Lines and regions

Classroom resources

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Students will be able to

• I can graph lines such as x = 3 and y = −2.
• I can graph an inequality such as y < 2x + 1 as a region, with the right boundary (dashed or solid).
• I can test a point to decide whether it is in a region.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min

Before class

Do the Run board and the Make task yourself first. Students need Lines and their rules (gradient and intercept).

The one idea: an inequality is every point that makes it true; the line is its boundary, dashed when the line's points are left out (< and >) and solid when they are in (≤ and ≥). The common mistake is the wrong boundary style; it is planted in Ana's working.

Running the lesson

1. Predict: take a show of hands. Students who chose yes compared 4 with 1, not with 2 × 1 + 1.
2. Run: ask students to read out a point in the region and check it aloud.
3. Investigate: model the boundary-then-test routine with (0, 0); ask when (0, 0) would be a bad test point (when it is on the line).
4. Modify: before the vertical line task, ask what every point on x = −2 has in common.
5. Make: collect students' m and c and test C in each.

When students get stuck

Answers to the checks

1. in or out?: Is the point $(1, 4)$ in the region $y < 2x + 1$?
   No. Put x = 1 and y = 4 in: is 4 < 3? No. (1, 4) is above the line, outside the region.
2. into the region: Drag $P$ into the shaded region $y < 2x + 1$. Check its coordinates in the inequality. Was your prediction right about $(1, 4)$?
   
3. Boundary and test: For $y \ge x - 2$, is the boundary line solid or dashed?
   Solid: ≥ includes the points on the line
   ≤ and ≥ include the boundary (solid); < and > do not (dashed).
4. Find the mistake: Ana graphs $y > x + 1$. Click the line where her working goes wrong.
   line 2
   Line 2: > does not include the line itself, so the boundary is dashed. Her test and shading are right.
5. Two regions at once: Two regions: $x \ge 3$ and $y < 2$. Drag $P$ to a point that is in both.
   
   Any point with x at least 3 and y below 2, such as (4, 0).
6. Vertical and horizontal: Drag $P$ to where $x = -2$ meets $y = 3$.
   
   (−2, 3): on x = −2 (x is −2) and on y = 3 (y is 3).
7. Read the region: Which inequality is shaded?
   $y \ge -x + 2$
   The line y = −x + 2 (down 1 for each step right, through (0, 2)), solid, shaded above: y ≥ −x + 2.
8. On a solid boundary: The region is $y \le -x + 4$. Drag $P$ to a point on its boundary line.
   
   Points on y = −x + 4, such as (1, 3), are in the region: ≤ includes the line.
9. On a dashed boundary: Is $(2, 5)$ in the region $y > 2x + 1$?
   No: 5 > 5 is false, so it is on the dashed boundary, not in the region
   2 × 2 + 1 = 5, and 5 > 5 is false: the point is on the boundary, which > leaves out.
10. No x: Edge case: which points are in $y < 3$?
    Every point below the horizontal line y = 3, whatever its x
    x can be anything; y must be less than 3: the half below the dashed line y = 3.
11. Test a point: Which point is in $y \le 3x - 2$?
    $(2, 1)$
    (2, 1): 1 ≤ 3 × 2 − 2 = 4. True.
12. Solid or dashed: Which has a dashed boundary?
    $x < -1$
    < and > leave the line out: dashed.
13. Vertical: Which line is vertical?
    $x = 4$
    x = 4: every point has x = 4, so the points stack straight up: a vertical line.
14. Review: m and c: Review: what are the gradient and y-intercept of $y = 3x - 2$?
    Gradient 3, y-intercept −2
    y = mx + c with m = 3 and c = −2.
15. fit the region: Set $m$ and $c$ so the region $y \le mx + c$ contains $A(0, 0)$ and $B(2, 1)$ but not $C(0, 3)$.
    
    For example m = 1, c = 0: y ≤ x contains (0, 0) and (2, 1), and 3 ≤ 0 is false for C.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Vertical and horizontal

Open task: any program that does what the step asks.

one change at a time › Read the region

Open task: any program that does what the step asks.

one change at a time › On a solid boundary

Open task: any program that does what the step asks.

one change at a time › On a dashed boundary

Open task: any program that does what the step asks.

one change at a time › No x

Open task: any program that does what the step asks.

fit the region

Open task: any program that does what the step asks.

Stretch and support

Stretch: Shade the region where y ≥ x and y < 4 and x > 0 (a triangle). What are its corners?

Support: Use the Run board: drag P and read whether the coordinates make the inequality true.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Lines and regions".

The idea: Graphing linear relations including inequalities: I can Students graph linear relations of the form y = mx + b and one-variable inequalities on the Cartesian plane, identifying key features.

What understanding looks like:
- Correctly graphs a linear relation from a rule or table.
- Identifies gradient and y-intercept from graph and equation.
- Correctly represents an inequality including the boundary line convention (solid vs dashed).

Mistakes students often make here:
- A student draws the boundary solid for < (Ana's mistake).
- A student shades the wrong side.
- A student draws x = 3 horizontally.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Lines and regions".

Exit ticket

Graph y ≥ −2x + 3 (describe the boundary and the side). Is (1, 1) in it?

Solid line through (0, 3), down 2 for each step right; shade above. (1, 1): 1 ≥ 1, true: on the boundary, so in.

What each step is for

• in or out?: Committing to whether (1, 4) satisfies y < 2x + 1 shows whether they substitute both coordinates, before the board shows the region.
• into the region: Dragging P into the shaded region and reading its coordinates shows that every point there makes the inequality true, and the boundary is not included for <.
• boundary, then test: Draw the boundary (dashed for < and >, solid for ≤ and ≥), test a point not on it, and shade the side that makes it true. Ana's mistake draws a strict inequality's boundary solid.
• one change at a time: One change at a time: vertical and horizontal lines, reading an inequality from its region, a point on a solid boundary, a point on a dashed one, then a region with no x in it.
• regions: Testing points, choosing boundaries and reading lines, mixed; the solid-dashed slip planted again; the review keeps gradient and intercept fresh.
• fit the region: Choosing m and c so a region contains some points and not others reverses the lesson: from points to the inequality.

Where it leads

Next: Solving with graphs, then (in Algebra) Inequalities, which solves one-variable inequalities and checks them graphically.

Where it leads: linear programming and simultaneous inequalities (Years 10 and up).