# 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 Stick-On Note Math Masters, p. 281 per partnership: one stick-on note, pad of unused stick-on notes, centimeter cube Students compare the volume of a single stick-on note to the volume of a centimeter cube. EXTRA PRACTICE 5-Minute Math 5-Minute 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 Follow-Up 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 10-centimeter 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 10-centimeter 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 half-gallons 1 half-gallon = 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 half-gallon = 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 Follow-Up (Student Reference Book, p. 195) WHOLE-CLASS 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 2-dimensional 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 3-dimensional have length, width, and thickness, so they enclose a certain amount of space. The amount of surface inside a 2-dimensional 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 3-dimensional 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 3-dimensional 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

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 five-day 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

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