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

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

    1. AThe pink circle
    2. BThe purple square
    3. CThe blue triangle
    4. DThere is no shape 25
    ✏️ 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