At a glance. One lesson of about 60 minutes. It assumes gradient from two points and substitution into a formula. Students find an equation from two points, from a point and a gradient, and from a parallel or perpendicular line, and convert between forms.
Calculator: no calculator. The numbers are chosen to work by hand, so no calculator is needed.
What students will learn
Know
- Gradient-intercept form y = mx + c; point-gradient form y − y₁ = m(x − x₁); standard form ax + by + d = 0 with whole numbers.
- A line is fixed by a gradient and a point, or by two points.
- Parallel lines share a gradient; perpendicular gradients multiply to −1.
Understand
- Different-looking equations can describe the same line, and checking a point is the test.
- The best form depends on the information given.
- Standard form has a convention so that each line has one standard equation.
Do
- Find an equation from two points.
- Rearrange between the three forms.
- Find the equation of a parallel or perpendicular line through a point.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding and converting equations of lines (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record the three forms for several settings and describe how the coefficients relate to rise, run and c.
- Criterion C. Communicating: setting out m, then substitution to find c, then a check with the second point (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: a fare rule in the hook and the taxi fare in question 8.
The one thing to do with the figure
Use points to settle the hook. The point (0, 4) is on all four signs, so one point cannot tell them apart. The point (1, 6) is on A, B and C, but on sign D 6 + 2 = 8, not 4, so D is a different line.
Then move the sliders and ask: “What happens to the standard form when the gradient is a fraction?” The coefficients become whole numbers by multiplying through.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: which signs are the same fare rule? Test points. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B or C, return to a point test. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | x = 4; 3; 7. |
|---|---|
| Hook | Signs A, B and C are the same line y = 2x + 4; sign D is y = −2x + 4, a different line. |
| Worked examples | y = 2x + 3; 2x − 3y − 12 = 0; y = −3x + 7. |
| Faded example | y = −2x + 5. |
| 1. Writing standard form | A. B changes the sign of the x term; C changes the sign of the constant; D swaps the 2. |
| 2. Reading the intercept | −4. The point has x = 0. |
| 3. Substituting a point | −3. 7 − 10. |
| 4. A line through two points | 7. y = 3x + 7. |
| 5. Gradient from standard form | −0.5. −2 ÷ 4. |
| 6. A parallel line | 17. y = 4x + 5. |
| 7. A perpendicular line | 5. y = −x/2 + 5. |
| 8. A linear model from two points | 4150. naira, from y = 250x + 400. |
| 9. An equation from two points | C. A has the wrong sign; B has the wrong intercept; D inverts the gradient. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Changed a sign, or swapped the coefficient. | Test the point (0, −6) in each choice, then (3, 0). |
| 4 in question 2 | Lost the sign. | At x = 0, y = −4, so c = −4. |
| 3 or 17 in question 3 | Subtracted the wrong way round, or added. | Write 7 = 10 + c and solve in one line. |
| 3 or 1 in question 4 | Gave the gradient, or treated x as positive. | Write the substitution with brackets: 4 = 3 × (−1) + c. |
| 0.5, −2 or 3 in question 5 | Lost the sign, inverted, or gave c. | Isolate y with every step visible: 4y = −2x + 12, then divide by 4. |
| 11 or 12 in question 6 | Used the old intercept, or none. | Write the new equation fully before evaluating. |
| −5 or 1 in question 7 | Used the parallel gradient, or the old intercept. | Ask: what is the gradient of the new line? Then substitute. |
| 3750 or 5250 in question 8 | Left out the fixed charge, or assumed proportion. | Ask: what does a 0 km trip cost? Is that zero? |
| A, B or D in question 9 | Wrong sign, wrong intercept, or inverted gradient. | Test both points in each choice. |
Questions worth asking
- How can you tell two equations are the same line? Listen for: rearrange one into the other, or test several points.
- Which form do you use when you know a point and a gradient? Point-gradient; or y = mx + c and substitute.
- Why does standard form need whole numbers? To give one agreed form for each line.
- What does c mean in the fare model? The fixed charge, the cost when the distance is 0.
Differentiation
A way in for anyone stuck
- Give a flow chart: two points, gradient, substitute, check with the other point.
- Use a table of x and y for the line and look at the differences before writing the gradient.
- Practise rearranging with integer coefficients before fractions.
Stretch
- Find the standard form of the line through (3, 1) and (−5, 5) with whole-number coefficients.
- Find the equation of the perpendicular bisector of the segment from (0, 0) to (8, 4).
Students learning in English as an additional language
“Intercept” means the place where a line crosses an axis. “Standard form” is a fixed way of writing an equation. “Rearrange” means change the equation to a different but equal form. “Substitute” means replace a letter with a number.
| Word | Plain meaning | Example |
|---|---|---|
| intercept | the point where a line crosses an axis | (0, c) |
| gradient-intercept form | y = mx + c, showing gradient and y-intercept | y = 3x − 1 |
| point-gradient form | y − y₁ = m(x − x₁), from a point and a gradient | y − 2 = 3(x − 1) |
| standard form | ax + by + d = 0 with whole numbers | 2x − y = 0 |
| substitute | replace a letter with a value | x = 2 |
| linear model | a straight-line rule for a real situation | cost = 50 plus 30 per hour |
Sentence starters for explaining aloud or in writing:
- The gradient is … so the equation starts …
- I can check by substituting the point …
- These two equations are the same because …
Two things not to say
“Use y = mx + c every time.” Say instead: choose the form that matches what you know.
“Move it over and change the sign.” Say instead: subtract the same thing from both sides.
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.
- InterceptWrite down the y-intercept of y = 3x − 8.
−8. - GradientFind the gradient of the line through (1, 3) and (5, 11).
2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA water tank in Kampala holds 500 litres and loses 20 litres each hour. Write the volume V after t hours.
V = 500 − 20t. - Earlier work (Year 9)Solve 2x − 7 = 11.
x = 9.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that (2, 5) lies on the line 3x − y − 1 = 0 and find another point with whole-number coordinates.
6 − 5 − 1 = 0, and (3, 8) also works. - Working backwardsA line in standard form 3x + ky − 12 = 0 passes through (2, 3). Find k.
6 + 3k − 12 = 0, so k = 2.
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. “Show me the steps” gives the same states in words.
No device? Squared paper is enough: plot two points, join them and read off the gradient and intercept.