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 |

Figure 2: Factoring out a 5 |

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 |

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.