--- title: "Lines and regions" subtitle: "Linear relations · Lines and their rules · Lesson 8" --- # Lines and regions **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A delivery zone, a speed limit, a budget: often the answer is not one point but a whole region of points that work. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Put $x = 1$ and $y = 4$ into $y < 2x + 1$. Is it true? :::keep-together **Is the point $(1, 4)$ in the region $y < 2x + 1$?** ☐ Yes: 4 is bigger than 1 ☐ No: 4 is not less than 2 × 1 + 1 = 3 ☐ It is on the boundary line My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** No. Put x = 1 and y = 4 in: is 4 < 3? No. (1, 4) is above the line, outside the region. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Boundary, then test :::keep-together ### Boundary and test Here is how to graph an inequality. 1. $y < 2x + 1$ — The boundary is the line $y = 2x + 1$. 2. $\text{dashed}$ — $<$ does not include the line's points (for $\le$ or $\ge$, draw it solid). 3. $(0, 0):\; 0 < 2 \times 0 + 1$ — Test a point not on the line. $0 < 1$ is true. 4. $\text{shade the side with } (0, 0)$ — True: shade the test point's side. (False: shade the other side.) 5. $x = 3,\; y = -2$ — One letter: $x = 3$ is vertical (every point has $x$ = 3); $y = -2$ is horizontal. Solid or dashed for $y \ge x - 2$? **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 ::: :::teacher-note **Answer:** Solid: ≥ includes the points on the line ≤ and ≥ include the boundary (solid); < and > do not (dashed). ::: :::keep-together ### Find the mistake **Ana graphs $y > x + 1$. 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). ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: > does not include the line itself, so the boundary is dashed. Her test and shading are right. ::: :::typical ## 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.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Regions :::keep-together ### Test a point **Which point is in $y \le 3x - 2$?** ☐ $(1, 3)$ ☐ $(2, 1)$ ☐ $(0, 1)$ ::: :::teacher-note **Answer:** $(2, 1)$ (2, 1): 1 ≤ 3 × 2 − 2 = 4. True. ::: :::keep-together ### Solid or dashed **Which has a dashed boundary?** ☐ $x < -1$ ☐ $y \ge 2x$ ☐ $x \le -1$ ::: :::teacher-note **Answer:** $x < -1$ < and > leave the line out: dashed. ::: :::keep-together ### Vertical **Which line is vertical?** ☐ $x = 4$ ☐ $y = 4x$ ☐ $y = 4$ ::: :::teacher-note **Answer:** $x = 4$ x = 4: every point has x = 4, so the points stack straight up: a vertical line. ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** Gradient 3, y-intercept −2 y = mx + c with m = 3 and c = −2. ::: ## Make: fit the region Find two different answers. What do they have in common? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Graph y ≥ −2x + 3 (describe the boundary and the side). Is (1, 1) in it?** {{lines-3}} ::: :::teacher-note **Answer:** Solid line through (0, 3), down 2 for each step right; shade above. (1, 1): 1 ≥ 1, true: on the boundary, so in. :::