Introduction:
In today’s episode, we’re talking about comparison problems: what they are, why they’re tricky, and how we can help students build strategies to solve them.
We’re gonna start today’s episode with a problem. Omar made 22 baskets during practice. He made 9 more baskets than Eli. How many baskets did Eli make?
This is a comparison problem and there are 3 types of comparison problems we can use with students. And they are all hard!
What Are Comparison Problems?
Comparison problems are about figuring out how much more or less one quantity is compared to another.
3 types of comparison problems:
- Difference Unknown: The problem asks for the difference between two quantities.
Example: Jenna has 8 apples, and Liam has 5 apples. How many more apples does Jenna have than Liam?
- Quantity Unknown: The problem provides the difference and one quantity, the compare quantity in unknown.
Example: Jenna has 3 more apples than Liam. Liam has 5 apples. How many apples does Jenna have? - Referent Unknown: The problem provides the difference and the compare quantity, asking for the referent (referring to) quantity.
Example: Jenna has 3 more apples than Liam. Jenna has 8 apples. How many apples does Liam have?
How Students Might Solve These Comparison Problems
Discuss Different Solution Strategies:
- Direct modeling:
- Students draw or use counters to represent both quantities and visually compare them.
- Mention how this builds a strong foundation for conceptual understanding.
- Counting up or counting down:
- Students count up from the smaller number to the larger number or count down to find the difference.
- Using number lines:
- Explain how visualizing on a number line helps students see the distance between numbers.
- Invented algorithms:
- Students might decompose numbers to make the problem easier (e.g., “If I take 5 from 8, that’s just 3 left.”).
Misconceptions to Watch For:
- Students treat comparison problems as part-part-whole problems.
- Mixing up the language
- Not making sense of the problem and what the quantities mean in context.
Planning for Comparison Problem Types in the Classroom
- Consider the sticky points of the problem.
- What I mean by that is, what will YOUR students likely not understand? What do you think they might do that will trip them up as they are solving. Noting these sticky points will help you PLAN for the questions and moves yo’ull make tos support while maintaining their agency.
- Plan the questions you’ll ask. As I said, we want students to continue to problem solve and grapple without our support taking over their thinking. We want to maintain their agency which means they continue to stay in the driver seat. We are just there like the navigation to give them tips and ideas of where they might go.
- “How do you know which number is bigger?”
- “Have you ever something like this before?”
- “What does that number mean in the problem?”
- “What is the problem asking you to figure out?”
- Intentional sequencing of problems:
- Do two, the same problem type. One on Tuesday and the same problem type on Wednesday. This will give students a chance to apply what they learned the previous day to a new context.
- Also, in Step 5 of word problem workshop, Reflect– sometimes we reflect by giving One More Problem. That means you might keep the exact same problem, but just change the numbers. This gives students time to reflect on what they learned in the discussion and apply it to One More Problem! In fact, I wrote all about this and gave examples in the Word Problem Workshop book that is currently in production and should be available for pre-order soon. If you’re not already on the waitlist for the book you can join here:
MonaMath.com/BookWaitlist
