CC Investigation 4: Measurement


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1 Content Standards 6.G., 6.G. CC Investigation : Measurement Mathematical Goals At a Glance PACING 3 days DOMAIN: Geometry Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes, and show that the volume is the same as found by multiplying the prism s edge lengths. Apply the formulas V = lwh and V = bh to solve realworld problems involving the volumes of right rectangular prisms with fractional side lengths. Represent threedimensional figures using nets made of rectangle and triangles. Use nets of threedimensional figures to solve realworld problems involving surface area. Teaching Notes Show students examples of rectangular prisms and pyramids. To represent a threedimensional shape on a flat page, such as in a book, you must draw a twodimensional representation of that shape. Read the introduction with students and review the dimensions of threedimensional figures; length, width, and height. Point out that threedimensional objects consist of such parts as faces, edges, and vertices, and that by turning the object it can have the appearance of different twodimensional shapes. To begin the instruction on surface area, show students a box. Rotate the box to several positions. Review the shape on each side of the box. Ask students to explain how to find the area of one side. Review the formula for area of a rectangle with the class. Show the class a sheet of wrapping paper. Ask the class if they believe there is enough paper to cover the box. Let the students explain how they can find this information. Then develop a definition for surface area using the faces of the box to show the total area that must be covered by the wrapping paper. Use unit cubes in an exercise to give the students a handson experience with cubic measurement and volume. Show that the cubes can be rearranged into numerous configurations without the volume changing. Note that the solid shapes used in this investigation are all right shapes: the bases of the rectangular and triangular prisms are aligned vertically; the apexes of the pyramids are directly above the centers of the bases. Vocabulary net prism surface area volume Measurement 35
2 Problem. Before Problem., present various models of prisms and pyramids to students and review the faces that make up the figures. Ask: What twodimensional shapes make up the faces of a rectangular prism? (6 rectangles) What twodimensional shapes make up the faces of a square pyramid? ( square, triangles) During Problem. A, ask: What twodimensional shapes make up Ashley's net? (6 rectangles) What are the dimensions of each rectangle in the net? ( in. in.; in. in.; in. in.; in. in.; in. in.; in. in.) What do you notice about how some of the rectangles' sizes are related? (There are 3 pairs of equal rectangles.) During Problem. B, ask: What twodimensional shapes make up the faces of the figure? ( triangles and 3 rectangles) What do you notice about the two triangular faces? (They are the same size and shape.) What are the base and height lengths of the triangles? (b = 6 in.; h = in.) Problem. During Problem. A, ask: How many blocks long is the base of Ashley s model? (0) How many blocks wide is the base? (6) How can you find the number of blocks in the base layer? (Multiply 0 6.) During Problem. B, ask: How can you find the volume, in cubic inches, of one of Brandon s blocks? (Multiply.) 6 Could Brandon have used Ashley s blocks to make his model? (No; he couldn t model 6 3 in. using only in. blocks.) Summarize To summarize the lesson, ask: What shapes are the faces of a cube? (6 squares) What faces does a square pyramid have? ( square, triangles) How can you find the surface area of a prism? (Find the area of each side of the prism and add the areas.) How can you find the volume of a rectangular prism? (Multiply its base times its height, or its length times its width times its height.) How is finding the volume of prisms with edge lengths that are fractions or mixed numbers like finding the volume of a prism with integer side lengths? (Use the same formulas to find the volume.) 36 Common Core Teacher s Guide
3 Assignment Guide for Investigation Problem., Exercises 5 Problem., Exercises 6 5 Answers to Investigation Problem. A.. rectangular prism. Find the area of each rectangular piece of the net and then add the areas to find the total area of the net. 3. A = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) = = 00 in.. The area of the net is equal to the surface area of the threedimensional figure that can be made with the net. B.. in. in.. Find the area of each piece of the net and then add the areas to find the total area of the net. 3. A = ( ) + ( 0) + ( 6) + + = (6)() (6)() + + = 3 in. Problem. A.. 60 blocks. 600 blocks 3. 5 in. 3 in. 5 in. in. in. 0 in. 6 in. 6 in. 6 in.. V = lwh = in V = lwh = = 75 in The volume of a block that measures in. on each side is in. 3, so one of Ashley s blocks has a volume that is that volume. 7. The volume of the model is the product of the number of blocks and the volume of each block: B.. blocks wide; 3 blocks long. A = lw = in blocks = 3,0 blocks 5. V = Bh = 6 in V = Bh = ( ) = 50 5 in The volume of a block that measures in. on each side is in. 3, so one of Brandon s blocks has a volume that is that volume. 6. The volume of the model is the product of the number of blocks and the volume of each block: ,0. 6 Measurement 37
4 Exercises. Sample (not to scale):. 6 rectangles 3. triangles and 3 rectangles. square and triangles 5. Sample (not to scale): 6. Sample (not to scale): in. 0. m. 3 cm. ft 3. a. ft 6 ft, ft 6 ft, ft 6 ft, ft 6 ft, ft ft, ft ft b. ft, ft, 7 ft, 7 ft, ft, ft c. ft. No; Surface Area = = 79 in. ; 79 > a. square b. No, you also need the height of a triangular face m 3 7.,7 cm 3. 9 in ft in in in a. inch blocks b. 0 blocks c. 35 in. 3. A 5. a. blocks b. 3 in. in. in. c. 3 3 in. 3 d. Possible answers: in. in.; in. in.; in. 9 in. 3 in. 3 in.; in. in.; 3 Common Core Teacher s Guide
5 Name Date Class Additional Practice Investigation. Seth needs to cover the box below with fabric so that it can be used as a prop in the school play. Seth has 375 in. of fabric. Measurement 5 in. in. 5 in. a. Draw a net of the box. Name the shapes that make up the faces of the box. b. Does Seth have enough fabric to cover this box? Explain how you know. c. What is the longest rectangular box with a square base that measures 5 in. on each side that Seth could wrap with his fabric? Explain how you found your answer.. A tea company sells a flavored tea in the rectangular boxes shown below. Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. 6 cm cm 6 cm cm cm 7.5 cm 0. cm a. Without calculating, which box do you think holds the most tea? Explain your answer. b. What is the volume of each box? How does this compare to your prediction? c. Which box of tea would you expect to cost the least? Explain your reasoning. cm 5 cm Measurement 39
6 Name Date Class Skill: Find the Surface Area Investigation Measurement Find the surface area... cm 5 cm 3 cm cm 6 in. in. 3 in m 5 cm cm 0 cm.5 m.5 m 6 cm Skill: Find the Volume Find the volume in. 35 in. 9 in in. in. in. in. in. 3 3 yd in. 7 yd yd 3 Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. 0 Common Core Teacher s Guide
7 Name Date Class CheckUp Investigation. Jimmy needs to wrap this rectangular box for a birthday present. Measurement 3 in. in. 5 in. a. Draw a net of the box and label the dimensions. b. Jimmy has 600 in. of wrapping paper. Is this enough for him to wrap the box? Explain how you found your answer.. Jimmy has more presents he needs to wrap. The other presents each come in its own box with the shape shown below. Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. in. 6 in. 0 in. 6 in. a. Draw a net of the box and label the dimensions. b. How many of these boxes could Jimmy wrap with 600 in. of wrapping paper? Explain how you found your answer. Measurement
8 Name Date Class CheckUp (continued) Investigation 3. Latoya is going to build a rectangular storage shed for her backyard. The shed will have a volume of 3 ft 3. Latoya sketched this net of the shed. Measurement 9 ft A B C D 6 ft E F 6 ft a. The base of the shed is the face labeled C. What is the height of the shed? Explain how you found your answer. b. Latoya will paint the inside walls and ceiling of the shed, but not the floor. She finds cans of paint that will cover ft and cost $5.9 each. She also finds a gallon of paint that will cover 350 ft and costs $9.99. Which should Latoya buy? Explain your answer.. Latoya finds plans for another rectangular storage shed that has dimensions of 0 ft 5 ft 7.5 ft. If she wants the shed with more storage space, which shed should she build? Explain your choice. Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Common Core Teacher s Guide
9 CC Investigation : Measurement A net is a twodimensional model that can be folded into a threedimensional figure. Prisms are threedimensional figures that have two congruent and parallel faces that are polygons, such as rectangles or triangles. The rest of a prism s faces are parallelograms. You can use nets of rectangular and triangular prisms to find their surface areas. DOMAIN: Geometry Problem. Ashley cuts nets from poster board and folds them to make threedimensional models of buildings. A. Ashley first cuts out the net shown at the right.. What threedimensional figure can she fold from this net?. Ashley knows that she can use the formula A = lw to find the area of a rectangle, where l represents the rectangle s length and w represents its width. Explain how to find the total area of the net. 3. What is the area of the net?. How is the area of the net related to the surface area of the prism? in. in. in. in. in. in. in. B. Ashley wants to model the roof of a building. The triangular faces of the figure are parallel right triangles.. Draw a net of the figure. Label the lengths of the sides.. Explain how you can use the net to find the figure s surface area. in. 0 in. 6 in. in. 3. What is the surface area? Remember that the formula A 5 bh is used to find the area of a triangle with a base b and a height h. Measurement Notes Measurement ()
10 Remember that you can find the volume of a rectangular prism by multiplying the area of its base by its height. Use one of these formulas: V = Bh, where B is the area of the base, and h is the height, or V = lwh, where l and w are the length and width of the base, and h is the height. Problem. Ashley and Brandon use various cubic blocks to model some of the buildings near their school. A. Ashley s blocks are inch on each side. She starts her model by making a 5inch by 3inch rectangular base with the blocks.. How many blocks does she use? 5 in. 3 in.. Ashley adds 9 more layers to the base layer. How many blocks does she use in all? 3. What are the length (l), width (w), and height (h) of the model, in inches?. Use the formula V = lwh to find the volume, in cubic inches, of one of Ashley s blocks. Show your work. 5. Use the formula V = lwh to find the volume, in cubic inches, of Ashley s model. Show your work. 6. How is the volume of block related to the volume of a block that measures in. on each side? 7. Look at the volume of each block, the number of blocks used, and the volume of the model. How are these numbers related? Common Core Additional Investigations Notes () Common Core Teacher s Guide
11 B. Brandon measures his blocks to be inch on each side. He models a building in the shape of a rectangular prism that is in. wide, 3 in. long, and 6 3 in. high.. How many blocks wide and long is the base of Brandon s model?. What is the area, in square inches, of the model s base? 3. How many blocks tall is the model?. How many blocks did Brandon use for the model? 5. Use the formula V = Bh to find the volume, in cubic inches, of one of Brandon s blocks. Show your work. 6. Use the formula V = Bh to find the volume, in cubic inches, of Brandon s model. Show your work. 7. How is the volume of block related to the volume of a block that measures in. on each side?. Look at the volume of each block, the number of blocks used, and the volume of the model. How are these numbers related? Exercises. Draw the threedimensional figure modeled by this net. Name the twodimensional shapes used to make the faces of each object. Tell how many there are of each shape.. a rectangular prism, such as a shoebox 3. a triangular prism, like a tent. a pyramid, like the structures built by the Egyptians Measurement 3 Notes Measurement (3)
12 Draw a net for each figure Find the surface area of each figure in. in. m 6 m 0 m 5 m in... 9 cm 6 ft 9 cm 9 cm 6 ft 3. A container has two rectangular ends that measure ft by 6 ft, and another side that has a length of ft. a. What are the measurements of each of the faces of the container? b. What are the areas of all of the faces of the container? c. What is the total surface area of the container?. Keira has 750 square inches of wrapping paper. Her package is shaped like a right rectangular prism that is 5 inches long, inches wide, and inches high. Does she have enough paper to cover her package? Explain. 3 ft Common Core Additional Investigations Notes () Common Core Teacher s Guide
13 5. The pyramid at the right has four faces that are congruent triangles. a. What shape is the base of the pyramid? b. If you know just the length of a side of the base, do you have enough information to find the surface area of the pyramid? Explain. Find the volume of each rectangular prism m 7. m in. in. m 3 in. cm cm ft cm 6 ft 3 ft For Exercises 0, each rectangular prism is built with in. blocks. Find the length, width, height, and volume of the prism Measurement 5 Notes Measurement (5)
14 3. a. What size cubic blocks would you use to make a rectangular prism that is in. by 3 in. by in.? Explain your choice. b. How many blocks would you need? c. Give the volume of the model in cubic inches.. Multiple Choice Which set of dimensions describes the rectangular prism with the greatest volume? A. 3 in. by in. by 5 in. B. 3 in. by 3 in. by 3 in. C. in. by in. by in. D. in. by in. by 7 in. 5. Megan uses 6 cubic blocks to make a rectangular prism 9 blocks long and 3 blocks tall. Each block measures inch on each side. a. How many blocks wide is the prism? b. What are the prism s dimensions in inches? c. What is the prism s volume, in cubic inches? d. Megan uses all of the bricks to make a new prism that is 6 bricks tall. Give 3 possible sets of dimensions of the base of the prism Megan could have made. 6 Common Core Additional Investigations Notes (6) Common Core Teacher s Guide
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