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.

Revealio! The Hidden Negative Number

Subtraction is a difficult concept to grasp when introducing negative numbers. I had difficulty coming up with real-world story problems that had negative numbers either as an answer or part of the expression. Thinking in terms of negative numbers is so difficult because the first math concept we learn as young kids is that numbers exist to count physical, tangible objects. Count the number of oranges. Ten. What's left if all of them are taken away? Zero. Take away one more and what do you have? ....But how can you take away something that's gone? Placing a numeric value on something that can't be seen is like the magic trick where the magician pulls a rabbit out of an empty hat. Something is there that has value, but you can't see it!

But there is an old math trick that I was taught in middle school that reveals the hidden negative number, and I'll refer to it as the change-the-sign trick. Whenever there is a minus sign for subtraction, change it to a plus sign for addition, and then change the sign of the number after it. So a positive number becomes a negative, and a negative number becomes a positive. It's a great way to show students that they've been computing negative numbers all along; it has just been hiding! But how do I, as a future math teacher, show how this trick works, so my students don't rely on memorization to do math, but instead rely on their understanding?

In the teaching class I'm taking, our professor created a huge number line on the floor of our classroom, and we had to come up with ways to demonstrate how we add and subtract integers. The way that seemed less like following a bunch of rules was the one that was used and shown in the reading, The Intersection of Language and Mathematics, by Patricia E. Swanson from Mathematics Teaching in the Middle School, Vol 15, No. 9, May 2010, pg. 516-523. The first number is the person's starting point, the sign of the second number determines the direction they are facing - if it's a negative, face the negative numbers; if it's a positive, face the positive numbers - and the number tells how much they have to move to get to the answer. Lastly, the operation tells the person which way they move - if it's addition, the person moves forward; if it's subtraction, they move backward.

What makes this version the best to me is that the steps are less like rules because they are intuitive and use concepts students already use when they add and subtract whole positive numbers. Facing the direction of the sign of the second number doesn't require memorization - if the sign is negative, it makes sense to face the negative direction on the number line. As the reading discusses, students think of addition as gaining and subtraction as taking away. So, it makes sense for the students to move forward to gain, and move backward to take away. So, I've come up with a lesson using these steps to help my future students understand how to compute negative numbers and how the change-the-sign trick works. Students can either have their own number lines to follow along or these can be done on the blackboard as a whole class. (Figures created using GeoGebra.)

First, I would begin by explaining the steps on the number line and show them which direction the arrow points, or faces, according to the sign of the second number, as shown in the figure below.

Figure 1: Number Line Showing Direction of the Second Number

Next, I would use a simple math equation using subtraction that the students would easily know the answer to, like 5-2=3. I would demonstrate the answer using the steps: start at 5, second number is positive, so the arrow is pointing (facing) in the positive direction, and the operation is subtraction, so we are moving backwards. 

Figure 2: 5-2 = 3

  

Next, I'll show the students the change-the-sign trick. We have a minus sign, so we turn into a plus sign. Consequently, we have to change the sign of the second number from a positive to a negative. Then we'll go through the steps on the number line again to show that the answer is the same: start at 5, but this time the second number is negative, so the arrow is facing the negative direction, and the operation is addition, so we are moving forward. 

Figure 3: 5+-2 = 3

  

Next, I would show them the reflexivity property for adding numbers, and how it still holds true for adding negative numbers. So, first I would show them the property using positive numbers, as shown in the two figures below.

Figure 4: 2+5 = 7

Figure 5: 5+2 = 7

Then, I would use negative numbers, as shown in the two figures below:

Figure 6: -2+-5 = -7

Figure 7: -5+-2 = -7

But what happens when the change-the-sign trick isn't used? To answer this, I would demonstrate what happens using the two figures below. As we can see, the answer changes when we flip the numbers and then subtract, which means that reflexivity is not a property of subtraction.

Figure 8: -5-2 = -7

Figure 9: 2-(-5) = 7

Furthermore, I can show them how subtracting a negative number becomes adding a positive number by using the change-the-sign trick and comparing Figure 9 to Figure 10.

Figure 10: 2+5 = 7

If I have time and if the students seem to get comfortable with negative numbers, I would then show them how the number line, with the negatives on the left of zero and the positives on the right, as shown is Figure 1, is a convention. Technically, we could flip the number line, as shown in Figure 11 and the steps we have been using would still work, as shown in Figure 12 and Figure 13. This helps illustrate how math is consistent and not dependent on certain scenarios.

Figure 11: The Flipped Number Line Showing Direction of the Second Number

Figure 12: -2+-5 = -7 on Flipped Number Line

Figure 13: -2-5 = -7 on Flipped Number Line

Lastly, I would assess my students. In my teaching class, we've learned the steps on how to create a lesson plan by learning what students already know and forming a lesson around what they need to learn. The last step is Assessment, which is when I give them some activity to see how much they've learned and understand.

To assess their understanding of integers, I would show them a series of number lines, like the two below, and ask them to write the two equations that could represent what's shown. The red circle in the figures indicate the first, starting number. If they answer 7 + -15 = -8 and 7 - 15 = -8 for Figure 14, then I know they understand that subtracting a positive number is the same as adding a negative number. If they answer -3 - (-9) = 6 and -3 + 9 = 6 for Figure 15, then I know they understand that subtracting a negative number is the same as adding a positive number. Which is the point of learning the change-the-sign trick!

Figure 14: 7-15 = -8

Figure 15: -3 - (-9) = 6