"I didn't know I had to tell you."
Here's a collection of small math facts that people I know have had "Aha!" moments with. In a way the ideas are obvious, but only once you understand.
A coefficient is multiplication.
What does 2x mean? It means we have 2 of something called x. I've found myself reminding kids that adding something like 2x + 4x is just like adding apples. If you have 2 apples and 4 more apples, you get 6 apples. So 2x + 4x is 6x. But 2x also means 2 times x. To me these are the same. Having two of something means multiplying by 2. But for some people these feel like separate things until someone tells them: 2x = 2 times x.
Fractions are division
This one cam up perhaps because my kids helped in the kitchen so young. They encountered 1/2 cup and 2/3 cup and 1/2 tsp before they could read. So fractions were things all on their own. Parts of a real object whether a cup, a teaspoon or a pizza. And division was something you did to a number or a pile of small objects. (Or sometimes pictures of pie or pizza). I thought we had talked about the connection, but I guess the idea that 1 divided by 2 is 1/2 didn't automatically connect that all fractions are a description of division.
Squaring a number forms a square
A teacher I know told me that when she was a student she was taught to call multiplying a number by itself squaring the number, but never knew where the word came from. It was much later that she made the connection to a physical square. Here is 3x3=32
When we square a number, we build a square with that number as the side length.
Put it in parentheses when you substitute
When you have an expression with a variable like x2 + 2 and you want to know the value when for a particular x, you substitute. So for x = 3, 32 + 2 = 11. Or if x =-5, -5 squared is 25, then add 2, to get 27. Or symbolically (-5)2 + 2 = 25+2 = 27. Those parentheses matter. We want to be multiplying 2 negative numbers and getting a positive, not multiplying 2 positive numbers and making the answer negative. My daughter discovered in her high school math class, that I had never taught her the "rule" that you always put a value in parentheses when you substitute. That's because I only put in the paretheses when they make a difference. For me, it's about what x is. We have to act on all of x, if it's negative keep it negative. For her though it's a symbolic rule. Either way we get to the right answer.