**Lower Elementary**

*Question: *Bill has 6 pencils, Sarah has 8 pencils, Jim has 4 pencils, Tamara has 6 pencils, and Phillip has 3 pencils. If the boys combined their pencils and the girls combined their pencils. Who has more pencils, the boys or the girls?

*Answer: *The girls

*Solution: *Grouping the boys together and the girls together we have;

Boys: Bill – 6, Jim – 4, and Phillip – 3

Girls: Sarah – 8 and Tamara – 6

Adding together the total number of pencils the boys have we get 13 (6 + 4 + 3) and adding together the total number of pencils the girls have will give us 14 (8 + 6). We can see that the girls have 1 more pencil than the boys so the girls have the most pencils.

**Upper Elementary**

*Question: *Sally has 2 quarters and 1 nickel. How much more money does she need to buy a 99¢ ice cream cone? Without using half dollars, what are the least number she will need to buy the ice cream cone?

*Answer: *44¢; 10 coins: 3 Quarters, 1 Dime, 2 Nickels, and 4 Pennies

*Solution: *Sally has 2 quarters and 1 nickel or 55¢ (25¢ + 25¢ + 5¢). To find how much more money she needs we find the difference from 55¢ up to 99¢.

55¢ to 95¢ is 40¢

95¢ to 99¢ is 4¢

40¢ + 4¢ = 44¢

Next we find how the least number of coins she needs without using a half dollar. To make 44¢ with the least number of coins we would need 1 quarter, 1 dime, 1 nickel, and 4 pennies (25¢ + 10¢ + 5¢ + 4¢). To get 44¢ we would need 7 coins, but Sally already has 3 coins (2 quarters and 1 nickel), so altogether Sally will need 10 coins to buy the ice cream cone.

**Middle School**

*Question: *Hannah has a bag of coins consisting of dimes and nickels. In this bag there are 8 more dimes than nickels. If the coins in the bag totaled $3.35, how many of each coin does Hannah have?

*Answer: *17 Nickels and 25 Dimes

*Solution: *Letting n stand for the number of nickels Hannah has in her bag and because the number of dimes is 8 less than the number of nickels, we can say that the number of dimes in the bag is n + 8.

Because quantity × denomination equals the total amount,

n(5¢) + (n + 8)(10¢) = 335¢

5n + 10n + 80 = 335

15n = 255

n = 17

Because the number of dimes is 8 more than the number of nickels in the bag there are 25 dimes in the bag.

This problem can also be done by the guess and check method. A reasonable guess would by 10 nickels and 18 dimes. That would equal $1.30, which is too low. If we were to go with 20 nickels and 28 dimes we would have $3.80, which is closer buy too high. Eventually the student should get to 17 nickels and 25 dimes.

**Algebra and Up**

*Question: *Find the area of the shaded region.

*Answer: *36π cm^{2} – 54√3 cm^{2} or about 19.57 cm^{2}

*Solution: *Because a part is equal to the whole minus the other part, to solve this problem we will need to find the area of the hexagon, and subtract it from the circle.

We can find the area of the hexagon by first drawing in another radius to complete a triangle.

Because a hexagon is made up of 6 equilateral triangles, if we find the area of 1 triangle we can find the area of the whole hexagon. In an equilateral triangle all sides are the same and all 3 angles are 60°. To find the area of the triangle we need to find the height (h).

Cutting the triangle in half leaves us with a special 30-60-90 triangle with a base of 3 cm.

The side across from the 60° angle (h) is equal to the side across from the 30° angle times √3 or 3√3 cm.

Now that we have the base and height of the equilateral triangle we can find the area.

A_{t} = ½bh = ½(6)(3√3) = 9√3 cm^{2}

A_{h} = 6(9√3) = 54√3 cm^{2}

Because we know the area of the hexagon, to solve this problem we need to find the area of the circle then subtract the area of the hexagon from the area of the circle.

A_{c} = πr^{2} = π6^{2} = 36π cm^{2}

A = 36π cm^{2} – 54√3 cm^{2} ≈ 19.57 cm^{2}