Volume of Rectangular Prisms Objective To provide experiences with using a formula for the volume of rectangular prisms.


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1 Volume of Rectangular Prisms Objective To provide experiences with using a formula for the volume of rectangular prisms. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Use formulas (l w h or B h) to calculate the volumes of rectangular prisms. [Measurement and Reference Frames Goal 2] Define the and of a rectangular prism. [Geometry Goal 2] Explore the properties of rectangular prisms. [Geometry Goal 2] Write number sentences with variables to model volume problems. [Patterns, Functions, and Algebra Goal 2] Key Activities Students discuss the difference between volume and area. They define and for a rectangular prism and develop a formula for the volume of a rectangular prism. Students use a formula to find volumes and to model a rectangular prism with a given volume. Ongoing Assessment: Informing Instruction See page 751. Key Vocabulary volume cubic unit rectangular prism face (of a rectangular prism) (of a rectangular prism) Associative Property of Multiplication Materials Math Journal 2, pp. 321, 322, and Activity Sheet 8 Student Reference Book, pp Study Link 9 7 scissors transparent tape 36 centimeter cubes slate Math Boxes 9 8 Math Journal 2, p. 323 Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 3. [Number and Numeration Goal 5] Study Link 9 8 Math Masters, p. 279 Students practice and maintain skills through Study Link activities. READINESS Analyzing Prism Nets for Cubes Math Masters, pp. 280 and 429 Students investigate solid geometry concepts using cube diagrams. ENRICHMENT Finding the Volume of One StickOn Note Math Masters, p. 281 per partnership: one stickon note, pad of unused stickon notes, centimeter cube Students compare the volume of a single stickon note to the volume of a centimeter cube. EXTRA PRACTICE 5Minute Math 5Minute Math, pp. 52 and 214 Students calculate the volumes of prisms. Advance Preparation For Part 1, make models of the open boxes on Activity Sheet 8 to illustrate where to fold and where to tape the patterns for Boxes A and B. Have students save the boxes for use in Project 9. Teacher s Reference Manual, Grades 4 6 pp , Lesson
2 Getting Started Mental Math and Reflexes Students practice rounding numbers. Have them write their answers on their slates. Suggestions: Round 654 to the nearest hundred. 700 Round 4, to the nearest whole number. 4,655 Round to the nearest tenth Round to the nearest hundredth Round 67% of 100 to the nearest ten. 70 Round to the nearest tenth. 8.0 Mathematical Practices SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7, SMP8 Content Standards 5.NBT.4, 5.NBT.7, 5.MD.3a, 5.MD.3b, 5.MD.4, 5.MD.5a, 5.MD.5b Math Message Write 2 questions that can be answered by reading Student Reference Book, page 195. Study Link 9 7 FollowUp Ask volunteers to present the area formulas for a triangle and for a parallelogram and the number models they used to solve the Study Link Problems. Project Note For more practice finding the volume of rectangular prisms, see Project 9: Adding Volumes of Solid Figures. Volume and Capacity Volume The volume of a solid object such as a brick or a ball is a measure of how much space the object takes up. The volume of a container such as a freezer is a measure of how much the container will hold. Volume is measured in cubic units, such as cubic inches (in. 3 ), cubic feet (ft 3 ), and cubic centimeters (cm 3 ). It is easy to find the volume of an object that is shaped like a cube or other rectangular prism. For example, picture a container in the shape of a 10centimeter cube (that is, a cube that is 10 cm by 10 cm by 10 cm). It can be filled with exactly 1,000 centimeter cubes. Therefore, the volume of a 10centimeter cube is 1,000 cubic centimeters (1,000 cm 3 ). To find the volume of a rectangular prism, all you need to know are the length and width of its and its. The length, width, and are called the dimensions of the prism. You can also find the volume of another solid, such as a triangular prism, pyramid, cone, or sphere, by measuring its dimensions. It is even possible to find the volume of an irregular object such as a rock or your own body. Capacity We often measure things that can be poured into or out of containers such as liquids, grains, salt, and so on. The volume of a container that is filled with a liquid or a solid that can be poured is often called its capacity. Capacity is usually measured in units such as gallons, quarts, pints, cups, fluid ounces, liters, and milliliters. The tables at the right compare different units of capacity. These units of capacity are not cubic units, but liters and milliliters are easily converted to cubic units: 1 milliter = 1 cm 3 1 liter = 1,000 cm 3 1 cm 3 1,000 cm 3 Measurement length U.S. Customary Units 1 gallon (gal) = 4 quarts (qt) 1 gallon = 2 halfgallons 1 halfgallon = 2 quarts 1 quart = 2 pints (pt) 1 pint = 2 cups (c) 1 cup = 8 fluid ounces (fl oz) 1 pint = 16 fluid ounces 1 quart = 32 fluid ounces 1 halfgallon = 64 fluid ounces 1 gallon = 128 fluid ounces Metric Units width The dimensions of a rectangular prism 1 liter (L) = 1,000 milliliters (ml) 1 milliliter = 1_ 1,000 liter 1 liter = 1,000 cubic centimeters 1 Teaching the Lesson Math Message FollowUp (Student Reference Book, p. 195) WHOLECLASS DISCUSSION Ask students to pose their questions for the class. Use the questions and responses to discuss the difference between area and volume. To support English language learners, write the key ideas on the board. Emphasize the following points: Shapes that are 2dimensional are flat. The surfaces they enclose take up a certain amount of area, but they do not have any thickness, so they do not take up any space. Shapes that are 3dimensional have length, width, and thickness, so they enclose a certain amount of space. The amount of surface inside a 2dimensional shape is the area of the shape. Area is measured in square units, such as square inches, square feet, square centimeters, and so on. The amount of space enclosed by a 3dimensional shape is the volume of the shape. Volume is measured in cubic units, such as cubic inches, cubic feet, cubic centimeters, and so on. The symbol used to indicate square units is the abbreviation of the unit name with a superscript 2, for example, in 2, ft 2, cm 2, m 2, and so on. For cubic units, the symbol is the abbreviation of the unit name with a superscript 3, for example, in 3, ft 3, cm 3, m 3, and so on. To support English language learners, discuss the meanings of the word volume. Students might have heard the word used in contexts involving sound level or a book. Emphasize that volume is also used in the mathematical context involving 3dimensional shapes. ELL 1 milliliter = 1 cubic centimeter Student Reference Book, p _214_EMCS_S_SRB_G5_MEA_ indd 195 3/8/11 5:00 PM 748 Unit 9 Coordinates, Area, Volume, and Capacity
3 Defining Base and Height for Rectangular Prisms (Math Journal 2, p. 321) WHOLECLASS DISCUSSION As a class, read the introduction on journal page 321. Have students study the figures and then write their own definitions for the and of a rectangular prism. Ask volunteers to share their definitions with the class. Use their responses to reinforce the following points: A rectangular prism is formed by six flat surfaces, or faces, that are rectangles. To support English language learners, compare the common meaning and the mathematical meaning of the word face. Any of the six faces of the prism can be a, but the face that the prism sits on is often chosen as a. The (for a given ) is the shortest distance between the and the opposite face. ELL Date 9 8 Time Rectangular Prisms A rectangular prism is a geometric solid enclosed by six flat surfaces formed by rectangles. If each of the six rectangles is also a square, then the prism is a cube. The flat surfaces are called faces of the prism. Bricks, paperback books, and most boxes are rectangular prisms. Dice and sugar cubes are examples of cubes. Below are three different views of the same rectangular prism Study the figures above. Write your own definitions for and. Base of a rectangular prism: Sample answer: Any face of the prism; usually the face the prism sits on Height of a rectangular prism: Sample answer: The shortest distance between the and the opposite face Examine the patterns on Activity Sheet 6. These patterns will be used to construct open boxes boxes that have no tops. Try to figure out how many centimeter cubes are needed to fill each box to the top. Do not cut out the patterns yet. Answers vary. 2. I think that centimeter cubes are needed to fill Box A to the top. 3. I think that centimeter cubes are needed to fill Box B to the top. Math Journal 2, p. 321 Links to the Future _EMCS_S_G5_MJ2_U09_ indd 321 2/22/11 5:18 PM The names and properties of prisms will be revisited in more detail in Unit 11. There are two types of prisms, right and oblique. (See below.) When the faces of a prism are all rectangles, it is a right rectangular prism. When the faces are not all rectangles, it is an oblique rectangular prism. Oblique prisms are not considered in Fifth Grade Everyday Mathematics. NOTE You may want to remind students that to be a prism, a polyhedron must have at least one pair of parallel and congruent faces, which can be called s. Rectangular prisms have 3 pairs of parallel congruent faces. Therefore, when calculating the volume of a rectangular prism, any face can be chosen as the. Right rectangular prism Oblique rectangular prism Adjusting the Activity ELL Have students identify objects in the classroom that have the shape of rectangular prisms. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Developing a Formula for Volume (Math Journal 2, pp. 321, 322, and Activity Sheet 8; Student Reference Book, pp. 196 and 197) PROBLEM SOLVING WHOLECLASS Algebraic Thinking Give 24 centimeter cubes to each student. Ask students to tear out Activity Sheet 8 from the back of their journals and follow the directions on journal page 321 to answer Problems 2 and 3. Lesson
4 When students have completed the problems, display your prepared models as guides for folding and taping the boxes. Direct students to do the following: 1. Cut out the pattern for Box A, fold it on the dashed lines, and tape it to make an open box. 2. Cover the bottom of Box A with one layer of centimeter cubes. Ask: How many cubes are in this layer? 8 3. Put a second layer of cubes on top of the first layer. Ask: How many cubes are in the box now? Put a third layer on top of the second layer. Ask: How many cubes are in the box now? 24 What is the volume of the box? 24 cm 3 5. Cut out the pattern for Box B, fold it on the dashed lines, and tape it to make an open box. 6. Cover the bottom of Box B with one layer of cubes. Ask: How many cubes are in this layer? 9 Ask students to answer the following questions without putting any more cubes in the box: How many layers of cubes are needed to fill the box? 5 NOTE Traditionally, linear measures are represented in formulas by lowercase letters, and 2 and 3dimensional measures are represented by uppercase letters. For example, b is used to represent the length of the of a rectangle, and B is used to represent the area of the of a prism. Date 9 8 Time Volume of Rectangular Prisms Write the formulas for the volume of a rectangular prism. B is the area of the (l w ). h is the from that. V is the volume of the prism. V = B h or V = l w h How many centimeter cubes in all are needed to fill the box? 45 How did you find the answer? Sample answer: By multiplying the number of cubes in 1 layer by the number of layers. What is the volume of the box? 45 cm 3 Point out that the number of cubes in one layer is the same as the number of square centimeters in the (l w), and that the number of layers is the same as the in centimeters of the box. Ask students how this information might be used in a formula to find the volume of the box. Multiply the area of the by the of the box. This formula is written as V = B h, where V represents the volume, B represents the area of the, and h represents the from that. Ask students to write this formula at the top of journal page 322. Have students refer to pages 196 and 197 of the Student Reference Book as needed. Find the volume of each rectangular prism below in. 6 cm 4 in. 3 cm 4 in. 4 cm 80 in 3 V = V = 72 cm cm 7 cm 8 in. 4 in. 6 in. 7 cm 343 cm 3 V = V = 192 in ft 5 cm 3 ft 4 cm 6 ft 2.5 cm 90 ft 3 V = V = 50 cm 3 Math Journal 2, p _EMCS_S_MJ2_G5_U09_ indd 322 3/22/11 12:42 PM 750 Unit 9 Coordinates, Area, Volume, and Capacity
5 Another Formula for Volume of a Rectangular Prism (Student Reference Book, pp. 196 and 197) WHOLECLASS Ask students to review pages 196 and 197 of the Student Reference Book. Give each student 12 more centimeter cubes so each student has 36 cubes. NOTE If you do not have 36 centimeter cubes per student, ask students to build the rectangular prisms in partnerships. Ask students to use the centimeter cubes to do the following: 1. Build a rectangular prism with a that is 4 cubes long and 3 cubes wide, and that is 1 cube high. Ask: Describe a way you could determine the volume of the rectangular prism. Sample answer: Multiply to obtain 12 cm 3. What do the 4, 3, and 1 represent in the prism? The length, width, and 2. Build a second 4by3by1 rectangular prism. Place it on top of the original rectangular prism. Ask: How could you determine the volume of the new rectangular prism? Sample answers: Multiply to obtain 24 cm 3 ; multiply 12 2 to get 24 cm Build a third 4by3by1 layer. Place it on top of the existing rectangular prism. Ask students to write a number model using three factors to find the volume of the new prism = 36 cm 3 Now ask the following question: Suppose you know the number of cubes in the length, width, and of a rectangular prism. How could this information be used to write another formula for finding the volume of a rectangular prism? Multiply: length width (l w h); V = l w h. Have students write this formula at the top of journal page 322, next to the first formula they derived for finding the volume of a rectangular prism. Discuss why both formulas will result in the same volume. Conclude by asking students to explain how they could use the new formula to find the volume of a cube. Sample answers: Replace each sidelength letter in the formula with the length of a side of the cube. You could rewrite the formula as V = s 3, where s is the length of a side of the cube. Measurement Volume of a Geometric Solid You can think of the volume of a geometric solid as the total number of whole unit cubes and fractions of unit cubes needed to fill the interior of the solid without any gaps or overlaps. Prisms and Cylinders In a prism or cylinder, the cubes can be arranged in layers that each contain the same number of cubes or fractions of cubes. Find the volume of the prism. 8 cubes in 1 layer 3 layers 3 layers with 8 cubes in each layer makes a total of 24 cubes. Volume = 24 cubic units The of a prism or cylinder is the shortest distance between its s. The volume of a prism or cylinder is the product of the area of the (the number of cubes in one layer) multiplied by its (the number of layers). Pyramids and Cones The of a pyramid or cone is the shortest distance between its and the vertex opposite its. If a prism and a pyramid have the same size and, then the volume of the pyramid is onethird the volume of the prism. If a cylinder and a cone have the same size and, then the volume of the cone is onethird the volume of the cylinder. same same area 182_214_EMCS_S_SRB_G5_MEA_ indd 196 same same area Student Reference Book, p. 196 The area of the Pacific Ocean is about 64 million square miles. The average depth of that ocean is about 2.5 miles. So, the volume of the Pacific Ocean is about 64 million mi mi, or 160 million cubic miles. Note For a right rectangular prism with side lengths l, w, and h units, the volume V can be found using the formula V = l w h. Volume of a Rectangular or Triangular Prism Measurement Volume of a Prism Area of a Rectangle Area of a Triangle V = B h A = b h A = 1_ 2 (b h) V is the volume, B is the area A is the area, b is the length A is the area, b is the length of the, h is the of of the, h is the of of the, h is the of the prism. the rectangle. the triangle. Find the volume of the rectangular prism. Step 1: Find the area of the (B). Use the formula A = b h. length of the rectangular (b) = 8 cm of the rectangular (h) = 5 cm area of (B) = 8 cm 5 cm = 40 cm 2 Step 2: Multiply the area of the by the of the rectangular prism. Use the formula V = B h. area of (B) = 40 cm 2 of prism (h) = 6 cm volume (V) = 40 cm 2 6 cm = 240 cm 3 The volume of the rectangular prism is 240 cm 3. 8 cm 6 cm 5 cm 4/4/11 5:01 PM Find the volume of the triangular prism. Step 1: Find the area of the (B). Use the formula A = 1_ (b h). 2 length of the triangular (b) = 5 in. of the triangular (h) = 4 in. area of (B) = 1_ (5 in. 4 in.) = 10 in.2 2 Step 2: Multiply the area of the by the of the triangular prism. Use the formula V = B h. area of (B) = 10 in. 2 of prism (h) = 6 in. volume (V) = 10 in. 2 6 in. = 60 in. 3 The volume of the triangular prism is 60 in in. 5 in. 6 in. Find the volume of each prism. Include the unit in each answer yd 10 cm 3 yd 2 yd 10 cm 10 cm Check your answers on page ft 8 ft 12 ft Student Reference Book, p _214_EMCS_S_SRB_G5_MEA_ indd 197 3/8/11 5:00 PM Lesson A
6 Building a Rectangular Prism with a Given Volume PARTNER Algebraic Thinking Ask students to use centimeter cubes to build the shape of a rectangular prism with a volume of 24 cm 3. Ask: What are the possible areas for the (B) and the length of the (h) from that? B = 24 cm 2, h = 1 cm; B = 12 cm 2, h = 2 cm; B = 8 cm 2, h = 3 cm; B = 6 cm 2, h = 4 cm; B = 4 cm 2, h = 6 cm; B = 3 cm 2, h = 8 cm; B = 2 cm 2, h = 12 cm; B = 1 cm 2, h = 24 cm Ask students to use the cubes to make a rectangular prism that has a length of 3 cm, a width of 2 cm ( of 6 cm 2 ), and a of 4 cm. Explain that the volume can be found as follows: V = (l w) h = (3 2) 4 Now ask students to carefully rotate their rectangular prism so that it now has a length of 4 cm, a width of 2 cm ( of 8 cm 2 ), and a of 3 cm. Explain that the volume can be found as follows: V = (l w) h = 3 (2 4) Ask: What do the two rectangular prisms have in common? Sample answer: The dimensions of the prisms have the same three measures, and each has a volume of 24 cm 3. Guide students to conclude that if the measures of the three dimensions of two rectangular prisms are the same, the volumes are the same. Then point out that this is an illustration of the Associative Property of Multiplication, which means that changing the grouping of factors does not change the product. So, (3 º 2) º 4 = 3 º (2 º 4) 6 º 4 = 3 º 8 24 = 24 Have students repeat this activity by comparing the volumes of two rectangular prisms that have dimensions of 2 cm, 3 cm, and 5 cm, but different s. Finding the Volumes of Rectangular Prisms (Math Journal 2, p. 322) INDEPENDENT Algebraic Thinking Have students complete journal page 322 by calculating the volume of six rectangular prisms from the given dimensions. 750B Unit 9 Coordinates, Area, Volume, and Capacity
7 Ongoing Assessment: Informing Instruction Watch for students who are not correctly matching the given dimensions to the formula variables. Have them make and complete a table for the problems on journal page 322, such as the following: Problem V B h 1 80 in cm cm in ft cm Ongoing Learning & Practice Date Solve. Math Boxes a b Find the least common denominator for the fraction pairs. a. 2_ 7 and 1_ 3 21 b. 5_ 8 and 4_ c. 3_ 8 and 4_ d. 2_ 5 and 2_ 15 3 e. 4_ 16 and 6_ f. 5_ 15 and 2_ c d Use the graph to answer the questions. 124 a. Which day had the greatest attendance? Friday Time b. What was the total attendance for the fiveday period? 90 Math Journal 2, p _EMCS_S_G5_MJ2_U09_ indd Complete the What s My Rule? table, and state the rule Rule Elena received the following scores on math tests: 80, 85, 76, 70, 87, 80, 90, 80, and 90. Find the following landmarks: maximum: 90 minimum: 70 range: 20 mode: 80 mean: 82 Number of Tickets Sold in out ,500 2, , Movie Theater Attendance M Tu W Th F Day of the Week 2/22/11 5:18 PM Math Boxes 9 8 (Math Journal 2, p. 323) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson The skills in Problems 2 and 5 preview Unit 10 content. Writing/Reasoning Have students write a response to the following: Explain the strategy you used to solve Problem 1c and explain your reasoning. Answers vary. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 3 Use Math Boxes, Problem 3 to assess students abilities to find common denominators. Students are making adequate progress if they correctly identify the least common denominators. [Number and Numeration Goal 5] Study Link 9 8 (Math Masters, p. 279) INDEPENDENT Name Date Time STUDY LINK 9 8 Volumes of Cube Structures The structures below are made up of centimeter cubes nd layer 1st layer Area of = cm 2 Volume of first layer = cm 3 45 Volume of entire cube structure = cm 3 Study Link Master 1st layer st layer 8 Area of = cm Volume of first layer = cm Volume of entire cube structure = cm 3 Home Connection Students find the volume of cube structures. 9 Area of = cm 2 Volume of first layer = cm 3 9 Volume of entire 27 cube structure = cm 3 1st layer Area of = cm 2 Volume of first layer = cm 3 Volume of entire 56 cube structure = cm 3 Practice 3_ 5. 3_ 5 1_ = ,840 / 4 = 3, = 8. 4_ 5 5_ 6 = 4_ 960 6, or 2_ 3 Math Masters, p _497_EMCS_B_MM_G5_U09_ indd 279 2/22/11 6:06 PM Lesson
8 Name Date Time 9 8 Unfolding Prisms If you could unfold a prism so that its faces are laid out as a set attached at their edges, you would have a flat diagram for the shape. Imagine unfolding a cube. There are many different ways that you could make diagrams, depending on how you unfold the cube. Which of the following are diagrams that could be folded to make a cube? Write yes or no in the blank next to each diagram Teaching Master No Yes Yes No 3 Differentiation Options READINESS Analyzing Prism Nets for Cubes (Math Masters, pp. 280 and 429) PARTNER 5 15 Min To stimulate students ability to visualize, name, and describe geometric solids, have students look at diagrams to determine which shapes can and cannot be folded into cubes. Read and discuss the introduction as a group. As students choose which of the diagrams can be folded into a cube, provide inch grid paper so students may check their work. Discuss students solutions. Math Masters, p _497_EMCS_B_MM_G5_U09_ indd 280 ENRICHMENT PARTNER Finding the Volume of 2/22/11 6:06 PM One StickOn Note (Math Masters, p. 281) 5 15 Min Name Date Time 9 8 Comparing Volume What is the volume of one stickon note? In other words, how much space is taken up by a single stickon note? How does the volume of a stickon note compare to the volume of a centimeter cube? 1. An unused pad of stickon notes is an example of what shape? Rectangular prism 2. Estimate the volume of one stickon note. Teaching Master Sample answer: About 0.75 cm 3 Answers vary. 3. Calculate the volume of one stickon note. Volume = Record your strategy. To apply students understanding of how to find the volume of a rectangular prism, have partners compare the volume of a single stickon note and that of a centimeter cube. Give each partnership one stickon note, one unused pad of stickon notes, and one centimeter cube. Ask students to estimate how the volume of the single stickon note compares with the volume of the cube. Have students record their strategies and solutions on Math Masters, page 281. When partners have completed the page, ask them to present their solutions. NOTE One approach to finding the volume of a single stickon note would be to measure the dimensions and find the volume of the unused pad of stickon notes. A single stickon note would represent a fraction of the pad. EXTRA PRACTICE 5Minute Math SMALLGROUP 5 15 Min 4. Use a formula to calculate the volume of one centimeter cube. Volume = 1 cm 3 To offer students more experience with calculating the volumes of prisms, see 5Minute Math, pages 52 and 214. Write the number sentence for this calculation. Volume = (1 cm 1 cm) 1 cm = 1 cm 3 5. Explain how the volume of one stickon note compares with the volume of one centimeter cube. Answers vary. Math Masters, p _497_EMCS_B_MM_G5_U09_ indd 281 2/23/11 4:20 PM 752 Unit 9 Coordinates, Area, Volume, and Capacity
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