6.AT.4 Lesson 4-2-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
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4.2 Small Group · Group 2

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

Visual Models · Fractions, Decimals, and Percents · Percent · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How can one value be written as a fraction, a decimal, AND a percent?
1To turn a fraction into a percent, build an equivalent fraction with a denominator of 100: ask “what do I multiply the denominator by to get 100?” and multiply the numerator by that same number. The new numerator is the percent — 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. A player scored 3/4 of the points. Percent means out of 100, so I want an equivalent fraction with 100 on the bottom.
  2. What do I multiply the denominator 4 by to get 100? 4 × 25 = 100, so my scale factor is 25.
  3. I have to multiply the top by that same 25: 3 × 25 = 75. So 3/4 = 75/100.
  4. 75 out of 100 is 75%. And 75/100 as a decimal is 0.75, because hundredths sit two places after the point.
  5. So 3/4 = 75/100 = 0.75 = 75%. All three name the same score.
3Second Model — Try it together — then prove it
  1. Let's change 1/5 into a fraction out of 100, then a decimal and a percent.
  2. What do we multiply the denominator 5 by to get 100? (20)
  3. Now multiply the numerator by that same 20: 1 × 20 = ? (20) So 1/5 = 20/100.
  4. 20 out of 100 is what percent? (20%) And as a decimal? (0.2)
  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
  • Fractions, Decimals, and Percents (Fracciones, decimales y porcentajes) — Three equivalent ways to name the same part of a whole.
  • Percent (Porcentaje) — A way to compare a number to 100, shown with the % sign.
  • Decimal (Decimal) — A number with a dot, like 0.5, that shows a part less than one.
  • Equivalent (Equivalente) — Having the same value, just written a different way.
  • Benchmark (Referencia) — A familiar number you use to guess, like 50% or 1/2.
  • Discount (Descuento) — Money taken off the first price to make it cheaper.
5Watch out
  • A common mistake in Relate Fractions, Decimals, and Percents is treating the numerator and denominator as the percent's digits — turning 3/5 into "35%" by just reading the 3 and the 5. Percent means out of 100, so the fix is to build an equivalent fraction with denominator 100: 5 × 20 = 100, so multiply the top by 20 too — 3/5 = 60/100 = 60%. A second version of this mistake is scaling only ONE part of the fraction (writing 3/5 = 3/100 or 60/5). Whatever you multiply the denominator by, you must multiply the numerator by the same number. When a denominator will not reach 100 evenly (8, 3, 7), divide the numerator by the denominator instead and then multiply by 100.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 NUMBER LINE

    Place each value on the number line: A = 3/8, B = 0.6, C = 85%

    ✏️ Workspace & Solution Steps
  2. 2 FILL TABLE

    Complete each row. Build the fraction out of 100 first — that column hands you the percent.

    FractionEquivalent fraction out of 100DecimalPercent
    2/5
    5/8
    ✏️ Scratchpad / Reasoning
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Your arcade scoreboard reads 0.25 of the level cleared. What percent and fraction match this score?

    1. A25% and 1/4
    2. B2.5% and 1/4
    3. C25% and 1/5
    4. D250% and 1/40
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    A player powered up 3/5 of the energy bar. Which decimal and percent name the same amount?

    1. A0.6 and 60%
    2. B0.35 and 35%
    3. C0.6 and 6%
    4. D0.65 and 65%
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Convert 3/5 to a percent.

    1. A60%
    2. B35%
    3. C53%
    4. D65%
    Rewrite each fraction
    Work it out
    Simplify
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Level 2 Extension — Find the Decimal-to-Percent Mistake

    1. 1Convert 0.4 to a percent:0.4
    2. 2Move the decimal point:0.4 → 0.04
    3. 3Write the percent:0.04 = 4%
    4. 4Final answer:0.4 = 4%

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

    ✏️ Corrected Mathematical Work & Explanation
🗣️ 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