Problem-Solving Strategies
Master systematic approaches to solve any mathematical problem on the ParaPro test
Learning Objectives
The Problem-Solving Process
Polya's Four-Step Method
George Polya's classic approach provides a systematic framework for solving any mathematical problem.
Step 1: Understand the Problem
- • What is being asked?
- • What information is given?
- • What are the conditions or constraints?
- • Can you restate the problem in your own words?
Step 2: Devise a Plan
- • Choose a strategy or combination
- • Think of similar problems you've solved
- • Break complex problems into parts
- • Consider what operations to use
Step 3: Carry Out the Plan
- • Implement your chosen strategy
- • Show all your work clearly
- • Be patient and persistent
- • If stuck, try a different approach
Step 4: Look Back
- • Check your answer
- • Does it make sense?
- • Can you verify it another way?
- • Could you solve it differently?
Alternative Problem-Solving Models
STAR Method
- Search the problem
- Translate to math
- Answer the problem
- Review the solution
IDEAL Method
- Identify the problem
- Define the terms
- Examine options
- Act on a plan
- Look at results
Key Problem-Solving Strategies
Essential techniques for the ParaPro test
1. Draw a Picture or Diagram
Visual representations help clarify relationships and make abstract problems concrete.
Example Problem
A rectangular garden is twice as long as it is wide. If the perimeter is 36 feet, what are the dimensions?
Solution:
Draw rectangle, label width as w, length as 2w
Perimeter = 2w + 2(2w) = 6w = 36
w = 6 feet, length = 12 feet
2. Make a Table or Chart
Organizing information systematically reveals patterns and relationships.
Example Problem
A copy shop charges $5 setup fee plus $0.10 per copy. How many copies can you make for $20?
| Copies | Cost |
|---|---|
| 50 | $5 + $5 = $10 |
| 100 | $5 + $10 = $15 |
| 150 | $5 + $15 = $20 ✓ |
3. Look for a Pattern
Many mathematical problems involve sequences or repeating relationships.
Example: Find the next three numbers: 2, 5, 11, 23, 47, ...
5 - 2 = 3 → 11 - 5 = 6 → 23 - 11 = 12 → 47 - 23 = 24
Pattern: differences double each time
Next: 95, 191, 383
4. Work Backwards
Start with the final result and reverse the operations to find the starting value.
Example Problem
Sarah had some money. She spent half of it on books, then spent $10 on lunch. She has $15 left. How much did she start with?
End with $15 → Before lunch: $15 + $10 = $25
Before books: $25 × 2 = $50
Sarah started with $50
5. Guess and Check
Make educated guesses and refine them based on the results.
Example: Two numbers add to 15 and multiply to 56. What are they?
Try 5 and 10: 5×10 = 50 ✗
Try 6 and 9: 6×9 = 54 ✗
Try 7 and 8: 7×8 = 56 ✓
6. Use Logical Reasoning
Apply logic to eliminate possibilities and narrow down solutions.
Example Problem
Anna, Ben, and Carlos finished 1st, 2nd, and 3rd. Anna didn't win. Ben finished before Carlos. Who came in each place?
• Anna didn't win → she's 2nd or 3rd
• Ben finished before Carlos
Answer: Ben 1st, Anna 2nd, Carlos 3rd
Choosing the Right Strategy
Match problem types to strategies
| Problem Type | Recommended Strategies |
|---|---|
| Geometry/spatial problems | Draw a picture |
| Problems with many numbers | Make a table Find patterns |
| Multi-step problems | Work backwards |
| Optimization problems | Guess and check |
| Logic puzzles | Logical reasoning |
Checking Your Work
Verification methods
Substitution Check
Put your answer back into the original problem
- • Works for equations
- • Verifies word problems
- • Catches calculation errors
Estimation Check
See if your answer is reasonable
- • Round numbers first
- • Do mental math
- • Compare to estimate
Different Method
Solve the problem another way
- • Use alternate strategy
- • Work forwards if you worked backwards
- • Should get same answer
Unit Analysis
Check that units make sense
- • Distance in length units
- • Time in time units
- • Money with proper decimals
Common Errors to Avoid
Watch out for these mistakes
Misunderstanding the problem
Solving for the wrong thing or missing key information
✓ Prevention: Reread carefully, underline what's asked
Giving up too quickly
Stopping when the first approach doesn't work
✓ Prevention: Try multiple strategies, take breaks
Calculation mistakes
Arithmetic errors that lead to wrong answers
✓ Prevention: Show all work, double-check calculations
Not checking the answer
Accepting unreasonable or impossible results
✓ Prevention: Always verify your solution makes sense
Practice Problems
Test your skills
Number Pattern
In a sequence, the first term is 3, and each term is 4 more than the previous. What is the 25th term?
Logical Reasoning
Maya finished before Carlos but after Kim. David finished before Kim. Who won the race?
Work Backwards
If I multiply a number by 3, subtract 7, then divide by 2, I get 11. What's my number?
Draw a Diagram
A snail climbs a 10-foot wall. Each day it climbs 3 feet, but slides back 2 feet at night. On which day will it reach the top?
Multiple Strategies
In a parking lot, there are motorcycles (2 wheels) and cars (4 wheels). There are 25 vehicles and 70 wheels total. How many motorcycles?
Show Answers
1. 99 - Pattern: nth term = 3 + 4(n-1), so 25th = 3 + 4(24) = 99
2. David won - Order: David → Kim → Maya → Carlos
3. 29/3 or 9⅔ - Work backwards: 11×2 = 22, 22+7 = 29, 29÷3 = 9⅔
4. Day 8 - After 7 days at height 7, day 8 climbs to 10 (top!)
5. 15 motorcycles - Solve: m + c = 25, 2m + 4c = 70 → m = 15
Key Takeaways
Systematic Approach: Follow Understand → Plan → Solve → Check
Multiple Strategies: Have a toolkit of different approaches
Persistence Pays: If one method doesn't work, try another
Verification Matters: Always check that your answer makes sense
Continue Learning
Ready to Practice?
Test your problem-solving skills with our practice tests
