#### The Beauty of Math Patterns: Exploring Mathematical Sequences

Explore the fascinating world of math patterns, from Fibonacci sequences to fractals. Read our blog to learn more.

**Child Ready Answers**

First Grade: 11 + 12 = ___

There are two ways to think about 11 + 12, both require mastery of doubles facts. Think about the double, then add 1 or take away 1. So, for 11 + 12 start with 11 + 11 and add 1. 11 + 11 is 22, so 11 + 12 is 23. Or start with 12 + 12 and take away 1. 12 + 12 is 24, so 11 + 12 is 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: If 2 candies cost 5¢, how many candies can you buy for 35¢?

14 candies. To solve, reason in groups. here are 7 nickels (5¢) in 35¢. So, 2 candies,
7 times is 14 candies.

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 ^{1}⁄_{8}. That's ^{3}⁄_{8}. After the first 3 eighths, there are 5 more eighths left in the game. In other words, ^{5}⁄_{8} of the game is left.

Seventh Grade: How much is 6½% of 250?

Percent means “’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.

Pre-Algebra: If *a* = 5, *b* = 2 and *c* = 7, evaluate 3*a*^{2} + 5*b*(*c* – 4).

105. Substitute the values into the expression. Try to use mental math whenever possible.
75 + 10(3) = 105.

Algebra: Solve for *x*: -3(2*x* + 7) = 39.

*x* = -10. Before diving in and distributing the -3, take a moment and see if a mental math approach would work. Dividing both sides by -3 leaves 2*x* + 7 = -13. Subtract 7 from both sides to get 2*x* = -20. Divide both sides by 2 to get *x* = -10.

Geometry: 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 3^{2} + 4^{2} = *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.

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