Algebra

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The Red Queen finds the explanations for algebra infuriating:

• Algebra is a  math course taught in high school or grade school where we learn to use symbols to hold numbers and it places an emphasis on work with Equations.
• Algebra is a set of objects with two binary operations, addition and multiplication.

To walk away from an argument we don't want, we might ask her if it is acceptable to say the first definition refers to Elementary Algebra.

Observe the following math equations:

3+0=3

5+0=5

8+0=8

If we write a+0=a with 'a' being a symbol to hold any number, we are generalizing an idea seen in the three above equations to something that will work for all possibilities.  Surely that would please her highness.

For a game that might catch your fancy,  look at the following equation: a + 4 = b

There are two ways of thinking it:

1) We put a number into one of the two variables and the other variable immediately becomes a calculated value.

a=2 makes (a,b) go to (2,6)

a=3 makes (a,b) go to (3,7)

2) All possibilities for (a,b) exist at the same time.  We might build a number-line of points for a and say that at each point there is a value for b.  We might make a graph of a section of this number-line.

Algebra asks the question, "if we start with an equation, what things can we do and still maintain the equality?"

a+5 = b

(add two to each side)

a+7 = b+2

8a+6 = 4b

(divide each side by two)

4a+3 = 2b

If everything above is correct you can pick value a value for 'a,' use the first equation to calculate the 'b' (based on your 'a') and then you can put that (a,b) pair into the second equation. 

1) a+5=b

2)a+7=b+2

3)if a=3, then b=8 -- check that 3+7=10 and 8+2=10, we still have a equality

We're now going to focus on some specifics

Associativity

We have associativity of addition "(a+b)+c=a+(b+c)" and associativity of multiplication "(ab)c=a(bc)".  These are axioms so we don't have to prove them, but we do have two games for you to consider...

(3+4)+5 = 3 + (4+5)

On a piece of paper do three marks ||| and then a short distance away do four marks |||| and then a short distance away do five marks |||||.  If you count them, you see you have 12 marks.  with a red pen circle the 3 group and the 4 group.  Addition is binary so you can only add two numbers at a time.  That red circle says that the first thing you will do is add 3 and 4.  With a blue pen circle the 4 group and the 5 group.  That says to first do 4+5.  We now have two ways to do the addition, as shown in (3+4)+5=3+(4+5).  In some way this game shows the axiom (a+b)+c=a+(b+c) to be true.  We didn't say "prove", we said "show".

We encourage rabbit students to play games with algebra.  

Appendix A

The new student might be puzzled by early comments about Algebra.  "Algebra is algebra, right?"

Nope.  Algebra is like ice cream at Baskin & Robbins.  There are dozens of flavors.  There's an algebra, Boolean Algebra where they do everything with two values: True and False.