At a glance. One lesson of about 60 minutes. Students find both intercepts of ax + by = d by setting each variable to zero, convert to y = mx + c, find the gradient as −a/b, graph from intercepts and handle the vertical case b = 0.
Calculator: no calculator. Nothing here needs one. All the divisions are exact or give simple fractions.
What students will learn
Know
- General form is ax + by = d.
- x-intercept: put y = 0, x = d/a. y-intercept: put x = 0, y = d/b.
- The gradient is −a/b, and b = 0 gives a vertical line.
Understand
- Each form of a line is best for a different job: y = mx + c for gradient and intercept, general form for intercepts, point-slope form when you know a point.
- The two intercepts are enough to draw a line, and also enough to find its gradient (unless the line passes through the origin, where both intercepts are the same point).
- A constraint such as a budget is naturally in general form.
Do
- Find both intercepts from general form.
- Convert between general form and y = mx + c.
- Write general form from two intercepts and use it in context.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: intercepts and conversion (questions 1 to 9).
- Criterion B. Investigating patterns: the figure. Ask students to fix a and b and vary d, then describe what happens to the gradient and the intercepts, and to say why.
- Criterion C. Communicating: choosing and justifying a form for a task, and setting out the substitution clearly.
- Criterion D. Applying mathematics in real-life contexts: a market budget and a purchase of beans and rice (the hook and question 6).
The one thing to do with the figure
Ask for both intercepts before showing them. Set a, b and d, and have students say the two intercepts aloud as d ÷ a and d ÷ b before the readouts update. Then ask for the gradient and compare with −a ÷ b.
Drag b to 0 and ask what the line is, and why it cannot be written as y = mx + c.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the market budget; ask for the two extremes. |
| 15 min | The figure: intercepts, then gradient, then b = 0. |
| 10 min | Key idea; the worked example settles the hook; students finish 4x − 3y = 12. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | Gradient −2 and y-intercept 5; x = 4; y = 3 − 2x. |
|---|---|
| Worked example | (15, 0) and (0, 10); gradient −2/3. |
| Faded example | (0, −4); gradient 4/3. |
| 1. Intercepts from general form | B. A swaps the intercepts; C uses d itself; D adds minus signs. |
| 2. y-intercept from general form | 6. 30 ÷ 5. |
| 3. Rearranging to y = mx + c | 3. y = −0.5x + 3. |
| 4. Gradient from general form | 1.5. y = 1.5x − 6. |
| 5. x-intercept with a negative c | −6. −24 ÷ 4. |
| 6. A budget line | 15. 200r = 3000. |
| 7. Area from the intercepts | 20. Legs 4 and 10. |
| 8. General form from m and c | B. A has the wrong sign on y; C the wrong sign on the right-hand side; D swaps a and b. |
| 9. General form through two intercepts | A. Check both points in each option: only 3x + 2y = 12 fits both. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 10 in question 2; −8 in question 5 | Set the wrong variable to zero. | Ask which axis the intercept is on and what is true of the other coordinate there. |
| C in question 1; 30 in question 2; −24 in question 5 | Gave d itself. | Ask what d is being shared by. |
| 6 or −0.5 in question 3 | Did not divide every term, or gave the gradient. | Write each step of making y the subject and divide every term. |
| −1.5, 3 or −6 in question 4 | Sign slip with a negative b, or gave a or c. | Rearrange line by line and keep the minus sign with the term. |
| 3000 or 30 in question 6 | Stopped before dividing, or gave an intercept. | Ask how many shillings are left and what one bag costs. |
| 40, 14 or −20 in question 7 | Forgot the half, added the legs, or used a negative length. | Sketch the triangle and label the legs as lengths. |
| A, C or D in question 8 | Sign errors or swapped coefficients. | Test the y-intercept and the gradient of each option. |
| B, C or D in question 9 | Did not check both points. | Put each point into each option; only one passes both. |
Questions worth asking
- Why is the gradient −a/b? Rearranging by = −ax + d divides by b. Listen for “the x term moves across and changes sign”.
- When is general form the best form? To find intercepts, or when the equation is a total of two things.
- What does b = 0 do? It removes y: a vertical line, no gradient.
- Can d = 0? Yes: the line passes through the origin and both intercepts are 0.
Differentiation
A way in for anyone stuck
- Provide a two-column table: set y = 0 and set x = 0, with the working boxes already laid out.
- Use equations with a, b and d positive first.
- Always check an intercept by putting it back in the equation.
Stretch
- Question 7, then: show that the area of the triangle for ax + by = d is d² ÷ (2ab) for positive a, b and d.
- Find the equation of the line through (6, 0) and (0, −4) in general form with whole numbers. (2x − 3y = 12.)
Students learning in English as an additional language
“General” here means the form that works for every line. Say “set y equal to zero” and point to the x-axis. “Intercept” means where the line crosses an axis.
| Word | Plain meaning | Example |
|---|---|---|
| general form | a line written as ax + by = d | 3x + 2y = 12 |
| intercept | where a line crosses an axis | (4, 0) on the x-axis |
| coefficient | the number multiplying a letter | the 3 in 3x |
| rearrange | change an equation so a different letter is on its own | 2y = 6 − x becomes y = 3 − x/2 |
| constraint | a limit on what can be chosen | a budget of 600 pesos |
Sentence starters for explaining aloud or in writing:
- Putting y = 0 gives x = …
- The gradient is −a ÷ b, which is …
- I chose this form because …
Two things not to say
“Cross-multiply and flip.” Rearranging is a sequence of equal operations on both sides.
“General form always has whole numbers.” The coefficients may be fractions; whole numbers are a convention.
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.
- InterceptsFind the intercepts of 2x + 5y = 10.
x-intercept 5, y-intercept 2. - RearrangingWrite 2x + y = 7 as y = mx + c.
y = −2x + 7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextAdult tickets cost 30 naira and child tickets 20 naira. A school raises exactly 600 naira, so 30a + 20s = 600, where a and s are the numbers sold. How many child tickets if no adult tickets are sold?
30. - Earlier work (Lesson 4)Find the equation through (1, 3) with gradient 2.
y = 2x + 1.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe line kx + 4y = 24 has x-intercept 6. Find k and its gradient.
k = 4; gradient −1. - ReasoningDo 2x + 3y = 6 and 2x + 3y = 12 ever meet?
No: the same gradient −2/3 and different intercepts, so they are parallel.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under it, “Show me the steps” gives the same states in words.
No device? Use squared paper and plot the two intercepts.