# Answers

First Grade: |
11 + 12 = ___First, we teach kids to count by 1s, then by 2s, then by 5s and 10s. When they can count by 10s, they can add 10s. 11 is one more than 10, and 12 is 2 more than 10. Add the 10s to get 20, then add 3 more to get 23. |

Second Grade: |
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ___Since 10s are easy to add up, find pairs that add up to 10. 1 plus 9 is 10, and 2 plus 8 is 10. It's a pattern. 3 plus 7 is 10, etc. Add up all the 10s (the four 10s from the pairs and the single 10) and you get 50, plus the 5 left over = 55. |

Third Grade: |
How much is 99 plus 99 plus 99?100s are almost as easy to add as 10s. Since 99 is one less than 100, adding three 99s gives you 3 less than 300 = 297. |

Fourth Grade: |
Count by 1¾ from 0 to 7.Think of adding money. Imagine you have $0. Then, add "one and three quarters" and you have 1¾ (or $1.75). Add the next 1¾ in two steps. First, add one, and you have 2¾ (or $2.75). Then, add 3 more quarters and you have 3½ (or $3.50). Then, do it again: One more is 4½ (or $4.50), plus three quarters is 5¼ (or $5.25). One more is 6¼ (or $6.25), plus ¾ more is 7 (or $7.00). So the answer is 0, 1¾, 3½, 5¼, 7. |

Fifth Grade: |
Which is greatest: ^{17}⁄_{18}, ^{23}⁄_{30}, or ^{18}⁄_{19}? Explain how you got your answer.A fraction shows what part of a whole. ^{23}⁄_{30} is out of the running because it isn't even close to a whole (1), whereas ^{17}⁄_{18} and ^{18}⁄_{19} are almost 1. When you divide something into more parts, each piece is smaller (think of cutting up a pie into a hundred pieces - each piece would be really small!). So, a piece of a pie with 19 pieces is smaller than a piece of pie with 18 pieces, so ^{18}⁄_{19} is bigger than ^{17}⁄_{18} because "the smaller the missing piece, the more that is left." |

Sixth Grade: |
Halfway through the second quarter, how much of the game is left?The game is divided into 4 parts, called "quarters." If we divide each quarter in half, we get 8 eighths. The first quarter is ^{2}⁄_{8}. Half of the next quarter is another ⅛. That's ⅜. After the first 3 eighths, there are 5 more eighths left in the game. In other words, ⅝ of the game is left. |

Seventh Grade: |
How much is 6½% of 250?Percent measures "'for each' 'hundred.'" There are two and a half hundreds in 250. So, it's 6½ for the first hundred, plus 6½ for the second hundred, plus half of 6½ (which is 3¼) for the fifty, or 6½ + 6½ + 3¼ = 16¼. |

Eighth Grade: |
On a certain map, 6 centimetres represent 25 kilometres. How many kilometres do 15 centimetres represent?The question tells us what 6 represents. 6, then, is our "measuring stick." We need to measure 15 in terms of our measuring stick, 6. 15 is two and a half 6s. So, it represents two and a half 25s, which is 62.5. So, 15 centimetres represents 62.5 kilometres on the map. |

Ninth Grade: |
When you take 3 away from twice a number, the answer is 8. What is the number?Replace the words "twice a number" with "something." It now reads, 'When you take 3 away from something, the answer is 8.' That's 11. Going back to the original question, we now know that "something" is 11, which is two times the number we are looking for. So our number is 5½. |

Functions: |
What is the absolute value of the point (3, 4)?Absolute value means "the distance from 0." So the question really is, "How far from 0 is the point (3, 4)?" The key to solving this problem is to realize that this distance [from 0 to (3, 4)] is the hypotenuse of a right triangle whose legs are 3 and 4. This can be visualized by dropping a perpendicular line from (3, 4) to the x-axis. The leg on the x-axis is 3, and the distance from the x-axis to the point is 4. So, using the Pythagorean theorem, a ^{2} + b^{2} = c^{2}, we get 32 + 42 = c^{2}. Solving for c, we get c^{2} = 9 + 16 = 25, so c = 5. So, the absolute value of the point (3, 4), its distance from 0, is 5. |