered with grass. to find this area, you can separate the shaded figure into rectangles and triangles. 80 ft 90 ft C B C 70 ft solve it 10 ft There are four right triangles that are all the same size, labeled. There is one larger rectangle, labeled B. There are two smaller rectangles that are the same size, labeled C. you can find the area of each part using the formulas for areas of triangles and rectangles. rea 5 1 (10)(0) sq ft rea B 5 (80)(90) 5 7,00 sq ft rea C 5 (70)(0) 5 1,400 sq ft The opposite sides of the figure appear to be parallel. What does this suggest about the figures? It suggests that I can draw a line that intersects opposite sides of the figure at right angles. How might drawing such a line help you find the area of this complex figure (or any complex figure)? It helps me decompose the figure into right triangles and rectangles. If one or more pairs of opposite sides were not parallel, could you find the area of the figures with the given information? Explain. Maybe. It would depend on which sides are not parallel. 05

19 t a glance Part 3: Guided Instruction Students find the area of the figure on page 190 and calculate the cost of the project described. Step By Step Students calculate the area of the complex figure on page 190. For Connect It, problem 10, guide students to see that they can think of the grassy area as an octagon within a rectangle. For problem 1, ask, What is the area of the large rectangle that contains the town park? [rea 5 10? ,800 sq ft] Direct students attention to the four unlabeled triangles within the large rectangle. sk, Can you find the area of these four triangles? Explain. [Yes. Their area is equal to the area of triangle because a diagonal line divides a rectangle into two equal triangles.] For Try It, tell students to assume that the figure shown has parallel opposite sides and also line symmetry. You may want to encourage students to use one method (addition or subtraction) to find the area and the other method to confirm their answer. SMP Tip: You may want to ask students to justify their reasoning at each step of their work and have other students respond if they do not agree. This provides an opportunity for them to practice constructing viable arguments and critiquing the reasoning of others (SMP 3). Part 3: guided instruction connect it Try it Solution 7 Solution: 68 sq cm; in this case, there is no advantage of one method over the other. now you will find the area of the shaded region and the cost of the project. 8 Look at Picture It. How can you figure out the base and height of the triangles? i can find the base by subtracting 80 ft from 10 ft and taking half to get 0 ft. i can find the height by subtracting 70 ft from 90 ft and taking half to get 10 ft. 9 How can you figure out the length and width of the smaller rectangles? one side is given as 70 ft. the other side is the same length as the base of the triangle, 0 ft. 10 Explain how to find the area of the shaded region. add the areas of the four triangles and the three rectangles. What is that area? Show your work. 4(100) 1 7,00 1 (1,400) 5 10,400 sq ft 11 What is the total cost of the project? Show your work. (10,400)($.50) 5 $6,000 you can also find area by subtracting. think about the shaded area as the area of the outer rectangle minus the areas of the four triangles at the corners. 1 Explain how you could find the area of the shaded region by subtracting. the rectangle is (10)(90) 5 10,800 sq ft. one triangle is (10)(0) sq ft. 1 the shaded area is 10,800 4(100) 5 10,400 sq ft. try it use what you just learned to find the area of the figure below. 13 Find the area of the hexagon shown below. 5 cm Find the area in two different ways. 68 sq cm 8 cm 1 cm Subtraction method: rea of large rectangle: 1? sq cm rea of four smaller triangles that are not part of the figure: 1? 4? 3.5? sq cm rea of rectangle area of triangles: sq cm ddition method: rea of center rectangle on figure: 5? sq cm rea of four right triangles: 1? 4? 3.5? sq cm rea of rectangle 1 area of triangles: sq cm ERROR LERT: Watch for students who use 7 cm as the height of the triangles. For the subtraction method, watch for students who calculate the area of the large rectangle as 5? sq cm instead of 1? sq cm.

20 Part 4: Guided Practice Part 4: guided Practice Part 4: guided Practice The student found the area of the figure by subtracting. study the model below. then solve problems What is the area of this figure? 5 Student Model 15 Jaime drew a snowflake on graph paper. What is its area in square units? Show your work. Possible student work: B B B B I think I can separate this snowflake into just two different types of figures. Look at how you could solve this problem by seeing the figure as a rectangle, square, and triangle. subtract the area of the square and the right triangle from the 4 area of square: (4)(4) 5 16 area of 4 triangles: 41 1 ()(1) square units Solution: Pair/share Could you find the area by counting? Pair/share What is another way to solve this problem? area of the large rectangle. area of the large rectangle: (4)(5) 5 0 area of the square: ()() 5 4 area of the triangle: 1 1 ()(3) 5 3 Solution: square units Which of these two kites will use more paper to make? Circle your answer. 30 in. 30 in. 0 in. B 40 in. The diagonals of a kite are perpendicular. Should I add or subtract to find the area of the figure? 14 The figure shows the floor plan of a carpeted area in front of a stage. How much 10 ft carpeting is needed to cover this area? Show your work. 15 ft 10 ft 15 ft a b c kite kite B They both use the same amount of paper. Pair/share Explain how you found the area. carpeted area 5 area of large rectangle area of smaller rectangle area of triangles area of rectangle sq ft area of triangle sq ft Solution: total area sq ft 0 ft D There is not enough information given. Possible student work: area of kite a: 1 1 (30)(15) sq in. area of kite b: 1 1 (40)(10) (40)(10) sq in. Rudy chose c as the correct answer. What did he do wrong? he simply added together the dimensions of each kite. Pair/share Do you have enough information to calculate the area? Explain t a glance Students practice solving problems that involve finding the area of complex figures. Step By Step sk students to solve the problems individually. Circulate and monitor students work, asking clarifying questions as needed. When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. Solutions Ex Use a 4 by 5 rectangle and subtraction method. 14 Solution: 5 sq ft; Students may add the areas of the rectangle ( ) and the two triangles ( ): sq ft. (DOK 3) 15 Solution: 0 sq units; Students may add the areas of the center square ( ) and the four triangles: 1 ( ); sq units. (DOK ) 16 Solution: ; Rudy s mistake was that he added the dimensions of each kite. Explain to students why the other two answer choices are not correct. B is not correct because kite has an area of 450 sq in., and kite B has an area of 400 sq. in. D is not correct because there is enough information given to find the areas of both kites. (DOK 3) 07

21 Part 5: Common Core Practice Part 5: common core Practice Part 5: common core Practice Solve the problems. 1 Find the area of the pentagon in the diagram below. in. in. in. 4 in. 3 in. 3 The figure shown on the right is composed of two squares. The smaller square has a side length of x. The side of the larger square measures twice that of the smaller square. Which expression accurately represents the area of the entire figure? Select all that apply. x 1 x 1 x 1 x 1 x 1 x 1 x B x 1 x C (x)(3x) x D x(3x) 1 x(x) E (x )(x) x cm B C D 15 sq in. 0 sq in. 5 sq in. 8 sq in. 4 The flag of Seychelles is shown below. The dimensions of the flag are 3 feet by feet. The lines drawn from the bottom left corner to the top and right sides of the flag divide those sides into 3 equal parts. What fraction of the flag is painted red? blue yellow red contractor needs to buy grass seed to cover a lawn. The dimensions of the lawn are shown below. One bag of grass seed covers an area of 600 square yards. Shade in the minimum number of bags of seed the contractor needs to buy to cover the entire lawn. white 90 yd Lawn 8 yd 18 yd 18 yd Porch 10 yd 14 yd 14 yd [not drawn to scale] Bags of grass seed green Possible student work: the area of the flag is 3() 5 6 sq ft. the area of the triangle that is painted blue and yellow ()() 5 sq ft. the area of the triangle that is painted white and green (3) 5 sq ft. the area of the triangle that is painted red sq ft. nswer 6 or 1 3 of the flag is painted red. self check Go back and see what you can check off on the Self Check on page t a glance Students solve problems involving complex figures that might appear on a mathematics test. Solutions 1 Solution: C; Decompose the regular trapezoid into two congruent right triangles and a rectangle. rea of right triangles 5 1 ()(4) 5 4; area of rectangle 5 (4) 5 8. rea of triangle at bottom: 5 1 (6)(3) 5 9; Total area: (4) sq in. (DOK ) Solution: Students shade 4 bags; Find the area of the lawn (1940 sq. yd) and divide by 600 to find the number of bags needed. Since you can t buy a partial bag, round up to the next whole number. (DOK ) 3 Solution: C; The area of the smaller square is x. The area of the larger square is (x), or 4x. The total area is 5x. The given expression simplifies to 5x. D; The total area is 5x. The given expression simplifies to 5x. (DOK ) 4 Solution: 6 or 1 ; see possible student work above. 3 (DOK ) 08

22 Differentiated Instruction ssessment and Remediation sk students to find the area of the complex figure at the right. [3 sq units] For students who are struggling, use the chart below to guide remediation. fter providing remediation, check students understanding. sk students to change the horizontal length of the entire figure to 16 units but keep all other measures constant. Then find the area of the figure. [40 sq units] If a student is still having difficulty, use Ready Instruction, Level 6, Lesson If the error is... Students may... To remediate... 4 square units have divided the figure horizontally and incorrectly used the area of a triangle formula to find each area. 36 square units have used subtraction, but only subtracted the triangles on the left. 48 square units not have used (bh) as 1 the formula for the area of the triangles. Examine the shape that results from dividing the figure horizontally. Guide them to see that it is not a triangle and that they would have to decompose it into a rectangle and two triangles. Review how to find the area of a complex figure using the subtraction method. Because they can do this to the whole figure, this method adds just one extra step. Review how to find the area of a triangle. Hands-On ctivity Draw and critique kites. Challenge ctivity Write a problem involving a composite figure. Materials: graph paper, rulers Have students look back at page 193, problem 16. Have student pairs redraw the kites on graph paper. Tell students that 1 square unit on the graph paper should correspond to 4 square inches. Then have pairs compare and critique each other s drawings. sk students which drawing they think is better for the problem: the drawing in the book or the drawings they made. sk them to explain their thinking. sk pairs of students to write a problem that involves a complex figure. Have them include the following: a precise sketch of the figure before and after decomposing it. a question that can only be answered after the area is found. (Refer students to the problem where they found the cost of planting grass.) a complete solution to the problem. Have pairs exchange problems with another pair and solve each other s problems. 09