Answer key · Unit 5

The linear function.

Each lesson’s practice questions, their correct answers and the mistakes they spot. Answers read off the Scene use the same setting the learner sees.

Misconception catalogue

Answer key 4

Lesson 1 of 4 · About 20 minutes

Slope

How much a line rises for every step to the right, why that is the same number anywhere on the line, and what a negative slope means.

Open the lesson

The 4 steps and their answers

Step 1 of 4
Open this step
Rise and run
Practice · Calculate
Question
On the line $y = 2x + 1$ ⁠, start at $x = 1$ ⁠ and move $2$ ⁠ steps to the right, to $x = 3$ ⁠. By how much has $y$ ⁠ risen?

Correct answer
4 ± 0.01
Setup, choices and feedback
The Scene at this step
Slope m
2fixed
Intercept n
1fixed
Run
2fixed
The line / Rise over run
The line
Accepted within
± 0.01
Computed from the Scene at this step’s setting
Y(pointX + run) − y(pointX)/nodes/graph/outputs/rise
Wrong answers it recognises
- 2 ± 0.01
  Misconception: Takes the slope itself as the rise, whatever the run: on y = 2x + 1, two steps right rise 2.
  Feedback from the catalogue
  The slope is the rise for one step to the right. Over $2$ ⁠ steps the line rises twice as much: $2 \cdot 2 = 4$ ⁠. In general, the rise is the slope times the run.
Feedback for a right answer
At $x = 1$ ⁠, $y = 3$ ⁠, and at $x = 3$ ⁠, $y = 7$ ⁠: a rise of $4$ ⁠ over a run of $2$ ⁠. That is $2$ ⁠ per step, exactly the slope $m = 2$ ⁠. Choose Rise over run under Look inside and see the triangle.

Hint
Work out $y$ ⁠ at both points: at $x = 1$ ⁠ and at $x = 3$ ⁠.

Worked solution
1. At the start: $x = 1$ ⁠, $y = 2 \cdot 1 + 1 = 3$ ⁠.
2. After $2$ ⁠ steps: $x = 3$ ⁠, $y = 2 \cdot 3 + 1 = 7$ ⁠.
3. The rise: $7 - 3 = 4$ ⁠.
In short: every step rises $2$ ⁠, and two steps rise $2 \cdot 2 = 4$ ⁠. If you answered $2$ ⁠, you gave the rise for one step, the slope.
Step 2 of 4
Open this step
Anywhere on the line
Practice · Predict, then look
Question
Run is now $1$ ⁠, and the rise is $2$ ⁠. What happens to rise divided by run when Run is $3$ ⁠?

Correct answer
- Correct It stays $2$ ⁠
Setup, choices and feedback
The Scene at this step
Slope m
2fixed
Intercept n
1fixed
Run
1fixed
The line / Rise over run
Rise over run
Choices
- Correct It stays $2$ ⁠
- Wrong It grows $3$ ⁠ times, to $6$ ⁠
  Misconception: Thinks a longer run gives a bigger slope.
  Feedback from the catalogue
  A longer step gives a bigger rise, but in the same proportion: $3$ ⁠ times the run, $3$ ⁠ times the rise. The ratio, the slope, stays the same number. That is what makes the line straight.
- Wrong It shrinks $3$ ⁠ times
  Feedback
  The run grows $3$ ⁠ times, but so does the rise: from $x = 1$ ⁠ to $x = 4$ ⁠ the line rises $6$ ⁠. $6 \div 3$ ⁠ is $2$ ⁠ again.
Once the prediction is locked in, the Scene changes
Run
3fixed
The Scene then shows: 2 · Rise ÷ run, the slope of the secant; NaN when run is 0/nodes/graph/outputs/secantSlope
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Run $3$ ⁠, rise $6$ ⁠, and $6 \div 3 = 2$ ⁠. On a line the ratio of rise to run is the same everywhere and for any length: that is the slope.

Worked solution
1. At $x = 1$ ⁠, $y = 3$ ⁠. After $3$ ⁠ steps, at $x = 4$ ⁠, $y = 2 \cdot 4 + 1 = 9$ ⁠.
2. The rise: $9 - 3 = 6$ ⁠. The run: $3$ ⁠.
3. Rise divided by run: $6 \div 3 = 2$ ⁠, the same number as before.
Every step adds $2$ ⁠ to $y$ ⁠, so $3$ ⁠ steps add $3 \cdot 2$ ⁠. Dividing by $3$ ⁠ brings you back to $2$ ⁠.
Step 3 of 4
Open this step
Negative slope
Practice · Choose
Question
The line on the board falls $3$ ⁠ for every $2$ ⁠ steps to the right. What is its slope?

Correct answer
- Correct $-1.5$ ⁠
Setup, choices and feedback
The Scene at this step
Slope m
-1.5fixed
Intercept n
3fixed
Run
2
The line / Rise over run
The line
Choices
Shown in a random order.
- Correct $-1.5$ ⁠
- Wrong $1.5$ ⁠
  Misconception: Drops the sign of a slope: a line that goes down gets a positive slope.
  Feedback from the catalogue
  A line that goes down as you move right “rises” by a negative amount, so its slope is negative. The sign is part of the answer: $1.5$ ⁠ and $-1.5$ ⁠ are two different lines, one going up and one going down.
- Wrong $-\tfrac{2}{3}$ ⁠
  Misconception: Divides run by rise instead of rise by run.
  Feedback from the catalogue
  The slope is the rise divided by the run: the change in $y$ ⁠ on top, the change in $x$ ⁠ underneath. The other way round gives the upside-down fraction, and a steep line suddenly looks flat.
- Wrong $-3$ ⁠
  Misconception: Takes the slope itself as the rise, whatever the run: on y = 2x + 1, two steps right rise 2.
  Feedback from the catalogue
  The slope is the rise for one step to the right. Over $2$ ⁠ steps the line rises twice as much: $2 \cdot 2 = 4$ ⁠. In general, the rise is the slope times the run.
Feedback for a right answer
A rise of $-3$ ⁠ over a run of $2$ ⁠: $m = -1.5$ ⁠. The line goes down, so the slope is negative. Choose Rise over run under Look inside and check.

Hint
The rise here is negative: $-3$ ⁠. Divide it by the run.

Worked solution
1. The run: $2$ ⁠ steps to the right.
2. The rise: the line falls $3$ ⁠, that is, it rises $-3$ ⁠.
3. The slope: $m = \dfrac{-3}{2} = -1.5$ ⁠.
$1.5$ ⁠ forgets that the line goes down. $-\tfrac{2}{3}$ ⁠ divides the run by the rise, the wrong way up.
Step 4 of 4
Open this step
Which is steeper?
Practice · Choose
Question
Which line is steeper: $y = -4x + 1$ ⁠ or $y = 2x + 1$ ⁠?

Correct answer
- Correct $y = -4x + 1$ ⁠
Setup, choices and feedback
The Scene at this step
Slope m
-4
Intercept n
1
Run
1
The line / Rise over run
Rise over run
Choices
Shown in a random order.
- Correct $y = -4x + 1$ ⁠
- Wrong $y = 2x + 1$ ⁠
  Misconception: Thinks a negative slope is always less steep than a positive one.
  Feedback from the catalogue
  Steepness is set by the size of the slope, without its sign. $-4$ ⁠ drops $4$ ⁠ for every step, and $2$ ⁠ rises only $2$ ⁠: the first is twice as steep. The sign only says which way.
- Wrong They are equally steep
  Feedback
  They cross the $y$ ⁠-axis at the same point, $(0, 1)$ ⁠, but one changes by $4$ ⁠ for every step and the other by only $2$ ⁠.
Feedback for a right answer
$y = -4x + 1$ ⁠ drops $4$ ⁠ for every step, and $y = 2x + 1$ ⁠ rises only $2$ ⁠. The size $|-4| = 4$ ⁠ is bigger than $2$ ⁠, so the first is steeper. Move Slope m between $-4$ ⁠ and $2$ ⁠ and compare.

Hint
Compare how much each line changes over one step, ignoring which way.

Worked solution
1. The slopes: $-4$ ⁠ and $2$ ⁠.
2. Steepness is the size without the sign: $|-4| = 4$ ⁠ and $|2| = 2$ ⁠.
3. $4 > 2$ ⁠, so $y = -4x + 1$ ⁠ is steeper. It simply goes down instead of up.
The sign only says which way the line tilts; $-4$ ⁠ is smaller than $2$ ⁠ as a number, but not as a steepness.

Lesson 2 of 4 · About 20 minutes

Where a line crosses the axes

Where a line crosses the y-axis and where it crosses the x-axis, what the number n does to the line, and how to find a line from its two crossing points.

Open the lesson

The 4 steps and their answers

Step 1 of 4
Open this step
Crossing the y-axis
Practice · Choose
Question
At which point does the line $y = 3x - 2$ ⁠ cross the $y$ ⁠-axis?

Correct answer
- Correct $(0, -2)$ ⁠
Setup, choices and feedback
The Scene at this step
Slope m
3fixed
Intercept n
-2fixed
The line / Where it crosses
The line
Choices
Shown in a random order.
- Correct $(0, -2)$ ⁠
- Wrong $(-2, 0)$ ⁠
  Misconception: Mixes up x and y: writes (n, 0) for the y-intercept, or gives a y-value when asked for x.
  Feedback from the catalogue
  $x$ ⁠ and $y$ ⁠ answer different questions: $x$ ⁠ is the input, how far across (for the candle, how many hours have passed), and $y$ ⁠ is the output, the height. A point is written $(x, y)$ ⁠. On the $y$ ⁠-axis $x = 0$ ⁠ always, so the line crosses it at $(0, n)$ ⁠; on the $x$ ⁠-axis $y = 0$ ⁠ always. Check which of the two the question asks for.
- Wrong $(0, 3)$ ⁠
  Misconception: Confuses m and n: reads the slope as the intercept or the other way round.
  Feedback from the catalogue
  In $y = mx + n$ ⁠, the number multiplying $x$ ⁠ is the slope $m$ ⁠, and the number on its own is $n$ ⁠, the height at which the line crosses the $y$ ⁠-axis. In $y = 12 - 1.5x$ ⁠ the slope is $-1.5$ ⁠, even though it is written last.
Feedback for a right answer
At $x = 0$ ⁠: $y = 3 \cdot 0 - 2 = -2$ ⁠. The point is $(0, -2)$ ⁠, which is exactly $n = -2$ ⁠. Choose Where it crosses under Look inside.

Hint
On the $y$ ⁠-axis, $x = 0$ ⁠. Substitute.

Worked solution
1. Every point on the $y$ ⁠-axis has the form $(0, y)$ ⁠, so substitute $x = 0$ ⁠.
2. $y = 3 \cdot 0 - 2 = -2$ ⁠.
3. The point is $(0, -2)$ ⁠. In $y = mx + n$ ⁠ it is always $(0, n)$ ⁠.
$(-2, 0)$ ⁠ writes the numbers the other way round: that is a point on the $x$ ⁠-axis. And $3$ ⁠ is the slope, not the height.
Step 2 of 4
Open this step
Moving n
Practice · Predict, then look
Question
The board shows $y = x + 1$ ⁠. Now change Intercept n from $1$ ⁠ to $-2$ ⁠. What happens to the line?

Correct answer
- Correct It moves down $3$ ⁠ units and stays parallel to itself
Setup, choices and feedback
The Scene at this step
Slope m
1fixed
Intercept n
1fixed
The line / Where it crosses
Where it crosses
Choices
- Correct It moves down $3$ ⁠ units and stays parallel to itself
- Wrong It turns and gets steeper
  Misconception: Thinks changing n turns the line, rather than sliding it up or down.
  Feedback from the catalogue
  $n$ ⁠ adds the same number to every $y$ ⁠, at every $x$ ⁠. So every point on the line moves up or down by the same amount, and the line slides parallel to itself. Only $m$ ⁠ sets the angle.
- Wrong It moves $3$ ⁠ units to the left
  Feedback
  Close! With a slope of $1$ ⁠, a line moved $3$ ⁠ down looks exactly like one moved $3$ ⁠ to the right. But $n$ ⁠ adds to $y$ ⁠, so it moves the line up and down. And left is the opposite direction.
Once the prediction is locked in, the Scene changes
Intercept n
-2fixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Every $y$ ⁠ goes down by $3$ ⁠, at every $x$ ⁠, so the line drops $3$ ⁠ parallel to itself. The slope did not change, and the $y$ ⁠-intercept moved from $(0, 1)$ ⁠ to $(0, -2)$ ⁠.

Worked solution
1. Before: $y = x + 1$ ⁠. After: $y = x - 2$ ⁠.
2. At every $x$ ⁠ the new $y$ ⁠ is $3$ ⁠ less: at $x = 4$ ⁠, for example, $5$ ⁠ became $2$ ⁠.
3. Every point drops $3$ ⁠, so the line drops $3$ ⁠ and stays parallel. The slope, $1$ ⁠, has not changed.
Only $m$ ⁠ sets the angle.
Step 3 of 4
Open this step
Crossing the x-axis
Practice · Calculate
Question
At which $x$ ⁠ does the line $y = 2x - 6$ ⁠ cross the $x$ ⁠-axis?

Correct answer
3 ± 0.01
Setup, choices and feedback
The Scene at this step
Slope m
2fixed
Intercept n
-6fixed
The line / Where it crosses
The line
Accepted within
± 0.01
Computed from the Scene at this step’s setting
The smaller root; NaN when there is none/nodes/graph/outputs/root1
Wrong answers it recognises
- -3 ± 0.01
  Misconception: Gets the sign of the x-intercept wrong: −n/m read as n/m.
  Feedback from the catalogue
  You find where the line crosses the $x$ ⁠-axis from $mx + n = 0$ ⁠: subtract $n$ ⁠ and divide by $m$ ⁠, so $x = -\dfrac{n}{m}$ ⁠. Check by substituting: in $y = 2x - 6$ ⁠, $x = 3$ ⁠ gives $0$ ⁠, but $x = -3$ ⁠ gives $-12$ ⁠.
- -6 ± 0.01
  Misconception: Mixes up x and y: writes (n, 0) for the y-intercept, or gives a y-value when asked for x.
  Feedback from the catalogue
  $x$ ⁠ and $y$ ⁠ answer different questions: $x$ ⁠ is the input, how far across (for the candle, how many hours have passed), and $y$ ⁠ is the output, the height. A point is written $(x, y)$ ⁠. On the $y$ ⁠-axis $x = 0$ ⁠ always, so the line crosses it at $(0, n)$ ⁠; on the $x$ ⁠-axis $y = 0$ ⁠ always. Check which of the two the question asks for.
Feedback for a right answer
$2x - 6 = 0$ ⁠, so $2x = 6$ ⁠ and $x = 3$ ⁠. The point is $(3, 0)$ ⁠. Choose Where it crosses under Look inside and see it.

Feedback for a wrong answer
Not yet. Substitute your answer: does $2x - 6$ ⁠ come out as $0$ ⁠?

Hint
On the $x$ ⁠-axis, $y = 0$ ⁠. Solve $2x - 6 = 0$ ⁠.

Worked solution
1. On the $x$ ⁠-axis, $y = 0$ ⁠: $2x - 6 = 0$ ⁠.
2. Add $6$ ⁠ to both sides: $2x = 6$ ⁠.
3. Divide by $2$ ⁠: $x = 3$ ⁠. The point is $(3, 0)$ ⁠.
4. Check: $2 \cdot 3 - 6 = 0$ ⁠. ✓
$-3$ ⁠ is a sign slip: $2 \cdot (-3) - 6 = -12$ ⁠. And $-6$ ⁠ is $n$ ⁠, the height where the line crosses the $y$ ⁠-axis.
Step 4 of 4
Open this step
Both crossing points
Practice · Reach the target
Question
Set Slope m and Intercept n so that the line crosses the $x$ ⁠-axis at $(2, 0)$ ⁠ and the $y$ ⁠-axis at $(0, -4)$ ⁠, then press Check.

Goal
- 2 ± 0.01
  At the start of the step: -1
- -4 ± 0.01
  At the start of the step: 1
Setup, choices and feedback
The Scene at this step
Slope m
1
Intercept n
1
The line / Where it crosses
Where it crosses
Read from the Scene at
The smaller root; NaN when there is none/nodes/graph/outputs/root1
Read from the Scene at
Y where the curve crosses the y-axis/nodes/graph/outputs/yIntercept
One setting that reaches it
Slope m
2
Intercept n
-4
Feedback for a right answer
$n = -4$ ⁠, and the slope is $\dfrac{0 - (-4)}{2 - 0} = 2$ ⁠: the line $y = 2x - 4$ ⁠. Check: at $x = 2$ ⁠, $y = 0$ ⁠.

Feedback for a wrong answer
Not yet. Set $n$ ⁠ from the $y$ ⁠-axis first, then change the slope until the line passes through $(2, 0)$ ⁠.

Hint
$n$ ⁠ is the height on the $y$ ⁠-axis. Between $(0, -4)$ ⁠ and $(2, 0)$ ⁠ the line rises $4$ ⁠ over a run of $2$ ⁠.

Worked solution
1. The line crosses the $y$ ⁠-axis at $(0, n)$ ⁠, so $n = -4$ ⁠.
2. The slope between $(0, -4)$ ⁠ and $(2, 0)$ ⁠: rise $0 - (-4) = 4$ ⁠, run $2 - 0 = 2$ ⁠, so $m = 2$ ⁠.
3. Set Slope m to $2$ ⁠ and Intercept n to $-4$ ⁠: $y = 2x - 4$ ⁠.
4. Check: $2 \cdot 2 - 4 = 0$ ⁠. ✓

Lesson 3 of 4 · About 20 minutes

A line through two points

How to get the slope from two points, how to find n after that, and how to write the equation of the line that passes through them.

Open the lesson

The 4 steps and their answers

Step 1 of 4
Open this step
The slope from two points
Practice · Calculate
Question
What is the slope of the line through $(1, 1)$ ⁠ and $(3, 5)$ ⁠?

Correct answer
2 ± 0.01
Setup, choices and feedback
The Scene at this step
Slope m
0.5fixed
Intercept n
0fixed
The line / Rise over run
The line
Accepted within
± 0.01
A fixed value in the question
Wrong answers it recognises
- 0.5 ± 0.01
  Misconception: Divides run by rise instead of rise by run.
  Feedback from the catalogue
  The slope is the rise divided by the run: the change in $y$ ⁠ on top, the change in $x$ ⁠ underneath. The other way round gives the upside-down fraction, and a steep line suddenly looks flat.
- -2 ± 0.01
  Misconception: Subtracts the points in different orders above and below: (y2 − y1)/(x1 − x2).
  Feedback from the catalogue
  When working out a slope, subtract in the same order top and bottom: $\dfrac{y_2 - y_1}{x_2 - x_1}$ ⁠. Subtract second minus first on top but first minus second underneath, and the sign flips.
Feedback for a right answer
$m = \dfrac{5 - 1}{3 - 1} = \dfrac{4}{2} = 2$ ⁠. In the next step you will move the line through both points.

Feedback for a wrong answer
Not yet. Work out how much $y$ ⁠ changed and how much $x$ ⁠ changed between the points, and divide the first by the second.

Hint
Rise: $5 - 1$ ⁠. Run: $3 - 1$ ⁠.

Worked solution
1. $(x_1, y_1) = (1, 1)$ ⁠ and $(x_2, y_2) = (3, 5)$ ⁠.
2. The rise: $y_2 - y_1 = 5 - 1 = 4$ ⁠.
3. The run: $x_2 - x_1 = 3 - 1 = 2$ ⁠.
4. $m = \dfrac{4}{2} = 2$ ⁠.
$0.5$ ⁠ is $\dfrac{2}{4}$ ⁠, run over rise. $-2$ ⁠ comes from subtracting in a different order top and bottom: $\dfrac{5 - 1}{1 - 3}$ ⁠.
Step 2 of 4
Open this step
Move the line
Practice · Reach the target
Question
Move the line so that it passes through both points, $(1, 1)$ ⁠ and $(3, 5)$ ⁠, then press Check.

Goal
- 2 ± 0.01
  At the start of the step: 0.5
- -1 ± 0.01
  At the start of the step: 0
Setup, choices and feedback
The Scene at this step
Slope m
0.5
Intercept n
0
The line / Rise over run
Rise over run
Read from the Scene at
B in standard form/nodes/graph/outputs/b
Read from the Scene at
C in standard form/nodes/graph/outputs/c
One setting that reaches it
Slope m
2
Intercept n
-1
Feedback for a right answer
$y = 2x - 1$ ⁠: at $x = 1$ ⁠, $y = 1$ ⁠, and at $x = 3$ ⁠, $y = 5$ ⁠. The green triangle stretches exactly between the points.

Feedback for a wrong answer
Not yet. If the line is parallel to the one between the points but passes beside them, the slope is right and only $n$ ⁠ needs to move.

Hint
You already know the slope: $2$ ⁠. Set it, then move Intercept n until the line touches the points.

Worked solution
1. The slope between the points is $2$ ⁠ (from the last step): set Slope m to $2$ ⁠.
2. Find $n$ ⁠ by substituting $(1, 1)$ ⁠: $1 = 2 \cdot 1 + n$ ⁠, so $n = -1$ ⁠.
3. Set Intercept n to $-1$ ⁠: $y = 2x - 1$ ⁠.
4. Check with the other point: $2 \cdot 3 - 1 = 5$ ⁠. ✓
Step 3 of 4
Open this step
Finding n
Practice · Calculate
Question
A line with slope $3$ ⁠ passes through the point $(2, 4)$ ⁠. What is its $n$ ⁠?

Correct answer
-2 ± 0.01
Setup, choices and feedback
The Scene at this step
Slope m
3fixed
Intercept n
0
The line / Rise over run
The line
Other settings
Points/nodes/graph/props/points = [2,4]
Accepted within
± 0.01
A fixed value in the question
Wrong answers it recognises
- 10 ± 0.01
  Misconception: Moves m·x to the wrong side when finding n: n = y + mx.
  Feedback from the catalogue
  From $y = mx + n$ ⁠ you isolate $n$ ⁠ by subtracting $mx$ ⁠ from both sides: $n = y - mx$ ⁠. With $(2, 4)$ ⁠ and $m = 3$ ⁠: $n = 4 - 6 = -2$ ⁠. Check: $3 \cdot 2 - 2 = 4$ ⁠.
- 4 ± 0.01
  Misconception: Mixes up x and y: writes (n, 0) for the y-intercept, or gives a y-value when asked for x.
  Feedback from the catalogue
  $x$ ⁠ and $y$ ⁠ answer different questions: $x$ ⁠ is the input, how far across (for the candle, how many hours have passed), and $y$ ⁠ is the output, the height. A point is written $(x, y)$ ⁠. On the $y$ ⁠-axis $x = 0$ ⁠ always, so the line crosses it at $(0, n)$ ⁠; on the $x$ ⁠-axis $y = 0$ ⁠ always. Check which of the two the question asks for.
Feedback for a right answer
$4 = 3 \cdot 2 + n$ ⁠, so $n = 4 - 6 = -2$ ⁠. Set Intercept n to $-2$ ⁠ and watch the line pass through the point.

Feedback for a wrong answer
Not yet. After substituting, you are left with an equation in $n$ ⁠ alone. Solve it.

Hint
Substitute $x = 2$ ⁠ and $y = 4$ ⁠ into $y = 3x + n$ ⁠.

Worked solution
1. The line is $y = 3x + n$ ⁠, and the point $(2, 4)$ ⁠ is on it.
2. Substitute: $4 = 3 \cdot 2 + n$ ⁠, that is, $4 = 6 + n$ ⁠.
3. Subtract $6$ ⁠: $n = -2$ ⁠.
4. Check: $3 \cdot 2 - 2 = 4$ ⁠. ✓ The line is $y = 3x - 2$ ⁠.
$10$ ⁠ comes from $n = 4 + 6$ ⁠: the $6$ ⁠ moved across without changing sign. And $4$ ⁠ is the point’s $y$ ⁠, not the height at which the line crosses the $y$ ⁠-axis.
Step 4 of 4
Open this step
The equation of the line
Practice · Choose
Question
What is the equation of the line through $(0, 2)$ ⁠ and $(4, 0)$ ⁠?

Correct answer
- Correct $y = -0.5x + 2$ ⁠
Setup, choices and feedback
The Scene at this step
Slope m
1
Intercept n
0
The line / Rise over run
The line
Other settings
Points/nodes/graph/props/points = [0,2,4,0]
Choices
Shown in a random order.
- Correct $y = -0.5x + 2$ ⁠
- Wrong $y = -2x + 2$ ⁠
  Misconception: Divides run by rise instead of rise by run.
  Feedback from the catalogue
  The slope is the rise divided by the run: the change in $y$ ⁠ on top, the change in $x$ ⁠ underneath. The other way round gives the upside-down fraction, and a steep line suddenly looks flat.
- Wrong $y = 0.5x + 2$ ⁠
  Misconception: Drops the sign of a slope: a line that goes down gets a positive slope.
  Feedback from the catalogue
  A line that goes down as you move right “rises” by a negative amount, so its slope is negative. The sign is part of the answer: $1.5$ ⁠ and $-1.5$ ⁠ are two different lines, one going up and one going down.
- Wrong $y = -0.5x + 4$ ⁠
  Misconception: Mixes up x and y: writes (n, 0) for the y-intercept, or gives a y-value when asked for x.
  Feedback from the catalogue
  $x$ ⁠ and $y$ ⁠ answer different questions: $x$ ⁠ is the input, how far across (for the candle, how many hours have passed), and $y$ ⁠ is the output, the height. A point is written $(x, y)$ ⁠. On the $y$ ⁠-axis $x = 0$ ⁠ always, so the line crosses it at $(0, n)$ ⁠; on the $x$ ⁠-axis $y = 0$ ⁠ always. Check which of the two the question asks for.
Feedback for a right answer
$n = 2$ ⁠, and $m = \dfrac{0 - 2}{4 - 0} = -0.5$ ⁠. The line drops half a unit for every step. Try building it with Slope m and Intercept n.

Hint
$(0, 2)$ ⁠ is on the $y$ ⁠-axis, so $n = 2$ ⁠. The slope: a rise of $0 - 2$ ⁠ over a run of $4 - 0$ ⁠.

Worked solution
1. $(0, 2)$ ⁠ is where the line crosses the $y$ ⁠-axis, so $n = 2$ ⁠.
2. The slope: $m = \dfrac{0 - 2}{4 - 0} = \dfrac{-2}{4} = -0.5$ ⁠.
3. The equation: $y = -0.5x + 2$ ⁠.
4. Check at $(4, 0)$ ⁠: $-0.5 \cdot 4 + 2 = 0$ ⁠. ✓
$-2x$ ⁠ turns the fraction upside down, and $0.5x$ ⁠ forgets that the line goes down.

Lesson 4 of 4 · About 20 minutes

Rate of change

The slope as a rate, how much something changes every hour, with units. A burning candle, when it runs out, and when two candles are the same height.

Open the lesson

The 4 steps and their answers

Step 1 of 4
Open this step
The candle
Practice · Calculate
Question
The candle is described by $y = 12 - 1.5x$ ⁠. What is the rate of change of its height, in centimetres per hour?

Correct answer
-1.5 ± 0.01
Setup, choices and feedback
The Scene at this step
Change per hour
-1.5fixed
Starting height
12fixed
The candle / When it burns out
The candle
Accepted within
± 0.01
Computed from the Scene at this step’s setting
B in standard form/nodes/graph/outputs/b
Wrong answers it recognises
- 1.5 ± 0.01
  Misconception: Drops the sign of a slope: a line that goes down gets a positive slope.
  Feedback from the catalogue
  A line that goes down as you move right “rises” by a negative amount, so its slope is negative. The sign is part of the answer: $1.5$ ⁠ and $-1.5$ ⁠ are two different lines, one going up and one going down.
- 12 ± 0.01
  Misconception: Confuses m and n: reads the slope as the intercept or the other way round.
  Feedback from the catalogue
  In $y = mx + n$ ⁠, the number multiplying $x$ ⁠ is the slope $m$ ⁠, and the number on its own is $n$ ⁠, the height at which the line crosses the $y$ ⁠-axis. In $y = 12 - 1.5x$ ⁠ the slope is $-1.5$ ⁠, even though it is written last.
Feedback for a right answer
$-1.5$ ⁠ centimetres per hour: every hour the height changes by $-1.5$ ⁠, that is, it goes down. The minus says the candle is getting shorter.

Hint
The rate of change is the slope: the number multiplying $x$ ⁠, with its sign.

Worked solution
1. Write it in the form $y = mx + n$ ⁠: $y = -1.5x + 12$ ⁠.
2. The rate of change is $m = -1.5$ ⁠.
3. The units: centimetres (of $y$ ⁠) per hour (of $x$ ⁠).
$1.5$ ⁠ without the minus says the candle grows. $12$ ⁠ is $n$ ⁠, the starting height, not the rate.
Step 2 of 4
Open this step
When does it run out?
Practice · Calculate
Question
After how many hours will the candle run out?

Correct answer
8 ± 0.01
Setup, choices and feedback
The Scene at this step
Change per hour
-1.5fixed
Starting height
12fixed
The candle / When it burns out
The candle
Accepted within
± 0.01
Computed from the Scene at this step’s setting
The smaller root; NaN when there is none/nodes/graph/outputs/root1
Wrong answers it recognises
- -8 ± 0.01
  Misconception: Gets the sign of the x-intercept wrong: −n/m read as n/m.
  Feedback from the catalogue
  You find where the line crosses the $x$ ⁠-axis from $mx + n = 0$ ⁠: subtract $n$ ⁠ and divide by $m$ ⁠, so $x = -\dfrac{n}{m}$ ⁠. Check by substituting: in $y = 2x - 6$ ⁠, $x = 3$ ⁠ gives $0$ ⁠, but $x = -3$ ⁠ gives $-12$ ⁠.
Feedback for a right answer
$12 - 1.5x = 0$ ⁠, so $1.5x = 12$ ⁠ and $x = 8$ ⁠ hours. Choose When it burns out under Look inside.

Feedback for a wrong answer
Not yet. Substitute your answer: does $12 - 1.5x$ ⁠ come out as $0$ ⁠?

Hint
The candle runs out when its height is $0$ ⁠: solve $12 - 1.5x = 0$ ⁠.

Worked solution
1. The candle runs out when $y = 0$ ⁠: $12 - 1.5x = 0$ ⁠.
2. Add $1.5x$ ⁠ to both sides: $12 = 1.5x$ ⁠.
3. Divide by $1.5$ ⁠: $x = 8$ ⁠.
4. Check: $12 - 1.5 \cdot 8 = 12 - 12 = 0$ ⁠. ✓
In words: $12$ ⁠ centimetres, $1.5$ ⁠ every hour, $12 \div 1.5 = 8$ ⁠ hours. $-8$ ⁠ is a sign slip, and a negative time would be before the candle was lit.
Step 3 of 4
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Two candles
Practice · Calculate
Question
The second candle is $y = 9 - 0.75x$ ⁠. After how many hours are the two candles the same height?

Correct answer
4 ± 0.01
Setup, choices and feedback
The Scene at this step
Change per hour
-1.5fixed
Starting height
12fixed
The candle / When it burns out
The candle
Other settings
Line2/nodes/graph/props/line2 = true
Accepted within
± 0.01
Computed from the Scene at this step’s setting
X where the curve meets the second line (the smaller, when two); NaN when they never meet or are the same line/nodes/graph/outputs/crossX
Wrong answers it recognises
- 6 ± 0.01
  Misconception: Mixes up x and y: writes (n, 0) for the y-intercept, or gives a y-value when asked for x.
  Feedback from the catalogue
  $x$ ⁠ and $y$ ⁠ answer different questions: $x$ ⁠ is the input, how far across (for the candle, how many hours have passed), and $y$ ⁠ is the output, the height. A point is written $(x, y)$ ⁠. On the $y$ ⁠-axis $x = 0$ ⁠ always, so the line crosses it at $(0, n)$ ⁠; on the $x$ ⁠-axis $y = 0$ ⁠ always. Check which of the two the question asks for.
Feedback for a right answer
$12 - 1.5x = 9 - 0.75x$ ⁠, so $3 = 0.75x$ ⁠ and $x = 4$ ⁠ hours. Then both are $6$ ⁠ cm tall. After that the second candle is the taller one.

Feedback for a wrong answer
Not yet. Where the two lines meet is the moment the heights are equal.

Hint
Set the heights equal: $12 - 1.5x = 9 - 0.75x$ ⁠.

Worked solution
1. The same height: $12 - 1.5x = 9 - 0.75x$ ⁠.
2. Add $1.5x$ ⁠ to both sides: $12 = 9 + 0.75x$ ⁠.
3. Subtract $9$ ⁠: $3 = 0.75x$ ⁠, so $x = 4$ ⁠.
4. Check: $12 - 1.5 \cdot 4 = 6$ ⁠ and $9 - 0.75 \cdot 4 = 6$ ⁠. ✓
If you answered $6$ ⁠, you gave the shared height rather than the time.
Step 4 of 4
Open this step
Units of slope
Practice · Choose
Question
On a graph of distance (in kilometres) against time (in hours), what units is the slope measured in?

Correct answer
- Correct Kilometres per hour
Setup, choices and feedback
The Scene at this step
Change per hour
-1.5
Starting height
12
The candle / When it burns out
When it burns out
Choices
Shown in a random order.
- Correct Kilometres per hour
- Wrong Hours per kilometre
  Misconception: Divides run by rise instead of rise by run.
  Feedback from the catalogue
  The slope is the rise divided by the run: the change in $y$ ⁠ on top, the change in $x$ ⁠ underneath. The other way round gives the upside-down fraction, and a steep line suddenly looks flat.
- Wrong Kilometres
  Feedback
  Kilometres are the units of $y$ ⁠ alone. The slope is a ratio: a change in $y$ ⁠ divided by a change in $x$ ⁠.
Feedback for a right answer
Kilometres divided by hours: kilometres per hour, that is, a speed. The slope of a distance-time graph is the speed.

Hint
The slope is rise divided by run: units of $y$ ⁠ per unit of $x$ ⁠.

Worked solution
1. The rise is measured in the units of $y$ ⁠: kilometres.
2. The run is measured in the units of $x$ ⁠: hours.
3. The slope is rise divided by run: kilometres per hour.
That is a speed: how many kilometres are added every hour. Just as with the candle, where the slope was centimetres per hour.

Across the unit · 11 misconceptions

Misconception catalogue

Each common mistake this unit recognises, in a few words. Open one for what the learner reads when an answer matches it, and where it is asked.

Index

Slope

Rise and run

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Anywhere on the line

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Negative slope

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Which is steeper?

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Where a line crosses the axes

Crossing the y-axis

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Moving n

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Crossing the x-axis

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Both crossing points

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

A line through two points

The slope from two points

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Move the line

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

Finding n

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution