Mathematics Success Grade 6

Transcription

1 Mathematics Success Grade 6 T683 [OBJECTIVE] The student will determine the volume of rectangular prisms with fractional edge lengths and solve problems in mathematical and real-world situations. [PREREQUISITE SKILLS] volume of prisms with whole number sides, area [MATERIALS] Student pages S357 S369 Centimeter cubes (50 cubes per pair) Foldable from Lesson 28 [ESSENTIAL QUESTIONS] 1. What does volume measure? 2. Why is volume measured in cubic units? 3. Explain how to find the volume of a rectangular prism with fractional edge lengths. [WORDS FOR WORD WALL] volume, rectangular prism, fractional edge lengths [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Algebraic Formula, Verbal Description, Concrete Representation, Pictorial Representation [WARM-UP] (IP, I, WG) S357 (Answers on T691.) Have students turn to S357 in their books to begin the Warm-Up. Students will find the area of rectangles to prepare for finding the volume of rectangular prisms. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review the answers as a class. {Algebraic Formula, Pictorial Representation} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [1-2 Days (1 day = 80 minutes) M, GP, WG, CP, IP, I] SOLVE Problem (GP, WG) S358 (Answers on T692.) Have students turn to S358 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to find the volume of prisms. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer}

2 T684 Mathematics Success Grade 6 Review Activity Volume of Rectangular Prisms (M, GP, CP, WG, IP) S358, S359 (Answers on T692, T693.) M, GP, WG, CP: Pass out the cubes to each pair of students. Use the following activity to complete the activity on S358 and S359 with your students. {Concrete Representation, Algebraic Formula, Verbal Description} MODELING Review Activity - Volume of a Rectangular Prism Step 1: Have students use cubes to create a rectangle that is 3 cubes long and 2 cubes wide. Partner A, what is the length? (3) Record. Partner B, what is the width? (2) Record. Partner A, what is the area of the base? (6) Record. Partner B, what is the height of the prism? (1) Record. Partner A, what is the total number of cubes? (6) Record. Step 2: Have students add another layer of cubes that is the exact same size. Partner B, what is the length? (3) Record. Partner A, what is the width? (2) Record. Partner B, what is the area of the base? (6) Record. Partner A, what is the height of the prism? (2) Record. Partner B, what is the total number of cubes? (12) Record. Have student pairs discuss and make a prediction about the total number of cubes that you will need to make a prism with a height of 3 layers. (Answers will vary.) Record. Step 3: Have students add another layer of cubes that is the exact same size. Partner A, what is the length? (3) Record. Partner B, what is the width? (2) Record. Partner A, what is the area of the base? (6) Record. Partner B, what is the height of the prism? (3) Record. Partner A, what is the total number of cubes? (18) Record. Step 4: Have student pairs discuss Question 6 and 7. Partner B, what pattern do you notice about the total number of cubes as another layer is added? (Each time a layer is added, the number of cubes increase by 6.) Record. Partner A, what does this mean about the volume of the prism as we add each layer? (This means that the volume increases by 6 cubic units for each layer.) Record. Partner B, what is the formula for the volume of a prism? (V= lwh or V = Bh) Record.

3 Mathematics Success Grade 6 T685 Have partners discuss what they know about the pattern of the cubes to make a prediction about the total number of cubes that you will need to make a prism with a height of 4 layers. Partner A, how many cubes will there will be for four layers of the prism? ( = 24: A prism with the same size base and a height of 4 would have a total number of 24 cubes.) Record. Partner B, how could we confirm our prediction using the formula? (V = lwh or V = Bh) Record. Teacher Note: For students that need to have more support as they review the volume of Rectangular prisms, use Steps 5 9 below to model Examples 1-4 on S3. Step 5: Have students look at Example 1 on S359. Remind students that to find the volume, they can multiply the length, width, and height of the figure. Label the length (5), the width (3), and the height (2). Remind them to always start by writing the formula and then substituting in the values. Explain that because the unit of measure of the prism in the example is not specified, students will use unit. The answer will be in units cubed (u 3 ). If students have difficulty understanding the units cubed, remind them that each unit is a cube. If they were to count the number of cubes in Example 1, there would be 30 cubes. Each cube is 1 cubic unit, so that will be a total of 30 cubic units. (If you used cm cubes, that measurement would be 30 cm 3.) Step 6: In the second column, model for students how to write the formula for the volume of a rectangular prism (V = lwh). Step 7: Look at Example 2 with students. Point out the length (20), width (6), and height (7). Model writing the formula and then substituting the values to find the volume in cubic millimeters. Also have students write the formula in the middle column. Step 8: Look at Example 3 with students. After reading the problem, underline each of the dimensions in the problem. Then model writing the formula and substituting the values to find the volume. Also have students write the formula in the middle column. Step 9: Look at Example 4 with students. After reading the problem, underline each of the dimensions in the problem. Ask students how they think you should find the volume, if you know the area of the base, and not the length and width. Remind them that area = length times width, so they need to multiply the base area by the height. (You can also refer back to the cube activity, where the area of the base was always the same, and the height changed when students added another layer.)

4 T686 Mathematics Success Grade 6 CP, IP, WG: Have students complete the problems on the chart on S359 to determine the volume of the figures in Problems 5 8. Review the answers as a whole group. {Pictorial Representation, Algebraic Formula, Verbal Description, Graphic Organizer} Volume of Prisms with Fractional Edge Lengths (M, GP, CP, WG, IP) S360, S361, S362 (Answers on T694, T695, T696.) M, GP, WG, CP: Students will use the cubes from the activity with centimeter cubes to apply the volume of the concrete and pictorial representations with fractional edge lengths. {Pictorial Representation, Algebraic Formula, Verbal Description, Graphic Organizer} MODELING Volume of Prisms with Fractional Edge Lengths Step 1: Let s use the same cubes that we used to create the prism on page S358. Use cubes to create a rectangle that is 2 cubes long and 2 cubes wide Partner A, what is the area of the bottom of your rectangular prism? (4 square units) Record. Partner B, what is the height of your rectangular prism? (1) Record. The length? (2) Record. The width? (2) Record. Partner A, how many cubes did it take to make your rectangular prism? (4 cubes) Record. Partner B, what is the volume of each section of the prism? (1 cubic unit) Record. Step 2: Let s look at the prism in a different way. Think about the rectangular base as having the measurements of 1 unit by 1 unit. [This means that the length of (2) cubes would equal (1) unit. If the length of (2) cubes is equal to (1) unit then the length of (1) cube is equal to ( 1 of a unit.] Record. 2) 1 unit Partner A, what are the measurements of the prism now? (Width: 1 unit length: 1 unit height: 1 unit) Record. 2

5 Mathematics Success Grade 6 T687 Step 3: On top of your cubes, add another layer of cubes that is the exact same size. Partner B, what are the measurements of the prism now? (Width: 1 unit length: 1 unit height: 1 unit) Record. Partner A, what is the total volume? [V = lwh V = (1)(1)(1) = 1 cubic unit] Record. Step 4: Have students turn to page S361. Partner B, what is the width of one of the cubes? 2 Partner A, what is the length of one of the cubes? 2 Partner B, what is the height of one of the cubes? ( 1 1 unit unit) Record. 2 Have students discuss the relationship between the volume of the whole cube and each individual cube. Partner A, describe the relationship. [The volume of the whole cube is 1 cubic unit. There are 8 smaller cubes in the large cube. The volume of each smaller cube is ( 1 Record. 8) of the volume of the larger cube.] Step 5: Can we use the volume formula in this situation? (Yes) Record. Partner B, how do we determine the volume of an individual cube? (Multiply the length times the width times the height.) Record. Partner A, what is the volume of one of the cubes? [ ( 1 2) ( 1 2) ( 1 2) = ( 1 8) cubic unit ] Let s look at another cube. Create a cube with three layers. Each layer should have a length of 3 small cubes and a width of 3 small cubes. Now let s look at the figure as having a length of 1 unit, a width of 1 unit and a height of 1 unit. Partner B, what is the width of one of the cubes? 3 What is the length of one of the cubes? 3 What is the height of one of the cubes? 3 1 unit 1 unit

6 T688 Mathematics Success Grade 6 Partner A, explain the relationship between the volume of the whole cube and each individual cube. [The volume of the whole cube is 1 cubic unit. There are 27 smaller cubes in the large cube. The volume of each smaller cube is ( 1 of the volume of the larger cube.] Record. 27) Partner B, what is the volume of one of the cubes? [( 1 3) ( 1 3) ( 1 3) = ( 1 27) cubic unit] Step 6: Have students turn to page S362 and look at Example 1. Have student pairs discuss how they can determine the volume of any rectangular prism. (multiply the length, width, and height of the figure) Partner A, are the dimensions of the prism given? (No) Have student pairs discuss strategies that they can use to determine the dimensions using the information given. Partner B, what information is given? (Each cube has a side length of 1 2 unit) Have students discuss how we can use that information to determine the length, width and height. Partner A, explain your thinking. (We can multiply the measure of the edge length by the number of cubes in each dimension.) Partner B, what is the length of the prism? [ 5 ( 1 2) = 5 2)] Record. Partner A, what is the width of the prism? [ 3 ( 1 2) = 3 2)] Record. Partner B, what is the height of the prism? [ 2 ( 1 2) = 2 2)] Record. Partner A, how can we determine the volume of the figure? (Use the volume formula: V = lwh.) Partner B, what is the volume of the prism? [ V = ( 15 8 ) = cubic units] Record. Step 7: Model Examples 2 and 3 using the directions from Step 6. CP, IP, WG: Have students complete Problems 4 6 in the graphic organizer with a partner. Review the answers as a whole group. {Pictorial Representation, Algebraic Formula, Verbal Description, Graphic Organizer}

7 Mathematics Success Grade 6 T689 Foldable (M, GP) Have students use the foldable from Lesson 28 to add the information about the Volume of a rectangular prism. {Algebraic Formula} MODELING Foldable Step 1: Open the square. Fold each corner into the center. Write Volume of Rectangular Prisms on one outside flap. Three outside flaps will be blank to be written on later. Fold all 4 corners into the center. Rectangular Prisms Volume Step 2: Pull up the flap that says Volume of Rectangular Prisms. On the triangle that sticks up, draw a rectangular prism, with the length, base, and height labeled. Underneath the prism, write the formula. On the square portion, draw another prism, with values for the dimensions. Find the volume using the formula. See the example foldable for the information. Step 3: Turn the graphic organizer over and write the information for Volume of a Rectangular Prism using fractional edge lengths. SOLVE Problem (WG, GP) S363, S364, S365, S366, S367 (Answers on T697, T698, T699, T700, T701.) Have students turn to S363 in their books. Remind students that the SOLVE problem on S363 is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Students will determine the volume of a rectangular prism with fractional edge lengths.) {SOLVE, Algebraic Formula, Verbal Description, Graphic Organizer} IP, CP, WG: Have students complete the SOLVE problems on S364- S367 {SOLVE, Verbal Description, Graphic Organizer, Algebraic Formula}

8