5.NF.B.4 Lesson 1-2-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

1.2 Small Group · Group 2

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

quantity · relationship · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Make sense, plan, check
1When we do math, we make sense of problems, develop a solution plan, check our progress, and try other strategies when we come to dead ends — 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 estimate the peas
  1. A bowl is full of peas — far too many to count one at a time. About how many peas are in the bowl?
  2. I do not count every pea — I look for a group I CAN count, like one small cluster of about 10.
  3. Then I ask how many of those clusters would cover the whole bowl. I count about 12.
  4. 12 clusters of about 10 is 12 × 10 = 120, so my estimate is about 120 peas — a reasoned answer, close enough to be useful.
3Second Model — Try it together — then prove it
  1. 76 = 100 + 5.76.
  2. 76 = 105 + 0.76.
  3. 76 = 110 − 4.24.
  4. 76 = 52.88 × 2, and 105.76 = 211.52 ÷ 2.
  5. Same number, five different breaks. Find one more of your own.
  6. Now prove it: say why that move had to work at all — not just that it did.
  7. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • quantity (cantidad) — An amount in a problem — a number with a meaning attached, like 540 meters.
  • relationship (relación) — How two quantities compare or connect to each other.
  • strategy (estrategia) — A plan for how to solve a problem before you start computing.
  • reasonable (razonable) — A solution that makes sense in the context of the problem.
  • persevere (perseverar) — To keep working on a problem, checking your progress and adjusting your plan when you get stuck.
5Watch out
  • Expecting a fraction of a number to be bigger than the number — 5/6 × 540 must be LESS than 540 because 5/6 is less than 1; only a fraction greater than 1, like 6/5, makes the product bigger.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Compute each building's height as a fraction of 540 meters.

    Building planFraction of 540 mHeight (m)
    Rio de Janeiro
    Tokyo
    Prototype
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each fraction of 540 to a shortcut that computes it.

    1. 9/10 of 540
    2. 6/5 of 540
    3. 5/6 of 540
    4. 1/2 of 540
    • A540 − 54
    • B540 + 108
    • C540 − 90
    • D540 ÷ 2
  3. 3 MULTIPLE CHOICE

    You calculate 5/6 × 540 and get 650 meters. What does the checking step of your plan tell you?

    1. AThe answer is fine because it is close to 540
    2. BThe answer is fine because 650 is a bigger number than 5/6
    3. CThe answer is unreasonable — 5/6 of 540 must be LESS than 540
    4. DChecking is only needed when the problem asks for it
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 4 MULTIPLE CHOICE

    To find 3/4 of 20, Sam divides by 4 and then multiplies by 3. Nia multiplies by 3 and then divides by 4. Both get 15. Which is easier for these numbers, and why?

    1. ASam's — 20 divides evenly by 4, so the numbers stay small
    2. BNia's — multiplying first is always easier than dividing first
    3. CThey cannot both be correct, so one of them made an error
    4. DSam's — because you must always divide before you multiply
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 OPEN RESPONSE

    Invent a route length and a fraction so that the student walks exactly 18 blocks. Your fraction may NOT be 1/2. Show that your example works.

    ✏️ Mathematical Justification & Response
  2. 6 OPEN RESPONSE

    When is it easier to divide first, and when is it easier to multiply first, when finding a fraction of a whole number? Give one example of each.

    ✏️ Mathematical Justification & Response
✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The mathematical model proves the solution because each step preserves quantitative equivalence.
BUT An estimate provides a quick benchmark, but an exact proof is required for precision.
SO The quantities follow the standard rule, so the final result is verified with certainty.
📐 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