6.AT.4 Lesson 4-5-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

4.5 Small Group · Group 2

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

Visual Models · Determine the Whole Given the Part and Percent · Part · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Working backwards to the whole
1Percent × whole = part. When the whole is unknown, divide the part by the percent in decimal form: whole = part ÷ decimal — 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 original price of the sweater
  1. Akela paid $42 for a sweater on sale, and that was 60% of the original price. The $42 is the part; the original price is the whole, which is 100%.
  2. On a double number line, I put $42 under 60%. To find a smaller step, I use 10%: if 60% is $42, then 10% is 42 ÷ 6 = $7.
  3. Now I count up to the whole: 100% is ten of those 10% steps, so 10 × $7 = $70.
  4. The equation says the same thing in one move: 0.6 × v = 42, so v = 42 ÷ 0.6 = 70. The original price was $70 — larger than the sale price, which is exactly what a discount should mean.
3Second Model — Try it together — then prove it
  1. Sheng's car is now worth $26,600, which is 70% of what his parents paid for it. Find the original price.
  2. Name the pieces first: $26,600 is the part, 70% is the percent, and the purchase price is the unknown whole.
  3. Write the percentage statement as an equation, using the decimal form: 0.7v = 26,600.
  4. Solve by dividing both sides by 0.7: v = 26,600 ÷ 0.7 = 38,000. The car cost $38,000 when they bought it.
  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
  • Determine the Whole Given the Part and Percent (Hallar el total dados la parte y el porcentaje) — Working backwards from a part and its percent to find the whole amount it came from.
  • Part (Parte) — The amount you already know — the piece of the whole that the percent describes.
  • Whole (Total) — The full amount the percent is measured against. The whole is always 100%.
  • Double number line (Recta numérica doble) — Two lines lined up — dollars on one, percents on the other — so matching marks show the same amount two ways.
  • Equation (Ecuación) — A math sentence with an equal sign. Here it says percent × whole = part, with the whole unknown.
  • Percent (Porcentaje) — A ratio comparing a number to 100. To compute with it, write it in decimal form: 60% = 0.6.
5Watch out
  • A common mistake when finding the whole is multiplying by the percent instead of dividing. For "$42 is 60% of the original price," computing 0.6 × 42 = $25.20 makes the original price SMALLER than the sale price — impossible for a discount. Since percent × whole = part, the whole is found by dividing: 42 ÷ 0.6 = $70. Whenever the percent is less than 100%, the whole must come out larger than the part, so a smaller answer is the tell.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    $42 is 60% of the original price. Complete the ratio table to reach 100%.

    PercentDollars
    60%
    10%
    100%
    ✏️ Scratchpad / Reasoning
  2. 2 BAR MODEL
    Total10%10%10%10%10%10%10%10%10%10%

    A sweater's sale price of $42 is 60% of the original price. Each section is 10%. Use the bar model to find the original price.

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

    A shop reports that this week's $60 in sales is 120% of last week's. Was last week better or worse, and by how much?

    1. AWorse — last week was $50, so sales rose by $10
    2. BBetter — last week was $72, so sales fell
    3. CWorse — last week was $48
    4. DYou cannot tell without knowing the number of items sold
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    To find the whole when $9 is 25% of it, Lena divides 9 by 0.25. Omar reasons that 25% is a quarter, so he multiplies 9 by 4. Whose method works?

    1. ABoth — dividing by 0.25 and multiplying by 4 are the same operation
    2. BOnly Lena — multiplying cannot undo a percent
    3. COnly Omar — you should never divide by a decimal
    4. DNeither — you need the whole before you can use either method
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    To find the whole from a part and a percent, you…

    1. ADivide the part by the percent in decimal form
    2. BMultiply the part by the percent
    3. CAdd the part and the percent
    4. DSubtract the percent from 100
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    Claim: “If the percent is less than 100, the whole is always bigger than the part.” Always, sometimes, or never true?

    1. AAlways — a percent under 100 describes less than one whole, so the whole must be larger
    2. BSometimes — it depends whether the numbers are money or counts
    3. CNever — the part and the whole are always equal
    4. DSometimes — it fails when the percent is under 10
    ✏️ 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