Learning intention LI
- Interpret dot circles as equal groups.
- Use different groupings to identify factor pairs.
- List all factors and justify whether a number is prime or composite.
Success criteria SC
- I can name the number represented by a dot circle and explain how I know.
- I can write “a groups of b” and match it to a × b = N.
- I can list all factors of N (2–32) without repeats.
- I can justify prime/composite using evidence from groupings.
Teacher tool shortcuts
Keyboard
Space new circle •
R reorder (if available)
Polypad reference: https://polypad.amplify.com/p#number-dots
Materials & setup Prep
- Project: Teacher Explore (Factor Circles tool).
- Optional: open Polypad Number Dots for live creation/variation.
- Mini whiteboards + markers + erasers.
- Student devices (optional): Student Quiz; Worksheet Creator printout.
Suggested launch numbers: 12, 14, 15, 18, 21, 24, 28 (mix prime/composite).
E Experience — Notice & Wonder (10–15 min)
Teacher actions Do
- Display a dot circle on the Teacher Explore tool (choose a composite first: e.g., 14 or 18).
- Ask students to silently record 2 notices and 1 wonder on mini whiteboards.
- Then open Polypad Number Dots and create a similar dot circle (or alter colours/groupings).
Notice & Wonder prompts Ask
- “What do you notice about the colours?”
- “What do you notice about the big circle and the smaller circles inside?”
- “How do you think the picture shows the number?”
- “What might the smaller circles represent?”
- “What do you wonder?”
Teacher talk moves Moves
- Wait time (3–5 seconds) before calling on anyone.
- Revoice: “So you’re saying the number might be the total dots…”
- Press for reasoning: “What in the picture tells you that?”
- Record & sort: build two columns on the board: “What the picture shows” vs “What that might mean”.
Bridge statement: “Today we’ll use these dot circles to explore factors — different equal-group ways to make the same number.”
L Language — build the vocabulary (10 min)
Key terms (display + co-construct)
- Factor: a number that divides evenly into N.
- Factor pair: two factors that multiply to make N (a × b = N).
- Prime: exactly two factors (1 and itself).
- Composite: more than two factors.
- Groups of: language for equal groups (multiplication structure).
Mini-whiteboard practice (fast cycles)
- Show a dot circle → students write: “The number is ___ because ___.”
- Then: “This is the same as ___ groups of ___.”
- Then: “Prime/composite because ___.”
Use quick “show me” checks. Call on 2–3 students to justify.
Teacher moves to strengthen reasoning
- Ask for a second way: “Can someone describe the grouping a different way?”
- Link representations: “Say it as ‘groups of’, then write it as multiplication.”
- Highlight structure: “Where do you see the number of groups? Where do you see the size of each group?”
P Pictorial — use the model to find factor pairs (15–25 min)
Core routine: “Read the picture”
- Identify the total number (N).
- Count the number of inner circles (groups).
- Count the number of dots in one inner circle (group size).
- Say it: “___ groups of ___.”
- Write it: “___ × ___ = N.”
Teacher explore sequence (suggested numbers)
- Composite with several pairs: 24 or 28 (use R reorder).
- Composite with fewer pairs: 14 or 15.
- Prime: 13 (contrast and generalise).
Record factor pairs on the board as you go (e.g., 24 = 1×24, 2×12, 3×8, 4×6).
Key generalisation: prime vs composite
Prime numbers show only one grouping circle (one inner circle arrangement), because the only factor pair is 1 × N.
Composite numbers can show more than one grouping (more factor pairs).
Teacher prompt: “What does it mean if we can’t find any new groupings after reordering?” → “There are no more factor pairs.”
Mini-whiteboard prompts during pictorial
- “Write two different ‘groups of’ statements for this picture.”
- “Is this number prime or composite? Justify using the picture.”
- “Predict the next reorder: what factor pair might appear?”
- “What factor pair have we not seen yet?”
Common misconceptions to surface
- Mixing up “number of groups” and “dots per group”.
- Thinking reorders change the total (they don’t).
- Forgetting 1 and N are always factors.
- Listing duplicate factors (e.g., repeating 2).
Teacher move: “Show me where you see the groups and where you see the group size in the picture.”
S Symbolic — connect to multiplication and factor lists (10–20 min)
Teacher-led modelling (I do / we do)
- Choose a number (e.g., 18) and model:
- “___ groups of ___” → “___ × ___ = 18”.
- Write the factor list in order: 1, 2, 3, 6, 9, 18.
- Ask: “How do we know we’ve got them all?” (pairs, symmetry around √N)
Student practice (you do)
- Give each student a target number (or pairs pick from 2–32).
- Students must produce:
- (1) two ‘groups of’ statements (or one if prime),
- (2) at least two multiplication equations,
- (3) full factor list,
- (4) prime/composite justification.
Teacher questions that strengthen generalisation
- “If a number is even, what do we immediately know about its factors?”
- “If a number has exactly one factor pair besides 1×N, what type of number could it be?”
- “How is a square number different (e.g., 16 has 4×4)?”
A Application — practice + transfer (15–30+ min)
Practice options using your tools
- Student Quiz (Tools tab): break 10 locks on Mild / Medium / Spicy.
- Worksheet Creator (Tools tab): fluency—list the factors of x (2–32).
How the Student Quiz works: Students choose a level, then complete 10 questions. Each question has a
3-step routine: (1) type the number shown in the dot circle, (2) add all
factor pairs as “___ groups of ___” and press Check, then (3) enter the full factor list
(each factor once). A lock only opens when Step 3 is correct; after 10 locks, students receive a completion code.
Suggested success target: Mild (all) → Medium (all) → Spicy (attempt).
Real-world contexts (choose 1)
- Seating: “We have 24 students—what rectangular seating plans are possible?”
- Packaging: “28 items—what equal box arrangements can we make?”
- Teams: “18 players—what equal training groups work?”
- Arrays/area: “32 tiles—what rectangles can we build?”
Require: at least 2 factor pairs + a reasoned recommendation.
Non-routine / challenge tasks
- Find a number from 2–32 with exactly 4 factors. Prove it.
- Find a number with the most factor pairs between 2–32. Justify.
- Pick two different numbers that have the same number of factors. Explain.
Assessment for learning AFL
Checks during the lesson
- Mini-whiteboard “show me” responses every 2–3 minutes during E/L/P.
- Ask students to justify using the picture (“Point to the groups”).
- Listen for language: “groups of”, “factor pair”, “prime”.
Exit ticket (2 minutes)
- Pick a number from 2–32. Write:
- all factors (in order),
- prime/composite + reason,
- one “groups of” statement (or “only 1×N” if prime).
Differentiation Support / Extend
Support
- Restrict number choices at first: 6, 8, 10, 12, 14, 15, 18.
- Provide a scaffold: “___ groups of ___ → ___ × ___ = ___”.
- Use paired reasoning: one student describes groups, partner writes equation.
Extension
- Numbers with special structure: squares (4, 9, 16, 25).
- Explain why primes show only one grouping circle (1×N) and no others.
- Create a “mystery number” clue set (e.g., “composite, exactly 6 factors”).
Teacher notes Notes
- Keep the focus on structure: equal groups → factor pairs, not just counting dots.
- When students make a claim, always follow with: “What in the picture shows that?”
- Use primes as a clear contrast: “Only one grouping circle exists because only 1×N works.”