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 unreasonable — 5/6 of 540 must be LESS than 540
    2. BThe answer is fine because it is close to 540
    3. CThe answer is fine because 650 is a bigger number than 5/6
    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