MPP.7 Lesson 10-5-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

10.5 Small Group · Group 2

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

repetition · pattern · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Repetition, pattern, and rhythm
1Math can be used to create beautiful designs. These designs often have repetition, patterns, and rhythm — and the pattern unit lets you predict what comes next — 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 use a pattern unit
  1. A border repeats: pink circle, purple square, blue triangle, pink circle, purple square, blue triangle, ...
  2. The pattern unit is 3 shapes long: circle, square, triangle.
  3. To predict shape number 12, I divide: 12 ÷ 3 = 4 with no remainder, so shape 12 ends a complete unit — it is the blue triangle.
  4. The next three shapes after any complete unit will be a pink circle, a purple square, and a blue triangle.
3Second Model — Try it together — then prove it
  1. The pattern unit is still 3 shapes long.
  2. 20 ÷ 3 = 6 complete units with a remainder of 2, because 6 × 3 = 18 and 20 − 18 = 2.
  3. A remainder of 2 means shape 20 is the second shape in the unit.
  4. The second shape is the purple square.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • repetition (repetición) — The same object, shape, or form repeated multiple times.
  • pattern (patrón) — A set of objects or shapes that are repeated and arranged in a regular order.
  • pattern unit (unidad del patrón) — The smallest chunk of a pattern that repeats.
  • rhythm (ritmo) — A design element created by varying the placement, size, orientation, or order of the same objects or shapes.
  • predictability (previsibilidad) — Being able to know what comes next because the design follows a rule.
5Watch out
  • Calling every repeated design a 'pattern.' One identical shape repeated is repetition; a pattern needs a unit of shapes repeating in a regular order; and rhythm varies the shapes' size and order, so it is neither.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    The border pattern repeats circle, square, triangle. Name the shape at each position.

    Position numberThinking with the unit of 3Shape
    7
    12
    20
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each vertex arrangement to its angle sum — and what it means for tessellation.

    1. 4 squares at a vertex
    2. 6 equilateral triangles at a vertex
    3. 3 regular hexagons at a vertex
    4. 3 regular pentagons at a vertex
    • A4 × 90° = 360° — no gaps
    • B6 × 60° = 360° — no gaps
    • C3 × 120° = 360° — no gaps
    • D3 × 108° = 324° — a gap remains
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    How is a design with rhythm different from a pattern?

    1. ARhythm uses shapes and a pattern does not
    2. BA pattern varies its shapes and rhythm never does
    3. CThere is no difference between them
    4. DRhythm varies size, orientation, and order, so it often lacks predictability; a pattern repeats in a regular order
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    The circle-square-triangle pattern continues. What is shape number 25?

    1. AThe purple square
    2. BThe blue triangle
    3. CThere is no shape 25
    4. DThe pink circle
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Which element of geometric design often LACKS predictability?

    1. ARhythm
    2. BRepetition
    3. CPattern
    4. DThe pattern unit
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    If a new row of circles were added to the rhythm painting, what would they look like? Explain your thinking.

    ✏️ 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