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# Lines and regions

Linear relations · Lines and their rules

<!-- Speaker note: Students graph vertical and horizontal lines and linear inequalities as regions, choosing a dashed or solid boundary and testing a point to decide which side to shade (AC9M8A02, node 1). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 13 min · Practise and Make 12 min). -->

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## Today

**We are learning to** y = 2x + 1 is a line; y < 2x + 1 is every point below it. Draw the boundary (dashed if points on it are not included, solid if they are), then test a point to see which side to shade. Vertical lines are x = a; horizontal lines are y = a.

- I can graph lines such as x = 3 and y = −2.
- I can graph an inequality such as y &lt; 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A delivery zone, a speed limit, a budget: often the answer is not one point but a whole region of points that work.

<!-- Speaker note: You can graph a line from its rule, such as $y = 2x + 1$. Now: what about $y < 2x + 1$? Every point whose $y$ is less than $2x + 1$ makes it true, and those points fill a whole **region** of the graph. You also meet lines with only one letter: $x = 3$ (vertical) and $y = -2$ (horizontal). -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Is the point $(1, 4)$ in the region $y &lt; 2x + 1$?**

- Yes: 4 is bigger than 1
- No: 4 is not less than 2 × 1 + 1 = 3
- It is on the boundary line

<!-- Speaker note: What happens: No. Put x = 1 and y = 4 in: is 4 &lt; 3? No. (1, 4) is above the line, outside the region. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose yes compared 4 with 1, not with 2 × 1 + 1. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Flines-and-regions%2Fstep-2&code=widget%3A+graph%0Ax%3A+-4%2C+5%0Ay%3A+-5%2C+8%0Ashade%3A+y+%3C+2x+%2B+1%0Apoint+P%3A+1%2C+4%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/lines-and-regions/?full=step-2#step-2)

<!-- Speaker note: Try a point on the dashed line too. Does it make $y < 2x + 1$ true? -->

<!-- Speaker note: Teacher: ask students to read out a point in the region and check it aloud. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* No. Put x = 1 and y = 4 in: is 4 &lt; 3? No. (1, 4) is above the line, outside the region.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Boundary, then test

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Flines-and-regions%2Fstep-3&code=widget%3A+graph%0Ax%3A+-2%2C+7%0Ay%3A+-4%2C+6%0Ashade%3A+x+%3E%3D+3%0Ashade%3A+y+%3C+2%0Apoint+P%3A+0%2C+4%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/lines-and-regions/?full=step-3#step-3)

<!-- Speaker note: What this step is for: 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. -->

<!-- Speaker note: Teacher: 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). -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Boundary and test

Here is how to graph an inequality.

Solid or dashed for $y \ge x - 2$?

$$y < 2x + 1$$

1) $\text{dashed}$ — $<$ does not include the line's points (for $\le$ or $\ge$, draw it solid).
2) $(0, 0):\; 0 < 2 \times 0 + 1$ — Test a point not on the line. $0 < 1$ is true.
3) $\text{shade the side with } (0, 0)$ — True: shade the test point's side. (False: shade the other side.)
4) $x = 3,\; y = -2$ — One letter: $x = 3$ is vertical (every point has $x$ = 3); $y = -2$ is horizontal.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write y < 2x + 1 and say "The boundary is the line y = 2x + 1." Line 1: write dashed and say "< does not include the line's points (for or , draw it solid)." Line 2: write (0, 0): 0 < 2 × 0 + 1 and say "Test a point not on the line. 0 < 1 is true." Line 3: write shade the side with (0, 0) and say "True: shade the test point's side. (False: shade the other side.)." Line 4: write x = 3, y = -2 and say "One letter: x = 3 is vertical (every point has x = 3); y = -2 is horizontal." -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Boundary and test

**For $y \ge x - 2$, is the boundary line solid or dashed?**

- Solid: ≥ includes the points on the line
- Dashed: points on the line are not included
- There is no boundary

* Answer: Solid: ≥ includes the points on the line

<!-- Speaker note: ≤ and ≥ include the boundary (solid); < and > do not (dashed). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Ana graphs $y &gt; x + 1$. Click the line where her working goes wrong.**

```
 1  Graph y > x + 1.
 2  Draw the line y = x + 1 as a solid line.
 3  Test (0, 0): is 0 > 0 + 1? No.
 4  Shade the side without (0, 0).
```

* Answer: line 2

<!-- Speaker note: Line 2: > does not include the line itself, so the boundary is dashed. Her test and shading are right. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Vertical and horizontal:** Each task changes one thing.
- **Read the region:** 
- **On a solid boundary:** 
- **On a dashed boundary:** 
- **No x:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/lines-and-regions/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: x = −2 is the vertical line; y = 3 the horizontal one. / On the line: y = 4 − x. The boundary is solid, so those points count. For students who need more support: Use the Run board: drag P and read whether the coordinates make the inequality true. For early finishers: Shade the region where y ≥ x and y < 4 and x > 0 (a triangle). What are its corners? When students get stuck: A student draws the boundary solid for < (Ana's mistake). Ask: Does a point on the line make y < 2x + 1 true? Fix: No (it makes the two sides equal), so the line is not included: dashed. / A student shades the wrong side. Ask: Which point did you test? Was the result true or false? Fix: True: shade the test point's side. False: the other side. / A student draws x = 3 horizontally. Ask: Name three points with x = 3. Fix: (3, 0), (3, 1), (3, −2): they line up vertically. -->

<!-- Speaker note: Teacher: before the vertical line task, ask what every point on x = −2 has in common. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Test a point

**Which point is in $y \le 3x - 2$?**

- $(1, 3)$
- $(2, 1)$
- $(0, 1)$

* Answer: $(2, 1)$

<!-- Speaker note: (2, 1): 1 ≤ 3 × 2 − 2 = 4. True. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Solid or dashed

**Which has a dashed boundary?**

- $x &lt; -1$
- $y \ge 2x$
- $x \le -1$

* Answer: $x &lt; -1$

<!-- Speaker note: < and > leave the line out: dashed. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Vertical

**Which line is vertical?**

- $x = 4$
- $y = 4x$
- $y = 4$

* Answer: $x = 4$

<!-- Speaker note: x = 4: every point has x = 4, so the points stack straight up: a vertical line. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: m and c

**Review: what are the gradient and y-intercept of $y = 3x - 2$?**

- Gradient −2, y-intercept 3
- Gradient 3, y-intercept 2
- Gradient 3, y-intercept −2

* Answer: Gradient 3, y-intercept −2

<!-- Speaker note: y = mx + c with m = 3 and c = −2. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: fit the region

Find two different answers. What do they have in common?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/lines-and-regions/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing m and c so a region contains some points and not others reverses the lesson: from points to the inequality. Hints, in order: (0, 0) in y ≤ mx + c means 0 ≤ c. / (0, 3) outside means c is less than 3. Then check B (2, 1): 1 ≤ 2m + c. -->

<!-- Speaker note: Teacher: collect students' m and c and test C in each. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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). -->