# Subtraction Reversal Game and Investigation

In this lesson, students start by playing a game with two digit subtraction. Patterns emerge, data is organized, and a “trick” (rule or algebraic equation) is discovered for quickly solving these special kinds of subtraction problems. I love this because it gives an experience of discovering a pattern and then describing it either with words or algebraically. I’ve mostly used this lesson with 2nd-4th graders, but it is highly differentiable. Finding a general solution involved quite a bit of elegant algebra.

Intro: Subtraction Reversal Game Instructions:

•Each player rolls a 10-sided die two times.
•Find the difference between the largest number you can make using both numbers and the smallest number you can make. For example: (3,5): 53-35= 18
•Whoever has the largest difference wins.
•Play several rounds, Recording your results on paper.
Organizing the Data:
Students form groups and share the results of their games. They will notice that certain numbers come up more than once. Some may even notice they are all multiples of 9, or that the digits add up to 9. Others may need to organize the data to see the patterns.
I ask the groups to come up with a way of combining all their data on one piece of paper. You will notice that certain ways of organizing the data better displays the patterns, this can be a point of discussion when groups share their way of organizing the data.
Below is an example of what a groups partially filled out table might look like:

0 (3,3), (4,4)
9 (4,5), (7,8)
18 (7,9), (2,4)
27
36
45
54 (9,3)
63
72
81
•What patterns do you notice?
•What other numbers would you like to try?
•Is there a way to be more systematic about which numbers you try?
Discovering and Describing “The Trick”

Students work to describe the pattern either in words or using algebraic expressions.

• Is there a rule you could write to describe the patterns you noticed?
• What’s the trick/rule for finding the difference quickly?
Research Poster Project
After students have enough time exploring the patterns, I have students create posters individually or in partners to share their discoveries. Below are a few examples of posters students have made.
Differentiation
Some students may only be ready to describe “a trick” using words. Others may be ready to write an algebraic expression to describe the trick. A deeper level would be to try to explain using place value, and/or algebra to explain why this trick always works for 2 digit subtraction when the numbers are reversed.
When students quickly come up with a rule for 2 digits, I let them work on the 3 digit version. For example 632-236. They often immediately make the conjecture that the answers will be multiples of 99, and that for 4 digits it will be 999. The 4-digit version gets a bit harder to predict because the middle two numbers interact, representing the place values algebraically helps with this.