PIZZA! PIZZA! TEACHER S GUIDE and ANSWER KEY


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1 PIZZA! PIZZA! TEACHER S GUIDE and ANSWER KEY The Student Handout is page 11. Give this page to students as a separate sheet. Area of Circles and Squares Circumference and Perimeters Volume of Cylinders and Rectangular Prisms Comparing Cost Teaching Suggestions begin on page 12. Lesson One: Area and Cost A. Area of Pizza (TC1) Triplets Jim, Joe, and Jeff want to order pizza for their upcoming triple birthday party. Since their money is limited, they decide to compare costs of pizzas to make sure they get the best value for their money. The brothers want a variety of pizzas and think having both round and square pizzas would be a good idea. So they first decide to compare sizes and surface areas of pizzas. Joe thinks he and his brothers need to get organized in their comparisons by making a table of their findings. So Joe creates tables that include sizes, shapes, and areas of common pizzas sold at their favorite restaurants along with price comparisons for supreme pizzas. On the next page, calculate the area of each size pizza. Round the areas to the nearest whole square inch. On your handout titled, PIZZA SIZE AND PRICE COMPARISONS, record your data in column 2, Area of Pizza. The following formulas may be helpful: Area of a circle = π r 2 Use 3.14 as the value of π If not using a scientific calculator. Area of a square = length x width Pizza! Pizza! Teacher Materials Page 1 of 13
2 Personal Pan Pizzas 6 inches 6 inches A = in 2 A = 36 in 2 Small Pizzas 9 inches 9 inches A = in 2 A = 81 in 2 Medium Pizzas 12 inches 12 inches A = in 2 A = 144 in 2 Large Pizzas 15inch 15 inches A = in 2 A = 225 in 2 Pizza! Pizza! Teacher Materials Page 2 of 13
3 B. Interpreting Data (TC2) Jim, Joe, and Jeff are surprised when they find out the areas and prices of the pizzas at their two favorite pizza restaurants. One restaurant specializes in the square pizzas listed on your handout, while the other restaurant bakes round pizzas. Write three observations they make regarding sizes, shapes, and/or prices of pizzas from the data on your handout. (Sample answers.) Observation 1: A 6inch round pizza is smaller than a 6inch square pizza but costs a dollar more. Observation 2: The area of each round pizza is always smaller than the area of the square pizza of the same dimension. The round pizza also costs more. Observation 3: Four 6inch square pizzas have the same area as one 12inch square pizza. C. Comparisons (TC2) Jeff, the oldest of the triplets, notices some relationships in the areas of a few of the pizzas. He thinks this will help the brothers decide which pizzas will be the best value. First, Jeff sees that two 12inch round pizzas are about equal in area to one 15inch square pizza. How do they compare in cost? The price of 2 12inch pizzas is almost double the cost of one 15inch pizza. Next, Jeff notices that the area of a 6inch square personal pan pizza goes into the area of a 12inch square pizza an exact number of times. How many 6inch square pizzas equal one 12inch square pizza? 4 How much would you pay for the 6inch pizzas? $31.96 Which is the better value if you compare the equivalent areas? Since a 12inch square pizza is $11.99 compared to the $31.95 price of the same amount of pizza in 4 6inch pizzas, the 12inch pizza is a much better value! Finally, Jeff gets carried away with his mental math expertise and notices that if he buys four 12inch square pizzas he will have the same amount of pizza as a certain number of 9inch round pizzas. He decides to let his brothers have a little mental exercise and figure out how many 9inch round pizzas equal four 12inch square pizzas. What do they get for an answer? It takes nine 9inch round pizzas to equal the same amount of pizza as four 12inch square pizzas Pizza! Pizza! Teacher Materials Page 3 of 13
4 Cost of the 12inch square pizzas? $47.96 Cost of the 9inch round pizzas? $ How do the prices compare? The four 12inch pizzas cost less than half the cost of nine 9inch pizzas even though they have the same area. D. $80.00 Budget (TC3) The brothers now must decide how much pizza they can buy on their budget of $ Since they are triplets born on the third day of the third month, they decide to get three slices of pizza for each guest. They plan to feed 15 guests plus themselves at the birthday party. How many slices of pizza do they need to purchase? 18 x 3 = 54 slices Since Jim, Joe, and Jeff want to have both round and square pizzas, find three combinations of pizzas within their $80.00 budget that could give them enough pizza for the party with only a few or no slices left over. You may or may not need all the rows on the tables. Order Combination 1 # of Pizzas Total # of Slices Size, Shape, and Cost per Pizza inch each inch each Total Cost per Size $31.98 $47.96 Total Slices: 56 Total Cost: $79.94 Order Combination 2 # of Total # of Slices Size, Shape, and Cost per Total Cost per Size Pizzas Pizza inch each $ inch $ inch 9.99 $ 9.99 Total Slices: 54 Total Cost: $77.94 Pizza! Pizza! Teacher Materials Page 4 of 13
5 Order Combination 3 # of Total # of Slices Size, Shape, and Cost Total Cost per Size Pizzas per Pizza inch each inch inch 9.99 $31.98 $29.98 $ 9.99 Total Slices: 54 Total Cost: $71.95 E. Calculating Area of Pizza Orders and Cost per Square Inch TC4 Calculate the area of pizza purchased by each order and the cost per square inch of pizza for each order. Show your work. Order Combination inch round: 177 x 2 = 354 in inch square: 113 x 4 = 452 in = 806 in 2 $ =.099 $.10 per square inch Order Combination inch round: 177 x 2 = 354 in inch square: 113 x 3 = 339 in inch square: 81 x 1 = 81 in = 774 in 2 $ = $.10 per square inch Order Combination inch round: 177 x 2 = 354 in inch square: 225 x 2 = 450 in inch square: 81 x 1 = 81 in = 885 $ = $.08 per square inch Which order is the best value? Order Combination 3 Pizza! Pizza! Teacher Materials Page 5 of 13
6 Lesson Two: Volume (TC5) A. The local pizza restaurants sell frozen pizzas to supermarkets throughout Washington. To ship the frozen pizzas, the pizzas are packed twelve to a carton. If the pizzas are one inch thick, what is the volume of the cylindrical and rectangular shipping cartons for the different size pizzas? Volume of a Rectangular Prism = length x width x height or Volume = (area of square pizza) x (height of carton) Volume of a Right Cylinder = πr 2 h or Volume = (area of round pizza) x (height of the carton) Volume of Shipping Cartons for 12 Frozen Pizzas Round to the nearest hundredth of an in 3. Square Pizzas Volume of Carton Round Pizzas Volume of Cylinder 6inch Square 432 in 3 6inch Round in 3 9inch Square 972 in 3 9inch Round in 3 12inch Square 1728 in 3 12inch Round in 3 15inch Square 2700 in 3 15inch Round in 3 Show work on the next page and record the data above. Pizza! Pizza! Teacher Materials Page 6 of 13
7 Computing Volumes for Pizza Cartons 12 Pizzas per Carton Hint: Height of all cartons will be 12 inches. 6inch Square 6 x 6 x 12 = 432 in 3 6nch Round 9 x 12 x π = in 3 9inch Square 9inch Round 9 x 9 x 12 = 972 in x 12 x π = in 3 12inch Square 12 x 12 x 12 = 1728 in 12inch Round 36 x 12 x π = in 3 15inch Square 15inch Round 15 x 15 x 12 = 2700 in x 12 x π = in 3 B. Compare the volume of a shipping carton for four 6inch square personal pan pizzas with the carton for one 12inch square pizza. The amount of pizza is the same for both, and all the pizzas are one inch thick. Show your work here. Four 6inch square: V = 6 x 6 x 4 = 144 in 3 $7.99 x 4 = $31.96 One 12inch square: V = 12 x 12 x 1 = 144 in 3 $11.99 Volume for four 6inch square pizzas?144 in 3 Volume for one 12inch square pizza? 144 in 3 How do the prices compare for the two orders? (Refer back to handout.) (Sample answer.) The cost of the four 6inch square pizzas is $4.00 less than three times the cost of one 12inch square pizza. Pizza! Pizza! Teacher Materials Page 7 of 13
8 Lesson Three: Circumference and Perimeter (TC6 ) Jeff was surprised when he noticed that the area of the square and round pizzas with the same width and diameter were so different. He wondered if there would be a relationship between the perimeter of the square pizzas and the circumference of the round pizzas. Find the circumference of the round pizzas and the perimeter of the square pizzas. Show your work in the boxes. Perimeter of a Square = Side + Side + Side + Side or Perimeter of a Square = 4S Circumference of a Circle = 2π r or Circumference of a Circle = dπ 6inch Square 6inch Round P = 4 x 6 = 24 inches C = 6π C = inches 9inch Square 9inch Round P = 4 x 9 = 36 inches C = 9π C = inches 12inch Square 12inch Round P = 4 x 12 = 48 inches C = 12π C = inches 15inch Square 15inch Round P = 4 x 15 = 60 inches C = 15π C = inches Which two pizzas measure about the same around the outside edge? The 12inch square pizza and the 15inch round pizza both measure 48 inches around the outside edge. Looking back on the handout, how do their areas compare? The area of the 15inch round pizza is 33 square inches greater than the area of the 12inch square pizza. Pizza! Pizza! Teacher Materials Page 8 of 13
9 A 6inch square pizza is half the perimeter of a 12inch square pizza, but their areas are quite different. Compare the areas of two 6inch square pizzas to one 12inch square pizza. The area of one 12inch square pizza is double the area of two 6inch square pizzas. The circumference of five 6inch round pizzas is about the same measurement as the two perimeters of which size pizza? five 6inch round: 19 x 5 = 95 inches two 12inch square: 2 x 48 = 96 inches What is the area of five 6inch round pizzas? 28 x 5 = 140 square inches What is the difference between the areas of these pizzas? The five 6inch round pizzas have about the same area as only one 12inch square pizza. Pizza! Pizza! Teacher Materials Page 9 of 13
10 Name: Date: Period: PIZZA SIZE AND PRICE COMPARISONS Personal Pan Pizzas (4 slices per pizza) Size & Shape Area of Pizza Price of Pizza 6inch round 28 in 2 $ inch square 36 in 2 $7.99 Small Pizzas (6 slices per pizza) Size & Shape Area of Pizza Price of Pizza 9inch round 64 in 2 $ inch square 81 in 2 $9.99 Medium Pizzas (8 slices per pizza) Size & Shape Area of One Slice Price of Pizza 12inch round 113 in 2 $ inch square 144 in 2 $11.99 Large Pizzas (12 slices per pizza) Size & Shape Area of Pizza Price of Pizza 15inch round 177 in 2 $ inch square 225 in 2 $14.99 Pizza! Pizza! Teacher Materials Page 10 of 13
11 TEACHING SUGGESTIONS Lesson One: Area and Cost Lesson One may take two class periods to complete. A suggested split in the lesson would be to complete sections AC on day one, then DE on day two. TC1 A. Area of Pizza 1. Concepts of diameter and radius. Using a circle diagram, draw a radius off the diameter to show the difference. 2. Area of a circle: Area = π r 2 3. Area of a rectangle: Area = length x width Some students may benefit by using graph paper to visualize the concept of area with the grid lines. Cutting out pizzas using graph paper with given diameters or side lengths and placing the round pizza over the square pizza will enable the student to compare the areas. 4. Concept of square inches (in 2.) Using the squares the graph paper and circles on or cut from graph paper, discuss the concept of area as square inches. 5. Calculator keys: x 2 and π Help students locate these important keys on their scientific calculators. Practice a few calculations to familiarize students with use of these keys. 6. Rounding decimals to nearest whole number. Rounding is used throughout the lesson. Students may need to review. Students need to use the approximation symbol to represent their answers. Example: A = in 2 TC2 B. Interpreting Data C. Comparisons 7. Students will be making observations and comparisons about area and cost data. Observations will vary in complexity. Lead a class discussion around observations guiding them into a more complex observation prior to letting them write three of their own. To practice comparisons, have students write comparisons three ways: difference between values, percents, and how many times one is greater/less than the other. Extension activities: Students may write comparisons using both area and cost. Students may write several comparisons using the data and challenge the class with guiding questions. As an additional pizza option, a pizza with an 18inch diameter could be added, though this is not a common pizza size. Pizza! Pizza! Teacher Materials Page 11 of 13
12 TC3 D. $80.00 Budget 8. Lead a sample pizza purchase discussion. Begin the example with a combination you know will cost more than $ Discuss strategies for the pizza combinations. Should you consider price, number of slices per pizza, or cost when choosing pizzas to order? 10. Extensions activities: How much pizza could you get for $80.00 if there is no limit to the area or number of slices purchased? What is the most expensive combination of pizzas to get 54 slices? If the restaurant gives a 10% discount if you purchase one pizza of each size, what combination would you buy and what would be the cost? TC4 E. Calculating Area of Pizza Orders and Cost per Square Inch 11. To calculate the cost per square inch of pizza, areas of each quantity and size of pizza must first be determined, and then totaled. Next, the total cost of the order must be divided by the total area of the order. Students will need to refer back to the tables on the handout. 12. Rounding practice will again enter into the final approximate answer for the cost per square inch. This time, the final cost will be below one dollar. Students may confuse the correct format for writing values less than one dollar. 13. Using your sample purchase over $80.00, model the steps for determining area of the order and cost per square inch. TC5 Lesson Two: Volume 14. To let students visualize volume of a box, construct a box with graph paper. Have students draw a net of a box, cut it out, and tape the sides together. Then fill the box with oneinch cubes. You will need to construct a box in advance to make sure that your dimensions will work for filling with cubes. 15. For the volume of a cylinder, fill a can with dry beans or rice. You could then fill the box with the volume of the cylinder if you want to also determine how many square inches filled the cylindrical can. 16. The concept of inches 3 may be introduced using the wooden blocks and/or drawings of cubes. Emphasize that now you are using three dimensions to determine volume, thus the use of inches 3. The 3 as the exponent represents the three dimensions: length, width, height. 17. For right cylinders, help students understand the idea of stacked round pizzas by stacking round objects. You may use quarters as the pizzas and fill a coin wrapper that represents the cylinder. 18. Model the calculations of volume of a rectangular prism and cylinder by walking students through the volume for the 6inch square and round pizzas. Pizza! Pizza! Teacher Materials Page 12 of 13
13 TC6 Lesson Three: Circumference and Perimeter 19. To visualize the concepts of circumference and perimeter, use string to measure the outer edge of round objects commonly found in a classroom. Then take the length and shape it into a square. 20. Planning in advance, shapes equivalent to the sizes of the pizzas in the problem could be cut out from sturdy cardboard or poster board. Students could use string to measure the outer edges of each. Next, students can compare their string measurements with the actual calculations of perimeter and circumference on page 9. How accurate were their string measurements? What factors affected the accuracy of the string measurements? 21. After completing the comparisons provided on page 9, have students write their own comparisons. Include data from the tables as well as the perimeter and circumference data on page 9. Pizza! Pizza! Teacher Materials Page 13 of 13
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