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:
•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.
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.
I’m going to try this today, as a sequel to A Billion Nickels. I’m considering it in Beta, and hoping my awesome students can help concretize and enrich the task.
Act 1: $30 million in 30 Days
How much does he need to spend per day?
How do you spend money without building Assets?
Does he do it?
Can it be done?
What would 30 million 1985 dollars be be adjusted for inflation?
Act 2: Take the Wimp Clause, or go for the $300 Million?
Students work in groups to figure out how they could spend $30 million in 30 days without accumulating assets. Using the internet and recording their plan on google spreadsheet.
Act 3: What does he do, does he win the $300 million?
“He Mailed It!”
Since our school aims to educate “change makers”, I wanted to put a twist on this plot line and ask students to think about how they could use $30 million to help people.