Lesson Exemplar — Mathematics 5
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. Open this one before you teach; file that one.
| Learning Area | Mathematics |
| Grade Level | Grade 5 |
| Term | Term 1, Week 7 |
| CG source | Quarter 1, competency 8, building on 7 (p. 44) |
| Lesson Title | A Part of a Part — Why Multiplying Can Make a Number Smaller |
| 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. 44), Grade 5 Quarter 1, content standard 3:
The learners should have knowledge and understanding of multiplication of fractions.
B. Performance Standards
By the end of the quarter, the learners are able to multiply fractions. (NA)
C. Learning Competencies and Objectives
Competency — CG Quarter 1, number 8, quoted verbatim:
8. multiply a fraction by a fraction.
Recalled from Week 6 — CG Quarter 1, number 7:
7. multiply fractions using models.
Objectives for this session. By the end of 45 minutes, at least 80% of learners can:
- Predict whether a fraction of a fraction will be larger or smaller than the factors, and commit to that prediction publicly
- Demonstrate, by folding a sheet of paper, that ½ × ½ = ¼
- State in their own words why multiplying by a number less than one makes a quantity smaller
- Produce and record the product for five fraction pairs using the fold
Objective 3 is the lesson. Objectives 1, 2 and 4 exist to make 3 unavoidable. A learner who leaves with the five products memorised and the reasoning missing has not met this lesson, and will meet division of fractions in Week 10 with nothing to stand on.
D. Content
A part of a part is smaller than both parts.
| Term | Meaning used this lesson |
|---|---|
| Of | The word that signals multiplication. Half of a half is ½ × ½ |
| Numerator | The count — how many parts we have |
| Denominator | The size — how many parts the whole was cut into |
| Product | The answer to a multiplication |
The class sentence, introduced at §D and repeated all week: "'Of' means multiply. A part of a part is smaller than both."
E. Integration
| Strand | How it appears in this lesson |
|---|---|
| EsP / GMRC | The fold is done in pairs. Both partners must agree on the prediction before either folds — disagreement has to be talked out, not overruled |
| EPP / TLE | Halving a recipe as the closing context |
| English | Of as a mathematical operator, contrasted with its ordinary use |
| 21st-century skills | Learners predict before computing and are held to the prediction in public |
II. LEARNING RESOURCES
Required — no cost.
| Resource | Quantity | Note |
|---|---|---|
| Scrap paper, roughly rectangular | 2 sheets per pair | Old newspaper, used bond paper, cut cartolina — anything foldable |
| Pencil or crayon in two colours | 1 pair per pair of learners | Two different marks matter more than two colours; hatching one way and the other way works in pencil alone |
| Chalk and board | — | The lesson runs complete on these |
Optional — if available.
| Resource | Where |
|---|---|
| Slide deck, slides 1–6 | slides.html |
| Browser game, Round 1 | game.html |
| Printable worksheet | worksheet.html |
| Wall chart for the classroom | wall-chart.html |
Prepared in advance: nothing. Cut or tear the paper during §C.1 in front of the class — the tearing is part of the point that the whole is arbitrary and it is the cutting that creates the fraction.
III. TEACHING AND LEARNING PROCEDURE
A. Activating Prior Knowledge — 5 minutes (0:00 → 0:05)
Write on the board, and work them aloud with the class:
6 × 4 = 24
12 × 3 = 36
20 × 5 = 100
Then ask, and wait for the chorus:
"When you multiply, what happens to the number?"
The class will say "It gets bigger." Some will say "it gets much bigger."
Write it on the board in large letters, and draw a box around it:
┌─────────────────────────────┐
│ MULTIPLYING MAKES IT BIGGER │
└─────────────────────────────┘
Do not correct this. Do not hedge it. Do not add "sometimes." Write it exactly as the class said it, and leave it on the board for the next forty minutes. It comes down at §D, by a learner's hand, not yours.
Four years of whole-number multiplication have taught this, and it is true for every example the class has ever seen. It has to be disproved in front of them, with their own hands, or it survives the lesson.
B. Establishing Lesson Purpose — 6 minutes (0:05 → 0:11)
Write on the board, cleanly, below the box:
½ × ½ = ?
Say:
"Half, times a half. Do not compute it. I want your prediction only. Hands up — who thinks the answer is bigger than a half?"
Count the hands. Write the number on the board. Then:
"Who thinks it is smaller than a half?"
Count. Write that number too. Then, if anyone abstains:
"Who does not want to guess?"
Most classes vote bigger by a wide margin, and that is the desired outcome. Record the counts like this and leave them:
bigger: 24 smaller: 6 not sure: 3
Now say — and this is the whole framing of the lesson:
"Twenty-four of you say bigger. Six say smaller. One of those groups is going to be wrong, and in twenty minutes you are going to prove it yourselves. I am not going to tell you."
Do not resolve it. Do not hint. Move straight into §C.
If your class votes "smaller" by a majority — some will, especially if Week 6's model work landed hard — do not fake surprise. Say: "Most of you say smaller. Good. Can you prove it? Predicting right is not the same as knowing why." The lesson is unchanged; the burden simply moves from being convinced to being able to justify.
C. Developing and Deepening Understanding — 22 minutes (0:11 → 0:33)
C.1 Explicitation — the fold (7 min) (0:11 → 0:18)
Hold up one sheet. Tear it roughly rectangular in front of the class.
"This is one whole. Not one metre, not one kilo. One whole sheet."
Step 1. Fold it in half. Open it. Shade one half with hatching that runs one way.
"How much is shaded? Half. Everyone agrees."
Step 2. Now fold it in half the other way — across the first fold. Open it. Shade one half again, hatching that runs the other way.
"How much did I just shade? Half again. But half of what?"
Take answers. Someone will say "half of the whole thing." Someone will say "half of the half." Both have been shaded — that is exactly the ambiguity to sit in for a moment.
Step 3. Hold it up and point at the piece with both hatchings crossing.
"This piece is the half of the half. Count the pieces on the whole sheet. How many?"
Four.
"And how many have both marks?"
One.
"So a half of a half is one out of four. One quarter."
Write on the board:
½ × ½ = ¼
Now every pair does it themselves with their own sheet. Circulate. The instruction is: fold, shade, count, then say the sentence aloud to your partner.
C.2 Worked example — I do (4 min) (0:18 → 0:22)
New sheet. This time ½ × ⅓.
Narrate every step, and narrate the choice of step:
"I need thirds this time, so I fold into three. That is harder than folding in half — take your time, it does not have to be perfect. Shade one of the three. That is my one-third. Now I fold the other way in half, and shade one half. How many pieces altogether? Six. How many carry both marks? One. So half of a third is one sixth."
½ × ⅓ = ⅙
Then ask, before moving on:
"Is one sixth smaller than a half? Is it smaller than a third?"
Both. Say it explicitly. This is objective 3 arriving early, and it should.
C.3 Guided practice — we do (5 min) (0:22 → 0:27)
Together, on the board and on their sheets simultaneously: ⅔ × ½.
Ask the questions, do not give the steps:
| Ask | Expected | If they stall |
|---|---|---|
| "Which fraction do I fold first?" | Either — it does not matter | Say: "Try thirds first. We will check later whether the order changed anything." |
| "How many pieces will the sheet end up in?" | Six | Point at the folds: "Three this way, two that way." |
| "How many will carry both marks?" | Two | "Two of the three rows, one of the two columns — count them." |
| "So what is the product?" | 2/6 | — |
| "Is 2/6 in simplest form?" | No — ⅓ | Recall Grade 4 equivalent fractions |
⅔ × ½ = 2/6 = ⅓
Do this if there is time, and skip it if there is not: ask a pair who folded halves first and a pair who folded thirds first to hold their sheets side by side. Same picture, rotated. That is commutativity, seen rather than stated, and it costs thirty seconds.
C.4 Independent practice — you do (6 min) (0:27 → 0:33)
On the board:
Fold, shade, count, record. Two of these, then check with your partner.
| Fold this | Then this | Pieces altogether | Pieces with both marks | Product | Simplest form |
|---|---|---|---|---|---|
| ½ | ½ | 4 | 1 | 1/4 | ¼ |
| ½ | ⅓ | 6 | 1 | 1/6 | ⅙ |
| ⅔ | ½ | 6 | 2 | 2/6 | ⅓ |
| ¾ | ½ | 8 | 3 | 3/8 | ⅜ |
| ⅔ | ¾ | 12 | 6 | 6/12 | ½ |
(The table above is the teacher's copy. Learners get the first two columns only.)
Circulate with one question and one question only:
"Is your answer smaller than both the fractions you started with?"
That question does the diagnostic work of a whole marking pile. A learner who cannot answer it has counted pieces without understanding what they counted.
The last row is the one that matters. ⅔ × ¾ = 6/12 = ½ — the product is exactly a half, which is smaller than ⅔ and smaller than ¾, but is a "nice" number that tempts learners to think they have made an error. If a pair gets ½ and rubs it out, that is the moment to stop and have them explain their count to the class.
D. Making Generalizations — 7 minutes (0:33 → 0:40)
Return to the board. Point at the boxed sentence from §A, still standing.
"Twenty-four of you voted bigger. Read that box out loud."
Let them read it. Then:
"Is it true?"
Now — and this matters — call on a learner who voted "bigger" and hand them the chalk.
"You voted bigger. You have just folded five sheets. Come and fix it."
Let them cross it out and write the replacement themselves. Accept any wording that carries the idea; then sharpen it together to:
Multiplying by a number LESS THAN ONE makes it smaller.
A part of a part is smaller than both.
Check the generalisation holds, out loud, against all five results:
| Product | Smaller than both? | |
|---|---|---|
| ½ × ½ | ¼ | ✔ |
| ½ × ⅓ | ⅙ | ✔ |
| ⅔ × ½ | ⅓ | ✔ |
| ¾ × ½ | ⅜ | ✔ |
| ⅔ × ¾ | ½ | ✔ |
Then plant the seed for Day 3, and do not answer it:
"Tomorrow, one question. Nobody folded a sheet into twenty pieces today. What if I asked you for three-fifths times three-quarters? Think about it tonight."
Close by saying the class sentence together, twice.
E. Contingency — no electricity, no devices
This lesson has no digital dependency at all. No projector, no browser, no printout. The paper can be any scrap; if there is genuinely no paper, the entire lesson runs on the board with rectangles drawn in chalk and hatched twice — the count of pieces is identical. If there is no chalk, it runs on the ground with a stick.
The only thing that cannot be substituted is learners making the marks themselves. A demonstration the class watches will not shift the misconception. If materials are short, halve the class into pairs and then fours before you make it a demonstration.
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, no partner:
¾ × ½
1. Before you work it out — will the answer be
smaller than ¾ ? YES / NO
smaller than ½ ? YES / NO
2. Now find it. ¾ × ½ = ______
3. In one sentence: why? ____________________
Marking key.
| Item | Answer |
|---|---|
| 1 | YES and YES |
| 2 | 3/8 (⅜) |
| 3 | Any wording carrying "it is a part of a part" or "we are taking half of something, so it must be less than what we started with" |
Scoring, and what to do about it.
| Score | Reading | Action |
|---|---|---|
| 3/3 | Secure. Model and meaning both present | Enrichment on Day 2 — see Differentiation |
| 2/3, item 3 missing | Has the procedure, not the idea. The most common outcome, and the most dangerous | Not remediation — this learner looks fine. Require a sketch beside every answer on Days 2–4 |
| 2/3, item 2 wrong | Has the idea, slipped on the count | Nothing. Day 2's grid work fixes counting |
| 1/3 or 0/3 | Neither | Small group during Day 2 and Day 3 independent work. Back to halves and quarters only |
Item 3 is worth more than items 1 and 2 combined, and should be read first. A class where most learners get 2/3 by missing item 3 has not had a successful lesson, however good the arithmetic looks.
B. Teacher's Remarks
| Number of learners who met the objectives | ______ of ______ |
| Vote count at §B — bigger / smaller / unsure | ______ / ______ / ______ |
| Learners needing remediation | |
| Timing — what ran over or short | |
| Materials that worked / did not |
C. Teacher's Reflection
- Did leaving the wrong sentence on the board work, or did it confuse learners who were already right? This is the riskiest move in the lesson. If a learner who voted "smaller" looked unsettled for forty minutes, that cost is real and worth recording.
- Did the learner you called up at §D actually change their mind, or perform changing it?
- How many pairs rubbed out ⅔ × ¾ = ½ because it looked too neat?
- Who could count the pieces but not say why? Names. They are the ones Week 10 will catch.
- Was the paper adequate? If pairs spent more than a minute on folding thirds, pre-crease the sheets next cycle — the folding is not the competency.
Anticipated Misconceptions
| Misconception | Where it shows | What to do |
|---|---|---|
| "Multiplying makes it bigger" | §A and §B, by design | Nothing, until §D. It must be disproved by the learner's own hands |
| Counting only the second shading | §C.1 — learner says ½ × ½ = ½ | "Point at the piece with both marks. Only that one counts." |
| Adding instead of multiplying | ½ × ½ = 1 (or 2/4 read as "two quarters, so a half") | Go back to the sheet. "Show me the piece worth 1. There isn't one — the whole sheet is 1." |
| Multiplying the denominators but adding the numerators | ⅔ × ¾ = 5/12 | Count the doubly-shaded cells physically. Five cannot be produced by any count |
| Believing the order matters | Learner refolds because they "did it backwards" | The side-by-side comparison in §C.3 |
| Reading ⅔ × ¾ = ½ as an error | §C.4, last row | Have the pair explain their count to the class. Their count is right |
| "Of" means share out | §D onward | Deferred to Day 5, where it is taught directly. Do not open it today |
Differentiation
| Group | Adjustment | Same competency? |
|---|---|---|
| Below level | Halves and quarters only — the first two rows of §C.4, done twice rather than five rows once. Pre-creased sheets so the effort goes into counting, not folding | Yes — denominators reduced, reasoning identical |
| On level | As written | Yes |
| Above level | The fifth row first, then: "Find two fractions whose product is ⅙. Now find a different pair." Game Round 3 is pitched here | Yes — extended by depth |
| Language support | Of is said aloud and contrasted with everyday use before it is written. A learner may explain to their partner in the mother tongue before writing in English | Yes |
| Additional needs | Pre-folded sheets for fine-motor difficulty; enlarged rectangles; verbal exit ticket instead of written; peer scribe. Item 3 of the exit ticket may be spoken and transcribed | Yes |
The Rest of the Week
| Day | Focus | Competency | Where |
|---|---|---|---|
| 1 | A part of a part — the fold | 7 → 8 | This plan |
| 2 | The area model on a grid — products the fold cannot reach | 7, 8 | lesson-plan-ilaw.md |
| 3 | From the grid to the rule — numerator × numerator | 8 | lesson-plan-ilaw.md |
| 4 | Simplifying first; whole numbers and mixed numbers | 8 | lesson-plan-ilaw.md |
| 5 | "Of" in word problems; weekly check | 8 | lesson-plan-ilaw.md · assessment.md §6 |
Day 2 depends on today in one specific way. The closing question — "what if I asked you for three-fifths times three-quarters?" — is the opening of Day 2. If you run out of time and skip it, open Day 2 by folding ½ × ½ again and asking it then. Do not open Day 2 with the grid already drawn.
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