D Formulas and D Formulas and Geometry Geometry Circles and Solids Notes Valerie wants to make a mold she can later use to make candles. She decides to use a cylinder-shaped mold. For the base of the mold, she has cut a circle that has a 6-cm diameter. 6 cm diameter 13a If compasses are not available, a pencil with a short piece of string tied to it could be used to draw the circle. Remind students that the diameter is 6 cm, not the radius. 13b Students will need strips about 20 cm long to do this. They can cut this from notebook paper. 13c To visually demonstrate the relationship between diameter and circumference, you could use a piece of string the length of a diameter and show that it takes approximately 3 of these to reach around the circle. height overlap 1 cm base 13. a. Make an accurate drawing of a circle that is 6 cm in diameter. Use a compass. b. Use a strip of paper to find the size of the mantle of the mold. Allow at least 1 cm overlap to glue the mantle together. What are the measurements of the mantle without the overlap? Valerie used this formula for the mantle of her mold: circumference of a circle π diameter c. Explain why this formula makes sense. Fruit drinks come in cans of different sizes. Some cans are narrow and tall; others are wide and short. 14. a. What shapes are juice cans usually? b. Is it possible for cans in different shapes to contain the same amount of liquid? Reaching All Learners Hands-On Learning You might show different size cans to students and ask them to estimate the volume. Why would a manufacturer make a narrow and tall tin can as opposed to a wider can for paint? If you have an actual cylindrical carton available, show students what the net would look like before starting with problem 13. Toilet paper rolls are also useful. English Language Learners For problem 13b, you might want to explain what is meant by the mantle of the mold. 38 Building Formulas Solutions and Samples Hints and Comments 13. a. Check the diameter of the circle students made. Neatness of the drawing is important. Materials b. Note that the height of the mold is not important. Answers for the height will vary. Check the use of measurement units. Accept answers between 18 and 19 centimeters for the width. c. The formula gives Valerie the width of the rectangle she needs for the mantle of her mold. 14. a. Juice cans are usually the shape of a cylinder. b. Answers will vary. Most students will answer that it is possible. Juice cans that look completely different may contain the same amount of liquid. compass (one per student); sheets of blank paper (one per student) Overview Students investigate the shape of cans and use a formula to find the dimensions. About the Mathematics In this part of Section D, formulas for volume and area, which students were introduced to in the unit Reallotment, are reviewed and used in different contexts. The net of a mold for candles and the net of a juice can allow students to calculate the surface area of a cylinder. Planning Students may work on problems 13 and 14 in pairs or small groups. Section D: Formulas and Geometry 38T D Formulas and Formulas and Geometry D Geometry This juice can is made up of two circles and a rectangle. Notes 15a Before starting this page, compare and contrast the formulas for area and circumference. 15b If the volume formula is new to students, explain the terms Base and Height. 15c Students need to find out themselves that they need the circumference of a circle to find the area of the mantle. Or you may prefer to give students the formula for the circumference of a circle before they start working on problem 15. 15c Some students may need to roll a sheet of paper to model the rectangle and see that one dimension is the circumference of the base of the cylinder. The can shown in the drawing has a height of 15 cm. The diameter of the bottom is 7 cm. 15. a. Calculate the area of the bottom of the can. b. Calculate the volume of the can. Remember that the formula for the volume of any cylinder is: Volume area of Base Height c. What are the measurements of the rectangle that makes the sides of the can? d. The can is made of tin. How much tin (in cm2) is needed to make this can? This type of fruit juice is also available in cans that are twice as high. 16. a. Compare the amounts of fruit juice that each can contains. b. How do the surface areas of the cans compare? Be prepared to explain your answer without making calculations. 17. Suppose one can has double the diameter of another can. a. Do you think the amount of liquid that fits in the larger can will double? Give mathematical reasons to support your answer. b. What can you tell about the surface area of the larger can compared to that of the original can? Reaching All Learners Hands-On Learning If students find it difficult to answer the questions on this page, it may be helpful if you physically place one can on top of the other to see that the volume doubles if the height doubles. In order to show what happens if the diameter doubles, suggest that they draw circles to visualize the two cylinders. First they trace the end of a can to represent the smaller can. Then they use a compass to draw the larger circle. Extension After completing problem 15c, you might ask students why you should not use the table made for problem 5a (on page 35) to calculate (28.3 50) ÷ 2 39.15. Discuss the difference between this type of table (which relates one value to another by a formula) and a ratio table (which only includes equivalent ratios). 39 Building Formulas Solutions and Samples Hints and Comments 15. a. The area of the bottom of the can is 38.5 cm2, or 38.5 square centimeters. The radius is 7 2 3.5 cm. area π 3.5 3.5 Materials b. The volume is 577.5 cm3 or 578 cm3 or 577.5 (578) cubic centimeters. volume 38.5 15 c. The area of the rectangle is 330 cm2. One side of the rectangle has the measure of the height of the can, which is 15 centimeters. One side of the rectangle has the measure of the circumference of the can. Use the formula for the circumference of a circle: circumference π diameter or circumference 2 π radius circumference π 7 ≈ 22 centimeters area of the rectangle: 15 22 330 cm2 calculators (one per student) Overview Students calculate the surface area and volume of cans. They investigate the influence of a change in dimensions of the cans. d. Tin needed for the can: 330 38.5 38.5 407 cm2. Note that the bottom and the top of the can have the same area. 16. a. If the can becomes twice as high, the volume is multiplied by 2. Explanations will vary. Sample explanation: The volume of a can can be computed with: volume area of base height Now multiply height by two: volume area of base 2 height Some students may need to use examples of actual measurements. b. The total area changes but less than two times. Explanations will vary. Sample explanation: The area of the top and the bottom of the can does not change. The side of the rectangle that is the circumference of the circle does not change. The other side of the rectangle (the height) is multiplied by two, so the area of the rectangle is multiplied by two. 17. a. If the diameter of a can is doubled, the volume is not doubled but multiplied by four. Explanations will vary. Sample explanation: If the diameter of a can is doubled, the radius is also doubled. Since the base is a circle, the area of the circle will be multiplied by four. area1 π radius radius area2 π (2 radius) (2 radius) 4 π radius radius The height does not change, so the volume will be multiplied by four. Some students may need to use examples of actual measurements. b. The total area changes more than two times but less than four times. Explanations will vary. Sample explanation: The surface area of top and bottom is multiplied by four. The shortest side of the rectangle does not change. The longest side is multiplied by two, so the area of the rectangle is multiplied by two. Section D: Formulas and Geometry 39T
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