6.AT.4 Lesson 4-1-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
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4.1 Small Group · Group 2

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

Understand Percent · Percent · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — What a percent really says
1A percent is a ratio comparing a number to 100. The whole is always 100%, so a percent above 100 means MORE than one whole — 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 turn a vote into a percent
  1. Students voted on a mascot, and 4 out of every 10 voters chose the tiger. I want to know how many that is out of 100.
  2. "4 out of every 10" is a ratio, so I write an equivalent ratio with 100 as the second term. Multiplying both terms by 10 gives 40 out of 100.
  3. On a decimal grid of 100 squares, I shade 40 squares — one square for each percent.
  4. 40 out of 100 is written 40%. So 40% of the students voted for the tiger.
3Second Model — Try it together — then prove it
  1. Shanice's gray kitten weighs 4 ounces. The orange kitten weighs 1.5 times as much. What percent of the gray kitten's weight is the orange kitten's weight?
  2. The gray kitten is the whole, so its weight is 100%.
  3. Draw a tape diagram: one full tape is 100%. The orange kitten needs 1.5 tapes — one whole tape plus half of another.
  4. One whole is 100% and half a whole is 50%, so 100% + 50% = 150%. The orange kitten weighs 150% of the gray kitten's weight — more than one whole, which matches it being the heavier kitten.
  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
  • Understand Percent (Comprender el porcentaje) — A percent is a ratio that compares a number to 100, so every percent answers the question “how many out of 100?”
  • Percent (Porcentaje) — A ratio that compares a number to 100. The word means “per hundred,” and the symbol is %.
  • Per hundred (Por cada cien) — The meaning hidden inside the word percent: for every group of 100. It is why every percent can be written as a fraction over 100.
  • Decimal grid (Cuadrícula decimal) — A 10-by-10 grid of 100 squares. Shading squares shows a percent directly, one square per percent.
  • Part-to-whole (Parte a todo) — A comparison of one part against the whole amount. In a percent, the whole is always 100%.
  • Greater than 100% (Mayor que 100 %) — More than one whole. A percent above 100 means the amount is larger than the thing it is being compared to.
5Watch out
  • A common mistake with percent is scaling only one term of the ratio: reading 7 out of 10 as 7%. A percent compares to 100, so BOTH terms must be multiplied by the same factor — 10 × 10 = 100, so 7 × 10 = 70, giving 70%. Reading it as 7% describes seats that are nearly empty when they are nearly full. A second mistake is treating a percent as an amount: 150% of a 200-point goal is 1.5 × 200 = 300 points, not 150 points.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 MATCHING GAME

    Match each comparison to its percent.

    1. 4 out of 10
    2. 23 out of 100
    3. 1.5 times the whole
    4. the whole itself
    • A40%
    • B23%
    • C150%
    • D100%
  2. 2 FILL TABLE

    Write each ratio as an equivalent ratio out of 100, then as a percent.

    RatioOut of 100Percent
    4 out of 10
    7 out of 20
    3 out of 4
    ✏️ Scratchpad / Reasoning
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Which is exactly one whole?

    1. A100%
    2. B50%
    3. C10%
    4. D1%
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    A recipe makes 200% of what you need. That means it makes…

    1. ATwice as much as you need
    2. BHalf of what you need
    3. CExactly what you need
    4. DTwo servings total
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    In Mr. Alvarez's class 40% of 30 students play an instrument. In the chess club 60% of 15 members do. Which group has MORE musicians?

    1. AThe class, 12 against 9
    2. BThe chess club, because 60% is greater than 40%
    3. CThey tie, because both are close to half
    4. DYou cannot compare percents of different-sized groups
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    Nadia finds 30% of 60 by taking 10% three times: 6, 12, 18. Owen halves 60 to get 50%, then subtracts 20%. Whose method reaches 30% of 60?

    1. AOnly Nadia — 10% three times is exactly 30%
    2. BBoth, because 50% − 20% = 30%
    3. COnly Owen — benchmarks are faster than repeating 10%
    4. DNeither — 30% of 60 cannot be found with benchmarks
    ✏️ Workspace & Solution Steps
🗣️ 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