Covering & Surrounding

Transcription

1 Covering & Surrounding Two-Dimensional Measurement and Three-Dimensional Measurement Name: Hour:

2 Table of Contents Investigation 1 Investigation 1.1 page 3 Investigation 1.2 page 7 Investigation 1.3 page 11 Investigation 1.4 page 15 Investigation 1 Reflection.page 17 Investigation 2 Investigation 2.1 page 19 Investigation 2.2 page 24 Investigation 2.3 page 27 Investigation 2.4 page 31 Investigation 2 Reflection.page 34 Investigation 3 Investigation 3.1 page 36 Investigation 3.2 page 41 Investigation 3.3 page 44 Investigation 3.4 page 48 Investigation 3 Reflection.page 55 Investigation 4 Investigation 4.1 page 57 Investigation 4.2 page 62 Investigation 4.3 page 68 Investigation 4 Reflection.page 72 2

3 Investigation 1.1 When a customer places an order, the designers at MARS use square tiles to model possible floor plans. MARS receives the customer orders below. Experiment with square tiles and then sketch some designs on grid paper for the customer to consider. A. 1. Lone Star Carnivals in Texas wants a bumper-car ride that covers 36 square meters of floor space and has lots of rail sections. Sketch two or three possible floor plans. 3

4 2. Badge State Shows in Wisconsin requests a bumper-car ride with 36 square meters of floor space and 26 meters of rail sections. Sketch two or three floor plans for this request. B. The designers at MARS created four designs for bumper-car rides. 1. Find the area and perimeter of each bumper-car floor plan. Record your data in a table such as the one shown. You will use the Cost column of the table later in part 3. Bumper-Car Floor Plans Design Area Perimeter Cost A B 2. Which of the designs can be made from the same number of floor tiles? Will those designs have the same number of rail sections? Explain. 4

5 3. The designers at MARS charge $25 for each rail section and $30 for each floor tile. For the designs with the same floor area, which design costs the most? Which design costs the least? Explain. 4. Rearrange the tiles in Design B to form a rectangle. Can you make more than one rectangle? If so, are the perimeters the same? Explain. C. Riverview School orders a bumper-car ride in the shape of a rectangle for their fundraising festival. The MARS Company sends the school Designs I, II, and III. 1. What is the area of each design? Explain how you found the area. 5

6 2. What is the perimeter of each design? Explain how you found the perimeter. 1.1 Summary Focus Question: What are the formulas for finding the area and perimeter of a rectangle? Explain why they work. The dimensions of a rectangle are called length l and width w. Look for patterns throughout Problem 1.1 to help you answer the questions below. a. Use words to describe a formula for finding the perimeter of a rectangle. Write the formula using symbols. Explain why it works. b. Use words to describe a formula for finding the area of a rectangle. Write the formula using symbols. Explain why it works. c. Find the perimeter and area of a rectangle with a width of 6 centimeters and a length of 15 centimeters. 6

7 Investigation 1.2 The rangers in a national park want to build several storm shelters. The shelters must have 24 square meters of rectangular floor space. A. Experiment with different rectangles that have whole-number dimensions. Sketch each possible floor plan on grid paper. Record your data in a table such as the one started below. Look for patterns, and describe the data. Shelter Floor Plans Rectangle Length Width Perimeter Area 1 m x 24 m 1 m 24 m 50 m 24 m 2 24 m 2 24 m 2 24 m 2 24 m 2 7

8 B. Suppose the walls are made of flat rectangular panels that are 1 meter wide and have the needed height. 1. What determines how many wall panels are needed, area or perimeter? Explain your reasoning. 8

9 2. Which design would require the most panels? Explain. 3. Which design would require the fewest panels? Explain. C. 1. Use your table to make a graph, such as the one below, to compare length and perimeters of various rectangles with an area of 24 square meters. 9

10 2. Describe the shape of the graph. How do the patterns that you saw in your table show up in the graph? D. 1. Suppose you build a storm shelter with 36 square meters of rectangular floor space. Which design has the least perimeter? Which has the greatest perimeter? Explain your reasoning. 1.2 Summary Focus Question: For a fixed area, what are the shape and perimeter of the rectangles with the greatest and least perimeters? In general, describe the rectangle that has the greatest perimeter for a fixed, or unchanging, area. Describe the rectangle that has the least perimeter for a fixed area. 10

11 Investigation 1.3 Americans have over 78 million dogs as pets. In many parts of the country, particularly in cities, there are laws against letting dogs run free. Many people build pens so their dogs can get outside for fresh air and exercise. Suppose you have 24 meters of fencing to build a rectangular pen for a dog. A. 1. Experiment with different rectangles that have whole-number dimensions. Sketch each rectangle on grid paper. Record your data in a table such as the one started on the following page. Look for patterns, and describe the data. 11

12 Dog Pen Floor Plans Rectangle Length Width Perimeter Area 1 m x 11 m 1 m 11 m 4 m 11 m 2 4 m 4 m 4 m 4 m 12

13 2. Which rectangle has the least area? Which rectangle has the greatest area? 3. Which design would you choose to build a pen for your dog? Explain your reasoning. B. 1. Use your table to make a graph, such as the one below, to compare the lengths and areas of various rectangles with a perimeter of 24 meters. 13

14 2. Describe the shape of the graph. How do the patterns that you saw in your table show up in the graph? 3. How is this graph similar to the graph you made in Problem 1.2? How is it different? C. 1. Suppose you have 36 meters of fencing to surround a rectangular pen. Which rectangle has the least area? Which rectangle has the greatest area? Explain your reasoning. 1.3 Summary Focus Question: For a fixed perimeter, what are the shape and area of the rectangles with the greatest and least area? In general, describe the rectangle that has the least area for a fixed perimeter. Describe the rectangle that has the greatest area for a fixed perimeter. 14

15 Investigation 1.4 Brevort Township wants to find the area and perimeter of two of its most popular lakes by using a transparent grid. 75. a. Estimate the area of Loon lake. b. Estimate the area of Ghost Lake. 76. a. How could you find the perimeter of each lake? b. Estimate the perimeter of each lake. 15

16 77. Use your estimates to answer the following questions. Explain your reasoning. a. Naturalists claim that water birds need long shorelines for nesting and fishing. Which lake better supports water birds? b. Sailboaters and waterskiers want a lake with room to cruise. Which lake works better for boating and skiing? c. Which lake can better handle swimming, boating, and fishing all at the same time? d. Which lake has more space for lakeside campsites? 16

17 CS Investigation 1 Mathematical Reflections In this Investigation, you revisited strategies for finding the area and perimeter of a rectangle. You examined the areas and perimeters of figures made from square tiles. You also found that some arrangements of tiles have the same area but different perimeters. Other arrangements of the tiles have the same perimeter but different area. The following questions will help you summarize what you have learned. 1. Explain what area and perimeter of a figure means. 2. Describe a strategy for finding the area and perimeter of any twodimensional shape. 3. Describe how you can find the area of a rectangle. Explain why this method works. 17

18 4. Describe how you can find the perimeter of a rectangle. Explain why this method works. 5. Consider all the rectangles with the same area. Describe the rectangle with the least perimeter. Describe the rectangle with the greatest perimeter. 6. Consider all the rectangles with the same perimeter. Describe the rectangle with the least area. Describe the rectangle with the greatest area. 7. Explain how graphing relationships between length and perimeter or length and area helps explain patterns between area and perimeter. 18

19 Investigation 2.1 In Investigation 1, you studied rectangles and other figures that are examples of polygons. Polygon: Vertex: Draw one diagonal in the square to form two triangles. What is the area of each triangle? Is the perimeter of each of the triangles greater than, less than, or equal to 3 centimeters? Explain your thinking. Right angle: Perpendicular Lines: Base: Height: 19

20 For this Problem several triangles are drawn on grid paper. As you find the area of each triangle in this Problem, think about the patterns you observe that will help you write a formula for finding the area of any triangle. A. On page 22, six triangles labeled A-F are drawn on a centimeter grid. 1. Find the perimeter of each triangle. Describe the strategies you use. A: B: C: D: E: F: 2. Find the area of each triangle. Describe the strategies you use. A: B: C: D: E: F: 20

21 B. Look at triangles A-F again. Using the grid lines, draw the smallest possible rectangle around each triangle. Find the area of each rectangle you drew. Record your data in the table below with the areas of the triangles from Questions A, part (2). Design A B C D E F Area of rectangle (cm 2 ) Area of triangle (cm 2 ) Compare the area of the rectangle to the area of the triangle. Describe a pattern that tells how the two are related. C. 1. Use your results from Question B to explain why your formula works. 2. Use your formula to find the area of a triangle with a base of 8 inches and a height of 3 ½ inches. 21

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23 2.1 Summary Focus Question: What is a formula for finding the area of a triangle? 23

24 Investigation

25 A. 1. For each triangle, choose one side to use as the base. Use the transparent centimeter grid paper to help you find the area of each triangle. Explain how you found the area. Include any calculations that you used. 2. Find the area of each triangle in two other ways, using each of the other two sides as the base. Record the base, height, and area in a table. 25

26 2.2 Summary Focus Question: Does is make any difference which side is used as the base when finding the area of a triangle? B. 1. Does changing the side you choose as the base of a triangle change the area? Explain. 2. When finding the area of a triangle, are there advantages or disadvantages to choosing a particular side as the base? Explain. C. 1. Maria claims that if a triangle s height falls outside the triangle, the formula for area will not work. Is Maria correct? Explain. 26

27 Investigation

28 For each triangle in Questions A, draw a segment 6 centimeters long the grid paper. Use the segment a base for the triangle. A. 1. Sketch a right triangle with a height of 4 centimeters. 2. Sketch a different right triangle with a height of 4 centimeters. 3. Sketch an isosceles triangle with a height of 4 centimeters. An isosceles triangle is a triangle with exactly two equal sides. 4. Sketch a scalene triangle with a height of 4 centimeters. A scalene triangle is a triangle with no equal sides. 5. Find the area of each triangle that you made. 28

29 B. 1. What do these four triangles have in common? 2. Why do you think these four triangles can be called a triangle family? C. 1. Use the grid paper below to make a new triangle family that has a different base and height from the triangle family you have already made. What are the base, height, and area of each triangle in your triangle family? 29

30 2.3 Summary Focus Question: What can you say is true and what can you say is not true about triangles that have the same base and height? 30

31 Investigation 2.4 Designing Triangles Under Constraints In this Problem, you will use what you know about triangles to draw triangles that met given conditions (constraints). What conditions for a triangle produce triangles that have the same area? Do these triangles have the same shape? For each description, draw two triangles that are not congruent. (Two figures are congruent if they have the same size and the same shape.) If you cannot draw more than one triangle for each description, explain why. A. The triangles both have a base of 5 centimeters and a height of 6 centimeters. Do the two triangles have the same area? 31

32 B. The triangles both have an area of 15 square centimeters. Do the two triangles have the same perimeter? C. The triangles both have sides of length 3 centimeters, 4 centimeters, and 5 centimeters. Do the two triangles have the same area? D. The triangles are right triangles and both have a 30-degree angle. Do the two triangles have the same area? Do they have the same perimeter? 32

33 2.4 Summary Focus Question: What conditions for a triangle produce triangles that have the same are? Do they have the same shape? Explain. 33

34 CS Investigation 2 Mathematical Reflections In this Investigation, you discovered strategies for finding the areas and perimeters of triangles by relating them to what you know about rectangles. The following questions will help you summarize what you have learned. 1. Describe how to find the area of a triangle. Explain why your method works. 2. Describe how to find the perimeter of a triangle. Explain why your method works. 3. Does the choice of the base affect the area of a triangle? Does the choice of the base affect the perimeter of a triangle? Explain why or why not. 4. What can you say about the area and perimeter of two triangles that have the same base and height? Give evidence to support your answer. 34

35 5. How is finding the area of a triangle related to finding the area of a rectangle? How is finding the perimeter of a triangle related to finding the perimeter of a rectangle? 6. Draw a triangle that has the same area but is not congruent. 7. Find the area of this figure. 35

36 Investigation 3.1 What do you think the base and the height of a parallelogram mean? How can you mark and measure the base and height of the third parallelogram? 36

37 The centimeter grid on the next page shows six parallelogram labeled A-F. A. Find the perimeter of each parallelogram. Describe the strategies you use. A: B: C: D: E: F: B. Find the area of each parallelogram. Describe the strategies you use. A: B: C: D: E: F: 37

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39 C. 1. For each parallelogram, record the base, height, and area in a table. Describe any patterns you see in the data. Parallelograms Triangles Design Base (cm) Height (cm) Area (cm 2 ) Base (cm) Height (cm) Area (cm 2 ) A B C D E F 2. Draw one diagonal in each parallelogram as shown below. Fill in the table above recording the base, height, and area of each triangle you make. 39

40 3.1 Summary Focus Question: What is a strategy for finding the area of a parallelogram? Explain why the strategy works. 40

41 Investigation

42 For each parallelogram in Question B, part (1), follow the steps below. Draw a segment 6 centimeters long on the grid paper below. Use the segment as a base for the parallelogram. 42

43 B. 1. Sketch four different parallelograms with a height of 4 centimeters. 2. Find the area of each parallelogram. Why do you think these four parallelograms can be called a parallelogram family? 3.2 Summary Focus Question: What can you say about parallelograms that have the same height and base? 43

44 Investigation 3.3 For each description, draw two figures that are not congruent to each other. If you cannot draw a second figure, expain why. A. Draw two rectangles that each have an area of 18 square centimeters. If you can draw two different rectangles, do they have the same perimeter? Explain. 44

45 B. Draw two rectangles that are each 3 centimeters by 8 centimeters. If you can draw two different rectangles, do they have the same area? Explain. C. Draw two parallelograms that each have a base of 7 centimeters and a height of 4 centimeters. If you can draw two different parallelograms, do they have the same area? The same perimeter? Explain. 45

46 D. Draw two parallelograms that each have four 6-cm-long sides. If you can draw two different parallelograms, do they have the same area? Explain. E. Draw two parallelograms that each have an area of 30 square centimeters. If you draw two different parallelograms, do they have the same perimeter? Explain. 46

47 3.3 Summary Focus Question: Under what constraints will two or more parallelograms have the same area? Do these parallelograms have the same shape? Explain. 47

48 Investigation 3.4 You can design construction projects and artwork with the help of computer programs. To use some of these programs, it is helpful to describe polygons and other shapes by naming coordinates of key points. Points on a coordinate grid are labeled with a pair of points called coordinates. The first number, or x-coordinate, tells the horizontal distance from the y-axis. The second or y-coordinate tells the vertical distance from the x-axis. For example, vertex A on the pentagon below has coordinates (10,7). What are the coordinates of the following vertices? B C D E The Midway Amusement Rides Company (MARS) is working on new polygon designs for bumper-car floor plans. They use a computer program that places the polygons on a coordinate grid. A. The diagram on the next page shows four polygons on a coordinate grid. 1. Describe each triangle or quadrilateral as precisely as possible. 2. For each figure, give the coordinates of all vertices. 48

49 Figure Name of Polygon Vertices When is it possible to find the side lengths of a polygon using the coordinates? Find at least one example on each figure. For each example, use the coordinates to find the side lengths. Figure 1: Figure 2: Figure 3: Figure 4: 49

50 4. Find the area of each polygon. Figure 1: Figure 2: Figure 3: Figure 4: 5. If each figure is moved to a different location on the grid, what will change and what will not change? Explain. B. For each polygon listed in Question B, follow the steps below. Find all of the coordinates of the vertices of the polygon. Draw the polygon on a coordinate grid. Find the area of the polygon. 1. A square with vertices A(2,1), B(5,1), C(x 1,y 1 ) and D(x 2,y 2 ). C(, ) D(, ) Area: 50

51 2. An isosceles triangle with vertices P(1,3), Q(1,7), and R(12,y). R(, ) Area: 3. A rectangle with vertices E(9,3), F(9,7), G(4,7), and H(x,y). H(, ) Area: 4. A parallelogram with vertices J(3,2), K(6,4), L(6,11), and M(x,y). M(, ) Area: 51

52 3.4 Day 2 C. Below are four figures on a coordinate graph. One of the vertices in each figure is missing a coordinate. 1. For each figure, find the missing coordinate. 52

53 2. The bases of all the figures have the same length. Are the area the same? Explain. 3. Are the perimeters of the figures the same? Explain. D. Here are three different polygons on a coordinate grid. All three polygons could be broken down into triangles and/or rectangles. 1. Find the area of each polygon. Explain how you found each area. 2. After studying these polygons, Angie insists that all three polygons have the same area. Do you agree with her? Explain your reasoning. 53

54 3. Design another polygon that has the same area as the hexagon. 3.4 Summary Focus Question: How can you find the area of a polygon drawn on a coordinate graph? On grid paper? 54

55 CS Investigation 3 Mathematical Reflections In this Investigation, you developed strategies for finding the area and perimeter of parallelograms. The following questions will help you summarize what you have learned. 1. Describe how to find the area of a parallelogram. Explain why your method works. 2. Describe how to find the perimeter of a parallelogram. Explain why your method works. 3. Does the choice of the base change the area of a parallelogram? Does the choice of the base change the perimeter of a parallelogram? Explain why or why not. 4. What can you say about the shape, area, and perimeter of two parallelograms that have the same base and height? Give evidence to support your answer. 55

56 5. How is the area of a parallelogram related to the area of a triangle and a rectangle? How is the perimeter of a parallelogram related to the perimeter of a triangle and a rectangle? 6. Find the area of the shaded part of this figure. 4 ft. 2 ft. 2 ft. 7 ft. 7. Draw a rectangle with vertices E(3, 9), F(7, 9), G(7, 4), and H(x, y). What coordinate is H? What is the area of the rectangle? 8. What is the area and perimeter of this figure? 56

57 Investigation

58 A. On grid paper, draw at least three different nets that will fold into a box shaped like a unit cube. 1. What is the total area of each net, in square units? 58

59 2. Design a net that forms any other rectangular prism. B. An engineer at the Save-a-Tree packaging company drew the nets below. He lost the notes that indicated the dimensions of the boxes. Using a copy of the diagram above, Draw in fold lines. Cut out each pattern and fold it to form a box. 1. What are the dimensions of each box? P: Q: R: S: 59

60 2. How are the dimensions of each box related to the dimensions of its faces? 3. What is the surface area of each box? P: Q: R: S: 4. How many unit cubes does it take to fill each box? P: Q: R: S: 5. Design a net for a box that has a different shape than Box P but holds the same number of cubes as Box P. 60

61 4.1 Summary Focus Question: What is a strategy for finding the surface area of a rectangular prism? Explain why the strategy works. Did area change with any of the following nets? Explain. 61

62 Investigation

63 Design Length (cm) Width (cm) Height (cm) Volume (cm 3 ) Surface area (cm 2 ) Prism I Prism II Prism III 2. What is the volume of each prism? Record our answers in the table above and describe how you found it. 3. What is the surface area of each prism? Record your answers in the table above and describe how you found it. 63

64 B. Natasha and Kurt use different strategies to find the volume of a rectangular box. Kurt finds the volume by multiplying the length by the width by the height. He uses the diagram below to illustrate his strategy. Natasha finds the volume of a rectangular prism by multiplying the area of the base by the height. She uses the following diagram to illustrate her strategy. 1. Is either of these strategies correct for finding volume? Explain. 2. For each correct method, write a formula for finding the volume of a rectangular prism. Explain what each of the letters in your formulas represents. Compare your formulas. 64

65 3. Use your formulas to find the volume of a rectangular prism with a length of 12 centimeters, a width of 10 centimeters, and a height of 7 centimeters. 4. Dushane says that the formulas for Natasha s and Kurt s method do not work for cubes because the volume of a cube with side length l is l 3. Do you agree with Dushane? Explain your reasoning. 2. What is the volume of the box? 65

66 1. What is the volume of each prism? Explain how you found it. 2. What is the surface area of each prism? Explain how you found it. 3. If all the edges of each prism are taped (with no overlap), how much tape is needed for each prism? Explain. 66

67 4.2 Summary Focus Question: What is the strategy for finding the volume of a rectangular prism? Explain why the strategy works. 67

68 Investigation

69 A. 1. Describe the shape of each box including the shapes and dimensions of its faces. 2. Find the surface area of each box. Show your work. 3. Describe a method for finding the surface area of any box. 69

70 4. If the school wanted to seal all the edge with colored tape (with no overlap) to add some decoration, how much tape is needed for each box? How did you determine the amount? 5. Which design should the school submit? List both the advantage and disadvantages of your choices. B. Valley View Middle School submitted the two boxes at the right. 1. Describe how you could determine the surface area of each box, then find the surface area of each box. 2. Sketch a net that would fold up to make each box. 70

71 4.3 Summary Focus Question: What is a strategy for finding the surface area of a threedimensional object? Explain why the strategy works. 71

72 CS Investigation 4 Mathematical Reflections In this Investigation, you revisited strategies for finding the area and perimeter of a rectangle. You examined the areas and perimeters of figures made from square tiles. You also found that some arrangements of tiles have the same area but different perimeters. Other arrangements of the tiles have the same perimeter but different areas. The following questions will help you summarize what you have learned. 1. a. What information do you need to find the volume of a rectangular prism? Describe a strategy to find the volume of a rectangular prism. b. What information do you need to find the surface area of a rectangular prism? Describe a strategy to find the surface area of a rectangular prism. 72

73 2. a. Describe a strategy for finding the surface area of three-dimensional shapes made from rectangles and triangles. b. How does knowing the area of two-dimensional figures help you find the surface area of a three-dimensional shape? 73