Integers and algebra

Notice: Incomplete

Integers and algebra are useful tools and are the basis of all mathematics. This should be used to make the jump from the natural numbers to the integers and algebra.

The integers and letters themselves
To see how they work, let's get out some counters and containers. A green counter will be one and a red counter will be -1. Start with a green counter. Add another one and tell your child that that's now two counters. Add yet another one and tell them it's three. Repeat this for higher and higher numbers. Then do this several times daily until your child has remembered the number names up to a number of your liking.

When your child has remembered their numbers, then do this. Start with a red counter and say it's -1. Add another and say it's -2. And another and say it's -3. Do this to your child as before, until your child can remember the concept of 'negative'

When your child has remembered all of the integers, show them no counters and say it's zero. Then get containers and label it with letters.

Note: From here, in all sections from now on, I'll use certain letters as examples, but any letter-labelled container will do. In fact, it's probably best that you vary the container labels so your child doesn't associate a specific letter with a certain number. I recommend using letters at the start of the alphabet for containers your child can just pour in as many or as few as they want and letters at the end of the alphabet for containers your child has to figure out how many counters (and whether they should be red or green) to put in, as this is the convention among mathematicians (math-studiers).

Start with green containers marked with anything. Tell your child that all containers with the same letter must have the same number and colo(u)r of counters. For now, let your child choose how many should go into each. If your child wants to have 5 green counters in one, but 3 red ones in a different one, give a different container with different letters for them for that, as you must be consistent with variable ordering.

Say your container is marked x. Show your child a container. Say it's x. Show him two containers and say it's 2x. Show three and say it's 3x etc. Show them a red container marked x, and say it's -x. Show them two red containers and say it's -2x, then three containers is -3x etc. Repeat this with different letters.

Arithmetic
Arithmetic turns many numbers into one. There are four things: addition, subtraction, multiplication and division.

Addition
To add things, put them next to each other. Red counters cancel with green counters and vice versa. Red containers cancel with green containers marked with the same letter and vice versa.

Subtraction
To subtract something from something else, flip the colo(u)rs of all of the first 'somethings' from red to green and vice versa. Then add the modified somethings to the 'something else'.

Solving equations
To solve an equation means to find out how many counters to put into each container and whether the counters should be red or green.