6.NOS.6 Lesson 7-2-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

7.2 Small Group · Group 2

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

Visual Models · Rational Numbers on the Number Line · Rational number · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do you place rational numbers on a number line?
1To plot a rational number, find the two whole numbers it falls between, then split the space into equal parts — 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
  1. I will plot -2.5 on the number line.
  2. First, the whole numbers it falls between: -2.5 is between -2 and -3.
  3. The .5 means halfway, so I split the space between -2 and -3 in half.
  4. I place my dot exactly halfway between -2 and -3. That is -2.5.
3Second Model — Try it together — then prove it
  1. Now let's plot 0.75. Which two whole numbers is it between? It is between 0 and 1.
  2. What does 0.75 mean? It is 3/4 of the way from 0 to 1.
  3. So we split 0 to 1 into 4 equal parts and place the dot at the 3rd mark.
  4. Now prove it: say why that move had to work at all — not just that it did.
  5. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Rational Numbers on the Number Line (Números racionales en la recta numérica) — Every rational number has one exact spot on the number line. Writing it as a fraction, a decimal, or an integer does not change where it sits.
  • Rational number (Número racional) — A number that can be written as a fraction of two integers, with a bottom number that is not zero.
  • Fraction (Fracción) — A number that shows part of a whole, like 3/4.
  • Decimal (Decimal) — A number with a dot, like 0.5, that shows a part less than one.
  • Number line (Recta numérica) — A straight line where numbers are placed in order; numbers get smaller to the left and larger to the right.
  • Equivalent (Equivalente) — Having the same value, just written a different way.
5Watch out
  • A common mistake in Rational Numbers on the Number Line is treating the numerator and denominator of a negative fraction as two separate whole numbers instead of one value between two integers — for example, plotting -3/4 by going to -3 and then counting 4 more units to the left, landing (incorrectly) at -7, instead of recognizing that -3/4 = -0.75, which sits between 0 and -1. Before you submit an answer, find the two whole numbers your value falls between, then split that space into equal parts.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 NUMBER LINE

    Plot these rational numbers: -3.5, -1/4, 2.75, -2, 1 1/2

    ✏️ Workspace & Solution Steps
  2. 2 FILL TABLE

    Convert between fraction and decimal form. Then plot on a mental number line to compare.

    FractionDecimalBetween Which Two Integers?
    3/4
    -7/2
    2 1/5
    ✏️ Scratchpad / Reasoning
  3. 3 MULTIPLE CHOICE

    On a number line, point P is located at -2.5. Which statement describes its location best?

    1. AAt -2
    2. BHalfway between -2 and -3
    3. CHalfway between 2 and 3
    4. DAt -3
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 4 MULTIPLE CHOICE

    Which list of numbers is ordered from LEAST to GREATEST?

    1. A3, 0, -2, -5
    2. B-2, -5, 0, 3
    3. C-5, -2, 0, 3
    4. D0, -2, 3, -5
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 ERROR ANALYSIS

    Find the Plotting Error

    1. 1Problem:Plot -3/4 on the number line
    2. 2Student thinking:-3/4 has a 3 and a 4, so I go to -3 on the number line and then count 4 more to the left
    3. 3Student answer:Point plotted at -7

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  2. 6 OPEN RESPONSE

    Name three different rational numbers between -1 and 0. Write each as both a fraction and a decimal. Explain how you know they are between -1 and 0.

    ✏️ Mathematical Justification & Response
🗣️ 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 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