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
2a^{2}^{
}+ 8a _{.}
2a^{2} + 11a + 12

Expanded “foil”

(10 + 2)(10 + 4)
100
+ 40 + 20 + 8
100 + 60 + 8

2a^{2}
+ 3a + 8a + 12
2a^{2} + 11a + 12

Area

100 + 40 + 20 + 8

2a^{2}
+ 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.