#### Parent Mathing Tip #9: Shapes

These tips give parents ideas for math exercises that will develop their child's math abilities. Today’s parent tip for “mathing” with your child focuses on shapes.

It's National Pizza Day! We're celebrating with some fun pizza word problems. Mmmm....who's hungry?

**Question:** Mark, Scott, and Christian ordered an extra-large pepperoni pizza. Mark and Christian both had 3 slices each and Scott had 4 slices of pizza. If there were 6 slices of pizza left, how many slices of pizza does the extra-large pizza start with?

**Answer: 16 slices**

**Solution:** To find the answer to this problem we start by adding all the slices of pizza eaten by Mark, Christian, and Scott. Mark had 3 slices, Christian had 3 slices, and Scott had 4 slices of pizza. Together they had 10 slices of pizza (3 + 3 + 4 = 10). Because there were 6 slices of pizza remaining, the pizza started with 16 slices.

**Question: **Albert made a pizza. Half of the pizza was cheese and the other half was pepperoni. Albert also wanted 1/3 of the pizza to have mushrooms, but he did not want any mushrooms to be on the cheese half of the pizza. What fractional part of the pizza was pepperoni only?

**Answer: 1/6**

**Solution:** Half of the pizza is cheese and the other half is pepperoni. Since mushrooms were not added to the cheese half of the pizza, they must have been added to the pepperoni half of the pizza. To find what part of the pizza is just pepperoni, subtract the fractional part that is pepperoni and mushrooms from the fractional part that is pepperoni. Pepperoni is half of the pizza and mushrooms were added to 1/3 of the pizza. 1/2 – 1/3. In order to subtract these fractions, they must have the same denominator, or same name. If we convert the fractions to sixths, we have 3/6 – 2/6 = 1/6. 1/6 of the pizza is pepperoni only.

**Question:** Pizza Palace sells medium pizzas that have 13 inch diameters and large pizzas that have 18 inch diameters. If a medium pizza costs $7.00 and a large pizza costs $14.00, which size pizza is the better buy?

**Answer: The medium pizza**

**Solution: **In order to find which is the better buy, we figure out which pizza gives the best value. That means which one has more pizza for the same amount of money spent. We start by finding the area of both pizzas.

The area of a circle is equal to pi times the radius squared (A =π x r2). We are given the diameter of each pizza, so we need to take half of that to find the radius of each. Half of 13 is 6.5 and half of 18 is 9.

So for the two pizzas:

A medium = π x 6.52 = 42.25π inches^{2}

A large = π x 92 = 81π inches^{2}

At this point we see that the price of the medium pizza is half the price of the large pizza. That means two medium pizzas would have the same cost as one large pizza. Because the area of the medium pizza is 42.25π inches^{2}, two medium pizzas would have a combined area of 42.25π inches^{2} + 42.25π inches^{2} = 84.5π inches^{2}. This area is greater than the are of one large pizza. That means you get more pizza for $14.00 if you order two medium pizzas than if you order one large pizza. So the medium pizza is a better buy.

Another way to look at this is to find the cost per square inch for each type of pizza. That means we take the cost of each pizza and divide by it’s area.

Medium pizza: $7.00/42.25π inches^{2} ≈ 0.053/inch^{2}

Large pizza: $14.00/81π inches^{2} ≈ 0.055/inch^{2}

Notice that because they both round to $0.06 we have to compare them to a fraction of a cent to see the better buy.

**Question:** A pepperoni pizza has a radius of 7 inches. Each pepperoni has a diameter of 1 inch. If the pizza is topped with 14 pepperonis, what is the area of the pizza that is not covered by pepperonis?

**Answer: 91π/2 or approximately 142.94 square inches**

**Solution:** First we need to find the area of the whole pizza. The pizza has a radius of 7 inches. The formula for the area of a circle is Area = π • radius2. 72 = 49, so the area of the whole pizza is 49π square inches. Each pepperoni has a diameter of 1 inch, meaning that each pepperoni has a radius of 1/2 an inch. The area of one pepperoni is Area = π • (1/2)2 = 1/4 • π = π/4 square inches. There are 14 pepperonis on the pizza, so the area of the pizza that is covered by pepperonis is 14 • π/4 = 7π/2 square inches. To find the area of the pizza that is not covered by pepperoni, subtract the amount that is covered by pepperoni from the total. 49π – 7π/2 = 91π/2. The area of the pizza that is not covered by pepperoni is 91π/2, or approximately 142.94 square inches.

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