6.AT.4 Lesson 4-3-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

4.3 Small Group · Group 2

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

estimate · benchmark percent · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Approximately — the problem is asking for an estimate
1Anchor the whole at 100%, mark benchmark percents, and place the amount between the landmarks it falls beside — 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 set up the double number line
  1. The whole is 6 GB — that is 100%.
  2. I mark the bottom line in benchmark jumps of 25%: 0%, 25%, 50%, 75%, 100%.
  3. Equivalent ratios give the top line: 25% of 6 GB is 1.5 GB, so the top reads 0, 1.5, 3, 4.5, 6.
  4. 4 GB used lands between 4.5 (75%) and 3 (50%), close to 4.5 — so about 70% of the storage is used.
3Second Model — Try it together — then prove it
  1. The Green-Team recycled 216 pounds of material; 8.6% was metal.
  2. Round both: 8.6% is about 10%, and 216 is about 220.
  3. 10% of 220 = 0.1 × 220 = 22. About 22 pounds of metal.
  4. Compatible numbers work too: 10% of 200 = 20. Both estimates agree — around 20 pounds.
  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
  • estimate (estimación) — A reasoned answer that is close enough to be useful, found without exact computation.
  • benchmark percent (porcentaje de referencia) — A friendly percent — 1%, 10%, 25%, 50%, 75%, 100% — used as a landmark for estimating.
  • compatible numbers (números compatibles) — Numbers close to the originals that are easy to compute with mentally.
  • double number line (recta numérica doble) — Two parallel number lines that pair amounts with their percents at the same positions.
  • the whole (el todo) — The quantity that counts as 100% in a problem — and it can change between questions.
5Watch out
  • Keeping the same whole for every question — when the problem says 'of the used storage,' the whole changes from 6 GB to 4 GB, and estimates anchored to the old whole are wrong before any arithmetic happens.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Complete the benchmark table for 240.

    BenchmarkShortcutValue
    10% of 240
    1% of 240
    50% of 240
    25% of 240
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each percent problem to the benchmark combination that estimates it.

    1. 15% of 200
    2. 75% of 80
    3. 20% of 350
    4. 11% of 400
    • A10% + 5%: 20 + 10 = 30
    • B50% + 25%: 40 + 20 = 60
    • C10% doubled: 35 × 2 = 70
    • D10% + 1%: 40 + 4 = 44
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Your estimate for 8.6% of 216 was 22 pounds (using 10% of 220). Is the exact answer higher or lower than 22?

    1. ALower, because both 10% and 220 rounded UP
    2. BHigher, because rounding always shrinks the answer
    3. CExactly 22
    4. DImpossible to tell
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    1 MB equals 0.001 GB. Mail uses 300 MB. About what percent of Margaret's 6 GB is mail?

    1. AAbout 5%
    2. BAbout 50%
    3. CAbout 0.5%
    4. DAbout 30%
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Two students estimate 8.6% of 216. Ana uses 10% of 220 = 22. Ben uses 10% of 200 = 20. Whose estimate is closer to the exact answer, 18.576?

    1. ABen's — 20 is 1.4 away; Ana's 22 is 3.4 away
    2. BAna's — 22 is closer to 18.576
    3. CThey are equally close
    4. DNeither is a valid estimate
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    Estimate 47% of 312. Show which strategy you used — benchmark, rounding, or compatible numbers — and give your estimate.

    ✏️ Mathematical Justification & Response
✍️ 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