5.OA.B.3 Lesson 1-5-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

1.5 Small Group · Group 2

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

pattern · pattern rule · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — When we do math, we look for patterns and relationships
1We use the patterns and relationships we notice to solve problems, make generalizations, and check that our solutions are reasonable — 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 find the coin patterns
  1. A change jar is full of pennies, nickels, dimes, and quarters worth $53.50 in total. How many of each coin could there be?
  2. Each penny has a value of $0.01, so 100 pennies have a value of $1.00.
  3. Each nickel has a value of $0.05, so 100 nickels have a value of $5.00. Each dime has a value of $0.10, so 100 dimes have a value of $10. Each quarter has a value of $0.25, so 100 quarters have a value of $25.
  4. I multiply the number of each coin by its value. And I notice a generalization: because the jar's total, $53.50, ends in 0, the number of pennies can only be a multiple of 5.
3Second Model — Try it together — then prove it
  1. Try 200 quarters: 200 × $0.25 = $50.00.
  2. Add 25 dimes: 25 × $0.10 = $2.50, bringing the total to $52.50.
  3. Add 15 nickels: 15 × $0.05 = $0.75, and 25 pennies: 25 × $0.01 = $0.25.
  4. $50.00 + $2.50 + $0.75 + $0.25 = $53.50. It works — and 25 pennies is a multiple of 5, just as the pattern predicted. Now use the patterns to find a different combination.
  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
  • pattern (patrón) — Something that repeats or changes in a predictable way.
  • pattern rule (regla del patrón) — The instruction that tells how to get from one number in a pattern to the next.
  • generalization (generalización) — A statement that is true for a whole pattern, made by noticing repeated calculations.
  • table of values (tabla de valores) — A table that lists each step of a pattern so you can check that a solution is reasonable.
  • reasonableness (sensatez del resultado) — Whether an answer makes sense — checked as you work, with adjustments made when needed.
5Watch out
  • Reporting only the points from the streak (20 or 30) and forgetting to add the points each player had before the streak began.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Player A starts with 18 points and makes two-pointers. Complete the table.

    Two-pointers madeRule: 18 + 2 × shotsTotal points
    1
    5
    10
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each input-output table to its pattern rule.

    1. In: 1, 2, 3 → Out: 5, 10, 15
    2. In: 1, 2, 3 → Out: 3, 5, 7
    3. In: 2, 4, 6 → Out: 1, 2, 3
    4. In: 1, 2, 3 → Out: 2.5, 5, 7.5
    • AMultiply by 5
    • BDouble, then add 1
    • CDivide by 2
    • DMultiply by 2.5
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    If both players kept their streaks going, after how many MORE shots would Player B tie Player A?

    1. A2 more shots
    2. B1 more shot
    3. CThey can never tie
    4. D10 more shots
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    A sei whale measures 4.5 meters at birth and grows about 2.5 centimeters each day. About how many days until it reaches 15 meters?

    1. AAbout 420 days
    2. BAbout 600 days
    3. CAbout 42 days
    4. DAbout 4.2 days
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 OPEN RESPONSE

    Use the multiple-of-5 pattern to find a DIFFERENT combination of coins that totals $53.50. Show that it works.

    ✏️ Mathematical Justification & Response
  2. 6 OPEN RESPONSE

    After the streaks, Player A has 38 and Player B has 36. Player B started 12 points behind but nearly caught up. What about the two pattern rules explains that?

    ✏️ 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