--- title: "Plotting a rule" subtitle: "Linear relations · Rules, tables and graphs · Lesson 3" --- # Plotting a rule **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A table of numbers hides its pattern; a graph shows it at a glance. Plotting the pairs turns a rule into a picture you can read. :::teacher-note **In one sentence:** Students plot ordered pairs from a rule's table, across then up, see that rules adding the same amount each step give straight lines, and describe the line (AC9M7A05, node 3). **Students should be able to say:** I can plot ordered pairs from a table, across first and then up. I can describe a straight-line graph by which way it goes and how much for each step. I can tell from a table whether its points will lie on a straight line. **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 | $x$ | 0 | 1 | 2 | 3 | |---|---|---|---|---| | $y = 2x + 1$ | 1 | 3 | 5 | 7 | How much does $y$ go up each time $x$ goes up by 1? :::keep-together **The pairs of $y = 2x + 1$ are $(0, 1)$, $(1, 3)$, $(2, 5)$ and $(3, 7)$. If you plot them, what shape will they make?** ☐ A straight line going up ☐ Scattered points, no pattern ☐ A curve that gets steeper ☐ A flat line My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** A straight line going up. Each step of 1 across adds the same amount (2) to y, so every point is the same step up from the one before. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Plotting and describing :::keep-together ### Plotting a rule Here is how to plot a rule and describe its graph. 1. $y = 2x + 1: \; (0, 1), (1, 3), (2, 5), (3, 7)$ — Start with the table of ordered pairs. 2. $(0, 1): \text{ across } 0, \text{ up } 1$ — Plot each pair from $(0, 0)$: across first, then up. 3. $(1, 3), (2, 5), (3, 7)$ — Plot the rest the same way. 4. $\text{up } 2 \text{ for each step right}$ — Each step of 1 across, $y$ goes up 2: the same step every time. 5. $\text{a straight line}$ — So the points lie on a straight line. Join them with a ruler. 6. $\text{going up, starting at } 1$ — Describe it: it goes up 2 for each step right and crosses the $y$-axis at 1. **Which point is $(3, 2)$?** ☐ The point 3 across and 2 up ☐ The point 2 across and 3 up ☐ The point 3 across and 3 up ::: :::teacher-note **Answer:** The point 3 across and 2 up The first number is how far across (x); the second is how far up (y). (2, 3) is a different point. ::: :::keep-together ### Find the mistake **Sam plots the point $(2, 5)$. Click the line where his method goes wrong.** ``` 1 Plot the point (2, 5). 2 Start at (0, 0). 3 Go up 2. 4 Go across 5. 5 Mark the point there. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: the first number is x, how far across. Sam should go across 2, then up 5. Going up 2 and across 5 plots (5, 2). ::: :::keep-together ### Describing a graph **The table for $y = 10 - x$ is $(0, 10)$, $(1, 9)$, $(2, 8)$, $(3, 7)$. Which describes its graph?** ☐ A curve going down ☐ A straight line going up 1 for each step right ☐ A straight line going down 1 for each step right ::: :::teacher-note **Answer:** A straight line going down 1 for each step right y goes down by 1 each time x goes up by 1: the same step each time, so a straight line, going down. ::: :::typical ## New tables **Subtraction.** Each task loads a new table. Drag $A$ to $D$ to the pairs, across then up. **Negative inputs.** **Going down.** **Not a straight line.** Plot it, then look at the steps between the points. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Points and graphs :::keep-together ### Coordinates **A point is 4 across and 1 up from $(0, 0)$. What are its coordinates?** ☐ $(1, 4)$ ☐ $(4, 1)$ ☐ $(4, 4)$ ::: :::teacher-note **Answer:** $(4, 1)$ Across is x, up is y: (4, 1). ::: :::keep-together ### Straight or not? **Which table's points lie on a straight line?** ☐ (0, 2), (1, 3), (2, 6), (3, 11) ☐ (0, 2), (1, 5), (2, 8), (3, 11) ☐ (0, 1), (1, 2), (2, 4), (3, 8) ::: :::teacher-note **Answer:** (0, 2), (1, 5), (2, 8), (3, 11) y goes up 3 every time in the first table. In the others the steps change (1, 3, 5 and 1, 2, 4), so the points curve. ::: :::keep-together ### The next point **The points of a rule lie on a straight line through $(0, 3)$, $(1, 5)$ and $(2, 7)$. Where is the next point, at $x = 3$? Give its $y$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 9 Up 2 for each step right: 7 + 2 = 9. The point is (3, 9). ::: :::keep-together ### Review **Review: for $y = 4x - 1$, what is the output when the input is 2?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 4 × 2 − 1 = 8 − 1 = 7. The pair is (2, 7). ::: ## Make: your own straight line Choose where your line starts. Then place the other points so the line goes up 3 for each step to the right. Stretch: what rule $y = \ldots$ gives your points? Check it with all three. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Plot the pairs of y = 3x − 1 for x = 0 to 3. Describe the graph.** {{lines-3}} ::: :::teacher-note **Answer:** (0, −1), (1, 2), (2, 5), (3, 8): a straight line going up 3 for each step right, crossing the y-axis at −1. :::