Distribution Dilemma

Students at a middle school I was observing were simplifying equations, and I noticed a consistent issue among many of them; they did not know how to distribute properly. Many would distribute to the first number inside the parenthesis, but not the second. Others would distribute the outside number to both sets of parenthesis if there were more than one.  So, if I were a teacher and saw this issue with my students, how would I fix it? What is it about the concept the students aren't getting?

The underlying issue might be that the students don't understand the function of the parenthesis and how the number outside each parenthesis got there. Since distribution involves factors and multiplies of numbers, teaching that connection could help the students understand how distribution works, and understand the purpose of the parenthesis. As a way of teaching this concept, I will use a great problem solving method I learned from my professor, John Golden, called the work-backwards method. For example, lets take the problem 8x = 48. Even though 8 is being multiplied by x to get 48, 48 needs to be divided by 8 to get the solution for x, which is 6. In other words, we start with the answer and work backwards. And that's how I approached designing this lesson. I'm going to start with an expression in distributed form and then take out the common factor to show how the parenthesis groups together a set of numbers that has a common factor. This will show how distribution is factoring in reverse.

So lets start with a simple expression as shown in Figure 1. One common factor between 40 is 8. The number 4 is also correct, as well as 2. So, 8 is taken or factored out, as indicated by the arrows, and after dividing 40 and 72 by 8, 5 and 9 are left. The next step is crucial, which is explaining the parenthesis. Since I took out the 8, I need a way to show that the 5+9 was originally 40+72. So, I place the parenthesis around 5+9, right after the 8, which will indicate that the 8 gets distributed and gets multiplied by 5 and 9 to get 40+72. This setup tells you that the number outside the parenthesis gets distributed to the group of numbers inside that parenthesis only. In other words, the parenthesis groups different sets of numbers together respective to their distributor.

Figure 1: Factoring out an 8

The next expression will have 3 terms instead of 2. That way the students will see that more than 2 numbers can be within a parenthesis. As shown in Figure 2, we start with 20+45+55. To help the students visually with the parenthesis, I put them in at the first step. Then the 5 gets factored out, and 4+9+11 is left in the parenthesis.

Figure 2: Factoring out a 5

The next expression will show that not all numbers in the expression can be factored. As shown in Figure 3, a 3 gets factored out of 27 + - 60 - 9, but not 11. So the parenthesis group the left over 9 -20 -3 right after the factored out 3.

Figure 3: Factoring out a 3, but not 11

The next expression has 4 terms, which are factored out by 2, which is placed outside the parenthesis, and the left over 4+6+25+5 is inside the parenthesis.

Figure 4: Factoring out a 2

As discussed before some of these can be factored out in multiple ways. So, I repeated the expression in Figure 4, to show how the same expression can be re-written by re-grouping different sets of numbers by their different common factors. 4 can be factored out of 8 and 12, resulting in a 2+3 in one grouping (as indicated by the parenthesis), while 10 can be factored out of 50 and 10, which results in another grouping of 5 +1 (as indicated by the parenthesis).

Figure 5: Factoring out a 4 and a 10

As an evaluation after this mini-lesson on distribution and factors, I would give the students an expression with the variable x, like the one in Figure 6, for them to distribute and simplify. Hopefully, as Figure 6 indicates, they would distribute correctly. The simplified answer to Figure 6 is 1 - 13x. A good indication of how well this lesson worked at increasing student understanding would be seeing less errors with distribution.

Figure 6: Distribute 5 and 2

The figures in this blog post were created using GeoGebra.