**Lower Elementary:**

*Question: *Tasha went to the beach for the weekend. On Friday, she found 8 seashells. On Saturday, she found twice as many seashells as she did on Friday. On Sunday, she found half as many seashells as she did on Friday. How many seashells did Tasha find altogether?

*Answer: *28 seashells

*Solution: *Tasha found 8 seashells on Friday. She found twice as many seashells on Saturday. 8 doubled is 16. She found 16 seashells on Saturday. She found half as many seashells on Sunday as she did on Friday. Half of 8 is 4. She found 4 seashells on Sunday. Add them all together to find the total. 8 + 16 + 4 = 8 + 20 = 28. Tasha found 28 seashells altogether.

**Upper Elementary:**

*Question: *Besides 1 and myself, I only have 1 other factor. I am less than 100, but greater than 26. What number am I?

*Answer: *49

*Solution: *The number only has 1 factor besides 1 and itself. That means that the number must a number times itself. 5 × 5 = 25 and 10 × 10 = 100, so our possible choices for the remaining factor are 6, 7, 8, or 9. However, 6, 8, and 9 are each composite numbers, meaning that there would be more than 1 factor for each of those numbers. This means that there would be more than 1 factor for 6 × 6 = 36, 8 × 8 = 64, and 9 × 9 = 81. The only number left is 7. 7 × 7 = 49, and 49 has no other factors. So, 49 is the number that is greater than 26, less than 100, and only have 1 other factor besides 1 and itself.

**Middle School:**

*Question: *A dragon’s wingspan is proportional to its height. A baby dragon’s height is 5 feet and its wingspan is 7 feet. When the baby dragon grows to be an adult, its wingspan becomes 24.5 feet. What is the height of the adult dragon?

*Answer: *17.5 feet

*Solution: *Because the wingspan and the height are proportional, we can set up two proportions and set them equal to each other. One way to set up the proportion is:

baby dragon height/baby dragon wingspan = adult dragon height/adult dragon wingspan

Substitute in the values. Let H represent the height of the adult dragon.

5 feet / 7 feet = H feet / 24.5 feet

One way to solve this is to cross multiply.

7H = 122.5

Divide by 7.

H = 17.5

The height of the adult dragon is 17.5 feet.

Alternatively, notice that 7 × 3.5 = 24.5. So, multiplying 5 by 3.5 is another way to solve for H. 5 × 3.5 = 17.5. The height of the dragon is 17.5 feet.

**Algebra and Up:**

*Question: *A ladder is leaning against a building. The base of the ladder is 3 feet away from the base of the building and the ladder hits the building 4 feet above the ground. The ladder then slides and now hits the building 2 feet above the ground. How far away is the base of the ladder from the base of the building after it slides?

*Answer: *sqrt(21) which is approximately 4.58 feet

*Solution: *When the ladder is leaning against the building, it makes a right triangle with the leg measure of 3 feet and 4 feet. We will use the Pythagorean Theorem to find how long the ladder is.

a^{2} + b^{2} = c^{2}

Plug in 3 and 4 for a and b.

3^{2} + 4^{2} = c^{2}

Square the terms.

9 + 16 = c^{2}

Simplify.

25 = c^{2}

Take the square root.

5 = c

So the length of the ladder is 5 feet. When the ladder slides, it now becomes a right triangle with a hypotenuse of 5 feet and a leg measuring 2 feet. To find the measure of the other leg, which is also how far away from the base of the ladder is from the base of the building, we will use the Pythagorean Theorem again.

a^{2} + b^{2} = c^{2}

Plug in 2 and 5 for a and c, respectively.

2^{2} + b^{2} = 5^{2}

Square the terms.

4 + b^{2} = 25

Subtract 4 from each side.

b^{2} = 21

Take the square root.

b = sqrt(21) which is approximately 4.58 feet

After the ladder slid, the base of the ladder is now sqrt(21) feet, or approximately 4.58 feet, away from the base of the building.