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

7.7 Small Group · Group 2

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

vertex · polygon · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — The coordinates already know the side lengths
1When two vertices share a coordinate, the side between them is the absolute difference of the coordinates that differ — 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 find one side
  1. (-4.5, 3) and (5, 3) share y = 3, so the side between them is horizontal.
  2. Its length is the distance between the x-coordinates: |5 − (−4.5)| = 9.5 units.
  3. (5, 3) and (5, -3.75) share x = 5, so that side is vertical: |3 − (−3.75)| = 6.75 units.
3Second Model — Try it together — then prove it
  1. The shape is a rectangle, so the opposite sides match: two sides of 9.5 and two of 6.75.
  2. Perimeter = 2(9.5) + 2(6.75) = 19 + 13.5 = 32.5 units.
  3. That is why two of the terms are multiplied by 2 — each length appears on two sides.
  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
  • vertex (vértice) — A corner point of a polygon, named by an ordered pair on the coordinate plane.
  • polygon (polígono) — A closed figure made of line segments; on the plane you draw it by plotting the vertices and connecting them in order.
  • perimeter (perímetro) — The total distance around a polygon — the sum of all its side lengths.
  • distance between vertices (distancia entre vértices) — When two vertices share an x- or y-coordinate, the distance between them is the absolute difference of the other coordinates.
  • scale (escala) — What one grid unit stands for in the real situation.
5Watch out
  • Subtracting coordinates without the absolute value — from y = 3 down to y = −3.75 the distance is 3 + 3.75 = 6.75 units, not 3.75 − 3 = 0.75. Distances that cross zero are the whole reason the absolute value is there.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Find the distance between each pair of vertices.

    VerticesShared coordinateDistance (units)
    (−4, 3) and (5, 3)
    (5, 3) and (5, −4)
    (−2, −1) and (6, −1)
    ✏️ Scratchpad / Reasoning
SECTION 2 COMPUTATION & PROCEDURAL FLUENCY [Show Every Step]
  1. 2 COORDINATE GRID

    Plot the vertices of rectangle PQRS, then use the coordinates to find its perimeter.

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

    A rectangle has vertices (-3, 2), (4, 2), (4, -1), (-3, -1). What is its perimeter?

    1. A20 units
    2. B10 units
    3. C21 units
    4. D13 units
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  2. 4 MULTIPLE CHOICE

    Vertices (−2, 6) and (−2, −6): the distance between them is

    1. A12 units
    2. B0 units
    3. C4 units
    4. D36 units
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Maria is choosing between two store spaces on a grid where each unit is 1 foot and each grid line is 5 feet apart. Space A is a rectangle 25 ft by 40 ft; Space B is 30 ft by 30 ft. Which has more floor area?

    1. ASpace A — 1,000 sq ft vs 900 sq ft
    2. BSpace B — 900 sq ft vs 850 sq ft
    3. CThey are equal
    4. DSpace B — because a square always beats a rectangle
    ✏️ Workspace & Solution Steps
SECTION 4 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    Sheng plans a zoo visit using a map where exhibits sit at grid points. Explain how identifying WHAT the problem asks (his shortest route? total distance?) changes what you compute.

    ✏️ 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