6.AT.4 Lesson 4-4-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

4.4 Small Group · Group 2

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

Visual Models · Percent of a Number · Percent · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I find the percent of a number?
1Write the percent as a decimal (move the point 2 places left), then multiply it by the base: part = percent × base — 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
  1. The claw machine costs 60 tickets, and it is 25% off. I need 25% of 60.
  2. Step 1 — find the base (the whole): the number after “of” is 60.
  3. Step 2 — the percent as a decimal: 25% means 25 out of 100, so 25 ÷ 100 = 0.25.
  4. Step 3 — multiply: 0.25 × 60 = 15.
  5. The discount is 15 tickets. Check: 15 is about a quarter of 60, so that makes sense.
3Second Model — Try it together — then prove it
  1. Find 40% of 80. Step 1 — what is the base? (80, the number after “of”)
  2. Step 2 — 40% as a decimal? (40 ÷ 100 = 0.40)
  3. Step 3 — multiply: 0.40 × 80 = ? (32)
  4. Does 32 make sense? 40% is a little less than half, and half of 80 is 40. ✓
  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
  • Percent of a Number (Porcentaje de un número) — A part of a number found by multiplying the number by the percent written as a decimal.
  • Percent (Porcentaje) — A way to compare a number to 100, shown with the % sign.
  • Base (Base) — The whole amount you start with — the number right after the word “of.”
  • Part (Parte) — How much of the base (the whole) you get. It is your answer.
  • Equation (Ecuación) — A number sentence with an equal sign. The percent one is: part = percent × base.
  • Discount (Descuento) — Money taken off the first price to make it cheaper.
5Watch out
  • A common mistake in Find the Percent of a Number is multiplying by the percent number itself instead of its decimal form — for example, finding 30% of 80 by computing 30 × 80 = 2,400 instead of 0.30 × 80 = 24. Always divide the percent by 100 to get the decimal, then multiply that decimal by the base.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Complete each row to find the percent of a number.

    PercentBasePart
    10%
    75%
    35%
    8%
    ✏️ Scratchpad / Reasoning
  2. 2 BAR MODEL
    Total25%25%25%25%

    200 ÷ 4 = 50. Each 25% section = 50 points. So 25% of 200 = 50.

    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    The claw machine prize is 80 tickets and today it is 25% off. How many tickets do you save?

    1. A20
    2. B25
    3. C55
    4. D16
    0%50%100%
    Work
  2. 4 MULTIPLE CHOICE

    Your bowling score was 60 points. The bonus round adds 15% more points. How many bonus points is that?

    1. A9
    2. B15
    3. C45
    4. D75
    0%50%100%
    Work
  3. 5 MULTIPLE CHOICE

    A snack at the arcade costs $40. The sales tax is 5%. How many dollars is the tax?

    1. A$2
    2. B$5
    3. C$8
    4. D$20
    0%50%100%
    Work
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    A class has 32 students. 25% are in the math club, 50% are in the reading club, and 12.5% are in both. How many students are in each group? Can the total of all three groups be more than 32? Explain why or why not.

    ✏️ Mathematical Justification & Response
🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

Group Discussion Prompt: How does your visual model justify your mathematical solution?

🗣️ Partner A:
Partner A: "I modeled this by identifying the relationship and..."
👂 Partner B:
Partner B: "I agree with your step because the standard mathematical rule states..."
✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.
BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.
SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.
📐 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