# The Art of Problem Solving: Pre-Algebra

Today I learned how to make long problems easier. We focused on reviewing what I learned about addition then learned about multiplication. I’ve already done both, but this is practice to start the pre-algebra stuff. I learned the commutative property, the associative property, and the order of operations. My mom found a grammar error in the textbook. Derp!

In the **commutative property**, the numbers are moved around (“commuted”) on the two sides of the equation.

In the **associative property**, the numbers stay in the same place but are grouped (“associated”) differently.

1. **Addition is commutative!**

Let a and b be numbers. Then

A + b = b + a.

2. **Addition is associative!**

Let a, b, and c be numbers. Then

(A + b) + c = a + (b + c).

3. **Adding zero!**

let a be a number. Then

a + 0 = a

4. **Multiplication is commutative!**

let a and b be numbers. Then

ab = ba.

5. **Multiplication is associative!**

Let a,b, and c be numbers. Then

(ab)c = a(bc).

6. **Multiplying by 1!**

Let a be a number. Then

1a = a

7. **Order of Operations!**

Perform the operations in an expression in the following order.

P.E.M.D.A.S (Please Excuse My Dear Aunt Sally)

P = Parentheses

E = Exponents

M = Multiply

D = Divide

A = Add

S = Subtract

8. **Multiplication distributes over addition!**

Let a, b, and c be numbers. Then

A(b+c) = ab + ac and (b+c)a = ba + ca.

9. **Multiplying by zero!**

Let x be a number. Then

0x = 0.

Here is an awesome video from *The Art of Problem Solving* author on how to make long problems easier:

# The Solar System

As I told you yesterday, my Mom and I are making a solar system out of Papier mâché and 9 balloons. (One For each.) it rained last night, so they were still damp. We will add our second layer tomorrow. I decided once we made all three layers for the balloons, we would cut a hole in the bottom so we can see what is inside. For example, the sun would have tons of flame inside with a medium sized core. Jupiter would have a lesser core because it is made of mostly hydrogen and helium. The first four closest planets in to the sun are not made of hydrogen and helium. (Mercury, Venus, Earth, and Mars.) They would have a bigger core than Saturn or Jupiter because they are made of more hydrogen and helium.