6.NOS.9 Lesson 7-6-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

7.6 Small Group · Group 2

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

Distance on the Coordinate Plane · Distance · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do you find distance on the coordinate plane?
1If points are on the same side of zero, subtract; if they are on opposite sides of zero, add the absolute values — 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 find the distance from (-3, 2) to (4, 2). Both have y = 2, so they sit on the same horizontal line.
  2. The points are on opposite sides of zero, so I add the absolute values.
  3. |-3| = 3 and |4| = 4.
  4. I add: 3 + 4 = 7. The distance between them is 7 units.
3Second Model — Try it together — then prove it
  1. Now let's find the distance from (2, 3) to (2, 8). What do these points share? The same x = 2, so it is a vertical line.
  2. These points are on the SAME side of zero (both positive), so do we add or subtract? We subtract.
  3. The distance is |8 - 3| = 5 units. When points sit on the same side of zero, subtract to find the distance.
  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
  • Distance on the Coordinate Plane (Distancia en el plano de coordenadas) — The positive space between points, found from the absolute difference of coordinates on a horizontal or vertical line.
  • Distance (Distancia) — How far apart two points are. It is never negative.
  • Absolute value (Valor absoluto) — How far a number is from zero. It is never negative.
  • Horizontal distance (Distancia horizontal) — How far apart two points are going left or right.
  • Vertical distance (Distancia vertical) — How far apart two points are going up or down.
  • Coordinate plane (Plano cartesiano) — A grid with a line going across and a line going up to plot points.
5Watch out
  • A common mistake in Distance on the Coordinate Plane is subtracting the coordinates as if both points were on the same side of zero, instead of adding their absolute values when the points are on opposite sides. For example, to find the distance between (-6, 3) and (2, 3), a student might subtract 6 − 2 = 4, but since -6 and 2 are on opposite sides of zero, the absolute values must be added instead: |-6| + |2| = 6 + 2 = 8. The correct distance is 8 units, not 4. Before you submit, ask: 'Are these two points on opposite sides of zero — and if so, did I add the absolute values instead of subtracting?'
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Find the perimeter of each rectangle using its vertex coordinates.

    RectangleVerticesWidthHeightPerimeter
    Rect 1
    Rect 2
    ✏️ Scratchpad / Reasoning
SECTION 2 COMPUTATION & PROCEDURAL FLUENCY [Show Every Step]
  1. 2 COORDINATE GRID

    Plot these pairs and find each distance: A(-5, 3) to B(2, 3), and C(4, -2) to D(4, 4)

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

    A rectangle has corners (1, 2), (1, 5), (5, 5), and (5, 2). What is its perimeter?

    1. A7 units
    2. B12 units
    3. C14 units
    4. D10 units
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  2. 4 MULTIPLE CHOICE

    How far apart are the points (-2, 5) and (-2, -3)?

    1. A2 units
    2. B8 units
    3. C15 units
    4. D-8 units
    ✏️ Workspace & Solution Steps
SECTION 4 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 ERROR ANALYSIS

    Find the Distance Error

    1. 1Problem:Find the distance between (-3, 4) and (5, 4)
    2. 2Student thinking:Subtract: 5 - 3 = 2
    3. 3Student answer:The distance is 2 units

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

    ✏️ Corrected Mathematical Work & Explanation
  2. 6 OPEN RESPONSE

    A rectangle has vertices at (-4, -2), (3, -2), (3, 5), and (-4, 5). Find the length, width, perimeter, and area of the rectangle. Show your work using absolute value.

    ✏️ 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