At a glance. One lesson of about 60 minutes. It assumes Lessons 1 to 4. Students follow a short exploration from a question to a tested model and its limits, then evaluate it. The follow-up task is to plan their own: a question, the data they could collect and the family of model they expect.
Calculator: GDC optional. A GDC does the regression and displays the model, which is the Diploma workflow; the report must also show a by-hand check and an honest reading of the output, so the GDC is a tool and not the argument.
What students will learn
Know
- The stages: question, data, model, test, limits and reflection.
- A residual is data minus model; the sum of squares compares models.
- A prediction outside the data needs a reason.
Understand
- Two different models can fit the same readings and disagree outside them.
- Context knowledge helps choose between models that the data cannot separate.
- Precision is not accuracy.
Do
- Choose a family from ratios and fit by hand.
- Test with residuals and a new reading.
- Write an evaluation that states the limits.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: ratios, the model, residuals, the sum of squares (questions 1 to 7).
- Criterion B. Investigating patterns: the first figure. Ask students to find the factor and start that minimize the sum of squares, and explain how they did it.
- Criterion C. Communicating: the structure of the written exploration, with defined symbols and units; the evaluation sentence in the exit ticket.
- Criterion D. Applying mathematics in real-life contexts: cooling tea, a battery level, the limit of a model beyond the data.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Which setting do you think gives the smallest sum of squares?” and have pairs agree on a and b before trying it.
In the second figure, ask which model they would trust at 40 minutes and why, before showing the readouts.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: a cooling cup of tea. Collect what students would compute and what worries them. |
| 10 min | First figure and the key card: the stages. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask whether a plant gets shorter by itself. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. Set the planning task (the practical note) as homework. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | Exponential; 20; 19. |
|---|---|
| Hook | No: the differences are 19, 13, 10, 6, not constant; the ratios are close to 0.7. After an hour the model gives an excess of about 0.9 degrees, but it has not been tested beyond 25 minutes. |
| Worked example | E = 64 × 0.7t/5; 10.76 at 25 minutes against 11, a residual of 0.24. |
| Faded example | 64 × 0.76 = 7.5 degrees above the room. |
| 1. Reading a prediction outside the data | B. A treats the model as the plant; C blames the data; D claims a quadratic is always best. |
| 2. Predict the next value from a ratio | 36.9. 45 × 0.82. |
| 3. Residual of a new reading | 0.6. 31.0 − 30.4. |
| 4. Predict a later time from the model | 13.7. 100 × 0.8210. |
| 5. Compare two models by squared residuals | 0.27. 0.25 + 0.01 + 0.01. |
| 6. Where a quadratic model stops being sensible | 30. 3 ÷ 0.1. |
| 7. Excess temperature to temperature | 42.0. 20 + 21.95. |
| 8. Evaluating a model | A. B trusts it everywhere; C demands a perfect fit; D trusts the plain sum. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Trusts the model outside the data, blames the data, or claims a family is always best. | Ask: does a healthy plant get shorter? |
| 27, 35 or 8.1 in question 2 | Subtracts a difference, or uses the fraction lost. | Ask: is the pattern adding or multiplying, from the ratios? |
| −0.6 or 61.4 in question 3 | Reverses the subtraction, or adds. | Say: residual is what is left over, data minus model. |
| 820, 8.2 or 16.8 in question 4 | Multiplies by the hours, divides by them, or is one step short. | Count the steps from hour 0. |
| 0.29, 0.7 or 0.5 in question 5 | Takes the other model, adds without squaring, or lets residuals cancel. | Make a table of squares for each model. |
| 60, 3 or 5 in question 6 | Forgets the 2, gives b, or gives the lowest value. | Ask what the vertex says about when the curve turns round. |
| 22, 28.8 or 64.8 in question 7 | Gives the excess only, applies the factor to the room, or takes one step. | Write T = room temperature + excess. |
| B, C or D in question 8 | Trusts the model everywhere, demands zero residuals, or uses the plain sum. | Ask: what evidence is there that it will predict a new reading? |
Questions worth asking
- What would make you doubt a model that fits perfectly? Having as many parameters as readings, or silly values outside the data.
- Why test with a reading that was not used? A fit to the data will always look good on the data; only a new reading can surprise it.
- What does the vertex of the quadratic tell you about its use here? After the vertex it rises, so it contradicts the situation.
- How does the context help choose the model? It tells you what must happen far away, such as approaching the room temperature.
Differentiation
A way in for anyone stuck
- Give a planning grid with the five stages and a sentence starter for each.
- Provide the ratio calculations in a table, and ask for the family and the parameters only.
- Let students choose a context from a short list (cooling, a charging battery, a growing plant) before planning.
Stretch
- Show that the exponential through (0, 64) and (20, 16) has b = 0.25 for each 20 minutes, and find the factor for each 5 minutes.
- Design a short investigation of your own that would give data with constant ratios, and say what you would measure.
Students learning in English as an additional language
“Exploration” is a piece of mathematics you investigate and write about yourself. “Evaluate” means to say how good something is, with reasons. “Limitation” is a place where something stops working. “Excess” here means how far above a fixed value something is.
| Word | Plain meaning | Example |
|---|---|---|
| exploration | a piece of mathematics you choose, investigate and write up | a study of how traffic builds up outside a school |
| model | a function chosen to describe data | a rule that matches the readings of a bus's distance against time |
| residual | the data value minus the model value | a small positive number |
| limitation | a place where a model stops being reliable | a prediction far outside the data |
| evaluate | to judge how good something is, giving evidence | the last section of a report |
| excess | the amount above a fixed value | degrees above room temperature |
Sentence starters for explaining aloud or in writing:
- My question is … because ….
- I chose an exponential / quadratic / linear model because the ratios / differences are ….
- The model is reliable for … but not for … because ….
Two things not to say
“A perfect fit means the model is right.” It means only that the model passes through these readings.
“The GDC gives the model, so the work is done.” The work is the choice, the test and the evaluation.
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.
- RatioA plant's heights are 10, 12, 14.4. Find the ratio of the second to the first.
1.2. - ResidualA model gives 25 and the reading is 23. What is the residual?
−2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe excess of a drink is E = 50 × 0.8t with t in 5-minute steps. Find E after 15 minutes.
25.6. 50 × 0.512. - Earlier work (Lesson 3)y = 20 × 1.5t. First whole t with y at least 100.
4. 20 × 1.53 = 67.5 and 20 × 1.54 = 101.25.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA model for a falling quantity has a rising section after 40 minutes. Is that a reason to reject the whole model?
No: it may be fine inside the data; it is a limit on where to use it. - Working backwardsA model E = a × 0.5t/10 gives 12 at t = 20. Find a.
48. 12 ÷ 0.25.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
For the planning task, each student writes one question they could investigate, what they would measure, and which family they expect. No device is needed.