Arithmetic and Algebra, are They Different?

I have not had a lot of time to reflect this year since I have an overload (classes all day and no prep period).  But last Thursday I was at a meeting were they were discussing number/math talks (Making Number Talks Matter).  One of our tasks was to solve a problem then try to think about all the different ways a student might solve a problem, like 55 – 17 of which I thought of four ways to solve it however doing this problem with students the presenter stated that there are eight common ways to solve it.

All this got me thinking about arithmetic and algebra.  I know that algebra is very powerful and more powerful than arithmetic but are there some similarities.  Are arithmetic and algebra really that different?  Consider the following parallel.

Model	Arithmetic	Algebra
Algorithm	12  X  14     4  8 1  2  0 1  6  8
Algorithm Expanded	10  +  2    X   10  +  4          40     8 100  20         .  100 + 60 + 8	a + 4       X  2a + 3           3a + 12 2a2 +  8a         . 2a2 + 11a + 12
Expanded “foil”	(10 + 2)(10 + 4) 100 + 40 + 20 + 8     100 + 60 + 8	(a + 4)(2a + 3) 2a2 + 3a + 8a + 12   2a2 + 11a + 12
Area	100 + 40 + 20 + 8	2a2 + 3a + 8a + 12
Summary	Pair each digit of one number with digit of the other number	Pair each term of one binomial with each term of the other binomial
Extending It	123 X 456     100 + 20 + 3 X 400 + 50 + 6 (100 + 20 + 3)(400 + 50 + 6) Pair each digit of one number with digit of the other number	a + 2c + 3 X 4a + 5c + 6 (a + 2c + 3)(4a + 2c + 6) Pair each term of one trinomial with each term of the other trinomial

“Algebra” is really doing the exact same processes with numbers that we do with variables, which shouldn’t surprise you considering the variables are representing numbers.  Drawing connections between our algebraic concepts to numerical calculations is important to our students understanding of mathematics.  Many times we as educators will need to redraw these connections as students get farther and farther into the algebra courses.