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

10.3 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    Look at the table a different way: 1 disc → 1, 2 discs → 3, 3 discs → 7, 4 discs → 15, 5 discs → 31. Which describes the relationship between the number of discs and the number of steps?

    1. AThe steps are 1 less than a doubling pattern: 2, 4, 8, 16, 32 each minus 1
    2. BThe steps are the number of discs multiplied by itself
    3. CThe steps are 6 times the number of discs, plus or minus 1
    4. DThe steps are the two previous step counts added together
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    The table shows 1 disc → 1 step, 2 discs → 3 steps, 3 discs → 7 steps. What is the pattern rule for the number of steps when you add one more disc?

    1. ADouble the number of steps and add 1
    2. BAdd 2 to the number of steps
    3. CAdd the number of discs to the steps
    4. DMultiply the number of steps by 3
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    Use the pattern rule to complete the table: how many steps for a four-disc tower?

    1. A15 steps
    2. B14 steps
    3. C9 steps
    4. D28 steps
    ✏️ Workspace & Solution Steps
  4. 4 MULTIPLE CHOICE

    What is the minimum number of steps to solve a FIVE-disc Tower of Hanoi puzzle?

    1. A31 steps
    2. B15 steps
    3. C30 steps
    4. D35 steps
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 OPEN RESPONSE

    The 15 Puzzle is a sliding puzzle with 15 tiles in a frame 4 tiles high by 4 tiles wide. You arrange the tiles from 1 to 15 by sliding one tile at a time into the open space. Sometimes it is solvable and sometimes it is not. How would you make an argument about whether a particular 15 Puzzle is solvable, using what this lesson taught about being playful?

    ✏️ Mathematical Justification & Response
  2. 6 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:Why is finding a pattern MORE powerful than just solving the three-disc puzzle by hand?
    2. 2A classmate at our table answered:The pattern lets you skip the rules of the puzzle

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
📐 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.
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It