Lesson Exemplar — Mathematics 8
Term 1 · Week 7 · Day 1 · Number and Algebra
This is one day. For the whole of Week 7 in the format DepEd prescribes for filing, see
lesson-plan-ilaw.md. This document is the teaching script for Monday — every question, every anticipated wrong answer, every timing.
| Learning Area | Mathematics |
| Grade Level | Grade 8 |
| Term | Term 1, Week 7 |
| CG source | Quarter 1, competency 8 (p. 56) |
| Lesson Title | Three Answers, All Correct — What "Completely" Means |
| Duration | 45 minutes |
| Format | MATATAG Lesson Exemplar (DepEd) — not ILAW, see above |
I. CURRICULUM CONTENT, STANDARDS, AND LESSON COMPETENCIES
A. Content Standards
Quoted from the MATATAG Mathematics Curriculum Guide (DepEd, August 2023, p. 56), Grade 8 Quarter 1, content standard 3:
The learners demonstrate knowledge and understanding of special products for binomials, and factorization of polynomials.
B. Performance Standards
By the end of the quarter, the learners are able to factorize different types of polynomials. (NA)
C. Learning Competencies and Objectives
Competency — CG Quarter 1, number 8, quoted verbatim:
8. completely factor different types of polynomials (polynomials with common monomial factor; difference of two squares; quadratic trinomials, including perfect square trinomials).
Objectives for this session. By the end of 45 minutes, at least 80% of learners can:
- Produce at least one correct factorisation of 2x² − 8
- Verify by expanding that three different-looking factorisations are all correct
- State what completely factored means — no factor can be factored further
- Judge, for five given factorisations, whether each is complete, and finish those that are not
Objective 3 is the lesson, and no new method is taught today. The three factoring techniques arrive on Days 2, 3 and 4. What Day 1 buys is the reason a learner will bother to look at their answer a second time — and without that, all three techniques produce half-finished work in the examination.
D. Content
Factored is not the same as completely factored.
| Term | Meaning used this lesson |
|---|---|
| Factor (verb) | Write as a product |
| Completely factored | No factor can be factored any further |
| Common monomial factor | A term that divides every term |
The class sentence, introduced at §D and repeated all week: "Common factor first. Then look again."
E. Integration
| Strand | How it appears in this lesson |
|---|---|
| EsP / GMRC | Three learners' answers are used, and all three are called correct. Being partly right in public is made safe |
| English | Completely has a precise mathematical meaning, unlike its everyday use — said explicitly |
| 21st-century skills | Verifying your own answer instead of waiting to be marked |
II. LEARNING RESOURCES
Required — no cost.
| Resource | Quantity | Note |
|---|---|---|
| Chalk and the board | — | The lesson runs complete on these |
| Paper and pencil | 1 per learner | — |
Optional — if available.
| Resource | Where |
|---|---|
| Slide deck, slides 1–8 | slides.html |
| Browser game, Round 1 | game.html |
| Printable worksheet | worksheet.html |
| Wall chart | wall-chart.html |
| Algebra tiles cut from cardboard | for the below-level group — see Differentiation |
Prepared in advance: nothing.
III. TEACHING AND LEARNING PROCEDURE
A. Activating Prior Knowledge — 5 minutes (0:00 → 0:05)
Three expansions from last week, on the board. Ninety seconds, then take answers:
(x + 3)(x − 3) = (x + 4)² = 2(x − 1)(x + 5) =
| Answer | |
|---|---|
| (x + 3)(x − 3) | x² − 9 |
| (x + 4)² | x² + 8x + 16 |
| 2(x − 1)(x + 5) | 2x² + 8x − 10 |
Then:
"You can go forwards. Today we go backwards — and backwards has a complication that forwards does not."
B. Establishing Lesson Purpose — 6 minutes (0:05 → 0:11)
Write on the board, large:
Factor: 2x² − 8
Say:
"Two minutes. Everyone writes something. If you can only take out a 2, write that."
Circulate and look for the three answers. They will be there:
A. 2(x² − 4)
B. (2x − 4)(x + 2)
C. 2(x − 2)(x + 2)
Collect them and write all three on the board, side by side, without saying which is right. If one is missing, offer it yourself as "someone in another class gave me this one."
Then ask, and take a show of hands for each:
"Which of these is correct?"
Most classes vote for exactly one — usually C if a textbook has trained them, or A if not. Record the counts.
Do not reveal anything yet. The whole design of this lesson is that the class expects one of the three to be correct, and the answer is that all three are.
C. Developing and Deepening Understanding — 22 minutes (0:11 → 0:33)
C.1 Explicitation — expand all three (8 min) (0:11 → 0:19)
"There is one way to settle this and it does not involve me. Expand them."
Do all three on the board, with the class:
A. 2(x² − 4) = 2x² − 8 ✔
B. (2x − 4)(x + 2) = 2x² + 4x − 4x − 8 = 2x² − 8 ✔
C. 2(x − 2)(x + 2) = 2(x² − 4) = 2x² − 8 ✔
Stand back.
"All three are correct. Every one of them multiplies back to what we started with. So why does the textbook only accept one?"
Let the question sit. Take answers. Someone will get close to it: "the others can go further." Push on that:
"Show me. Take answer A. Can anything in it be factored again?"
x² − 4 → (x − 2)(x + 2). Yes.
"Take answer B. Anything?"
2x − 4 → 2(x − 2). Yes — and doing so turns B into C.
"Take answer C."
Nothing. x − 2 and x + 2 cannot go further, and 2 is a number.
C.2 Worked example — I do (4 min) (0:19 → 0:23)
Write the definition, boxed:
┌────────────────────────────────────────────────┐
│ COMPLETELY FACTORED means │
│ no factor can be factored any further. │
└────────────────────────────────────────────────┘
Then a second example, narrated:
"3x² − 27. Watch what I do first."
3x² − 27
= 3(x² − 9) ← common factor first
= 3(x − 3)(x + 3) ← look again
*"If I had not taken the 3 out first, I would have been factoring 3x² − 27 directly, which is harder and easier to get wrong. Common factor first. Then look again."*
C.3 Guided practice — we do (5 min) (0:23 → 0:28)
Together, and ask rather than tell: is 5x(x + 3) completely factored?
| Ask | Expected | If they stall |
|---|---|---|
| "Is there a common factor left inside the bracket?" | No | "Does anything divide both x and 3?" |
| "Can x + 3 be factored?" | No | "Two terms — is it a difference of squares? No, it is a sum, and neither is a square" |
| "Is 5x a single term?" | Yes | — |
| "So?" | It is complete | — |
Then a contrast: x(2x − 18).
"Same shape. Is this one complete?"
No — 2x − 18 has a common factor of 2. It becomes 2x(x − 9).
This pair is the heart of the guided practice. Two expressions that look identical in structure, one finished and one not. Learners who judge by shape rather than by checking get one of them wrong.
C.4 Independent practice — you do (5 min) (0:28 → 0:33)
On the board:
For each: is it completely factored? If not, 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) |
(Teacher's copy. Learners get the first column only.)
Circulate with one question:
"Can anything in there be factored again?"
Rows 1 and 2 have the same final answer from different starting points. That is worth pointing out at the end — there is one complete factorisation, and every incomplete route leads to it.
D. Making Generalizations — 7 minutes (0:33 → 0:40)
Return to the three answers from §B, still on the board, and the vote counts beside them.
"How many of you voted for exactly one of these?"
Nearly everyone.
"You were all right about it being correct, and all wrong about the others being wrong. Correct is not the question. Finished is the question."
Now build the working rule together, and write it where it will stay all week:
1. Common factor? → take it out FIRST
2. LOOK AGAIN → can any bracket factor further?
3. Check → expand it back
Say the class sentence together, twice: "Common factor first. Then look again."
Then plant the week, and do not answer it:
"This week you learn three ways to factor. Tomorrow, the one that has to come first — and the reason it has to come first is on the board already."
E. Contingency — no electricity, no devices
This lesson has no digital dependency. It needs a board wide enough to hold three factorisations side by side — that side-by-side layout is the mechanism, so if the board is small, use the width and write small rather than working them one at a time.
If there is no paper, §C.4 runs as a whole-class vote: read each row, learners show one finger for complete and two for not, and one learner is asked to justify each.
IV. EVALUATING LEARNING: FORMATIVE ASSESSMENT AND TEACHER'S REFLECTION
A. Evaluating Learning — 5 minutes (0:40 → 0:45)
Exit ticket. On a quarter-sheet, individually:
A learner writes: 3x² − 27 = 3(x² − 9)
1. Is this correct? YES / NO
2. Is it COMPLETELY factored? YES / NO
3. If not, finish it. 3x² − 27 = ______________
4. One sentence: what is the difference between
"correct" and "completely factored"?
________________________________________________
Marking key.
| Item | Answer |
|---|---|
| 1 | YES — it expands back correctly |
| 2 | NO |
| 3 | 3(x − 3)(x + 3) |
| 4 | Any wording carrying "correct means it multiplies back; complete means nothing can be factored again" |
Scoring, and what to do about it.
| Score | Reading | Action |
|---|---|---|
| 4/4 | Secure. The distinction is held | Enrichment on Day 2 |
| 3/4, item 4 missing | Can apply it, cannot state it | Acceptable at this stage. Require the sentence again on Day 3 |
| Item 1 answered NO | The most serious outcome, and it looks like caution. The learner has decided incomplete means wrong | Correct individually and immediately. This learner will not trust their own partial work, and factoring is done in stages |
| 2/4 or below | The distinction has not landed | Small group on Days 2 and 3. Judging already-factored expressions only — no factoring from scratch |
Item 1 is the one to read first, and a NO is worse than a blank. A learner who thinks 3(x² − 9) is wrong has learned the opposite of the lesson: they will abandon a correct intermediate step instead of building on it.
B. Teacher's Remarks
| Number of learners who met the objectives | ______ of ______ |
| Vote at §B — A / B / C | ______ / ______ / ______ |
| Learners needing remediation | |
| Timing — what ran over or short | |
| Did all three answers actually come from the class? |
C. Teacher's Reflection
- Did the class believe "all three are correct"? Some learners find it genuinely destabilising that mathematics admits more than one right answer here. If the room went quiet rather than curious, spend the first two minutes of Day 2 on it again.
- Was answer B offered by anyone, or did you have to supply it? B is the interesting one — it is the only answer that looks fully factored and is not.
- How many wrote NO to exit-ticket item 1? That number matters more than the total score.
- Is the three-step rule still on the board at the end of the day? It needs to survive to Friday.
Anticipated Misconceptions
| Misconception | Where it shows | What to do |
|---|---|---|
| Only one factorisation can be correct | §B, by design | Expand all three. The arithmetic settles it, not the teacher |
| Incomplete means wrong | Exit ticket item 1 | Correct immediately. Factoring proceeds in stages and every stage is correct |
| A common factor of 2 "does not count" | Answer B left as final | 2x − 4 = 2(x − 2). A numerical common factor is still a factor |
| Complete means "as many brackets as possible" | Learner writes 2(x−2)(x+2) as (2)(x−2)(x+2) and asks if it is more complete | Both are the same. The 2 is a factor whether or not it has brackets round it |
| x² + 4 factors like x² − 4 | Will appear on Day 3, but sometimes today | Defer it explicitly: "Hold that question until Wednesday — it is a good one." Write the learner's name on the board beside it |
| Factoring is guessing | Learners wait to be told | Every answer is checkable by expanding. Nobody needs permission to know if they are right |
| The common factor must be a number | 5x² + 15x → 5(x² + 3x), stopping there | The bracket still has an x in every term. Push: "Look again." |
Differentiation
| Group | Adjustment | Same competency? |
|---|---|---|
| Below level | Rows 1, 3 and 4 of §C.4 only. Algebra tiles for x² − 4: the tiles physically form a rectangle whose sides are (x − 2) and (x + 2) | Yes — the judgement demanded is identical |
| On level | As written | Yes |
| Above level | After §C.4: "Factor x⁴ − 16 completely. How many steps?" (Three, and it ends at (x − 2)(x + 2)(x² + 4).) Game Round 3 is pitched here | Yes — extended by depth |
| Language support | Completely is defined before it is used, and contrasted with the everyday sense of "totally" | Yes |
| Additional needs | Enlarged board work; pre-printed expansion frames; verbal exit ticket; peer scribe | Yes |
The Rest of the Week
| Day | Focus | Type | Where |
|---|---|---|---|
| 1 | Three answers, all correct — what "completely" means | all | This plan |
| 2 | Common monomial factor — always first | CMF | lesson-plan-ilaw.md |
| 3 | Difference of two squares — and why a sum does not factor | DOTS | lesson-plan-ilaw.md |
| 4 | Quadratic trinomials, including perfect squares | trinomials | lesson-plan-ilaw.md |
| 5 | Choosing the method; weekly check | all | lesson-plan-ilaw.md · assessment.md §6 |
Day 2 depends on today for one thing only: the three-step rule on the board. If §D ran short and the rule did not get written, write it at the start of Day 2 before anything else.
Part of the E-turo MATATAG artifact set. Companion files:
lesson-plan-ilaw.md (the full week, ILAW format) ·
syllabus.md · assessment.md ·
worksheet.html · slides.html ·
game.html · offline-activities.md ·
class-record.html