What is Constant Difference?
When we use constant difference, we add or subtract the same amount to/from both numbers in a subtraction expression. When we do this, the difference remains constant. For example, 9-5=4. But if we add one to both the 9 and 5, we now have 10-6. The difference is still 4.
Or we could add 10 to the minuend and subtrahend, to make 19-15, The difference is still 4.
Why Is Constant Difference Useful?
Imagine being able to solve an expression like 172-98 mentally in just a few seconds! We can easily do this using constant difference. For this example, let’s add 2 to both. Now we have 174-100. We can solve this simply in our heads!
How would you solve this one using constant difference: 386-194?
I solved it by adding 6 to each, to make 392-200.
Building Conceptual Understanding of Constant Difference: How To Teach This Strategy
Conceptual understanding of this strategy is essential. We want our students to be able to connect the conceptual understanding to their procedural understanding to ensure that this strategy is not simply a memorized series of steps, but rather an understanding. Here are some of my favorite ways to model this strategy.
Number lines are an excellent way to show this. In this example, the orange number rod represents the difference, which is 10. That 10 (the difference) is going to stay constant, no matter where we place it on the number line. The picture below shows how we can slide that difference up and down the number line. So 22-12 has the same difference as 23-13, or 24-14, or even 31-21. We can connect this concrete idea to equations, which are more abstract, to help our students develop the understanding behind this strategy.
Another way to think of this is with ages. If we have two people, ages 35 and 12, what is the difference in their ages? It’s 23 years. Next year, on this exact day, they will be 36 and 13. What is the difference in their ages? Still 23 years! There will always be a 23 year difference between them on this day each year.
Another way to model this is through the heights of real people. Try standing two students beside one another and discussing what the difference in their heights is. If they both grow 2 cm, does the difference in their heights change or stay the same? It stays the same! What if only one person grows? The difference in their heights changes. So as long as they both grow the exact same amount the difference in their heights will stay constant.
One last way to model this is with a concrete manipulative like unifix cubes. Make two towers and ask students what the difference in heights is. Add one block to each tower. Now what’s the difference? Take two blocks away from each tower – does the difference stay the same? What if we add a block to only one tower? How does that affect the difference?
Considering Different Options
As with any math strategy, flexibility is key. We don’t ever want our students to think there is only one way to think about this strategy. Let’s consider some different ways to use constant difference for this problem: 162-85. I’ve showed several different options. Which one is most effective for you?
Subtracting a Friendly Number vs Subtracting FROM a Friendly Number
When working with this strategy flexibly, it’s important to consider whether it’s more effective to subtract a friendly number or to subtract FROM a friendly number. Which one gives you the least amount of chance for error? Personally, I prefer to make the subtrahend into the friendly number. To me, it’s easier to subtract a friendly number rather than to subtract from a friendly number. Many students will make the minuend friendly instinctively, but it’s important to have the conversation to bring students’ attention to the fact that there are other ways to think about it that may be more effective.
Using Constant Difference to Reduce Chances of Errors
We can also use constant difference to reduce the chances of errors when regrouping across zeros (if your curriculum requires your students to learn the standard algorithm). In this example, instead of doing the calculations for 1000-346, we can subtract 1 from each to make 999-345. Now no regrouping is required and the chance of making an error is reduced.
This is a great strategy to incorporate into your number talks routine. Don’t have a number talks routine yet? Here’s how you can get started.
See this strategy on Instagram below:
