ILAW Lesson Plan — Mathematics 8
Term 1 · Week 7 · Number and Algebra · five sessions × 45 minutes
| Learning Area | Mathematics |
| Grade Level | Grade 8 |
| Term / Week | Term 1, Week 7 |
| CG source | Quarter 1, competency 8 (p. 56) |
| Week Title | Look Again — Factoring Polynomials Completely |
| Duration | 5 × 45 minutes |
| Format | ILAW — Intentions · Learning Experiences · Assessment · Ways Forward |
Two formats, two jobs. This is the week, in the format DepEd prescribes for the plan a teacher files. Its companion
lesson-plan.mdis Day 1 only, at teaching-script depth. Use this one to plan and to file; open that one before you teach Monday.
⚠️ The ILAW field list is reported, not read. DO No. 009, s. 2026 is understood to prescribe ILAW as the single lesson plan template for SY 2026–2027, replacing both the DLL and the DLP. The four section names are consistently reported; the exact sub-fields your division expects may differ. See
source-catalogue.md§8.
I — INTENTIONS
A. Curriculum anchor
Quoted from the MATATAG Mathematics CG (DepEd, August 2023, p. 56):
Content standard. The learners demonstrate knowledge and understanding of special products for binomials, and factorization of polynomials.
Performance standard. By the end of the quarter, the learners are able to factorize different types of polynomials.
Competency covered this week — quoted verbatim:
8. completely factor different types of polynomials (polynomials with common monomial factor; difference of two squares; quadratic trinomials, including perfect square trinomials).
Assumed from Weeks 5 and 6: multiplying binomials, and the special product patterns.
✅ Verified against the curriculum guide committed at
docs/research/source/.
B. What learners will be able to do, by day
| Day | Intention — by the end of the session, at least 80% of learners can… | Type |
|---|---|---|
| 1 | Show that a polynomial can have more than one correct factorisation, and identify which one is complete | all |
| 2 | Take out the common monomial factor first, every time, and say why it must come first | CMF |
| 3 | Recognise and factor a difference of two squares, including after a common factor is removed | DOTS |
| 4 | Factor quadratic trinomials, and recognise a perfect square trinomial on sight | trinomials |
| 5 | Choose the right method unprompted, and check by expanding | all |
Day 1 teaches no new method at all. It exists to establish what the word completely means. A class that starts on the common monomial factor on Monday will factor correctly all week and stop one step early in the examination, because nobody ever told them there was another step.
C. Key idea and vocabulary
Factored is not the same as completely factored.
| Term | Meaning used this week |
|---|---|
| Factor (verb) | Write as a product |
| Completely factored | No factor can be factored any further |
| Common monomial factor (CMF) | A term that divides every term — take it out first |
| Difference of two squares | a² − b² = (a − b)(a + b). Note: difference, never sum |
| Perfect square trinomial | a² + 2ab + b² = (a + b)². Recognisable, not just factorable |
The class sentence, repeated all week until it is automatic: "Common factor first. Then look again."
D. Integration
| Strand | How it appears |
|---|---|
| EPP / TLE | Areas of bordered rectangles — the term performance task |
| EsP / GMRC | Day 1's three answers are all offered by learners and all treated as contributions |
| English | Completely as a mathematical word with a precise meaning, unlike its everyday use |
| 21st-century skills | Self-checking: every factorisation is verified by expanding it back |
L — LEARNING EXPERIENCES
Every session is 45 minutes and runs complete with chalk alone. Digital artifacts are listed where they help; none is required.
Day 1 · Three answers, all correct
Full teaching script:
lesson-plan.md. Summary only here.
| Phase | Min | What happens |
|---|---|---|
| Activating prior knowledge | 5 | Expand three products from Week 6: (x+3)(x−3), (x+4)², 2(x−1)(x+5). Confirm fluency going forwards before asking for backwards. |
| Establishing purpose | 6 | Board: factor 2x² − 8. Collect answers. Three will appear: 2(x² − 4), (2x − 4)(x + 2), and 2(x − 2)(x + 2). Write all three. |
| Developing understanding | 22 | Expand all three. Every one gives 2x² − 8. "All three are correct. So why do textbooks only accept the last?" Because in the first, x² − 4 factors further; in the second, 2x − 4 has a common factor of 2. Only the third is finished. |
| Making generalisations | 7 | Land the definition: completely factored means no factor can be factored again. Then the strategy: take the common factor out first, and it will not happen to you. |
| Evaluating | 5 | Exit ticket: is 3x² − 27 = 3(x² − 9) completely factored? Justify. |
Independent practice, without devices — for each, say whether it is complete and finish it:
| Given | Complete? | Completely factored |
|---|---|---|
| 2(x² − 4) | no | 2(x − 2)(x + 2) |
| (2x − 4)(x + 2) | no | 2(x − 2)(x + 2) |
| 3(x² − 9) | no | 3(x − 3)(x + 3) |
| 5x(x + 3) | yes | 5x(x + 3) |
| x(2x − 18) | no | 2x(x − 9) |
Day 2 · Common factor first
| Phase | Min | What happens |
|---|---|---|
| Recall | 5 | Yesterday's five, from memory. |
| Purpose | 5 | Board: 12x⁴ − 18x². "Try it without taking a common factor out first." Let them struggle for two minutes, then do it the other way. |
| Developing | 24 | The CMF is the GCF of the coefficients times the lowest power of each variable. Worked: 5x² + 15x → 5x(x + 3). 6x³ − 9x² → 3x²(2x − 3). 12x⁴ − 18x² → 6x²(2x² − 3). 4x² + 8x + 12 → 4(x² + 2x + 3) — and the bracket does not factor further, which is worth showing. Then: always look again at what is left. |
| Generalising | 6 | Why first? Because it makes every later step smaller, and because forgetting it is the single commonest way to leave an answer incomplete. |
| Evaluating | 5 | Exit ticket: factor 2x³ − 18x completely. (2x(x − 3)(x + 3) — two steps.) |
Worked set (teacher's copy):
| Polynomial | CMF | Completely factored |
|---|---|---|
| 5x² + 15x | 5x | 5x(x + 3) |
| 6x³ − 9x² | 3x² | 3x²(2x − 3) |
| 12x⁴ − 18x² | 6x² | 6x²(2x² − 3) |
| 4x² + 8x + 12 | 4 | 4(x² + 2x + 3) |
| 2x³ − 18x | 2x | 2x(x − 3)(x + 3) |
Day 3 · Difference of two squares
| Phase | Min | What happens |
|---|---|---|
| Recall | 5 | Three CMFs, timed. |
| Purpose | 5 | Board: (x − 3)(x + 3) expanded — the middle terms vanish. "Run it backwards." |
| Developing | 23 | a² − b² = (a − b)(a + b). Worked: x² − 9, x² − 16, 4x² − 25 (a = 2x), 9x² − 16 (a = 3x). Then the combination that catches everyone: 2x² − 18 — common factor first, then DOTS → 2(x − 3)(x + 3). Then the warning: x² + 9 does not factor. A sum of two squares has no factorisation over the numbers this class knows. |
| Generalising | 7 | Two things must both be true: both terms are perfect squares, and the sign between them is minus. |
| Evaluating | 5 | Exit ticket: factor 3x² − 27 completely, and state whether x² + 4 factors. |
Worked set (teacher's copy):
| Polynomial | a, b | Completely factored |
|---|---|---|
| x² − 9 | x, 3 | (x − 3)(x + 3) |
| x² − 16 | x, 4 | (x − 4)(x + 4) |
| 4x² − 25 | 2x, 5 | (2x − 5)(2x + 5) |
| 9x² − 16 | 3x, 4 | (3x − 4)(3x + 4) |
| 2x² − 18 | after CMF | 2(x − 3)(x + 3) |
| x² + 9 | — | does not factor |
Day 4 · Trinomials, and the perfect squares
| Phase | Min | What happens |
|---|---|---|
| Recall | 5 | Three DOTS, timed. |
| Purpose | 5 | Board: x² + 5x + 6. "Two numbers that multiply to 6 and add to 5." |
| Developing | 23 | The product-and-sum method: x² + 5x + 6 → (x + 2)(x + 3). x² − 7x + 12 → (x − 3)(x − 4). x² − 4x − 12 → (x − 6)(x + 2) — signs differ, and this is where errors cluster. Then perfect square trinomials: x² + 6x + 9 → (x + 3)², x² − 10x + 25 → (x − 5)². Recognise them by the pattern: first and last terms are squares, middle is twice the product of their roots. Then the combined case: 2x² + 10x + 12 → CMF first → 2(x + 2)(x + 3). |
| Generalising | 7 | A perfect square trinomial is not a separate method — it is a trinomial whose two factors happen to be the same. Spotting it saves time; missing it costs none. |
| Evaluating | 5 | Exit ticket: factor x² − 4x − 12 and x² + 8x + 16, naming which is a perfect square. |
Worked set (teacher's copy):
| Polynomial | Two numbers | Completely factored |
|---|---|---|
| x² + 5x + 6 | 2 and 3 | (x + 2)(x + 3) |
| x² − 7x + 12 | −3 and −4 | (x − 3)(x − 4) |
| x² − 4x − 12 | −6 and 2 | (x − 6)(x + 2) |
| x² + 6x + 9 | 3 and 3 | (x + 3)² — perfect square |
| x² − 10x + 25 | −5 and −5 | (x − 5)² — perfect square |
| x² + 8x + 16 | 4 and 4 | (x + 4)² — perfect square |
| 2x² + 10x + 12 | after CMF | 2(x + 2)(x + 3) |
| 4x² − 12x + 9 | 2x, −3 | (2x − 3)² — perfect square |
Day 5 · Choosing the method, and the weekly check
| Phase | Min | What happens |
|---|---|---|
| Recall | 5 | The class sentence, then one of each type. |
| Purpose | 4 | Board: eight polynomials in no order. "Do not factor them. Sort them." |
| Developing | 16 | Build the decision routine together, in this order: (1) common factor? take it out. (2) two terms? try difference of two squares. (3) three terms? product and sum. (4) look again at every bracket. Then work six mixed problems, each checked by expanding. |
| Weekly check | 15 | Ten items — assessment.md §6. Written Works. |
| Closing | 5 | Day 1 exit tickets returned. Learners write one line: "On Monday I thought ______. Now I know ______." |
The decision routine, as it goes on the board:
1. Common factor? → take it out FIRST
2. Two terms? → difference of two squares (minus only)
3. Three terms? → two numbers: product and sum
4. LOOK AGAIN → can any bracket factor further?
5. Check → expand it back
A — ASSESSMENT
A. Formative — not graded, used to steer
| Day | Evidence gathered | What it tells you |
|---|---|---|
| 1 | Which of the three answers each learner offered | Whether the class already senses "complete" or has no concept of it |
| 1 | Exit ticket — is 3(x² − 9) complete? | The single best predictor of Week 7 success |
| 2 | Exit ticket — 2x³ − 18x, two steps | Whether "look again" survived one day |
| 3 | Exit ticket — 3x² − 27 and does x² + 4 factor | Whether difference is being read, or just "two squares" |
| 4 | Exit ticket — the sign case x² − 4x − 12 | Where the errors actually are |
| 5 | The sorting task, before any factoring | Whether method choice is reasoned |
The single most useful check all week is requiring the expansion. It is the only self-check in algebra that needs no answer key, and a learner who does it habitually will never hand in an incomplete factorisation without noticing.
Also available all week: the Teacher Panel in game.html reports per-round
accuracy per learner in under a minute.
B. Summative — graded, per DO No. 015, s. 2026
Mathematics is a Key Stage 3 core learning area: Written Works 20% · Performance Tasks 50% · Examinations 30%.
| Instrument | Component | Where |
|---|---|---|
| Worksheet §A–C, and the Day 5 weekly check | Written Works | worksheet.html · assessment.md §6 |
| Summative Test 2 — end of Week 10, 20 items | Examinations (30%) | assessment.md §4 |
| Term 1 Examination — 40 items | Examinations (40%) | assessment.md §5 |
| Group Activity 1 and 2 | Performance Tasks | rubrics in assessment.md §7 |
| The Tile Border (term task) | Performance Tasks | rubric in assessment.md §7 |
Recording and transmutation: class-record.html.
C. Success criteria for the week
- Factor any polynomial of the three named types completely
- Take the common monomial factor out first, without being told
- Recognise that a sum of two squares does not factor
- Check by expanding, unprompted
Criterion 4 is the one that predicts Grade 9. Solving a quadratic by factoring requires the factorisation to be right and complete, and there is no partial credit for a root that does not exist.
W — WAYS FORWARD
A. Remediation — for learners who answered "yes" to the Day 1 exit ticket
Do not re-teach the methods. Re-teach look again, concretely:
- Give only already-factored expressions and ask one question: can any bracket go further? No factoring from scratch at all, for a whole session.
- Algebra tiles for x² − 4 and x² − 9, physically rearranged into a rectangle. The rectangle is the factorisation.
- Ten minutes during Day 3 and Day 4 independent work, small group.
B. Enrichment — for learners secure by Day 2
- x⁴ − 16. Three steps: (x² − 4)(x² + 4) → (x − 2)(x + 2)(x² + 4). And x² + 4 stops.
- "Why does a sum of two squares not factor? Try to find factors and say what goes wrong."
- Factoring by grouping — four terms, two at a time. Not in the Grade 8 competency, but the natural next question and a Grade 9 prerequisite.
- Game Round 3 from Day 2.
C. Differentiation held all week
| Group | Adjustment | Same competency? |
|---|---|---|
| Below level | Coefficients of 1; small squares only (4, 9, 16, 25); algebra tiles available | Yes — numbers reduced, structure identical |
| On level | As written | Yes |
| Above level | x⁴ − 16, the sum-of-squares question, grouping, Round 3 from Day 2 | Yes — extended by depth |
| Language support | Completely explained as a mathematical term before it is used in a task | Yes |
| Additional needs | Enlarged print; pre-printed expansion grids; verbal exit ticket; peer scribe | Yes |
D. Teacher's reflection
To be completed after Day 5.
| Learners meeting the week's intentions | ______ of ______ |
| How many offered 2(x² − 4) on Day 1? | ______ of ______ |
| Named for remediation | |
| Did showing that all three answers are correct help, or unsettle? | It is the riskiest move in the week |
| Which day ran over, and what was cut | |
| Who is expanding to check without being told? | Names. Those learners are ready for Grade 9 |
| Did anyone ask about x² + 9 before Day 3? | That question is worth stopping for whenever it comes |
E. Next week
Week 8 pairs competencies 9 and 10 — problems involving special products and factors, and simplifying rational algebraic expressions. Simplifying a rational expression is factoring twice and cancelling, so if §C criterion 1 is not met for most of the class, Week 8 will not work at all. Spend Day 1 of Week 8 on factoring instead and compress the problems; Week 9 already carries three competencies and cannot absorb more.
Part of the E-turo MATATAG artifact set. Companion files:
lesson-plan.md (Day 1, Lesson Exemplar format) ·
syllabus.md · assessment.md ·
worksheet.html · slides.html ·
game.html · offline-activities.md ·
class-record.html