--- title: "What m does to a line" subtitle: "Linear relations · Lines and their rules · Lesson 6" --- # What m does to a line **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Ramps, roofs, ski runs and phone bills all have a steepness: how much more for each step along. In a rule like y = 3x + 1, one number decides it. :::teacher-note **In one sentence:** Students vary m in y = mx + 1 on a graph board and find that m is how far the line goes up for each step right (AC9M8A04, node 1). **Students should be able to say:** I can describe what happens to the line y = mx + c when m gets bigger, is 0, or is negative. I can read m from a rule as how far the line goes up for each step right. I can find m for a line through (0, 1) and another point. **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 $$y = x + 1 \quad\longrightarrow\quad y = 3x + 1$$ Think about two points on each line: where $x = 0$, and where $x = 1$. Which one moves? :::keep-together **The line $y = x + 1$ has $m = 1$. If $m$ becomes 3, so the rule is $y = 3x + 1$, what happens to the line?** ☐ It moves up by 2, and is just as steep ☐ It gets less steep ☐ It gets steeper, and still crosses the $y$-axis at 1 ☐ It moves 2 to the right My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** It gets steeper. When x = 0, y is still 1 (3 × 0 + 1 = 1), so the line still crosses the y-axis at 1, but now y goes up 3 for each step right instead of 1. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Reading m :::keep-together ### Up for each step right Here is how to read $m$ from a rule, by working out $y$ one step at a time. 1. $y = 3x + 1$ — How far does this line go up for each step right? Start with the rule. 2. $x = 0: \; y = 3(0) + 1 = 1$ — Work out $y$ where $x$ is 0. 3. $x = 1: \; y = 3(1) + 1 = 4$ — One step right: $x$ goes from 0 to 1. 4. $4 - 1 = 3$ — $y$ went up by 3. 5. $x = 2: \; y = 3(2) + 1 = 7, \; 7 - 4 = 3$ — Another step right: up 3 again. 6. $m = 3$ — Each step right adds one more lot of 3, so $y$ goes up by $m$, the number in front of $x$. **In $y = 5x + 1$, how far does the line go up for each step of 1 to the right?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 5. Each step right adds one more lot of 5 to 5x, so y goes up by 5 each time. ::: :::keep-together ### Find the mistake **Lee decides which line is steeper. Click the line where his reasoning goes wrong.** ``` 1 Which line is steeper: y = 2x + 4 or y = 3x + 1? 2 y = 2x + 4: when x = 0, y = 4 3 y = 3x + 1: when x = 0, y = 1 4 4 is bigger than 1, so y = 2x + 4 is steeper. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: 4 and 1 are where the lines cross the y-axis, not how steep they are. Steepness is the number in front of x: 3 is more than 2, so y = 3x + 1 is steeper. ::: :::keep-together ### Further along **On the line $y = 3x + 1$, how far does $y$ go up from $x = 5$ to $x = 6$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 At x = 5, y = 16; at x = 6, y = 19. Up 3, the same as from x = 0 to x = 1: m is the same all along the line. ::: :::typical ## Change m on purpose **Steeper.** Each task puts a point $T$ on the board. Find the $m$ that takes the line through $T(2, 9)$. Change only $m$. **Flat.** Now take the line through $T(3, 1)$. What does the line do? **Down.** $T(2, -3)$ is below the start of the line. Which way must it go? **Less than 1.** Take the line through $T(4, 3)$. The slider now moves in halves. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Read the line :::keep-together ### From the rule **What is $m$ in $y = 4x - 2$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4 m = 4: the line goes up 4 for each step right. ::: :::keep-together ### Which way? **Which line goes down from left to right?** ☐ $y = 0x + 4$ ☐ $y = 3x - 2$ ☐ $y = -2x + 5$ ::: :::teacher-note **Answer:** $y = -2x + 5$ Only a negative m makes a line go down: y = −2x + 5 goes down 2 for each step right. The −2 in y = 3x − 2 is c, so that line still goes up 3 for each step; y = 0x + 4 is flat. ::: :::keep-together ### From a point **The line $y = mx + 1$ passes through $(3, 7)$. What is $m$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2 From (0, 1) to (3, 7) is up 6 over 3 steps right, so up 2 for each step: m = 2. ::: :::keep-together ### Review **Review: work out $y$ when $x = 4$ in $y = 3x - 5$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 3 × 4 − 5 = 12 − 5 = 7. ::: ## Make: your own line Choose your own steepness with the slider. Then use counting (1 right, $m$ up) to find a point on your line and drag $P$ there. Try a negative $m$ too. Stretch: without the board, give a point on $y = 3x + 1$ where $x = 10$. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Without plotting, describe the line y = −2x + 1: where does it cross the y-axis, which way does it go, and how steep is it?** {{lines-3}} ::: :::teacher-note **Answer:** It crosses the y-axis at 1 and goes down 2 for each step to the right (m = −2). :::