Lesson Exemplar — Mathematics 6

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.

I. CURRICULUM CONTENT, STANDARDS, AND LESSON COMPETENCIES

A. Content Standards

Quoted from the MATATAG Mathematics Curriculum Guide (DepEd, August 2023, p. 48), Grade 6 Quarter 1, content standard 4:

The learners should have knowledge and understanding of the four operations with different combinations of fractions, whole numbers, and mixed numbers.

B. Performance Standards

By the end of the quarter, the learners are able to perform the four operations with different combinations of fractions, whole numbers, and mixed numbers. (NA)

C. Learning Competencies and Objectives

Competency — CG Quarter 1, number 11, quoted verbatim:

11. obtain products that result from multiplying different combinations of fractions, whole numbers, and mixed numbers.

Assumed secure from Grade 5 (competency 8): multiply a fraction by a fraction.

Objectives for this session. By the end of 45 minutes, at least 80% of learners can:

1. Attempt 2½ × 1⅓ and commit to an answer in public
2. Draw and label the four pieces of a 2½ × 1⅓ rectangle, and total them
3. State why multiplying the whole parts and the fraction parts separately loses two pieces
4. Obtain the same answer by the improper-fraction route, and recognise it as the same answer

Objective 3 is the lesson. A learner who leaves with the improper-fraction procedure and no rectangle behind it will apply it correctly this week and revert in Week 9, when division of mixed numbers offers the same tempting shortcut.

D. Content

Make everything a fraction first. Then there is only one rule.

The class sentence, introduced at §D and repeated all week: "Make it a fraction. Then multiply. Then check the size."

E. Integration

II. LEARNING RESOURCES

Required — no cost.

Optional — if available.

Prepared in advance: nothing. The rectangle is drawn live at §C.2 — drawing it slowly, in front of the class, is what makes the two missing pieces visible.

III. TEACHING AND LEARNING PROCEDURE

A. Activating Prior Knowledge — 5 minutes (0:00 → 0:05)

Three Grade 5 products on the board. Timed — sixty seconds, then take answers:

⅔ × ¾  =            ⅗ × ½  =            ⅞ × ⅔  =

Confirm the rule aloud, in the class's own words: top times top, bottom times bottom, then simplify. Then:

"That rule has worked all of last year. Today we find out how far it goes."

Do not spend more than five minutes here. If the class is shaky on these, the lesson still runs — the rectangle at §C.2 does not depend on fluency — but note the names.

B. Establishing Lesson Purpose — 6 minutes (0:05 → 0:11)

Write on the board, cleanly and large:

2½ × 1⅓ = ?

Say:

"New shape. There are whole numbers in the way now. Work it out — you have two minutes, and I want an answer from everyone, even a guess."

Circulate. Most of the class will produce 2⅙, by multiplying 2 × 1 = 2 and ½ × ⅓ = ⅙ and putting them together. Some will produce 3⅓ correctly. A few will convert to improper fractions because they remember being told to.

After two minutes, take answers and write every distinct one on the board without judgement:

2⅙          3⅓          2 5/6        ...

Then:

"Three different answers. At most one of them is right. Put your hand up for the one you chose."

Count each. Write the counts beside them. Then, pointing at 2⅙:

"Most of you chose this. It comes from a sensible idea — do the wholes, do the parts, put them together. We are going to find out whether the idea is sensible or only comfortable."

Do not resolve it. Move to §C.

Thank the learner who offered 2⅙ out loud, by name. The whole lesson hangs off that answer being available to work on. A class that learns wrong answers get used, not mocked, will keep offering them — which is the only way you ever find out what they think.

C. Developing and Deepening Understanding — 22 minutes (0:11 → 0:33)

C.1 Explicitation — the rectangle (6 min) (0:11 → 0:17)

"A product is an area. 2½ × 1⅓ is the area of a rectangle two-and-a-half wide and one-and-a-third tall. Let us draw it."

Draw it large, and draw it in this order — the order is the whole point:

1. A rectangle. Mark the width 2½ along the bottom, the height 1⅓ up the side.
2. Now cut the width where the whole part ends: a vertical line separating the 2 from the ½.
3. Now cut the height where the whole part ends: a horizontal line separating the 1 from the ⅓.

Stand back and ask:

"How many pieces?"

Four. Let the class count them. Then, and this is the moment:

"When you multiplied 2 × 1, which piece was that? Come and point at it."

"When you multiplied ½ × ⅓, which piece was that? Come and point."

"So which pieces did nobody multiply?"

The two long thin ones. Shade them.

"Those pieces are real. They have area. Your answer left them out."

C.2 Worked example — I do (5 min) (0:17 → 0:22)

Label and compute all four, writing each inside its piece:

Now total them on the board, showing the common denominator:

2 + ½ + ⅔ + ⅙
= 12/6 + 3/6 + 4/6 + 1/6
= 20/6
= 3⅓

Return to the board where 2⅙ is still written.

"Two and one sixth. Three and one third. The difference is the two pieces you left out — a half and two thirds, which is seven sixths. Add seven sixths to two and one sixth."

2⅙ + 7/6 = 13/6 + 7/6 = 20/6 = 3⅓. The arithmetic closes exactly, and it should be shown closing.

C.3 Guided practice — we do (6 min) (0:22 → 0:28)

"Four pieces every time is a lot of drawing. There is a shorter way, and you already know it."

Ask, do not tell:

Write the two routes side by side and draw a box round the fact that they agree:

   four pieces:  2 + ½ + ⅔ + ⅙  =  20/6  =  3⅓
   improper:     5/2 × 4/3       =  20/6  =  3⅓

Point at the 20/6 in both lines. Not just the same final answer — the same intermediate value. The improper route is not a different method that happens to agree; it is the four pieces, added by the arithmetic instead of by the picture.

C.4 Independent practice — you do (5 min) (0:28 → 0:33)

On the board:

Convert, multiply, simplify. Show the improper form.

(Teacher's copy. Learners get column 1 only.)

Circulate with one question:

"Is your answer bigger or smaller than the number you started with? Should it be?"

Rows 2, 3 and 5 all give whole numbers, and that is deliberate. Learners distrust a whole number arriving out of fraction work and will rub it out. If you see an eraser, stop and ask them to explain their working to you — it will be correct. Row 3 also quietly introduces the whole-number case (3 = 3/1) a day before Day 2 teaches it, so that Day 2 is a confirmation rather than a new rule.

D. Making Generalizations — 7 minutes (0:33 → 0:40)

Back to the three answers still on the board from §B. Ask the class to decide, by hands, which survives. Then have a learner who voted 2⅙ come and cross it out and write beside it why:

"2⅙ misses two pieces."

Now build the general statement together. Accept any wording, then sharpen to:

A mixed number is ONE number, not two.
Make it a fraction first. Then there is only one rule.

Test the generalisation against the five practice rows out loud — every one of them was solved by exactly the same three steps, whatever shape the numbers came in.

Then plant Day 2, and do not answer it:

"Tomorrow: 4 × 2⅗. There is no fraction at all in the first number. Does the rule still work? Think about it tonight."

Close by saying the class sentence together, twice.

E. Contingency — no electricity, no devices

This lesson has no digital dependency. The rectangle needs chalk; the practice needs paper. If there is no squared paper, a freehand rectangle works — the four pieces do not need to be to scale, and arguably teach better when they are not, because learners cannot count squares and must reason instead.

If there is no paper at all, the whole lesson runs on the board with learners coming up in pairs to compute and write one piece each. That takes five minutes longer; drop §C.4 to three rows.

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:

1½ × 2¼

1. Before working it out — should the answer be
   more than 3, or less than 3?     MORE / LESS

2. Write both numbers as improper fractions.
   1½ = ______      2¼ = ______

3. Now find the product.            = ______

4. One sentence: why is the answer NOT 2⅛ ?
   ________________________________________

Marking key.

Scoring, and what to do about it.

Item 4 is worth more than items 1–3 combined, and should be read first. 2⅛ is exactly the part-by-part answer for this problem (2 × 1 = 2, ½ × ¼ = ⅛), so a learner who cannot say what is wrong with it has not met objective 3 however tidy items 2 and 3 are.

B. Teacher's Remarks

C. Teacher's Reflection

• Did the two shaded pieces actually land? The test is whether anyone said "oh" out loud at §C.1. If the room stayed quiet, the rectangle was watched rather than understood — draw it again on Day 2 with different numbers before starting.
• Did showing 2⅙ + 7/6 = 3⅓ help, or was it one step too many? It is the most compressible part of §C.2 if time is short.
• How many learners rubbed out a correct whole-number answer at §C.4? Names.
• Was the learner who offered 2⅙ comfortable being used? If not, the framing needs work before the next time a wrong answer is needed — and it will be needed most weeks.

Anticipated Misconceptions

Differentiation

The Rest of the Week

Day 2 opens with today's closing question — does the rule still work when one number has no fraction in it? If §D ran out of time and you did not ask it, ask it at the start of Day 2 before showing anything.

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