MPP.7 Lesson 10-3-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

10.3 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

puzzle · pattern · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Playing with math means playing with patterns
1We play with math by using our imagination and by looking for patterns or relationships that help us think logically — a pattern can tell us the answer to a puzzle we have not even played yet — and you can say why it is true, and where it would stop being true.
  • You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.
2Worked Example — Watch me imagine the solution first
  1. Before moving anything, I describe what the solved puzzle will look like: the discs in the same order — smallest on top to largest on bottom — but on a different rod.
  2. Then I play the small cases: a one-disc tower takes 1 move, and a two-disc tower takes 3 moves.
  3. A three-disc tower takes at least 7 moves. Small cases are how mathematicians play.
3Second Model — Try it together — then prove it
  1. Discs: 1, 2, 3 — Steps: 1, 3, 7. What is happening from each row to the next?
  2. Double the number of steps and add 1: 1 doubled plus 1 is 3, and 3 doubled plus 1 is 7.
  3. So 4 discs take 7 × 2 + 1 = 15 steps, and 5 discs take 15 × 2 + 1 = 31 steps.
  4. The pattern just solved a puzzle we never touched.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • puzzle (rompecabezas) — A problem designed to be played with — it has rules, a goal, and room to imagine solutions.
  • pattern (patrón) — Something that repeats or changes in a predictable way.
  • pattern rule (regla del patrón) — A rule that tells you how to get the next value in a pattern from the one before it.
  • predict (predecir) — To say what will happen before it happens, using a pattern or reasoning.
  • organize (organizar) — To arrange information — often in a table — so patterns become easier to see.
5Watch out
  • Doubling the previous number of steps but forgetting to add 1 — answering 14 instead of 15 for four discs, or 30 instead of 31 for five.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Use the doubling pattern to complete the table of fewest moves.

    DiscsPattern: double the previous, add 1Fewest moves
    3
    4
    6
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each tower to the calculation that gives its fewest moves.

    1. 4-disc tower
    2. 5-disc tower
    3. 6-disc tower
    4. Any n-disc tower
    • A2 × 7 + 1 = 15
    • B2 × 15 + 1 = 31
    • C2 × 31 + 1 = 63
    • DDouble the (n − 1)-disc answer, add 1
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Why is finding a pattern MORE powerful than just solving the three-disc puzzle by hand?

    1. AThe pattern predicts the answer for towers you have never played, like five or six discs
    2. BThe pattern lets you skip the rules of the puzzle
    3. CPatterns make the discs move by themselves
    4. DA pattern proves the puzzle is impossible
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Will the fewest number of steps for a Tower of Hanoi puzzle always be an ODD number?

    1. AYes — doubling any whole number gives an even number, and adding 1 makes it odd
    2. BNo — every other tower size gives an even number of steps
    3. CYes — but only because 7 happens to be odd
    4. DNo — the steps alternate between odd and even
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    A five-disc tower takes 31 steps. Using the pattern rule, how many steps would a SIX-disc tower take?

    1. A63 steps
    2. B62 steps
    3. C36 steps
    4. D32 steps
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    Before solving, the lesson asked you to DESCRIBE what the solution will look like. Why is imagining the ending a useful way to start a puzzle?

    ✏️ Mathematical Justification & Response
✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The mathematical model proves the solution because each step preserves quantitative equivalence.
BUT An estimate provides a quick benchmark, but an exact proof is required for precision.
SO The quantities follow the standard rule, so the final result is verified with certainty.
📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim
State your direct mathematical answer/claim with units.
E Evidence
Cite exact numbers, calculations, dimensions, or graph data.
R Reasoning
Explain the mathematical theorem, property, or definition connecting evidence to claim.
✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

✏️ Author Workspace & Complete Solution Key
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It