Volume of Pyramids and Cones


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1 Volume of Pyramids and Cones Objective To provide experiences with investigating the relationships between the volumes of geometric solids. 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 to find the volume of geometric solids. [Measurement and Reference Frames Goal 2] Compare the properties of pyramids, prisms, cones, and cylinders. [Geometry Goal 2] Describe patterns in relationships between the volumes of prisms, pyramids, cones, and cylinders. [Patterns, Functions, and Algebra Goal 1] Key Activities Students observe a demonstration in which models of geometric solids are used to show how to find the volumes of pyramids and cones. Students then calculate the volumes of a pyramid and a cone. Materials Math Journal 2, p. 379 Study Link 11 3 Math Masters, pp. 334 and 440 for demonstration: 1 piece card stock, food can, dry fill slate calculator Playing Rugs and Fences Math Journal 2, p. 380 Math Masters, pp Class Data Pad scissors Students practice calculating the perimeters and areas of polygons. Math Boxes 11 4 Math Journal 2, p. 381 Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1. [Number and Numeration Goal 2] Study Link 11 4 Math Masters, p. 335 Students practice and maintain skills through Study Link activities. READINESS Finding the Areas of Concentric Circles Math Masters, p. 336 per partnership: crayons or colored pencils (yellow, orange, and red) Class Data Pad (optional) Students compare the areas of different circular regions. ENRICHMENT Measuring Regions Math Masters, p. 336 Students investigate the areas of concentric circle regions in relation to their boundaries. EXTRA PRACTICE 5Minute Math 5Minute Math, pp. 144, 147, and 229 Students identify the properties of geometric solids. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 142 Students define and illustrate the term volume of a cone. Advance Preparation For Part 1, copy Math Masters, page 334 on card stock. Cut out the templates; score the dashed lines; fold them so that the markings are on the inside of the shapes; and tape the sides together completely to seal the seams. If you cannot copy onto card stock, tape the master to card stock, cut along the solid lines for each pattern, and then draw the corresponding dashed lines. Copy Math Masters, page 440. Cut out the cone template. Curl it into position by lining up the two heavy black lines and the sets of dotted gray lines. Seal the cone along the seams inside and out. You will need a 15 or 16oz food can with the top removed. Place the cone in the can so its tip touches the base of the can. Locate the line on the inside of the cone that touches the can s rim. Cut along the line to remove the excess. You will need about 1 pound of dry fill: rice, sugar, or sand. Teacher s Reference Manual, Grades 4 6 pp. 185, 186, Unit 11 Volume
2 Getting Started Mental Math and Reflexes Have students complete each sentence by using the relationship between multiplication and division. 2 8 = 16 because 16 8 = = 6 because 6 2 = = 27 because 27 3 = = 48 because = = 25 because 25 5 = 5. 7 = 42 because 42 6 = = 20 because 20 5 = = 16 because 16 2 = = 100 because = 10. Math Message A rectangular prism and a cylinder each have exactly the same height and exactly the same volume. The base of the prism is an 8 cm 5 cm rectangle. What is the area of the base of the cylinder? Study Link 11 3 FollowUp Have students share their volume measurements with the class. 1 Teaching the Lesson Math Message FollowUp WHOLECLASS DISCUSSION Algebraic Thinking Ask volunteers to share their solution strategies. Since the prism and cylinder each have the same volume and height, the areas of their bases must also be equal. The area of the base of the prism is 40 cm 2 (8 cm 5 cm), so the area of the base of the cylinder must also be 40 cm 2. Exploring the Relationship between the Volumes of Prisms and Pyramids (Math Masters, p. 334) PROBLEM SOLVING WHOLECLASS Gather the class around a desk or table. Show them the prism and pyramid you have made. Turn the pyramid so that the apex is pointing down and show that, when the pyramid is placed inside the prism, the boundaries of their bases match and the apex of the pyramid will touch a base of the prism. The two solids you made will fit in this way because they have identical bases and heights. Have students guess how many pyramids filled with material it would take to fill the prism. Select a pair of volunteers to follow the procedure on the next page: Prism and pyramid patterns from Math Masters, page 334 Lesson
3 1. Fill the pyramid with dry fill so that the material is level with the top. Empty the material into the prism. 2. Fill the pyramid again and empty the material into the prism. 3. Repeat until the prism is full and level at the top. It will take about 3 pyramids of material to fill the prism. Students need not find the actual volumes of either the prism or the pyramid. It is enough for them to discover that about 3 pyramids of material fill the matching prism. Ask students to state the relationships between the volumes of the two shapes. The volume of the prism is 3 times the volume of the pyramid. The volume of the pyramid is the volume of the prism. 3 Exploring the Relationship between the Volumes of Cylinders and Cones (Math Journal 2, p. 379; Math Masters, p. 440) PROBLEM SOLVING WHOLECLASS Date 11 4 Time Volume of Pyramids and Cones 1. To calculate the volume of any prism or cylinder, you multiply the area of the base by the height. How would you calculate the volume of a pyramid or a cone? I would multiply the area of the base by the height and divide the product by 3. The Pyramid of Cheops is near Cairo, Egypt. It was built about 2600 B.C. It is a square pyramid. Each side of the square base is 756 feet long. Its height is feet. The pyramid contains about 2,300,000 limestone blocks. 2. Calculate the volume of this pyramid. ft 3 3. What is the average volume of one limestone block? 37 ft ft A movie theater sells popcorn in a box for $2.75. It also sells cones of popcorn for $2.00 each. The dimensions of the box and the cone are shown below. 3 in. Cone pattern from Math Masters, page 440 Student Page 7 in. 85,635,144 6 in. The shaded portion overlaps on the outside ft Algebraic Thinking Repeat the demonstration using a 15 or 16oz food can and the cone you have made. Turn the cone upside down and show that, when the cone is placed inside the cylinder, the boundaries of their bases match and the apex of the cone will touch a base of the cylinder. The two solids fit because they have identical bases and heights. Have students guess how many cones of material will fill the can. Expect that many students will correctly guess 3 because of the previous demonstration. Select another pair of volunteers to follow the same procedure as before. It takes about 3 cones of material to fill the cylinder. Ask students to complete Problem 1 on journal page 379: How would you calculate the volume of a pyramid or a cone? Then have students share their solution strategies. Emphasize the following points: Since the volume of a pyramid (cone) is the volume of a prism 3 (cylinder) with an identical base and height, you can calculate the volume of a pyramid (cone) by multiplying the area of the base of the pyramid (cone) by its height and then dividing the result by 3. A formula for finding the volume of a pyramid or cone is V = 3 B h. 9 in. $ Calculate the volume of the box. in 3 5. Calculate the volume of the cone. in 3 Try This 6. Which is the better buy the box or the cone of popcorn? Explain. The box is the better buy. 275 / cents per cubic inch for the box, and 200 / cents per cubic inch for the cone. 10 in. $2.00 Solving Volume Problems (Math Journal 2, p. 379) INDEPENDENT Algebraic Thinking Assign the rest of journal page 379. Circulate and assist. When students have completed the page, ask them to share their solution strategies. Emphasize the following points: Math Journal 2, p Unit 11 Volume
4 Problem 2 Date Student Page Time A rectangular prism with the same base and height as the Pyramid of Cheops would have a volume of B h = = 256,905,432 ft 3. The pyramid s volume is 3 as much, or 85,635,144 ft3. Problem 3 To find the average volume of a block, divide the total volume by the number of blocks: _ 85,635,144 = slightly more than 2,300,00 37 cubic feet per block. Problem 4 The popcorn box is a rectangular prism whose volume equals 189 in 3. Problem 5 A cylinder with the same base and height as the popcorn cone would have a volume of B h = (π 3 2 ) 10 = 283 in 3. The cone s volume is 3 as much, or 94 in3. Problem Rugs and Fences: An Area and Perimeter Game Materials 1 Rugs and Fences Area and Perimeter Deck (Math Masters, p. 498) 1 Rugs and Fences Polygon Deck (Math Masters, pp. 499 and 500) 1 Rugs and Fences Record Sheet (Math Masters, p. 501) Players 2 Object of the game Directions To score the highest number of points by finding the area and perimeter of polygons. 1. Shuffle the Area and Perimeter Deck, and place it facedown. 2. Shuffle the Polygon Deck, and place it facedown next to the Area and Perimeter Deck. 3. Players take turns. At each turn, a player draws one card from each deck and places it faceup. The player finds the perimeter or area of the figure on the Polygon card as directed by the Area and Perimeter card. If a Player s Choice card is drawn, the player may choose to find either the area or the perimeter of the figure. If an Opponent s Choice card is drawn, the other player chooses whether the area or the perimeter of the figure will be found. 4. Players record their turns on the record sheet by writing the Polygon card number, by circling A (area) or P (perimeter), and then by writing the number model used to calculate the area or perimeter. The solution is the player s score for the round. 5. The player with the highest total score at the end of 8 rounds is the winner. Math Journal 2, p. 380 The box is the better buy. Ask students to calculate a costvolume ratio for each container: _ 200 = 2.13 cents per cubic 94 inch for the cone, and _ 275 = 1.46 cents per cubic inch for the box Ongoing Learning & Practice Playing Rugs and Fences (Math Journal 2, p. 380; Math Masters, pp ) Algebraic Thinking Students practice calculating the perimeter and area of polygons by playing Rugs and Fences. Write P = perimeter, A = area, b = length of base, and h = height on the Class Data Pad. Ask students to define perimeter. The distance around a closed, 2dimensional shape Then have volunteers write the formulas for the area of a rectangle, A = b h, or A = bh a parallelogram, A = b h, or A = bh and a triangle A = (b h), or A = bh on the board or Class Data 2 2 Pad. Read the game directions on journal page 380 as a class. Partners cut out the cards on the Math Masters pages and play eight rounds, recording their score on Math Masters, page 501. See the margin for the area and perimeter of each figure. Polygon Deck B Polygon Deck C Card A P Card A P The area (A) and perimeter (P) of the polygons in Rugs and Fences Lesson
5 Date Solve. a. 1 3 of 36 b. 2 5 of 75 c. 3 8 of 88 Math Boxes d. 5 6 of 30 e of Lilly earns $18.75 each day at her job. How much does she earn in 5 days? Open sentence: d Solution: d $ Make a factor tree to find the prime factorization of º 16 2 º 2 º 8 2 º 2 º 2 º 4 2 º 2 º 2 º 2 º 2 Student Page Time 2. Find the volume of the solid. Volume B h where B is the area of the base and h is the height. 4. Solve. 3 units a. 2 c fl oz b. 1 pt fl oz c. 1 qt fl oz d. 1 halfgal fl oz e. 1 gal fl oz 6. Jamar buys juice for the family. He buys eight 6packs of juice boxes. His grandmother buys three more 6packs. Which expression correctly represents how many juice boxes they bought? Circle the best answer. A. (8 3) 6 B. 6 (8 3) C. 6 (8 3) area of base 30 units 2 Volume 90 units Math Journal 2, p Math Boxes 11 4 (Math Journal 2, p. 381) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 112 and The skill in Problem 5 previews Unit 12 content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 1 Use Math Boxes, Problem 1 to assess students ability to calculate a fraction of a whole. Students are making adequate progress if they correctly identify each of the five values. [Number and Numeration Goal 2] Study Link 11 4 (Math Masters, p. 335) INDEPENDENT Home Connection Students compare the volumes of geometric solids. 3 Differentiation Options Name Date Time STUDY LINK 11 4 Comparing Volumes Use,, or to compare the volumes of the two figures in each problem below cm 24 ft 6 cm 3 yd base is a square height of base 2 yd 5 m height 6 m 5 m 4. Explain how you got your answer for Problem 3. Because both pyramids have the same height, compare the areas of the bases. The base area of the square pyramid is 5 º 5 25 m 2. The base area of the triangular pyramid is 1 2 º 5 º 5, or 12.5 m2. Practice Study Link Master cm 8 yd height 6 m 6 cm 5 m 6 ft 6 cm height of base 5 m 7. 6 º , READINESS Finding the Areas of Concentric Circles (Math Masters, p. 336) 5 15 Min Algebraic Thinking To explore the relationship between radius and the area of circles, have students compare the areas of different circular regions. Read the introduction to Math Masters, page 336. Write the formula for finding the area of a circle: A = π r r or A = π r 2 on the board or the Class Data Pad, and then make the table shown below. Ask students to use the formula to find the area for each circle. Radius Area 1 in in 2 2 in in 2 3 in in 2 4 in in 2 5 in in 2 Have partners solve Problem 1. Suggest that they discuss their strategy before they begin. Circulate and assist. Math Masters, p Unit 11 Volume
6 When students have finished, have volunteers explain their solution strategies. Sample answers: In Problem 1, we know that the area for the red region is the same as the area of the third circle. The area for the orange region is the same as the area for the fifth circle minus the area for the fourth circle. The area for the red region is the same as the area for the orange region. ENRICHMENT Measuring Regions (Math Masters, p. 336) Min Name Date Time 11 4 Teaching Master Finding the Area of Concentric Circles Concentric circles are circles that have the same center, but the radius of each circle has a different length. The smallest of the 5 concentric circles below has a radius of 1 in. The next largest circle has a radius of 2 in. The next has a radius of 3 in. The next has a radius of 4 in., and the largest circle has a radius of 5 in. The distance from the edge of one circle to the next larger circle is 1 in. 1 in. 1 in. 1 in. 1 in. 1 in. orange yellow red To apply students understanding of area, have them modify the distance between concentric circles to enlarge or shrink regions. Have partners complete Problem 2 on Math Masters, page 336. When students have finished, have them share and discuss their solution strategies. For Problem 2a, I would make a table to record region areas and the distance between the circles. Then I would use guessandcheck to increase the radius of the circle for the yellow region until it is about twice the area of the red region. For Problem 2b, I would decrease the red region to make the areas of the yellow and orange regions equal. 1. Use colored pencils or crayons to shade the region of the smallest 3 circles red. Shade the region that you can see of the next circle yellow, and the region that you can see of the largest circle orange. Which region has the greater area, the red region or the orange region? They are equal. 2. a. How can you change the distance between the circles to make the area of the yellow region equal to the area of the red region? Explain your answer on the back of this page. b. How can you change the distance between the circles to make the area of the yellow region equal to the area of the orange region? Explain your answer on the back of this page. Math Masters, p. 336 EXTRA PRACTICE SMALLGROUP 5Minute Math 5 15 Min To offer students more experience with identifying the properties of geometric solids, see 5Minute Math, pages 144, 147, and 229. ELL SUPPORT Building a Math Word Bank (Differentiation Handbook, p. 142) Min To provide language support for volume, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the phrase volume of a cone and write words, numbers, symbols, or draw pictures that are related to the term. See the Differentiation Handbook for more information. Planning Ahead Be sure to collect the materials listed at the end of Lesson 113 before the start of the next lesson. Lesson
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