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 (TC-1)
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 r2
Use 3.14 as the value of π
Area of a square = length x width
If not using a scientific calculator.
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Teacher Materials Page 1 of 13
Personal Pan Pizzas
6 inches
6 inches
A = 28.27 ≈ 29 in2
A = 36 in2
Small Pizzas
9 inches
9 inches
A = 63.617 ≈ 64 in2
A = 81 in2
Medium Pizzas
12 inches
12 inches
A = 113.097 ≈ 113 in2
A = 144 in2
Large Pizzas
15-inch
A = 176.71 ≈ 177 in2
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15 inches
A = 225 in2
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B. Interpreting Data (TC-2)
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 6-inch round pizza is smaller than a 6-inch 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 6-inch square pizzas have the same area as one 12-inch square pizza.
C. Comparisons (TC-2)
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 12-inch round pizzas are about equal in area to one 15-inch
square pizza. How do they compare in cost?
The price of 2 12-inch pizzas is almost double the cost of one 15-inch pizza.
Next, Jeff notices that the area of a 6-inch square personal pan pizza goes into the
area of a 12-inch square pizza an exact number of times. How many 6-inch square
pizzas equal one 12-inch square pizza? 4
How much would you pay for the 6-inch
pizzas? $31.96
Which is the better value if you compare the equivalent areas? Since a 12-inch square
pizza is $11.99 compared to the $31.95 price of the same amount of pizza in 4 6-inch
pizzas, the 12-inch pizza is a much better value!
Finally, Jeff gets carried away with his mental math expertise and notices that if he
buys four 12-inch square pizzas he will have the same amount of pizza as a certain
number of 9-inch round pizzas. He decides to let his brothers have a little mental
exercise and figure out how many
9-inch round pizzas equal four 12-inch square pizzas. What do they get for an answer?
It takes nine 9-inch round pizzas to equal the same amount of pizza as four 12-inch
square pizzas
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Cost of the 12-inch square pizzas? $47.96 Cost of the 9-inch round pizzas? $107.91
How do the prices compare? The four 12-inch pizzas cost less than half the cost of
nine 9-inch pizzas even though they have the same area.
D. $80.00 Budget (TC-3)
The brothers now must decide how much pizza they can buy on their budget of $80.00.
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
Total Cost per Size
2
24
15-inch round
$31.98
@ 15.99 each
4
32
12-inch square
$47.96
@ 11.99 each
Total Slices:
56
Total Cost:
$79.94
Order Combination 2
# of
Total # of Slices
Pizzas
Size, Shape, and Cost per
Total Cost per Size
Pizza
2
24
15-inch round @ 15.99 each
$31.98
3
24
12-inch square @ 11.99
$35.97
1
6
9-inch square @ 9.99
$ 9.99
Total
54
Total Cost:
$77.94
Slices:
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Order Combination 3
# of
Total # of Slices
Size, Shape, and Cost
Pizzas
2
Total Cost per Size
per Pizza
24
15-inch round
$31.98
@ 15.99 each
2
24
15-inch square
$29.98
@ 14.99
1
6
9-inch square
$ 9.99
@ 9.99
Total
54
Total Cost:
$71.95
Slices:
E. Calculating Area of Pizza Orders and Cost per Square Inch TC-4
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 1
2 15-inch round: 177 x 2 = 354 in2
354 + 452 = 806 in2
4 12-inch square: 113 x 4 = 452 in2
$79.95 ÷ 806 = .099 ≈ $.10 per square inch
Order Combination 2
2 15-inch round: 177 x 2 = 354 in2
3 12-inch square: 113 x 3 = 339 in2
1 9-inch square: 81 x 1 = 81 in2
354 + 339 + 81 = 774 in2
$77.94 ÷ 774 = .10069 ≈ $.10 per square inch
Order Combination 3
2 15-inch round: 177 x 2 = 354 in2
2 15-inch square: 225 x 2 = 450 in2
1 9-inch square: 81 x 1 = 81 in2
354 + 450 + 81 = 885
$71.95 ÷ 885 = .08129 ≈ $.08 per square inch
Which order is the best value? Order Combination 3
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Lesson Two: Volume (TC-5)
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 = πr2h
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 in3.
Square Pizzas
Volume of Carton
Round Pizzas
Volume of Cylinder
6-inch Square
432 in3
6-inch Round
339.29 in3
9-inch Square
972 in3
9-inch Round
763.40 in3
12-inch Square
1728 in3
12-inch Round
1357.17 in3
15-inch Square
2700 in3
15-inch Round
2120.58 in3
Show work on the next page and record the data above.
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Computing Volumes for Pizza Cartons – 12 Pizzas per Carton
Hint: Height of all cartons will be 12 inches.
6-inch Square
6-nch Round
6 x 6 x 12 = 432 in3
9 x 12 x π = 339.29 in3
9-inch Square
9-inch Round
3
9 x 9 x 12 = 972 in
20.25 x 12 x π = 763.401 in3
12-inch Square
12-inch Round
12 x 12 x 12 = 1728 in
36 x 12 x π = 1357.168 in3
15-inch Square
15-inch Round
15 x 15 x 12 = 2700 in3
56.25 x 12 x π = 2120.575 in3
B. Compare the volume of a shipping carton for four 6-inch square personal pan pizzas
with the carton for one 12-inch square pizza. The amount of pizza is the same for both,
and all the pizzas are one inch thick. Show your work here.
Four 6-inch square: V = 6 x 6 x 4 = 144 in3
One 12-inch square: V = 12 x 12 x 1 = 144 in3
$7.99 x 4 = $31.96
$11.99
Volume for four 6-inch square pizzas?144 in3
Volume for one 12-inch square pizza? 144 in3
How do the prices compare for the two orders? (Refer back to handout.)
(Sample
answer.) The cost of the four 6-inch square pizzas is $4.00 less than three times the
cost of one 12-inch square pizza.
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Lesson Three: Circumference and Perimeter (TC-6 )
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
Circumference of a Circle = 2π r
or
or
Perimeter of a Square = 4S
Circumference of a Circle = dπ
6-inch Square
6-inch Round
P = 4 x 6 = 24 inches
C = 6π
9-inch Square
9-inch Round
P = 4 x 9 = 36 inches
C = 9π
12-inch Square
12-inch Round
P = 4 x 12 = 48 inches
C = 12π
15-inch Square
15-inch Round
P = 4 x 15 = 60 inches
C = 15π
C = 18.85 ≈ 19 inches
C = 28.27 ≈ 28 inches
C = 37.69 ≈ 38 inches
C = 47.12 ≈ 47 inches
Which two pizzas measure about the same around the outside edge? The 12-inch
square pizza and the 15-inch round pizza both measure 48 inches around the outside
edge.
Looking back on the handout, how do their areas compare? The area of the 15-inch
round pizza is 33 square inches greater than the area of the 12-inch square pizza.
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A 6-inch square pizza is half the perimeter of a 12-inch square pizza, but their areas are
quite different. Compare the areas of two 6-inch square pizzas to one 12-inch square
pizza.
The area of one 12-inch square pizza is double the area of two 6-inch square pizzas.
The circumference of five 6-inch round pizzas is about the same measurement as the
two perimeters of which size pizza?
five 6-inch round: 19 x 5 = 95 inches
two 12-inch square: 2 x 48 = 96 inches
What is the area of five 6-inch round pizzas? 28 x 5 = 140 square inches
What is the difference between the areas of these pizzas? The five 6-inch round pizzas
have about the same area as only one 12-inch square pizza.
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Name: ________________________ Date: _____________ Period: ______
PIZZA SIZE AND PRICE COMPARISONS
Personal Pan Pizzas (4 slices per pizza)
Size & Shape
Area of Pizza
Price of Pizza
6-inch round
28 in2
$8.99
6-inch square
36 in2
$7.99
Size & Shape
Area of Pizza
Price of Pizza
9-inch round
64 in2
$11.99
9-inch square
81 in2
$9.99
Small Pizzas (6 slices per pizza)
Medium Pizzas (8 slices per pizza)
Size & Shape
Area of One Slice Price of Pizza
12-inch round
113 in2
$13.99
12-inch square
144 in2
$11.99
Size & Shape
Area of Pizza
Price of Pizza
15-inch round
177 in2
$15.99
15-inch square
225 in2
$14.99
Large Pizzas (12 slices per pizza)
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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 A-C on day one, then D-E on day two.
TC-1 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 r2
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 (in2.)
• Using the squares the graph paper and circles on or cut from graph paper,
discuss the concept of area as square inches.
2
5. Calculator keys: x 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 = 113.097 ≈ 113 in2
TC-2
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 18-inch diameter could
be added, though this is not a common pizza size.
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TC-3 D. $80.00 Budget
8. Lead a sample pizza purchase discussion. Begin the example with a combination
you know will cost more than $80.00.
9. 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?
TC-4 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.
TC-5 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 one-inch 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 inches3 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 inches3. 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 6-inch square and round pizzas.
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TC-6 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.
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