Tricie Jackson
Strategy Overview
For what types of problems is this strategy helpful?
The strategy “solve an easier related problem” means you take a hard problem and change it into an easier problem that you can better understand or solve. Once you solve the easier version, you use what you learned to solve the original or harder problem.
How it works
Sometimes math problems feel too big or confusing. Instead of giving up, you:
- Change the numbers to smaller or simpler ones
- Solve the simpler problem
- Look for patterns or steps that you can apply to the original problem
- Go back and solve the original problem using what you discovered
For what types of problems is this strategy helpful?
- It makes hard problems less scary
- It helps you see patterns
- It builds confidence step by step
- It’s useful in math, science, and real-life problem solving
When is it NOT very useful?
- When the problem is simple enough
- When you are using logic or rules vs numbers
- When the pattern doesn’t work in the original problem
Common mistakes or misconceptions to watch out for:
- Thinking the easier problem is the real or final answer
- Changing the problem to much
- Don’t assume patterns stay the same
How does this strategy help students become better mathematical thinkers?
If you use this strategy, you stop looking at hard problems like they are impossible. You can learn how to break them down step by step and become your own mathematician!
Example Problems
Problem 1:
Sum It Up
What is the sum of the first one hundred whole numbers? (Johnson & Herr, 2001, Pg. 266)
- Hint: There are many ways to approach this problem!
Problem 2:
Checkerboard Squares
How many squares are there on a checkerboard? (Johnson & Herr, 2001, Pg. 271)
- Hint: It is more than 64!
Worked Examples + Think-Alouds
Here are some steps to remember when solving any mathematical problem!
YouTube video provided by MiaLearningChannelK-12 (Steps in Problem Solving)
Problem 1:
How to solve problem #1:
Problem 2:
How to solve problem #2:
Strategy:
Focus on the smaller checkerboards’ vs the large 8×8 checkerboard
because we know the answer is going to be more than 64 based upon our hint.
First, look for a pattern:
| Checkerboard Size | Calculation | Total Squares |
|---|---|---|
| 1 by 1 | 1 | 1 |
| 2 by 2 | 4+1 | 5 |
| 3 by 3 | 9+4+1 | 14 |
| 4 by 4 | 16+9+4+1 | 30 |
Now go back to the original problem:
You can write it out in expanded form as: (using product of 2)
8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2
= 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1
= 204
Student Self-Talk Questions
- What is the question asking me to find?
- Can I make this question smaller or simpler?
Teacher Notes and Instructional Tips
Lesson Launch Ideas
Sports Analogy: Would a basketball player practice making 500 free throws all at once on the first day? Students will probably answer no!
- You must learn technique first
- Start with fewer shots first
- Then gradually you make more shots. So as mathematicians we must first start with smaller problems before solving larger ones.
Teacher Questions & Anticipated Student Responses
Assessing/Advancing questions a teacher could ask:
- What is the main question asking you to find?
- Can you solve a simpler version of the same problem first?
- What stays the same in the easier problem, and what changes?
- What pattern do you notice in the easier problem?
- How can that pattern help you solve the original problem?
- Can you draw a picture, make a table, or use objects to model the easier problem?
- Does your strategy for the easier problem also work for the original one? Why or why not?
- What did you learn from the easier problem that you can use next?
- How can you check that your final answer makes sense?
Student Responses
Before Solving/ Teacher: What could you do if this problem feels too difficult?
Student Responses:
-
- “I can make the numbers smaller first.”
- “I’ll try an easier version of the same problem.”
- “I want to solve a simpler problem to see what happens.”
- “This problem is too big, so I’ll start with something I know I can solve.”
While Solving/ Teacher: How is the easier problem helping you?
Student Responses:
-
- “The smaller numbers helped me figure out the steps.”
- “I noticed a pattern that I can use.”
- “I solved the easy one first, and then I used the same strategy.”
- “Now I understand what the problem is asking.”
- “The easier problem showed me what operation to use.”
- “I know what to do now because I practiced on the smaller problem.”
Connecting to the Original Problem/ Teacher: How will you use the easier problem to solve the original one?
Student Responses:
-
- “I’ll use the same steps with the original numbers.”
- “The pattern stays the same even though the numbers are bigger.”
- “I just replace the small numbers with the real numbers.”
- “The easier problem gave me a plan.”
Differentiation Suggestions
Support:
Example (Support) Easier Related Problem #1:
Find the sum of the first 10 counting numbers.
Pair numbers: 1 + 10 = 11, 2 + 9 = 11, 3 + 8 = 11, 4 + 7 = 11, 5 + 6 = 11
There are 5 pairs of 11
(5 times 11 = 55)
Then apply the same idea to 1–100
50 pairs of 101
(50 times 101 = 5,050)
Extend:
Example (Extend) Ask students:
-
- “Would this strategy work for the first 50 counting numbers? Why?”
- “How could you find the sum of the first 1,000 counting numbers in just one calculation?”
- “Can you explain why there are exactly 50 pairs when adding the numbers from 1 to 100?”
* These activities help students move from recognizing a pattern to justifying and generalizing it.
Formative Assessment Tools
- Think-pair-share
- White board responses
- Exit ticket
- Turn and talk
- QRC
- DOL
Student Reflection Prompts
Reflecting Teacher: Why did this strategy work?
Student Responses:
- “It made the problem less confusing.”
- “I didn’t feel stuck anymore.”
- “Breaking it into a smaller problem helped me understand it.”
- “It helped me see the math more clearly.”
- “I could check my thinking before solving the harder problem.”
- “The easier problem gave me confidence to solve the original one.”
Student Reflection Examples:
- “At first, I didn’t know how to solve the problem because the numbers were large.”
- “I made the problem easier with smaller numbers and solved that first.”
- “I noticed a pattern, so I used the same steps on the original problem.”
- “The strategy helped me understand what to do instead of guessing.”
*These responses model the mathematical thinking teachers want students to develop simplifying a problem, identifying patterns, and transferring a successful strategy to a more complex situation.
What prior knowledge do students need?
- Recognizing patterns
- The use of the four basic operations of math
What classroom challenges commonly arise?
- Students may not know how to make the problem smaller
- Students are dependent on teacher guidance
- Students might create a problem not related to the original
- Students cannot explain their reasoning
Media Attributions
- Problem_Solving_-_Concept_Illustration © Digits.co.uk Images is licensed under a CC0 (Creative Commons Zero) license
