The Brain Freeze:
"There are decimals and pounds and melon dollars... do I divide 5 by 3.50?"
Action: Friendly Numbers
"What if cherries were $2 and he bought 4 lbs? \(2 \times 4 = \$8\). Then add the melon! Multiply price by lbs, then add."
Execute Real Numbers
Total = $19.00
\(4 \times 3.50 = 14\); \(14 + 5 = 19\) ✓
Pólya's Insight: Friendly numbers took 5 seconds, but completely eliminated the confusion!
Universal Actions 3 & 4
Action 03 Any Multi-Step Problem
Don't try to find the final answer in one leap. Ask: "What is ONE intermediate number I can calculate right now?"
Self-Question: "What is one small fact I can find first before the main question?"
Action 04 Any Mystery Starting Value
When you are told the ending result, start at the finish line. Do the opposite operations in reverse order to rewind to the start.
Self-Question: "What was the last step? How do I undo it?"
Chunking turns a giant 3-step MCAS challenge into three easy 1-step calculations.
Live Action Modeling
"A sweatshirt costs $40. It is on sale for 25% off. After the discount, a $2 sales tax is added. What is the final total?"
Chunk 1: Find Discount
$10 off
\(0.25 \times 40 = 10\)
Chunk 2: Sale Price
$30.00
\(40 - 10 = 30\)
Chunk 3: Add Tax
$32.00 total
\(30 + 2 = \$32\) ✓
Pólya's Rule: Never rush. Solve Chunk 1, then Chunk 2, then Chunk 3. You get 100% accuracy every time.
Pólya's Principle 4
1
Did they ask for the discount amount ($10) or the final cost ($32)? Always re-read the last sentence!
Prevents answering the wrong thing.
2
If a $40 shirt is on sale, could the answer be $320? Could a distance be negative? Common sense is math!
Catches decimal and sign slips.
3
Plug your final answer back into the question statement. Does everything check out perfectly?
Guarantees earned points.
Pólya's Rule: A problem is not solved until you have looked back and verified it.
MCAS Practice Mission
Questions & Actions Lab
1
Read the Self-Question prompt: Listen to the internal dialogue before you start calculating.
2
Take the Action: Sketch it, try friendly numbers, break it into chunks, or rewind from the end.
3
Complete the Look Back check: Check the box to verify your answer makes total sense!
8th Grade MCAS Mathematics • Metacognitive Toolkit George Pólya Principles in Practice
🗣️ Self-Question: "If it ended at 26, what opposite operations undo subtraction and multiplication?"
Rewind: Add 6, then divide by 4
Show your reverse steps from 26 back to the start:
Starting number:
Look Back: Test forwards: Does \((4 \times \text{number}) - 6 = 26\)?
05
Action: Test Known Benchmarks
Between which two consecutive whole numbers does \(\sqrt{45}\) lie on a number line? Explain using perfect squares.
🗣️ Self-Question: "What simple perfect squares do I know that are just below and just above 45?"
Test: \(5^2=25, 6^2=36, 7^2=49\)
Write the two perfect squares that trap 45:
\(\sqrt{45}\) is between: and
Look Back: Is 45 strictly between \(6^2\) (36) and \(7^2\) (49)?
06
Action: Estimate First
Streaming Plan A costs $12 per month. Streaming Plan B costs $8 per month plus a one-time $16 fee. After how many months will both plans cost the exact same amount?
🗣️ Self-Question: "Should it take 4 months or 40 months? How do I set both costs equal?"
Equation: \(12m = 8m + 16\)
Solve the equation step-by-step:
Equal after: months
Look Back: Plan A: \(12(m) = \$48\). Plan B: \(8(m) + 16 = \$48\) ✓
Pólya's Universal Reflection:
When an unfamiliar problem looks confusing on a test, which action will you try first: Sketching it, using friendly numbers, or breaking it into chunks? Why?
Pólya's Thinking Lab • Universal Questions & Actions Page 2 of 2 • Universal Problem Solving for MCAS
George Pólya's Thinking Guide • Teacher Manual Page 1 of 2 • Solutions & Action Analysis on Page 2
Solutions & Action Breakdown
Universal Actions Key
1 Challenge 01: Triangle Angle Sum (Action: Sketch It) Answer: 60 degrees
Question & Action: "What does this look like?" Sketch a triangle and label angles 50°, 70°, and \(\angle C\).
Execution: Three angles sum to \(180^\circ \implies 50 + 70 + \angle C = 180 \implies 120 + \angle C = 180 \implies \angle C = \mathbf{60^\circ}\).
Look Back: Check: \(50^\circ + 70^\circ + 60^\circ = 180^\circ\) ✓.
2 Challenge 02: Total Purchase (Action: Friendly Numbers) Answer: $19.00
Question & Action: "What if cherries were $3/lb?" Reveals: Multiply rate by pounds, then add the melon.
Execution: Cherries: \(4\text{ lbs} \times \$3.50 = \$14.00\). Total: \(\$14.00 + \$5.00 = \mathbf{\$19.00}\).
Look Back: Estimate: \(4 \times 3.5 = 14\); \(14 + 5 = 19\) ✓. Reasonable for fruit shopping!
3 Challenge 03: Multi-Step Discount (Action: Break Into Chunks) Answer: $32.00
Question & Action: "What is ONE number I can find first?" Chunk 1: Discount amount.
Execution: Chunk 1: \(25\% \text{ of } \$40 = \frac{1}{4}(40) = \$10\) discount. Chunk 2: \(\$40 - \$10 = \$30\) sale price. Chunk 3: \(\$30 + \$2\text{ tax} = \mathbf{\$32.00}\).
Look Back: Common error avoided: Did not subtract $2 tax; added tax to discounted price.
4 Challenge 04: Mystery Starting Number (Action: Rewind) Answer: 8
Question & Action: "If it ended at 26, how do I undo the last step?" Invert operations in reverse order.
Execution: End \(= 26\). Undo subtracting 6: \(26 + 6 = 32\). Undo multiplying by 4: \(32 \div 4 = \mathbf{8}\).
Look Back: Test forward: \((8 \times 4) - 6 = 32 - 6 = 26\) ✓.
5 Challenge 05: Square Root Number Line (Action: Known Benchmarks) Answer: Between 6 and 7
Question & Action: "What simple squares trap 45?" Test nearby integers.
Execution: \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), then \(\sqrt{36} < \sqrt{45} < \sqrt{49} \implies \mathbf{6\text{ and }7}\).
Look Back: Verified: 45 is closer to 49, so \(\sqrt{45} \approx 6.7\), safely between 6 and 7.
6 Challenge 06: Comparing Monthly Plans (Action: Estimate First & Solve) Answer: 4 months
Question & Action: "Ballpark estimate first?" Estimate around 4 months. Set costs equal: \(\text{Cost A} = \text{Cost B}\).
Execution: \(12m = 8m + 16 \implies 4m = 16 \implies m = \mathbf{4\text{ months}}\).
Look Back: Plan A: \(12(4) = \$48\). Plan B: \(8(4) + 16 = \$48\). Both cost exactly $48 ✓.
Formative Evaluation Rubric: Metacognitive Actions
Stuck / Passive (1 pt): Stares blankly; writes isolated random operations without taking an action (no sketch, no chunking).
Action-Oriented (2 pts): Takes a physical action (sketches, tests friendly numbers, or chunks); minor arithmetic slip.
Metacognitive Master (3 pts): Seamlessly asks self-questions; executes actions cleanly; independently verifies in Step 4.
George Pólya's Thinking Guide • Teacher Manual Page 2 of 2 • Universal Problem-Solving Mastery
