# How to teach algebraic expressions?

I don't know how to teach algebraic expressions to the students who have heard the term for the first time. $(a+b)^2$ results in $a^2+2ab+b^2$, we know that, but they ask me why it is so. I tried to make them understand by expanding the expression to $(a+b)*(a+b)$ and multiplying, but still some of them find it hard to understand. So how do I make them understand that? Are there any other ways?

• Would it help to let a=10 and b=1 and let them follow how the math of variables looks similar to multiplying numbers. Apr 23, 2014 at 2:18
• Can they understand $a(c+d)$ using the distributive property? If so, can they understand $(a+b)(c+d)$ using the distributive property? If so, you are describing a specific case of the latter (i.e., where $c = a$ and $d = b$). In addition, you could look up "lattice multiplication" (at least for $a,b >0$). Apr 23, 2014 at 2:23
• Do they have a good understanding of what variables are? May 10, 2014 at 7:32

Draw a square with sides of length $a+b$.

Then divide the square into $4$ parts: the $a\times a$ square, the $b\times b$ square, and the two $a\times b$ rectangles. Explain that to find the area of the square - that is, $(a+b)(a+b)=(a+b)^2$ - you have to add all the parts up: $$(a+b)^2=a^2+b^2+ab+ba=a^2+2ab+b^2$$ Then talk about the distributive property, and perhaps the meaning of the term "square".

• One way to visualize this type of problem is with algebra tiles. Here's an example resource: <illuminations.nctm.org/Activity.aspx?id=3482>. This is especially helpful for hands on learners. Apr 28, 2014 at 9:33

Difficulty with an advanced concept is often due to not grasping a prerequisite concept. Here are some prerequisite concepts that students should have grasped before trying to grasp expanding $(a+b)^2$...you could ask questions about these concepts to see if you need to review any of them before moving on:

• What are variables (ie, letters can be used to represent numbers that vary, like the dimensions of a room which vary as the architect draws and adjusts it)
• What is an expression (ie, $(a+b)^2$ says if we knew the actual numbers, we would add them and square the result)
• How to evaluate an expression when we know the actual variable values
• Can we do anything useful with expressions without knowing the actual variable values? (yes! eg, factor/expand/simplify to get an equivalent expression.)
• Bonus for best comprehension: As a cross-check whether two expressions are equivalent, evaluate them for arbitrary variable values (is this foolproof? how can we make it more robust?)

This particular expression makes sense as a numerical pattern:

• $1003^2 = \ \ 1006009$
• $2003^2 = \ \ 4012009$
• $3003^2 = \ \ 9018009$
• $4003^2 = 16024009$

• $1004^2 = \ \ 1008016$

• $\ldots$

You can look at these numbers and ask students to figure out the pattern. They may find it cool that the pattern gives a short cut for calculating $2014^2$.

If they find it cool, or if you've given them drills on this, you can show them $(a+b)^2=a^2+2ab+b^2$ as a good way of writing down the pattern. When $a$ is a multiple of 1000, it gives an especially good shortcut because you can see all the terms $a^2$, $2ab$, $b^2$ in the answer.

Then you can check what the pattern does with $103^2$, $203^2$, etc. The formula $a^2+2ab+b^2$ gives a good shortcut here too. And even for $a=11$ and $b=3$, you can check that $(a+b)^2=a^2+2ab+b^2$ is not a helpful shortcut, but still accurate.

This interpretation may appear contrived, but you can do something similar for any polynomial identity with positive coefficients. The underlying technique of guessing formulas from special cases of inputs, and testing them on other inputs, is surprisingly useful.

Here is one possible approach. (I also like the geometric rectangle method, but user1161 has already mentioned this.) The main idea is to forget about issues such as does the argument work for rational numbers in general, does the argument work for irrational numbers, etc. and simply focus on showing students how the rules arise in a natural way from what they know. Besides being more concrete for the students, this approach also gives them the tools to reconstruct a correct method of expanding, although I suspect very few will do this. (My experience is that those who would find it neat to know that one can discover the rules in this way are sufficiently "mathy" that they know the rules anyway.)

Notation: In what follows, ellipses (i.e. $\cdots$) denote the continuation of a finite sum, not the continuation of an infinite sum.

Begin by drilling the the distributive property with specific positive integer examples, such as:

$$(2)(2+3) \;\; = \;\; (2+3)+(2+3) \;\; = \;\; (2+2) + (3+3)$$ $$(3)(2+3) \;\; = \;\; (2+3)+(2+3)+(2+3) \;\; = \;\; (2+2+2) + (3+3+3)$$ $$(4)(2+3) \; = \; (2+3)+(2+3)+(2+3)+(2+3) \; = \; (2+2+2+2) + (3+3+3+3)$$

Now let's throw letters in and do it again:

$$(2)(a+b) \;\; = \;\; (a+b)+(a+b) \;\; = \;\; (a+a) + (b+b)$$ $$(3)(a+b) \;\; = \;\; (a+b)+(a+b)+(a+b) \;\; = \;\; (a+a+a) + (b+b+b)$$ $$(4)(a+b) \; = \; (a+b)+(a+b)+(a+b)+(a+b) \; = \; (a+a+a+a) + (b+b+b+b)$$

In general (throw another letter in), we see that

$$(n)(a+b) \; = \; (a+a+\cdots [n \; \text{many} \; a\text{'s}]) \; + \; (b+b+\cdots [n \; \text{many} \; b\text{'s}])$$

Thus (explain that $n$ many $a$'s added will be $na),$

$$(n)(a+b) \;\; = \;\; na + nb$$

More generally, we can now see how the following arises:

$$(n)(a+b+c+\cdots) \; = \; (a+a+\cdots [n \; \text{many} \; a\text{'s}]) \; + \; (b+b+\cdots [n \; \text{many} \; b\text{'s}]) \;+\; (c+c+\cdots [n \; \text{many} \; c\text{'s}]) \; + \; \cdots$$ Therefore, $$(n)(a+b+c+\cdots) \;\; = \;\; na + nb + nc + \cdots$$

By replacing $n$ with $A+B+C+\cdots,$ we can now make the jump to expanding multinomials:

$$(A+B+C+\cdots)(a+b+c+\cdots) \;\; = \;\; (A+B+C+\cdots)(a) \; + \; (A+B+C+\cdots)(b) \; + \; (A+B+C+\cdots)(c) \; + \; \cdots$$ $$= \;\; (a)(A+B+C+\cdots) \; + \; (b)(A+B+C+\cdots) \; + \; (c)(A+B+C+\cdots) \; + \; \cdots$$ $$= \;\; (aA+aB+aC+\cdots) \; + \; (bA+bB+bC+\cdots) \; + \; (cA+cB+cC+\cdots) \; + \; \cdots$$

At this point tell them how this possibly bewildering maze of symbols can be summarized by noticing each term in the first original factor gets paired (or shakes hands with) each pair in the second original factor. The rectangle approach in user1161's answer can also be used to get this. Let one side of the rectangle be $A+B+C+\cdots$ and a perpendicular side be $a+b+c+\cdots.$

So how do we expand $(a+b)(a+b)$? Like this:

$$(a+b)(a+b) \; = \; aa + ab + ba + bb \;=\; a^2 + 2ab + b^2$$

To continue this to more complicated expansions, well beyond what one would do in a classroom but what one might do for that rare student who is especially interested in math, see my answer to How to expand $(a_0+a_1x+a_2x^2+...a_nx^n)^2$?.

• This lower case upper case notation is great to read... but verbalizing it... that could be fun. I like your approach here, I think I'd never get past the numerical stage with students in the algebra class. But, using numerical examples to anchor algebraic rules to that which is already "known" is a wise practice in my estimation. At least it helped me when I was still learning algebra steps in my youth. Dec 7, 2021 at 1:46
• @James S. Cook: I recently wrote a Mathematics Stack Exchange answer that can be considered as a follow-up to this answer in addition to what I mention in the last sentence above --- Relation between coefficients of $(x+1)^n$ and choosing $k$ elements from a set ot size $n$? This comment is also to remind me to include this additional answer at the end above if I later wind up wanting to make other edits/changes here. Dec 7, 2021 at 8:41