Dear Family, Content Overview In this unit, we are reviewing the volume of rectangular prisms with whole number edge lengths, and analyzing the difference between surface area and volume along with the units used to measure them. The surface area of a solid figure is the total area of all its faces. Volume is the measure of the space that a three-dimensional figure occupies. The solid figure below is made of centimeter cubes. l l is the length. w is the width. h is the height. B is the area of the base. h w B Surface Area SA = (2 × 10) + (2 × 2) + (2 × 5) = 34 Surface Area = 34 cm2 Volume V = lwh or V = Bh V = 5 × 2 × 1 or V = 10 × 1 Volume = 10 cm3 Surface area is measured in square units. Volume is measured in cubic units. This unit also introduces the volume of rectangular prisms with fractional edge lengths. 1 V = Bh = 12 • 2__ = 30, 2 Volume = 30 unit3 3 1 1 V = lwh = 6__ • 2__ • 3 = 48__ , 4 2 2 3 Volume = 48__ unit3 4 If you need practice materials or if you have any questions, please call or write to me. Sincerely, Your child’s teacher Unit 6 addresses the following standards from the Common Core State Standards for Mathematics with California Additions: 6.G.1, 6.G.2, 6.G.4, 6.EE.2, 6.EE.2c, and all Mathematical Practices. UNIT 6 LESSON 1 What Is Volume? 147 Estimada familia: Un vistazo general al contenido En esta unidad, repasaremos cómo obtener el volumen de prismas rectangulares cuyos lados tienen longitudes expresadas en números enteros. También analizaremos la diferencia entre el área total y el volumen y examinaremos las unidades de medida usadas. El área total de un cuerpo geométrico es la suma del área de todas sus caras. El volumen es la medida del espacio que ocupa una figura tridimensional. l h a Ab Área total At = (2 × 10) + (2 × 2) + (2 × 5) = 34 Área total = 34 cm2 l es el largo a es el ancho h es la altura Ab es el área de la base Volumen V = lah ó V = Abh V = 5 × 2 × 1 ó V = 10 × 1 Volumen = 10 cm3 El área total se mide en unidades cuadradas. El volumen se mide en unidades cúbicas. En esta unidad también se presenta el volumen de prismas rectangulares cuyos lados tienen longitudes expresadas en fracciones. 1 V = Abh = 12 • 2__ = 30, 2 Volumen = 30 unidades3 3 1 1 V = lah = 6__ • 2__ • 3 = 48__ , 4 2 2 3 Volumen = 48__ unidades3 4 Si necesita material para practicar o si tiene preguntas, por favor comuníquese conmigo. Atentamente, El maestro de su hijo En la Unidad 6 se aplican los siguientes estándares auxiliares, contenidos en los Estándares estatales comunes de matemáticas con adiciones para California: 6.G.1, 6.G.2, 6.G.4, 6.EE.2, 6.EE.2c, y todos los de prácticas matemáticas. 148 UNIT 6 LESSON 1 What Is Volume? 6–1 ► Nets for Cubic Units and a Rectangular Prism Cut out the nets and form the solid figure. Unit Cube 1 unit 1 unit 1 unit 1 in. 1 in. 1 in. 1 cm 1 cm 1 cm UNIT 6 LESSON 1 What Is Volume? 149 6–1 ► Nets for Cubic Units and a Rectangular Prism (continued) 150 UNIT 6 LESSON 1 What Is Volume? 6–1 Content Standards 6.G.1, 6.G.4, 6.EE.2, 6.EE.2c Mathematical Practices MP.3, MP.6, MP.7 ► Cubic Units The volume of a solid figure is the amount of space occupied by the figure. Volume is measured in cubic units. Vocabulary volume unit cube centimeter cube inch cube cubic unit (unit3) cubic centimeter (cm3) cubic inch (in.3) 1. How can you measure the amount of space each of these rectangular prisms takes up? How much space is inside each of the rectangular prisms? How many unit cubes does it take to fill the rectangular prism? A unit cube is a cube with each edge 1 unit long. The volume of a unit cube is 1 cubic unit. It can be written 1 cubic unit or 1 unit3. V= 1 unit3 1 unit 1 unit 1 unit 2. Label the length, width, and height of the centimeter cube on the right. Write the volume of a centimeter cube in two ways. Write the volume of the cube in two ways. 3. inch cube UNIT 6 LESSON 1 4. meter cube 5. foot cube 6. yard cube What Is Volume? 151 152 UNIT 6 LESSON 1 What Is Volume? 6–2 ► Nets for Part of a Unit Cube Cut out the nets and form the solid figures. A C B UNIT 6 LESSON 2 Fractional Unit Cubes 153 ► Nets for Part of a Unit Cube (continued) 154 UNIT 6 LESSON 2 Fractional Unit Cubes 6–3 ► Prism Layers Cut out the nets and form the solid figures. UNIT 6 LESSON 3 Compose Rectangular Prisms with Fractional Edge Lengths 155 ► Prism Layers (continued) 156 UNIT 6 LESSON 3 Compose Rectangular Prisms with Fractional Edge Lengths Unit 6 1. Select True or False for each statement. 1a. Surface area and volume both measure the amount of space occupied by a solid figure. True False 1b. Surface area and volume are measured using the same unit. True False 1c. Surface area and volume are both attributes of solid figures. True False 2. Fill in the bubble next to the unit used to measure volume. square units cubic units linear units Explain your answer choice in words or with a labeled sketch. 3. The volume of a rectangular pyramid can be found using the formula V = Bh. Part A: Use the tiles to show a different formula you can use to find the volume of a rectangular prism. You will not need to use every tile. w h B l V= Part B: Explain how the two formulas are related, and why you can use either formula to find the volume of a rectangular prism. 4. The equation for the volume of the square prism is V = Mt. Label the prism. UNIT 6 TEST 157 Unit 6 5. Choose one value from each column to make the volume of the rectangular prism true. V = 24 ft3 2 ft w 3 5 9 10 l l 1 2 3 4 w 6. Use mental math to find the volume. V= 10 in. 5 in. 5 in. 7. The volume of a prism can be found by packing it with unit cubes of the appropriate edge lengths. Part A 1 ft 5 ft 4 1 ft 4 What is the longest edge length that would be appropriate for the unit cubes? Explain your answer. edge lengths Part B If the longest dimension of the prism was 1_1 feet, would the 8 length you chose in Part A be sensible? Explain. 158 UNIT 6 TEST Unit 6 8. Cecilia used layers to find the volume of the prism below. Part A Explain how to find the volume using layers. Part B Use layers to find the volume of the prism. 1 4 2 units 1 6 2 units 4 units V= cubic units 9. Find the volume. Show your work. 3 ft 4 2 ft 3 1 ft 2 V= 10. The dimensions on the tiles below are values for l and h. 1 in., identify those expressions that For a prism with w = 2__ 2 will produce a volume of 270 in.3 and those that will produce a volume of 300 in.3 9 in. • 12 in. 1 2__ in. • 48 in. 2 1 16 in. • 7__ in. 2 15 in. • 8 in. Volume: 270 in.3 UNIT 6 TEST 1 24 in. • 4__ in. 2 1 in. 2 2 6 in. • 18 in. 20 in. • 2 5__ in. 5 h l Volume: 300 in.3 159 Unit 6 11. Choose one value from each column to make the volume of the rectangular prism true. w 3 V = 24 4 ft3 h 1 1__ 4 3 3 1__ 4 4 2 5 1 2__ 4 6 h 2 3 ft 4 w Select Yes or No to indicate if the equation represents the volume for the prism with sides e. 12a. V = e • e • e Yes No 12b. V = 3e Yes No 12c. V = e3 Yes No 13. Substitute the dimensions shown below into the volume formula to find the volume of the prism. Show your work. 1 yd 3 3 yd 1 4 yd 2 V= 160 UNIT 6 TEST e e e Unit 6 14. Anthony labeled this rectangular prism. Part A t Select the equations that represent the volume of the prism. Mark all that apply. A V= b+ b+ t C V = b2t B V= b• b•t D V = b2 + t b b Part B Show your work. Find the volume of the prism if b = 3 cm and t = 7.5 cm. V= 15. Write an equation for the volume of the prism using the given variables. W V= c 16. A gift box has a volume of 199 in.3 The area of the base of 3 in.2 What is the height? the box is 49__ 4 h= 1 17. A small pool in the shape of a rectangular prism has a length of 6__ ft, 2 a width of 5 ft and a height of 24 in. Jeb says the volume of the pool is 780 ft3. Is Jeb correct? If he is, show how he may have found the volume. If not, explain what he did wrong and give the correct volume. V= UNIT 6 TEST 161 Unit 6 18. Choose Yes or No to indicate if the given dimensions represent a rectangular prism with a volume of 96 cubic inches. 18a. 2 in. • 6 in. • 8 in. Yes No 18b. 2 in. • 4 in. • 12 in. Yes No 18c. 4 in. • 12 in. • 4 in. Yes No 18d. 3 in. • 4 in. • 9 in. Yes No 18e. 4 in. • 4 in. • 6 in. Yes No 19. What is the volume of a rectangular prism with a base area of 52 square inches and a height of 14 inches? Show your work. V= 20. The volume of a rectangular prism is 6,037.5 cubic inches. The area of the base of the prism is 525 square inches. What is the height of the prism? Show your work. h= 21. John wants to place a layer of mulch on top of his 12 ft by 24 ft vegetable garden. If a 1.5 ft3 bag of mulch costs $4, how much will it cost to place a 3-inch layer of mulch on the garden? Show your work. $ 162 UNIT 6 TEST

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