Lesson Exemplar — Mathematics 10
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.
| Learning Area | Mathematics |
| Grade Level | Grade 10 |
| Term | Term 1, Week 7 |
| CG source | Quarter 1, competencies 9–10 (p. 65) |
| Lesson Title | Don't Decide Which Side — Test a Point |
| 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. 65), Grade 10 Quarter 1, content standard 3:
The learners demonstrate knowledge and understanding of quadratic inequalities in one variable and in two variables.
B. Performance Standards
By the end of the quarter, the learners are able to solve and graph the solutions of quadratic inequalities in one variable and in two variables. (NA)
C. Learning Competencies and Objectives
Competencies — CG Quarter 1, numbers 9 and 10, quoted verbatim:
9. solve quadratic inequalities in two variables.
10. determine the region of solutions of a linear or quadratic inequality in two variables.
Objectives for this session. By the end of 45 minutes, at least 80% of learners can:
- Graph the boundary of a quadratic inequality in two variables, using a broken curve for a strict inequality
- Choose a point not on the boundary, substitute it, and shade according to the result
- State that the solution of an inequality in two variables is a region, not a number or an interval
- Show that y > x² − 4 and x² − y < 4 describe the same region, by testing rather than by comparing their signs
Objective 4 is the lesson. Objectives 1–3 are the machinery. A learner who shades by reading the inequality sign will be right most of the time and will never know why they are sometimes wrong.
D. Content
The solution of an inequality in two variables is a region. You find it in two steps, and the second one is not optional.
| Term | Meaning used this lesson |
|---|---|
| Boundary | The curve or line you get by replacing the inequality sign with = |
| Broken boundary | For <** and **> — the boundary points are not solutions |
| Solid boundary | For ≤ and ≥ — the boundary points are solutions |
| Test point | Any point not on the boundary, substituted to decide which side to shade |
| Region of solutions | Every point that makes the inequality true — infinitely many |
The two-step routine, written on the board and left there all week:
1. Draw the BOUNDARY. Replace the sign with = and graph it.
Broken for < and >. Solid for ≤ and ≥.
2. TEST A POINT. Any point not on the boundary.
True → shade that side.
False → shade the other side.
The class sentence, introduced at §D: "Don't decide which side. Test a point."
E. Integration
| Strand | How it appears in this lesson |
|---|---|
| Araling Panlipunan | Zoning and setback rules define regions, not lines |
| Science | The coverage area of a signal, and the region where a reading is within tolerance |
| English | Strict and inclusive — the difference a single word makes to a boundary |
| EsP / GMRC | A learner who says "I shaded it because it looked right" and then checks anyway is doing the honest thing, not the slow thing |
II. LEARNING RESOURCES
Required — no cost. Chalk and the board; paper and pencil. Squared paper helps but is not required today — one parabola is drawn on the board and the class works from it.
Explicitly NOT used today: graphing calculators or graphing software. A tool that shades the region for you removes the only decision this lesson is about.
Optional. slides.html slides 1–9 · game.html Round 1 ·
worksheet.html · wall-chart.html
III. TEACHING AND LEARNING PROCEDURE
A. Activating Prior Knowledge — 5 minutes (0:00 → 0:05)
Last week's work, on the board:
Solve x² − 4 > 0
Let the class work it. The answer is x < −2 or x > 2, and it should be drawn as two rays on a number line with open circles at −2 and 2.
| Ask | Answer |
|---|---|
| "What kind of thing is that answer?" | Two intervals — a set of numbers |
| "Could you write down all of them?" | No — but you can describe them, and draw them |
| "How many variables were in the question?" | One. That is why a line was enough |
Leave the number line on the board. It gets pointed at again at 0:11.
B. Establishing Lesson Purpose — 6 minutes (0:05 → 0:11)
Write both of these on the board, side by side, and say nothing about them:
y > x² − 4 x² − y < 4
Then:
"Same, or different?"
Take a show of hands and write the two counts on the board. Do not settle it. Do not hint. Most classes split, and a good number will vote "different" because one says greater and one says less.
"We will settle that before the bell, and we will settle it without arguing about it."
C. Developing and Deepening Understanding — 22 minutes (0:11 → 0:33)
C.1 Explicitation — the boundary (7 min) (0:11 → 0:18)
Point back at the number line from §A.
"One variable, one line. Now there are two variables — x and y. So the answer cannot live on a line any more. It lives on the whole plane."
Draw axes. Then:
"Step one. Replace the sign with equals and graph what you get."
y = x² − 4
Build it with the class:
| Ask | Answer |
|---|---|
| "Where is the vertex?" | (0, −4) |
| "Where does it cross the x-axis?" | x² − 4 = 0, so x = −2 and x = 2 |
| "Where does it cross the y-axis?" | (0, −4) — the vertex, here |
Draw the parabola. Then, deliberately, draw it as a broken curve:
*"Broken, not solid. The sign is >, not ≥ — so the points on the curve itself are not solutions. The boundary marks the edge of the region without belonging to it."*
Write on the board: broken = <, > · solid = ≤, ≥
C.2 Worked example — the test point (5 min) (0:18 → 0:23)
"Step two. The curve cuts the plane in two. One side is the answer. Which one?"
Wait. Someone will say "above". Accept the suggestion without confirming it, then:
"Maybe. Let us not decide. Let us test."
Pick the origin and mark it clearly with a dot and the label (0, 0).
y > x² − 4
0 > 0² − 4
0 > −4 TRUE
"True. So (0, 0) is in the region — and the whole side it sits on is shaded."
Shade it. The origin is inside the parabola's opening, so the region shaded is the one containing the vertex's open side.
"Note what we did NOT do. We did not look at the sign and reason about it. We picked one point and asked it."
C.3 Guided practice — we do (5 min) (0:23 → 0:28)
Now the second inequality from §B. Same two steps, together, out loud.
| Ask | Expected | If they stall |
|---|---|---|
| "What is the boundary of x² − y < 4?" | Set it equal: x² − y = 4, so y = x² − 4 | "Get y by itself" |
| "Is that a new curve?" | No. It is the same parabola | — |
| "Broken or solid?" | Broken — the sign is < | — |
| "Test (0, 0)." | 0² − 0 = 0, and 0 < 4 — TRUE | "Substitute both, then compare" |
| "So which side?" | The same side. The one with the origin | — |
Put the two results next to each other:
| Inequality | Test at (0, 0) | Result | Region |
|---|---|---|---|
| y > x² − 4 | 0 > −4 | TRUE | the side with the origin |
| x² − y < 4 | 0 < 4 | TRUE | the side with the origin |
"They are the same inequality. One of them has a greater than in it and one has a less than, and it makes no difference at all — because the sign is not what decides the side. The test point decides the side."
Return to the vote from §B and write the correct answer next to the two counts. Let the class see which way it went.
C.4 Independent practice — you do (5 min) (0:28 → 0:33)
Test the point (0, −10) in both inequalities. Show your substitution.
| Inequality | Substitution | Result |
|---|---|---|
| y > x² − 4 | −10 > 0 − 4, i.e. −10 > −4 | FALSE |
| x² − y < 4 | 0 − (−10) = 10, and 10 < 4 | FALSE |
"False in both. (0, −10) is outside the region — and it is outside the same region. That is the second piece of evidence, and it points the same way as the first."
Circulate with one question:
"Where is your substitution written?"
A learner who has shaded but written no substitution has not done the lesson.
D. Making Generalizations — 7 minutes (0:33 → 0:40)
Build the routine together and write it where it will stay all week:
1. Draw the BOUNDARY. Replace the sign with = and graph it.
Broken for < and >. Solid for ≤ and ≥.
2. TEST A POINT. Any point not on the boundary.
True → shade that side.
False → shade the other side.
Then the one caution, which is Day 2's opening:
"Any point not on the boundary. The origin is easiest and it usually works. Tomorrow I will give you one where it does not — and I will not tell you which one."
Say the class sentence together, twice:
"Don't decide which side. Test a point."
E. Contingency — no electricity, no devices
This lesson has no digital dependency. One parabola on the board, two substitutions, one shaded region. If there is no paper at all, learners can call out substitutions and the class shades the board copy — which is arguably better, since the substitution is then spoken and everyone hears whether it was done.
IV. EVALUATING LEARNING: FORMATIVE ASSESSMENT AND TEACHER'S REFLECTION
A. Evaluating Learning — 5 minutes (0:40 → 0:45)
Exit ticket. On a quarter-sheet:
y ≥ x² − 9
1. Where does the boundary cross the x-axis? x = ______ and x = ______
2. Is the boundary broken or solid? ______________
3. Test the point (0, 0). Show your substitution.
______________________________ TRUE / FALSE
4. Which side is shaded — the side with (0, 0), or the other side?
____________________________________________________
Marking key.
| Item | Answer |
|---|---|
| 1 | x = −3 and x = 3 |
| 2 | Solid — the sign is ≥, so the boundary points are solutions |
| 3 | 0 ≥ 0² − 9, i.e. 0 ≥ −9 — TRUE |
| 4 | The side with (0, 0) |
Scoring, and what to do about it.
| Score | Reading | Action |
|---|---|---|
| 4/4 | Secure — boundary, convention and method | Enrichment on Day 2: give them x² − y ≤ 9 and ask if it is the same |
| 3/4, item 2 wrong | Has the method, not the convention | Two minutes on Day 2. It costs a mark on every graph, so fix it at once |
| 3/4, item 3 blank but item 4 right | The one to watch. Guessed the side and was correct | Not remediation — a requirement. Every graphed answer must carry a circled test point for the rest of the week |
| Item 1 wrong | The parabola itself is insecure | Grade 9 Quarter 3, not this week. Small group on zeros and vertex before any shading |
| 2/4 or below | Small group Days 2–3, linear boundaries only until the routine is automatic |
Read item 3 before item 4. A learner who left the substitution blank and still got item 4 right has a 50% chance of looking exactly like a learner who understood. The substitution is the evidence; the shading is not.
B. Teacher's Remarks
| Number of learners who met the objectives | ______ of ______ |
| How many left item 3 blank? | ______ of ______ |
| The §B vote — how did it split? | ______ same / ______ different |
| Learners needing remediation | |
| Timing — what ran over or short |
C. Teacher's Reflection
- Did the vote at §B actually stay unsettled until §C.3, or did I give it away?
- Did anyone object that testing one point cannot prove a whole region? That is a good objection and deserves a real answer: the boundary is where the expression changes sign, so it cannot change sign anywhere else — one point settles the whole side.
- Did I let a learner shade correctly without a substitution and say nothing?
Anticipated Misconceptions
| # | Misconception | Where it shows | Response |
|---|---|---|---|
| 1 | "> means shade above, < means shade below" | Fails on x² − y < 4 | The §B pair. Both are true at (0, 0), and one says < |
| 2 | The origin always works as a test point | Fails on y > −x | Day 2. Do not pre-empt it today — the surprise is the teaching |
| 3 | Broken and solid are decorative | Item 2 of the exit ticket | Ask which points a broken curve leaves out. Answer: exactly the ones on it |
| 4 | "Inside the parabola" is the answer | Fails on a downward parabola | Day 3, y < −x² + 4 |
| 5 | A region can be written as a list of points | Learners try to name four or five solutions and stop | Ask for the next one. And the next. The point lands quickly |
| 6 | Testing one point is not a proof | Raised by strong learners | It is a real question — see §IV.C. Answer it properly, do not brush it off |
| 7 | The boundary must be re-derived for each form | Learners graph x² − y = 4 as a new curve | Rearranging to y = x² − 4 is the same curve, and showing that is half of §C.3 |
Differentiation
| Group | Adjustment |
|---|---|
| Struggling with the parabola | Give the curve pre-drawn. The competency is the region, not the sketch |
| Struggling with substitution | Two-column layout: left side of the inequality, right side, then the comparison |
| On track | As written |
| Ahead | Give x² − y ≤ 4 — same curve, solid, same side. Then ask for a third way to write the same region |
| Well ahead | Hand them a shaded region and ask for an inequality that produces it. The inverse task is genuinely hard and there is more than one right answer |
The Rest of the Week
| Day | Focus |
|---|---|
| 1 | This lesson — the two-step routine, on one quadratic |
| 2 | Linear boundaries; solid versus broken; the origin trap |
| 3 | Quadratic boundaries both ways up; "inside" is not a direction |
| 4 | Is a given point a solution? — including points on the boundary |
| 5 | One region, many ways to write it; group activity; weekly check |
Part of the E-turo MATATAG artifact set. Companion files:
syllabus.md · lesson-plan-ilaw.md ·
assessment.md · offline-activities.md ·
slides.html · game.html ·
worksheet.html · wall-chart.html