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Reserve & Extensions • K-12
Mathematics is not about memorizing formulas and plugging in numbers. It is about thinking. When you face a problem you have never seen before, you need strategies -- systematic approaches that give you a way in, even when the path is not obvious. These strategies, formalized by mathematician George Polya in 1945, remain the gold standard for problem solving.
Many problems become clearer -- sometimes trivially easy -- once you draw a picture.
Problem: A snail is at the bottom of a 30-foot well. Each day it climbs 3 feet, but each night it slides back 2 feet. How many days does it take to reach the top?
Without a diagram, students often say: "Net progress = 1 foot per day, so 30 days." But draw it out:
After day 1: 3 ft up, slides to 1 ft. After day 2: up to 4, slides to 2. ... After day 27: up to 29, slides to 27. After day 28: climbs 3 to reach 30 -- at the top! No sliding back.
Answer: 28 days. The diagram reveals that on the final day, the snail reaches the top during the climb and does not slide back.
When you know the end result and need to find the starting point, work from the answer back to the beginning.
Problem: After doubling a number, adding 5, and then dividing by 3, you get 7. What was the original number?
Work backwards from 7:
The original number is 8.
Check: 2(8) = 16, 16 + 5 = 21, 21/3 = 7. Correct.
Try several cases and look for a pattern that lets you jump to the answer.
Problem: How many handshakes occur if 20 people each shake hands with every other person exactly once?
Start small:
| People | Handshakes | Pattern |
|---|---|---|
| 2 | 1 | 1 |
| 3 | 3 | 1 + 2 |
| 4 | 6 | 1 + 2 + 3 |
| 5 | 10 | 1 + 2 + 3 + 4 |
| n | n(n−1)/2 | 1 + 2 + ... + (n−1) |
For 20 people: 20(19)/2 = 190 handshakes.
If the problem is too complex, solve a simpler version first, then build up.
Make an educated guess, check it, and adjust. This is not random guessing -- each guess is informed by the previous result.
The most common problem-solving failure is skipping Step 1 (understanding the problem). Students rush to compute without knowing what they are looking for. Spend time re-reading the problem, identifying what is given and what is asked. Circle key words. Restate the problem in your own words. This step alone prevents most errors.
1. A frog is at the bottom of a 10-step staircase. It can jump 1 or 2 steps at a time. How many different ways can it reach the top? (Hint: look for a pattern starting with small numbers of steps.)
Use "look for patterns." 1 step: 1 way. 2 steps: 2 ways (1+1 or 2). 3 steps: 3 ways. 4 steps: 5 ways. This is the Fibonacci sequence! Each number is the sum of the two before it: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ways for 10 steps.
2. I am thinking of two numbers. Their sum is 25 and their product is 144. What are they? (Hint: guess and check.)
Guess and check: try 12 and 13. Sum = 25. Product = 156 (too high). Try 9 and 16. Sum = 25. Product = 144. The numbers are 9 and 16.
3. After tripling a number and subtracting 8, you get 19. Find the number. (Hint: work backwards.)
Work backwards from 19: before subtracting 8: 19 + 8 = 27. Before tripling: 27 / 3 = 9. Check: 3(9) − 8 = 27 − 8 = 19.
4. How many diagonals does a 10-sided polygon have? (Hint: start with simpler polygons and find a pattern.)
Triangle: 0 diagonals. Quadrilateral: 2. Pentagon: 5. Hexagon: 9. Pattern: n(n−3)/2. For n = 10: 10(7)/2 = 35 diagonals.
5. Three friends split a restaurant bill. The total after tax was $47.25. They want to leave a 20% tip on the pre-tax amount. Tax was 8%. How much does each person pay total? (Hint: work backwards to find the pre-tax amount.)
Work backwards: $47.25 includes 8% tax, so pre-tax amount = 47.25 / 1.08 = $43.75. Tip = 20% of $43.75 = $8.75. Total = $47.25 + $8.75 = $56.00. Each person pays 56.00 / 3 = $18.67 (rounding to nearest cent).