Offline Classroom Activities — Mathematics 8
Term 1 · Week 7 · Factoring Completely · CG Quarter 1, competency 8
Five activities: three individual, two group. Every one runs with no electricity, no devices and no printing budget.
| Activity | Time | Grouping | Best used | |
|---|---|---|---|---|
| I-1 | The Complete Check | 20 min | Individual | Day 1 — replaces the digital game |
| I-2 | Algebra Tiles | 25 min | Individual | Day 4 — the trinomial made physical |
| I-3 | How Many Steps? | 20 min | Individual | Day 4 — extension |
| G-1 | The Factor Wall | 35 min | Groups of 5 | Day 3 — consolidation |
| G-2 | The Expansion Relay | 40 min | Teams of 5–6 | Day 5 — application |
Making the algebra tiles (once, reused all of Quarter 1)
Cut from any cardboard — grocery boxes, folder offcuts, the back of a used notebook cover. One set per pair covers the class.
| Tile | Size | How many | Represents |
|---|---|---|---|
| Large square | about 6 cm × 6 cm | 2 | x² |
| Rectangle | 6 cm × 2 cm | 12 | x |
| Small square | 2 cm × 2 cm | 12 | 1 |
The only rule that matters: the rectangle's long side must exactly match the large square's side. If it does not, the tiles will not form a rectangle and the activity teaches nothing. Cut the large squares first and use one as the ruler for everything else.
Mark one face of each tile with chalk or pencil hatching for the negative version. Grade 8 needs negatives from Day 4.
Made once, these serve Weeks 5 to 9 and again in Grade 9.
INDIVIDUAL ACTIVITIES
I-1 · The Complete Check
Time 20 minutes · Grouping Individual · Use Day 1, replaces the digital game
No factoring from scratch. The learner only judges — which is exactly the skill Day 1 is for, and it is the skill that is missing when a class hands in half-finished work.
Materials
- Paper and pencil.
Instructions
For each: is it completely factored? If not, finish it. If yes, say how you know.
| Given | Complete? | Completely factored | |
|---|---|---|---|
| 1 | 2(x² − 4) | no | 2(x − 2)(x + 2) |
| 2 | (2x − 4)(x + 2) | no | 2(x − 2)(x + 2) |
| 3 | 3(x² − 9) | no | 3(x − 3)(x + 3) |
| 4 | 5x(x + 3) | yes | 5x(x + 3) |
| 5 | x(2x − 18) | no | 2x(x − 9) |
| 6 | (x − 1)(x + 1) | yes | (x − 1)(x + 1) |
| 7 | 4(x² + 2x + 3) | yes | 4(x² + 2x + 3) |
| 8 | x(x² − 16) | no | x(x − 4)(x + 4) |
(Teacher's copy. Learners get column 2 only.)
Item 7 is the hardest on the sheet and the most valuable. x² + 2x + 3 looks factorable — it is a trinomial with small coefficients — but no two whole numbers multiply to 3 and add to 2. Learners must be allowed to conclude "it stops here", and must not be trained to believe every trinomial factors. Most do not.
Item 6 is the mirror trap: it looks unfinished because it is short. It is complete.
The question to ask while circulating
"Can anything in there be factored again? Show me where you looked."
Differentiation
| Group | Adjustment |
|---|---|
| Below level | Items 1–5. Items 1 and 2 modelled first |
| On level | All eight |
| Above level | All eight, then: "Write an expression that is factored, looks complete, and is not. Swap with a partner." |
| Additional needs | Verbal answers accepted; the "finish it" step may be dictated to a peer scribe |
Assessment — 10 points
| Criterion | Points |
|---|---|
| Correct complete/not-complete judgement (½ each, 8 items) | 4 |
| Unfinished ones correctly finished (1 each, 5 items) | 5 |
| At least one "yes" justified rather than asserted | 1 |
Written Works.
I-2 · Algebra Tiles
Time 25 minutes · Grouping Individual (tiles shared in pairs) · Use Day 4
A trinomial factors when its tiles form a rectangle. The factors are the rectangle's sides. That is not a metaphor — it is what factoring is, and a learner who has built it once stops guessing.
Materials
- One tile set per pair
- Paper to record
Instructions
For each trinomial: lay out the tiles, arrange them into a rectangle, read off the sides.
| Trinomial | Tiles needed | Rectangle sides | Factored | |
|---|---|---|---|---|
| 1 | x² + 5x + 6 | 1 large, 5 rectangles, 6 small | (x + 2) by (x + 3) | (x + 2)(x + 3) |
| 2 | x² + 7x + 12 | 1, 7, 12 | (x + 3) by (x + 4) | (x + 3)(x + 4) |
| 3 | x² + 6x + 9 | 1, 6, 9 | (x + 3) by (x + 3) | (x + 3)² |
| 4 | x² + 4x + 4 | 1, 4, 4 | (x + 2) by (x + 2) | (x + 2)² |
| 5 | x² + 8x + 15 | 1, 8, 15 | (x + 3) by (x + 5) | (x + 3)(x + 5) |
(Teacher's copy. Learners get column 2 only.)
- Then the one that will not work: x² + 5x + 7. Try to build it. It cannot be done — there is no arrangement of 1, 5 and 7 tiles that makes a rectangle. Record what went wrong.
What items 3 and 4 show without being told
Both make a square, not just a rectangle. That is what "perfect square trinomial" means, and seeing it is worth more than the definition. Ask the two learners who finish first to hold their squares up.
Differentiation
| Group | Adjustment |
|---|---|
| Below level | Items 1, 3 and 4 — the smallest tile counts |
| On level | Items 1–5, then 6 |
| Above level | Add: "Build x² + 5x + 6 a different way. Can you?" (No — the rectangle is unique up to rotation, and that is why the factorisation is unique) |
| Additional needs | Pre-cut tile sets; the recording step may be photographed or dictated instead of written |
Assessment — 10 points
| Criterion | Points |
|---|---|
| Five rectangles built | 3 |
| Sides read off correctly (1 each) | 5 |
| Items 3 and 4 identified as squares | 1 |
| Item 6 attempted and the failure described | 1 |
Written Works.
I-3 · How Many Steps?
Time 20 minutes · Grouping Individual · Use Day 4, extension
The whole competency is the word completely, and this activity makes the number of steps the thing being counted.
Materials
- Paper and pencil.
Instructions
Factor each completely. Then write how many separate factoring moves it took. Taking out a common factor is one move. Applying difference of two squares is one move. Factoring a trinomial is one move.
| Polynomial | Steps | Completely factored | |
|---|---|---|---|
| 1 | x² − 9 | 1 | (x − 3)(x + 3) |
| 2 | 5x² + 15x | 1 | 5x(x + 3) |
| 3 | 2x² − 18 | 2 | 2(x − 3)(x + 3) |
| 4 | 5x² − 45 | 2 | 5(x − 3)(x + 3) |
| 5 | 3x² + 15x + 18 | 2 | 3(x + 2)(x + 3) |
| 6 | 2x³ − 18x | 2 | 2x(x − 3)(x + 3) |
| 7 | 4x³ − 4x | 2 | 4x(x − 1)(x + 1) |
| 8 | x⁴ − 16 | 2 | (x − 2)(x + 2)(x² + 4) |
(Teacher's copy. Learners get column 2 only.)
The two worth discussing at the end
- Item 8 is the only one where the second move applies to a bracket produced by the first rather than to what is left after a common factor. x⁴ − 16 → (x² − 4)(x² + 4) → and then only the first bracket goes further. x² + 4 stops, and learners must be able to say why.
- Item 2 is one step and item 3 is two, and they look equally simple. Nothing about the size of a polynomial predicts how many steps it needs — only looking again does.
Assessment
Open task, three bands:
| Band | Evidence |
|---|---|
| Extending | All eight complete and correctly counted; can explain why x² + 4 stops |
| Secure | All eight completely factored; step counts mostly right |
| Developing | The one-step ones correct; two-step ones stop after the first move |
GROUP / TEAM ACTIVITIES
G-1 · The Factor Wall
Time 35 minutes · Groups of 5 · Use Day 3, consolidation
Six groups, six categories, one wall chart the class builds and then uses for the rest of the quarter.
Materials
- One large sheet per group; pencil; tape.
Preparation
Write the assignments on the board.
| Group | Category | Polynomials |
|---|---|---|
| 1 | Common monomial factor | 5x² + 15x · 6x³ − 9x² · 12x⁴ − 18x² · 4x² + 8x + 12 |
| 2 | Difference of two squares | x² − 9 · x² − 16 · 4x² − 25 · 9x² − 16 |
| 3 | Trinomials, both signs positive | x² + 5x + 6 · x² + 7x + 12 · x² + 9x + 20 · x² + 8x + 15 |
| 4 | Trinomials with a negative | x² − 7x + 12 · x² − 9x + 20 · x² − 4x − 12 · x² + 2x − 15 |
| 5 | Perfect square trinomials | x² + 6x + 9 · x² − 10x + 25 · x² + 8x + 16 · 4x² − 12x + 9 |
| 6 | Two-step | 2x² − 18 · 3x² + 15x + 18 · 2x³ − 18x · 5x² − 45 |
Roles — assign, do not let them volunteer
| Role | Responsibility |
|---|---|
| Factorer A | Factors the first two. Does not check them |
| Factorer B | Factors the last two. Does not check them |
| Expander | Multiplies all four back out. Signs each one that matches |
| Completeness checker | Asks of every bracket: can this go further? Signs separately |
| Reporter | Presents and answers one question from another group |
Two separate signatures — one for correctness, one for completeness — is the whole design. They are different failures and they need different eyes.
Instructions
- 5 min — roles assigned, sheet ruled into four boxes.
- 18 min — factor all four.
- 7 min — Expander verifies; Completeness checker verifies. Disagreement sends it back to the Factorer.
- 5 min — sheets taped up in category order, Reporters read their category.
Answer key
| Group | Answers |
|---|---|
| 1 | 5x(x + 3) · 3x²(2x − 3) · 6x²(2x² − 3) · 4(x² + 2x + 3) |
| 2 | (x − 3)(x + 3) · (x − 4)(x + 4) · (2x − 5)(2x + 5) · (3x − 4)(3x + 4) |
| 3 | (x + 2)(x + 3) · (x + 3)(x + 4) · (x + 4)(x + 5) · (x + 3)(x + 5) |
| 4 | (x − 3)(x − 4) · (x − 4)(x − 5) · (x − 6)(x + 2) · (x + 5)(x − 3) |
| 5 | (x + 3)² · (x − 5)² · (x + 4)² · (2x − 3)² |
| 6 | 2(x − 3)(x + 3) · 3(x + 2)(x + 3) · 2x(x − 3)(x + 3) · 5(x − 3)(x + 3) |
Group 1's fourth item, 4(x² + 2x + 3), stops after one move — the bracket does not factor. Group 1 will assume they have missed something. They have not, and their Reporter should say so explicitly to the class, because every other group's items go all the way.
Managing 45+ learners at fixed desks
Groups are whoever is within reach; the sheet passes along the row. With more than six groups, split category 6 in two — it is the largest and the most instructive.
Assessment — group rubric, 12 points
| Criterion | 3 | 2 | 1 |
|---|---|---|---|
| Accuracy | All four factor correctly | Three correct | Two or fewer |
| Completeness | All four fully factored, including any second move | One left incomplete | Two or more incomplete |
| Two-signature check | Both checkers signed independently, with visible working | One checker only | No verification |
| Roles | Every member did their own role only | Roles blurred once or twice | One or two did everything |
Performance Tasks.
G-2 · The Expansion Relay
Time 40 minutes · Teams of 5–6 · Use Day 5, application
Teams factor, then hand their work to a different team to expand and judge. Being marked by a peer who wants to find an error is a far better check than marking your own.
Materials
- Paper and pencil. Nothing else.
Preparation
None. Write the four rounds on the board as you reach them.
Instructions
Round 1 — 8 min. Factor. Each team factors the same four polynomials completely:
2x² − 32 x² + 10x + 25 3x² − 12x x² − x − 20
Round 2 — 8 min. Swap and audit. Papers pass one team to the left. The receiving team must, for each of the four:
- expand it back and say whether it is correct
- judge whether it is completely factored
- award 2 points for correct and complete, 1 for correct but incomplete, 0 for wrong
Round 3 — 10 min. Return and repair. Papers come back with the score. Teams fix anything that lost a point, and may challenge a mark — a challenge is settled by expanding on the board.
Round 4 — 14 min. Set a trap. Each team writes one polynomial of its own that it thinks will be factored incompletely by another team. Swap, solve, score. A team scores 2 if its trap catches the other team and 0 if the trap was badly built — if it does not factor at all, or if it factors in one step.
Answer key — Round 1
| Polynomial | Steps | Completely factored |
|---|---|---|
| 2x² − 32 | 2 | 2(x − 4)(x + 4) |
| x² + 10x + 25 | 1 | (x + 5)² |
| 3x² − 12x | 1 | 3x(x − 4) |
| x² − x − 20 | 1 | (x − 5)(x + 4) |
Only the first needs two moves, and it is the one most teams will leave as 2(x² − 16). That is worth 1 point, not 0, and saying so out loud before Round 2 begins matters: the whole point of the week is that incomplete is correct but unfinished, not wrong.
Why Round 4 is worth the fourteen minutes
Building a trap requires knowing exactly where the second step hides. A team that produces 4x² − 36 has understood the week. A team that produces x² + 7 has not — it does not factor at all, and scores nothing.
Managing 45+ learners at fixed desks
Rows are teams and papers pass along the row and back. Nobody moves. With more than eight teams, run Round 4 with half the teams setting traps and half solving, then reverse.
Differentiation
| Group | Adjustment |
|---|---|
| Below level | Rounds 1–3 only, with the first polynomial modelled |
| On level | All four rounds |
| Above level | In Round 4, require the trap to need three moves (e.g. 2x⁴ − 32) |
| Language support | The audit may be discussed in the mother tongue; the written verdict is in English |
| Additional needs | Expanding may be done with a pre-printed grid; the Reporter may present from notes |
Assessment — group rubric, 12 points
| Criterion | 3 | 2 | 1 |
|---|---|---|---|
| Round 1 accuracy | All four correct and complete | Three complete | Two or fewer |
| Audit quality | Every expansion shown, every judgement correct | Most correct | Judgements asserted without expanding |
| Repair | Everything lost in Round 2 fixed correctly | Most fixed | Little changed |
| The trap | Factors, needs two or more moves, and caught the other team | Valid trap, not caught | Does not factor, or one step only |
Performance Tasks.
Which activity to reach for
| If your class… | Run |
|---|---|
| Has no devices at all | I-1 on Day 1 and I-2 on Day 4 — together they replace the game |
| Still stops one step early after Day 3 | I-1 again. Judging is the skill, not factoring |
| Cannot see why a trinomial factors at all | I-2. The rectangle is the explanation |
| Finished early and is restless | I-3, then the x⁴ − 16 question |
| Needs the term performance-task mark | G-1 or G-2 — both rubric-scored |
| Has one 40-minute slot and nothing prepared | G-2. The only preparation is four polynomials on the board |
Part of the E-turo MATATAG artifact set. Companion files:
lesson-plan.md · lesson-plan-ilaw.md ·
syllabus.md · assessment.md ·
slides.html · game.html ·
worksheet.html