Complications with representing constant difference on an open number line

Representing student thinking during a number string is complex. Certain strategies are particularly challenging to represent. For addition and subtraction, representing constant difference and compensation can both be challenging, for different reasons. I will tackle compensation in another post. For today, let’s look at what makes constant difference tricky to represent.

When you represent subtraction on a number line, the same subtraction problem looks different based on strategy. If a student solves 60 – 25 by removing 25, the number line looks different than if a student solves it by adding on to 25 until they reach 60. Take a minute and draw out those two solutions.

The first, removal, looks like this. The answer is where you finally land on the number line.

If a student solves it by adding on thirty-five to 25, the number line looks like this.

Now the answer is the difference between two numbers on the number line.

As Jennifer DiBrienza writes in this post on “where is the answer,” kids need to discuss this tricky situation so they can see that both represent the same problem, just different strategies.

Constant difference, as the name implies, is a strategy based on understanding subtraction as difference, not removal. When a student uses constant difference, they find an equivalent problem to the one they are faced with, one with an equivalent distance. A student could solve 76 – 39 by changing the problem to 77 – 40.

Number line representations of constant difference are extremely powerful. But a typical number string for constant difference often begins with kids proposing removal strategies. Then, as facilitator, if you may end up puzzled as to why constant difference isn’t emerging. It may be because your number lines are showing removal, not difference.

Here are a couple of ways that you could address this.
1. Design a constant difference number string that begins with doubles. Krista Rasmussen, a graduate of Chapman University and student in one of my classes, designed the following constant difference number string,

50 – 25

51 – 26

49 – 24

59 – 34

159 – 134

1159 – 1134

It begins with 50 – 25, which works for two reasons. One it is a known fact, and also because it is a double (25 + 25), it will have the same representation whether kids use removal or difference. As she began teaching the string, she was able to add onto her first representation. While not all students used constant difference, some were encouraged to try it because the numbers that she used were friendly, and elicited the idea of a constant difference. Her board at the end of the string looked like this.

2. Use a micro context. You can add context to a number string, linking it to investigations you have already done (Ages and Timelines is a great context for subtraction and constant difference), or simply tell a little story that gets kids into the situation. You can design these micro contexts to help kids think about subtraction as difference. One great context (used in the Ages and Timelines Context for Learning unit) is ages. For the problem 60 – 25, you could tell students that your mom is 60 and your sister is 25. What is the difference in their ages?

Another good context for difference is a join, change unknown problem, which encourage kids to add on (for more about problem types in CGI, read the new version of the CGI classic Children’s Mathematics). You could begin the string with a story problem, telling the students that you have 25 dollars, and you want to buy something that is 60 dollars. How much more money will you need? Students will typically solve this kind of story problem with addition, The next problem can be 60 – 25, which allows you to make a nice representation of subtraction as difference. Then, when your next problem is 61 – 26, your number line will support kids to think about difference as equivalence.