9.2. Formulas for Volume. Are You Ready? Lesson Opener Making Connections. Resources. Essential Question. Texas Essential Knowledge and Skills

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1 9.2 Formulas for Volume? Essential Question How can you use formulas to find the volume of rectangular prisms? How can you use formulas to find the volume of rectangular prisms? Lesson Opener Making Connections Invite students to tell you what they know about rectangular prisms. Draw on the board or show students a rectangular prism. What is this solid? (a rectangular prism) What everyday objects have this shape? (Accept all reasonable responses.) What ways do you know to measure this shape? (possible answers: height, length, width) How do you think we can measure the space inside this rectangular prism? (Accept all reasonable responses.) Using the Digital Lesson Use blocks to model and discuss how the cubes represent the measurements of the truck bed. Make sure that students understand that the length, width, and height of 1 cube represents 1 foot. Learning Task What is the problem the students are trying to solve? Connect the story to the problem. What is the length of the truck bed? (5 feet) What is the width of the truck bed? (4 feet) What is the height of the truck bed? (2 feet) What do you need to find? (the amount of dirt the truck bed can hold) Literacy and Mathematics Have students write a problem similar to the one stated in which someone is trying to find the volume of a rectangular prism using their own example and measurements. Write the word volume on the board and explain its meaning in this context. Elicit from students other meanings of the word volume. Have them use the word in sentences to demonstrate the different meanings. Texas Essential Knowledge and Skills Algebraic Reasoning 5.4.G Use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l w h, V = s s s, and V = Bh) Also 5.6.A, 5.6.B MATHEMATICAL PROCESSES 5.1.B Use a problem-solving model 5.1.D Communicate mathematical ideas and reasoning 5.1.E Create and use representations Are You Ready? Access Prior Knowledge Use the Are You Ready? 9.2 in the Assessment Guide to assess students understanding of the prerequisite skills for this lesson. Vocabulary cubic units, volume Go to Multimedia eglossary at thinkcentral.com Materials centimeter cubes For the student Resources Interactive Student Edition provides students with an interactive learning environment! Math on the Spot Video Tutor itools Virtual Manipulatives Soar to Success Math Online Intervention For the teacher Digital Management Center organizes program resources by TEKS! eteacher Edition Online Assessment System Lesson A

2 Unlock the Problem Students will use centimeter cubes to replicate Margo s model. Tell students that the base of the rectangular prism that Margo is building is the rectangle that the bottom layer covers. Have students measure the height and connect that measurement to the number of layers. What is the length and width of the base in the model? Length is 4 cm and the width is. In the formula V = Bh, what does the B stand for? the area of the base, l w What is the area of the base? 4 times 2 is 8. The area of the base is 8 sq cm. Students should understand that volume is the amount of space occupied by a solid figure. Reinforce that by adding layers, you are adding height. Each layer has the same area.? 9.2 Essential Question Formulas for Volume Unlock the Problem How can you use formulas to find the volume of rectangular prisms? Connect The base of a rectangular prism is a rectangle. You know that area is measured in square units, and that the area of a rectangle can be found by multiplying the length and the width. Volume is the measure of the amount of space a solid figure occupies. Volume is measured in cubic units, such as cubic feet or cu ft. When you build a prism and add each layer of cubes, you are adding a third dimension, height. Margo is modeling a building using 1-centimeter cubes. The model has a rectangular base and a height of the 5 cubes. What is the volume of the rectangular prism that Margo built? Use centimeter cubes to show Margo s model. Count the total number of cubes after each layer. Use the information to complete the table. Height (in layers) Volume (in cubic centimeters) What multiplication pattern do the numbers in the table show? Multiply the number of layers by 8 to get the volume. Algebraic Reasoning 5.4.G Also 5.6.A, 5.6.B MATHEMATICAL PROCESSES 5.1.B, 5.1.D, 5.1.E The area of the base is _ 12 sq units. 8 Why do we multiply by 8? Each layer has an area of 8 square centimeters. You can find the volume of a prism in cubic units by multiplying the number of square units in the base shape by the number of layers, or its height. Write a formula for finding the volume of a rectangular prism. Use B for the area of the base and h for the height. V = Bh So, the volume of Margo s rectangular prism is _ 40 cubic centimeters. Module English Language Learners Leveled Activities Beginning: Activity 20 Intermediate: Activity 40 Advanced: Activity 58 Advanced High: Activity 60 ELPS 1.A.1, 3.G.2, 4.C.3 4.F.6, 4.G.2, 4.G.4 2.C.2, 4.C.3, 4.F.9 2.I.3, 4.F.7, 4.G.3 thinkcentral.com for the ELL Activity Guide containing these leveled activities. ELL Language Support ELPS 2.C.4, 3.F.2 Strategy: Model Concepts Materials: cubes Visual / Spatial Whole Class Build a small rectangular prism with cubes. Trace the outline of the base on a piece of paper and cut out the shape. Display the rectangle and say that it represents the base. Point out that area is flat. Ask, What word do we use to describe the inside of a rectangle? area Hold up the prism. What word do we use to describe the inside of a prism? volume Point out how volume is different than area. It is measured in cubes instead of squares. 355 Module 9

3 Relate Height to Volume You can use the formula for the area of a rectangle to rewrite the formula for the volume of a rectangular prism. What are the dimensions of the base of the box? and 4 cm What operation can you use to find the area of the base shape? multiplication Relate Height to Volume Read the problem with students. Toni stacks cube-shaped beads that measure 1 centimeter on each edge in a storage box. The box can hold 6 layers of 24 cubes with no gaps or overlaps. What is the volume of Toni s storage box? One Way Use V = Bh. The volume of each bead is _ 1 cubic cm. The storage box has a base with an area of _ 24 square cm. The height of the storage box is _ 6 centimeters. The volume of the storage box is 24 ( _ _ ), or _ cubic cm. Base area Another Way Use length, width, and height. The base is a rectangle. Replace B with an expression for the area of the base shape. V = B h B = base area; l = length; w = width V = (_ l _ w ) h Base area The base has a length of _ 6 centimeters and a width of _ 4 centimeters. The height is _ 6 centimeters. The volume of the storage box is ( _ _ ) _, or _ _, or _ cubic cm. Base area So, the volume of Toni s storage box is _ 144 cubic cm. 356 Enrich Visual / Spatial Partners Materials: cubes, centimeter grid paper, ruler Math Talk Mathematical Processes Describe one way you can check if the volume you calculated using the formulas is correct. Possible description: I can use centimeter cubes to build a model and count the cubes to check my answer. Ask students to describe and illustrate the difference between 10 cm, 10 square cm, and 10 cubic cm. Ask students to make a model to display each measure. Have students share their models with a partner. How are their representations different? How are they similar? One Way Remind students that each edge of a bead measures 1 centimeter, so (1 1) 1 gives a volume of 1 cubic centimeter. How did you find the area of the base of the storage box? Possible answer: the problem says that one layer holds 24 cubes, so the area of the base is 24 square centimeters. Another Way How is this method different from the first way? Possible answer: instead of multiplying the base and the height, I am multiplying the length, width, and height. How does using this method affect the volume? Possible answer: the volume remains the same because the same numbers are multiplied, just in a different order. Math Talk Mathematical Processes Use Math Talk to focus on students understanding of how to use models to determine the volume. COMMON ERRORS Error Students may confuse the units for linear measurement, area, and volume. Example The height of the prism is 5 sq ft. The area of the base is 24 cu ft. 5 ft The volume of the prism 4 ft is 120 sq ft. 6 ft Springboard to Learning Draw a line and label it 3 units. Remind students that distance is measured in units. Draw a 3 by 4 unit rectangle and show that its sides are measured in units. Divide the rectangle into unit squares. Emphasize what the squares cover and that area is measured in square units. Finally, demonstrate that cubes have length, width, and height and take up space that is called volume, and volume is measured in cubic units. 10 cm 10 sq cm Go to Go to thinkcentral.com for additional enrichment activities in the Enrich Activity Guide. 10 cu cm Lesson

4 Share and Show Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard Quick Check Share and Show Find the volume. 1. The length of the rectangular prism is. 2. The width is. So, the area of the base is. 6 in. The height is. So, the volume of the prism is sq in. 120 cu in. 6 in. IF THEN a student misses the checked exercises Differentiate Instruction with RtI Tier 1 Lesson 56 cu cm 6 in. 1 in. 2 in. 12 cu in. Problems In Exercise 4, students use reasoning to determine how height affects volume of a rectangular prism when the area of the base remains unchanged. In Exercise 5, students should use centimeter cubes to build a prism. They can count the cubes in their model for length and width of the base and realize that the number of cubes in the length, l, times the number of cubes in the width, w, gives the area of the rectangle that forms the base, B. The number of cubes in the height, h, times the area gives the total number of cubes, or the volume, V. Have students construct several rectangular prisms to see that this relationship is always true. For Exercise 6, have students make notes to organize their thinking. Have them list the length, width, and height of the dog, then add 12 inches to the length and width and 6 inches to the height to find the dimensions for the crate. 4. Connect What happens to the volume of a rectangular prism if you double the height? Give an example. The volume is doubled. Examples will vary. 5. Write Math Explain how the two formulas for the volume of a rectangular prism V = Bh and V = l w h are related. Use a model with centimeter cubes to justify your reasoning. See margin for description of modeling. Possible explanation: The area of the base is l w. So, I can write B instead of l w in the formula V = l w h to get V = Bh. 6. Multi-Step Rich is building a travel crate for his dog, Thomas, a beagle-mix who is about 30 inches long, 12 inches wide, and 24 inches tall. For Thomas to travel safely, his crate needs to be a rectangular prism that is about 12 inches greater than his length and width, and 6 inches greater than his height. What is the volume of the travel crate that Rich should build? The crate should be 42 inches long, 24 inches wide, and 30 inches high. V = or 30,240 cubic inches. Module 9 Lesson Math on the Spot Video Tutor LESSON 56 RtI Tier 1 Lesson 56 Enrich 53 Volume of Rectangular Prisms OBJECTIVE Find the volume of rectangular prisms G, 5.6.B Aquarium Volumes on Display Enrich 53 Through the Math on the Spot Video Tutor, students will be guided through an interactive solving of this type of H.O.T. problem. Use this video to also help students solve the H.O.T. problem in the Interactive Student Edition. With these videos and H.O.T. problems, students will build skills needed in the TEXAS assessment. Math on the Spot videos are in the Interactive Student Edition and at thinkcentral.com. 357 Module 9 Jorge wants to find the volume of this rectangular prism. He can use cubes that measure 1 centimeter on each side to find the volume. Step 1 The base has a length of 2 centimeters and a width of 3 centimeters. Multiply to find the area of the base. Base = 2 3 Base = 6 sq cm Step 2 The height of the prism is 4 centimeters. Add the number of cubes in each layer to find the volume. Think: Each layer has 6 cubes = 24 unit cubes Volume = 24 cu cm Step 3 Multiply the base and the height to check your answer. Volume = 6 4 Volume = 24 cubic centimeters So, the volume of Jorge s rectangular prism is 24 cubic centimeters. Find the volume cu cm 20 cu ft 2 in. 32 cu in ft 5 ft 2 ft 54 cu cm Geometry and Measurement cm 4 cm A pet store is selling aquariums. You have been asked to find the volumes of the aquariums in cubic inches. After finding the volumes, answer the questions that follow. 15-Gallon Aquarium 25-Gallon Aquarium Enrich 2 28 in. 12 in. 3, in. 23 in. $35 55-Gallon Aquarium 13, in. $85 1. A customer says he has a rectangular aquarium with a length of 37 in., a width of 19 in., and a height of 19 in. What is the volume of the aquarium? E in. 12. $ Gallon Aquarium 28.8 in The gallon sizes above are approximate. To which size is the customer s aquarium closest? 13,357 in gallon 3. Stretch Your Thinking What is the volume of the 25-gallon aquarium in cubic feet? Round to the nearest tenth. 3.6 ft 3 6, ,489.6 in Which of the four aquariums has the best value, based on the advertised gallon sizes? Explain your answer. The 55-gallon aquarium has the best value because it has the lowest unit cost at $1.55 per gallon. $175

5 Daily Assessment Task Fill in the bubble completely to show your answer. 7. Sandra orders a box of cube-shaped beads. The beads are neatly stacked in layers, with the same number of beads in each layer. Each layer is 12 beads across and 7 beads wide. There are 4 layers. How many beads does she receive? Mathematical Processes 2 Daily Assessment Task 1 Can students use formulas to find the volume of rectangular prisms? 3 A 23 beads C 76 beads B 336 beads D 88 beads 8. Which equation can you use to find the number of cubes in the rectangular prism? IF NO THEN Soar to Success Math Warm-Up 49.28, A V = C V = YES Enrich 53 Homework and Practice Lesson 9.2 B V = D V = 2 (11 8 5) 9. Multi-Step How many cubic units of material were used to make the two rectangular prisms? A 144 cubic units C 8,640 cubic units B 60 cubic units D 204 cubic units TEXAS Test Prep Coach In the Test Prep exercise, if students selected: A They multiplied 5 x 5 x 5. B They found the area of the base. C They multiplied length times width times height incorrectly. 358 TEXAS Test Prep 10. What is the volume of the rectangular prism at the right? A 125 cu in. C 155 cu in. B 35 cu in. D 175 cu in. 7 in.? Essential Question Write Math How can you use formulas to find the volume of a rectangular prism? Possible answer: I can replace the unknowns in the formula with measurements of the rectangular prism to find its volume. Differentiated Centers Kit Games Games Triple Play Students practice finding the volume of rectangular prisms. Activities What s in the Box? Students complete blue Activity Card 12 by finding volume using unit cubes. Activities Inner Space Students complete orange Activity Card 12 by finding the volume of rectangular prisms using a centimeter ruler. Lesson

6 5 Homework and Practice 9.2 Formulas for Volume Algebraic Reasoning 5.4.G Also 5.6.A, 5.6.B MATHEMATICAL PROCESSES 5.1.B, 5.1.D, 5.1.E Lesson Check Fill in the bubble completely to show your answer. TEXAS Test Prep Find the volume. 9. What is the volume of the rectangular prism shown below? 10. Hannah used centimeter cubes to build the model shown below cu cm 1 in. 3 in cu in. 8 cm Which equation can you use to find the volume of Hannah s model? A V = cm 90 cu cm 125 cu in. A B C D 48 cu cm 84 cu cm 288 cu cm 384 cu cm B V = (6 7) + 3 C V = (6 + 7) 3 D V = cm 72 cu cm 6. 3 in. 3 in. 36 cu in. 11. A shipping clerk packs a box of cube-shaped notepads. He packs 8 layers of notepads with 8 rows of 6 notepads in each layer. How many notepads does the clerk pack? A 336 C 96 B 112 D The number cubes the fifth-grade math classes use are packed into a box. When the box is full, it has 5 rows of 4 cubes with 6 layers of cubes. Mrs. Benson sees that one layer of cubes is missing from the box. How many number cubes are in the box? A 96 C 90 B 100 D A small refrigerator fits into a cabinet that measures 2 feet wide, 2 feet deep, and 4 feet high. What is the volume of the cabinet? 8. Mr. Otis built a storage shed. The shed has a length of 5 meters, a width of 3 meters, and a height of 4 meters. His goal was for the shed to have a volume greater than 50 cubic meters. Did Mr. Otis meet his goal? Explain. 16 cu ft Yes; The volume of the shed is 60 cubic meters. 60 > 50. Module 9 Lesson Multi-Step Alexis uses toy blocks to build a model of a building. Each toy block is 1 cubic inch. The first three floors of the model are made up of 6 rows of 4 blocks. Floors four through eight are made up of 4 rows of 4 blocks. What is the volume of the model? A 104 cu in. C 88 cu in. B 152 cu in. D 136 cu in Multi-Step Aaden is packing boxes into a carton that is 8 inches long, 8 inches wide, and 4 inches tall. The boxes are 2 inches long, 1 inch wide, and 1 inch tall. How many boxes will fit into the carton? A 128 C 64 B 256 D 512 Homework and Practice Use the Homework and Practice pages to provide students with more practice on the concepts and skills of this lesson Module 9