6.NOS.7 Lesson 7-8-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

7.8 Small Group · Group 2

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

Ordered Pairs in All Four Quadrants · Quadrant · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do you plot points in all four quadrants?
1The signs of (x, y) tell you the quadrant: (+,+) is I, (-,+) is II, (-,-) is III, (+,-) is IV — 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 (-3, -4). I start at the origin (0, 0).
  2. The x is -3, which is negative, so I move 3 units to the LEFT.
  3. The y is -4, which is negative, so I move 4 units DOWN. I place my dot.
  4. Both coordinates are negative, so the point is in Quadrant III.
3Second Model — Try it together — then prove it
  1. Now let's plot (-5, 2). The x is -5, so which way do we move? Left 5 units.
  2. The y is 2, which is positive, so which way? Up 2 units.
  3. Left and up lands us in which quadrant? Quadrant II.
  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
  • Ordered Pairs in All Four Quadrants (Pares ordenados en los cuatro cuadrantes) — Points written as (x, y) whose positive or negative coordinates place them in one of four quadrants.
  • Quadrant (Cuadrante) — One of the four parts of a coordinate grid, numbered I to IV.
  • Negative coordinate (Coordenada negativa) — A coordinate less than zero. It means left or below the center point.
  • Reflection (Reflexión) — A flipped, mirror image across a line.
  • Axis (Eje) — One of the two number lines that frame a graph: the x-axis runs across, the y-axis runs up and down.
  • Integer (Número entero) — Whole numbers and their opposites, like -2, -1, 0, 1, 2.
5Watch out
  • A common mistake in Ordered Pairs in All Four Quadrants is matching each sign to the wrong axis — for example, seeing (-2, 5) and thinking "the x is negative so I move DOWN, and the y is positive so I move RIGHT," which lands the point in Quadrant IV. But the x-coordinate always controls left/right, never up/down: negative x means LEFT (not down), and positive y means UP (not right). So (-2, 5) is actually left 2, up 5 — Quadrant II. Before you submit, check that you used the x-coordinate for left/right and the y-coordinate for up/down, and never swapped them.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    For each ordered pair, write the signs of x and y and name the quadrant (or axis).

    Pointx signy signQuadrant or Axis
    (6, -2)
    (-3, -5)
    (0, 7)
    (-1, 8)
    (4, 0)
    ✏️ Scratchpad / Reasoning
SECTION 2 COMPUTATION & PROCEDURAL FLUENCY [Show Every Step]
  1. 2 COORDINATE GRID

    Plot one point in each quadrant: (3, 2), (-4, 1), (-2, -3), (5, -4)

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

    A point (a, b) has a < 0 and b < 0. In which quadrant does it lie?

    1. AQuadrant I
    2. BQuadrant II
    3. CQuadrant IV
    4. DQuadrant III
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Reflecting the point (3, -6) across the x-axis gives which point?

    1. A(3, 6)
    2. B(-3, -6)
    3. C(-3, 6)
    4. D(6, 3)
    ✏️ Workspace & Solution Steps
SECTION 4 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 ERROR ANALYSIS

    Find the Quadrant Error

    1. 1Problem:Identify the quadrant for (-3, -5)
    2. 2Student thinking:The x is -3, so I go left 3. The y is -5, so I go up 5.
    3. 3Student answer:(-3, -5) is in Quadrant II

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

    ✏️ Corrected Mathematical Work & Explanation
  2. 6 OPEN RESPONSE

    A point starts in Quadrant I at (4, 6). Describe a move that puts it in Quadrant III. What must happen to both coordinates? Can you give the exact new coordinates after your move?

    ✏️ 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