At a glance. One lesson of about 60 minutes. Students use m as the gradient and c as the y-intercept, find an x-intercept by putting y = 0, graph from m and c with fractional gradients, and meet horizontal and vertical lines.
Calculator: no calculator. Nothing here needs one. Every value is a whole number, a half or a simple fraction, and the skill is the structure of the equation.
What students will learn
Know
- In y = mx + c, m is the gradient and c is the y-intercept.
- The x-intercept is found by putting y = 0, which gives x = −c ÷ m.
- A horizontal line is y = c; a vertical line is x = k and has no gradient.
Understand
- Changing c slides a line without changing its steepness; changing m turns it about the y-intercept.
- A gradient is a recipe for a step: across the run, up the rise.
- The order of terms does not change which number is m and which is c.
Do
- Read m and c from an equation in any order of terms.
- Graph a line from its gradient and y-intercept, including fractions.
- Find an x-intercept and write the equation of a horizontal or vertical line.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: reading and using m and c (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Ask students to find, by experiment, what must be true of two lines for them to be parallel, and to state the rule.
- Criterion C. Communicating: explaining why c does not change steepness and why x = 4 has no gradient (question 9 and the warning).
- Criterion D. Applying mathematics in real-life contexts: the taxi fare, read as m = 400 naira per km and c = 500 naira, and the emptying tank (the hook and question 4).
The one thing to do with the figure
Change one slider at a time and ask for a prediction. Keep m fixed and move c: “what stays the same?” Then keep c fixed and move m: “what stays the same?” Students should say the steepness stays when c moves, and the y-intercept stays when m moves.
Go to m = 0 and ask what the equation becomes. Then ask whether any setting of the sliders can draw a vertical line. The answer is no, which is the reason x = 4 needs its own form.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the fare 400x + 500; predictions about which number does which job. |
| 10 min | First figure: one slider at a time, predicting each time. |
| 10 min | Key idea; horizontal and vertical lines; the second figure. |
| 10 min | Worked example on y = −2x + 3; students finish the faded one. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 3; x = 3; −1. |
|---|---|
| Worked example | The points are (0, 3), (1, 1), (2, −1), (3, −3). |
| Faded example | The x-intercept is x = 4. |
| 1. Reading m and c | B. A swaps m and c; C uses c as the gradient; D calls c the x-intercept. |
| 2. Gradient from an equation | −3. y = −3x + 4. |
| 3. Using a gradient and the y-intercept | −6. y = −3 × 4 + 6. |
| 4. What the x-intercept means | 20. 80 − 4t = 0, so t = 20. |
| 5. Gradient from the intercept and a point | 2. Rise 6 over run 3. |
| 6. x-intercept from y = mx + c | −12. 0.25x = −3, so x = −12. |
| 7. Gradient from the two intercepts | 0.4. Rise 2 over run 5. |
| 8. A missing gradient from a point | −4. −5 = 2k + 3, so 2k = −8. |
| 9. Why a vertical line is not y = mx + c | B. A confuses vertical with horizontal; C thinks it crosses the y-axis; D doubts that it is straight. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Swapped m and c, or called c the gradient. | Ask which number is multiplied by x, and what happens when x = 0. |
| 4 or 3 in question 2 | Took the first number, or lost the sign. | Have them rewrite the equation as y = mx + c before reading it. |
| −12, 18 or 6 in question 3 | Left out c, ignored the sign of m, or gave the y-intercept. | Write y = mx + c with the numbers first, then put x = 4 in. |
| 80, −20, 4 or 76 in question 4 | Gave c, kept the sign from −80, gave m, or took one minute off. | Ask what V is when the tank is empty, then write 0 = 80 − 4t and solve it line by line. |
| 6, 0.5 or −2 in question 5 | Gave the rise, inverted the fraction, or gave c. | Ask for the rise and the run in words, then divide. |
| 3, −3 or 12 in question 6 | Gave c, gave the opposite of c, or lost the sign. | Write 0 = 0.25x + 3 and solve it line by line. |
| 2.5, −0.4 or −2 in question 7 | Inverted, flipped the sign, or gave c. | Ask whether the line rises or falls between the two intercepts. |
| 4, −1 or −2.5 in question 8 | Lost the sign, added the 3, or ignored the 3. | Write −5 = 2k + 3 and ask which side the 3 must leave. |
| A, C or D in question 9 | Confused vertical with horizontal, or thought the line is not straight. | Mark x = 4 on a grid and ask what the run is between two of its points. |
Questions worth asking
- If I make c bigger, does the line get steeper? No: it slides up. The steepness is m. Listen for “parallel”.
- What is the gradient of y = 7? 0: the line is horizontal. Listen for “there is no x term, so m = 0”.
- Where does y = 3x cross the y-axis? At (0, 0), because c = 0.
- Would you trust y = 400x + 500 for 100 km? Maybe: the fare is a model, and it may change with traffic and distance. This comes back in Lesson 7.
Differentiation
A way in for anyone stuck
- Give a pre-drawn grid and ask students to plot only the y-intercept first, then the next point.
- Use integer gradients before fractions, and positive before negative.
- Provide the sentence frame: “It crosses the y-axis at …, and goes … across and … up.”
Stretch
- Question 7, then: write the equation of the line through the two intercepts (0, −2) and (5, 0). (y = 0.4x − 2.)
- Show that the line y = mx + c crosses the x-axis at x = −c ÷ m, and say what goes wrong when m = 0.
Students learning in English as an additional language
“Intercept” is where a line crosses an axis. “Gradient” and “slope” mean the same thing. Say “crosses the y-axis at” with a finger on the graph.
| Word | Plain meaning | Example |
|---|---|---|
| gradient | the steepness of a line, written m | m = 2 in y = 2x + 1 |
| y-intercept | the point where a line crosses the y-axis, (0, c) | (0, −1) for y = 2x − 1 |
| x-intercept | the point where a line crosses the x-axis, where y = 0 | (4, 0) for y = 0.5x − 2 |
| equation of a line | a rule that every point on the line follows | y = 3x + 2 |
| parallel | never meeting, always the same distance apart | y = 2x + 1 and y = 2x − 5 |
| horizontal | going straight across, with gradient 0 | y = 3 |
Sentence starters for explaining aloud or in writing:
- The gradient is … and the line crosses the y-axis at …
- To find the x-intercept I put y = 0 and solve …
- It cannot be written as y = mx + c because …
Two things not to say
“c is where the line starts.” A line has no start; c is where it crosses the y-axis.
“The bigger number is the gradient.” The gradient is the number multiplying x, whatever its size.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ReadingState m and c for y = 5x − 7.
m = 5, c = −7. - Horizontal lineWrite the equation of the horizontal line through (−3, 5).
y = 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA phone battery shows B = 90 − 6h per cent after h hours. After how many hours is it empty?
15 hours. - Earlier work (Lesson 2)Find the gradient of the line through (2, 9) and (6, 1).
−2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA line has gradient 1.5 and crosses the x-axis at x = 4. Where does it cross the y-axis?
c = −6: 0 = 1.5 × 4 + c. - ReasoningTwo lines, y = 2x + k and y = −x + 9, cross the y-axis at the same point. Find k.
k = 9.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use squared paper: plot the y-intercept, then step by the gradient.