At a glance. One lesson of about 60 minutes. It assumes plotting points, fractions and decimals, and substitution. Students find a gradient from two points, express it as a fraction, decimal and percentage, test for parallel lines and find perpendicular gradients.
Calculator: no calculator. The numbers are chosen to work by hand with fractions and simple decimals, so no calculator is needed.
What students will learn
Know
- Gradient is rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁).
- Parallel lines have equal gradients; perpendicular gradients satisfy m₁ × m₂ = −1.
- A horizontal line has gradient 0; a vertical line has no gradient.
Understand
- Steepness depends on rise and run together, not on the rise alone.
- A perpendicular line turns the rise and run through a right angle, which swaps them and changes one sign.
- A general rule for gradients works for every line, with a stated exception for vertical lines.
Do
- Find a gradient from two points.
- Convert a gradient to a decimal or a percentage.
- Use equal gradients for parallel and the product of −1 for perpendicular.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding and using gradients (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Students record pairs of gradients for perpendicular lines and state the relationship.
- Criterion C. Communicating: setting out rise, run and gradient with correct signs (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the ramps in the hook and the mountain road in question 3.
The one thing to do with the figure
Ask for the second line before showing it. Set rise 1, run 3 and ask for the gradient of the line at right angles, on a mini-whiteboard. Then show it. Repeat with rise 3 and run 1, and then with a negative rise.
Ask: “What happened to the rise and the run?” They swapped places and one changed direction. Then set rise 0 and ask why the product rule has nothing to say.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: which ramp is steeper? Collect votes and reasons. |
| 12 min | Figures, 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 the second figure. |
| 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 | 0.375; 1/10; 4. |
|---|---|
| Hook | Ramp A is steeper: 1/4 = 0.25 and 2/9 is about 0.22. |
| Worked examples | −2; parallel (both gradients 2); −3/2. |
| Faded example | 1/5. |
| 1. A perpendicular gradient | A. B flips only; C changes the sign only; D is parallel. |
| 2. Gradient from two points | 4. 12 ÷ 3. |
| 3. Gradient as a percentage | 12. % (0.12 × 100). |
| 4. A negative gradient | −2.5. −10 ÷ 4. |
| 5. A perpendicular gradient as a decimal | −1.25. −1 ÷ 0.8. |
| 6. Parallel lines | 5. (k − 1) ÷ 2 = 2. |
| 7. A perpendicular line through the origin | −24. k ÷ 6 = −4. |
| 8. A right angle at a vertex | 6. k ÷ (−2) = −3. |
| 9. Choosing a perpendicular line | B. A flips only; C changes the sign only; D is parallel. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Flipped only, changed the sign only, or gave the parallel gradient. | Multiply 5/2 by the choice. Does it give −1? |
| 0.25 in question 2 | Gave run over rise. | Write rise over run with the words above the numbers. |
| 0.12 in question 3 | Did not convert to a percentage. | Ask: 0.12 of what? Per hundred is 12. |
| 2.5 or −0.4 in question 4 | Lost the sign, or inverted. | Say whether the line rises or falls, then check the sign of the answer. |
| 1.25 or −0.8 in question 5 | Reciprocal without the sign, or sign without the reciprocal. | Test by multiplying: 0.8 × the answer must be −1. |
| 2 or 4 in question 6 | Gave the gradient, or dropped the 1. | Write (k − 1) ÷ 2 = 2 and solve in two steps. |
| 24 or 1.5 in question 7 | Lost the sign, or made the lines parallel. | Ask: which gradient does the new line need? Find that first. |
| −6 or −3 in question 8 | Sign slip on the run, or gave the gradient. | Write the run of AC as −2 and keep the minus sign in the division. |
| A, C or D in question 9 | Flipped only, changed sign only, or chose a parallel line. | Find the gradient of each option and multiply by the gradient of L. |
Questions worth asking
- Why is a vertical line's gradient not 0? Listen for: the run is 0 and division by zero is undefined; a horizontal line has rise 0.
- Why do perpendicular gradients have opposite signs? One line rises as it goes right and the other falls.
- How can a gradient be written as a percentage, and when is that useful? Multiply by 100; road signs and ramps.
- Do two lines with the same gradient have to be the same line? No: they are parallel, and the same line only if they share a point.
Differentiation
A way in for anyone stuck
- Give a card with rise over run and the order of subtraction written out.
- Use squared paper and draw gradient triangles on lines before using the formula.
- Practise perpendicular gradients with fractions only before decimals.
Stretch
- Prove that if two lines have gradients m and −1/m then they are perpendicular, using a gradient triangle turned through 90 degrees.
- Find the gradients of the sides of the triangle with vertices (0, 0), (4, 2) and (−1, 2) and decide whether it has a right angle.
Students learning in English as an additional language
“Gradient” means how steep a line is. “Rise” is the vertical change and “run” is the horizontal change. “Perpendicular” means at a right angle. “Parallel” means always the same distance apart.
| Word | Plain meaning | Example |
|---|---|---|
| gradient | how steep a line is: rise divided by run | up 3 for 5 across |
| rise | the change in the y value | up |
| run | the change in the x value | across |
| parallel | lines with the same gradient that never meet | railway tracks |
| perpendicular | lines at a right angle to each other | corner of a page |
| reciprocal | one divided by the number | 1/7 for 7 |
Sentence starters for explaining aloud or in writing:
- The rise is … and the run is …, so the gradient is …
- These lines are parallel because …
- The perpendicular gradient is … because …
Two things not to say
“Just flip it and change the sign.” Say instead: the product must be −1, so find the number that makes it.
“The gradient is the number in front of x” only: that works when the line is written as y = mx + c.
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.
- GradientFind the gradient of the line through (0, 3) and (4, 11).
2. - PerpendicularFind the gradient perpendicular to −1/3.
3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA footpath in Lisbon climbs 7 m over 80 m horizontally. Write the gradient as a percentage.
8.75%. - Earlier work (Year 9)Write the equation of the line with gradient 3 through (0, −2).
y = 3x − 2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the points (0, 0), (2, 1) and (6, 3) are on one straight line.
Both gradients are 1/2. - Working backwardsThe line through (1, 2) and (k, 8) is perpendicular to the line through (0, 0) and (3, 1). Find k.
Gradient −3, so 6 ÷ (k − 1) = −3 and k = −1.
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. “Show me the steps” gives the same states in words.
No device? Squared paper and a ruler are enough.