At a glance. One lesson of about 60 minutes. Gradient and intercept from two points on the line, prediction, interpolation and extrapolation, and checking that a value is possible.
Calculator: no calculator. No calculator is needed: the arithmetic is small multiples and one division.
What students will learn
Know
- y = mx + c from a line of best fit.
- Interpolation is inside the data range; extrapolation is outside.
- A prediction must be a possible value.
Understand
- The model is only reliable where the data support it.
- The intercept may mean nothing in context.
Do
- Find m and c from two points.
- Predict y from x and x from y.
- Judge a prediction by how far it is from the data.
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 y = mx + c (questions 2 to 7).
- Criterion B. Investigating patterns: students test the line at ages in and out of the data and describe what changes.
- Criterion C. Communicating: each prediction is stated with a reliability comment.
- Criterion D. Applying mathematics in real-life contexts: height and age, fuel use, seedling growth.
The one thing to do with the figure
Ask for a prediction at ages 6, 12 and 30 before moving the slider. Then ask them to rank the three by how much they trust them.
Move to 40 and let the number speak: 318 cm. Ask what went wrong; the line did nothing wrong, it was used far beyond its data.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the line at 40. |
| 12 min | The figure; rank predictions by trust. |
| 12 min | Worked examples; gradient from two points. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 2; 22; the mean x and mean y. |
|---|---|
| Worked examples | y = 6x + 78. 129 cm and 174 cm; 129 is more reliable. |
| Faded example | y = 8x + 12. |
| 1. Gradient from two points | A. B inverts the fraction; C adds instead of subtracting. |
| 2. Predicting from y = mx + c | 80. |
| 3. Predicting from y = mx + c | 51. |
| 4. Finding the gradient | 4. |
| 5. Finding the intercept | 9, so the line is y = 4x + 9. |
| 6. Interpolating | 39. |
| 7. Solving for x | 20.3 years. |
| 8. How far past the data | 44 weeks. |
| 9. Which prediction to trust | A. B and C both extrapolate. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B or C in question 1 | Inverts the fraction or adds. | Ask: is gradient rise over run or run over rise? |
| 37, 32, 60 in question 2 | Adds, or drops the gradient or the constant. | Write 5 × 12 on its own line first. |
| 28, 24, 36 in question 3 | Same slips with different numbers. | Substitute with brackets. |
| 24, 6, 0.25 in question 4 | Rise only, run only, or inverted. | Draw the rise and run triangle. |
| 12, 17, 25 in question 5 | Subtracts or adds the wrong amount. | Ask what 4 × 3 is part of. |
| 33.3 or 122 in question 7 | Leaves out the 78 or the division. | Undo the equation in reverse order. |
| 52 or 43 in question 8 | Reports the week or is off by one. | Ask: how many weeks after week 8? |
Questions worth asking
- How do you decide whether to trust a prediction? Listen for “is x inside the data, and is the answer possible?”
- Why use points from the line and not from the dots? Listen for “the dots are scattered; the line is the model”.
- Does c always mean something in the context? Listen for “only if x = 0 is inside the data or makes sense”.
Differentiation
A way in for anyone stuck
- Provide a rise and run triangle template and a two-step frame for gradient and intercept.
- Give the equation already found and ask only for predictions.
- Mark the measured range on the x axis in colour.
Stretch
- Find a context where extrapolating would be reasonable, and explain what makes it so.
- A line has gradient −3 and passes through (2, 40). Find y when x = 20 and say whether that can be right for a quantity that cannot be negative.
Students learning in English as an additional language
Pair “interpolation” with “inter” (between) and “extrapolation” with “extra” (outside).
| Word | Plain meaning | Example |
|---|---|---|
| gradient | how much y changes for one step in x | 6 cm each year |
| intercept | the value of y where the line meets x = 0 | 78 |
| interpolation | predicting inside the range of the data | age 8.5 |
| extrapolation | predicting outside the range of the data | age 40 |
| prediction | a value worked out from a line or model, not measured | 120 cm at age 7 |
Sentence starters for explaining aloud or in writing:
- The gradient is … divided by …
- This is interpolation/extrapolation because the x value is … the data range.
- The prediction is/is not reasonable because …
Two things not to say
“The line is the truth.” It is a model of the dots.
“It is wrong because the answer is silly.” Say the line was used outside its data.
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.
- Predicty = 3x + 12. Find y when x = 10.
42. - GradientFind the gradient through (2, 8) and (6, 20).
3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- EquationA line passes through (2, 8) and (6, 20). Find its equation.
y = 3x + 2. - TypeData cover x from 5 to 20. Is x = 12 interpolation or extrapolation? And x = 40?
Interpolation; extrapolation.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA line gives a negative number of hours of sleep at x = 30. What does that tell you?
The line cannot be used that far outside its data. - From Lesson 3A line through the mean point (4, 55) has gradient 5. Find y at x = 8.
75.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard and by touch; “Show me the steps” describes four states in words.
On paper: a printed graph and ruler, with the measured range shaded.