At a glance. One lesson of about 60 minutes. It assumes Lessons 1 to 3 and quadratic functions from Year 10. Students choose a family from first differences, second differences or ratios, fit by hand, check with a GDC regression, compute residuals and sums of squares, and judge when a model can be trusted.
Calculator: GDC optional. The GDC regression is the Diploma habit and saves time, but a student who has not fitted by hand cannot tell whether a screen is sensible. The by-hand fit comes first and the GDC is a check.
What students will learn
Know
- Constant first differences mean linear; constant second differences mean quadratic; constant ratios mean exponential.
- For a quadratic, a is half the second difference when x goes up by 1.
- A residual is data minus model.
Understand
- The table tells you which family to ask the GDC for.
- Residuals can cancel, so the sum of squares is used to compare fits.
- A model is a description of data, and it can fail outside it.
Do
- Find first and second differences and ratios.
- Fit a line, a quadratic or an exponential by hand.
- Compute residuals and compare two models by the sum of squares.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: differences, ratios, a from the second difference (questions 1 to 7).
- Criterion B. Investigating patterns: the first figure. Ask students to design a data set that has constant second differences and then to test it.
- Criterion C. Communicating: stating which row is constant and writing the model with the parameters in context.
- Criterion D. Applying mathematics in real-life contexts: visits to a website, a data set where the model fails, an extrapolation.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Which row will be constant?” for data set 2 before it appears. Then look at data set 4 and ask what they would do if no row is constant.
In the second figure, ask whether residuals that add to zero always mean a good fit. Then ask why the sum of the residuals alone cannot be the measure of fit.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: visits to a website. Collect the checks students would do. |
| 10 min | First figure and key idea: first differences, second differences, ratios. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, write the first differences on the board and ask if they are equal. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace, with the GDC for those who want to check. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 13; 18; 28. |
|---|---|
| Hook | The differences 12, 16, 20, 24, 29 are not constant, so it is not a line. The second differences 4, 4, 4, 5 are nearly constant, so a quadratic; worked example 2 shows 141 is 1 above it. |
| Worked example | y = 2x2 + 10x + 40; 208 visits at day 7. Residual at day 5 is 1. |
| Faded example | a = 1, c = 10, b = 2; 58. |
| 1. Choosing the family from differences | C. A needs constant first differences; B needs constant ratios; D ignores the constant second difference. |
| 2. Predict with a linear model | 57. 7 + 5 × 10. |
| 3. a from the second difference | 5. 10 ÷ 2. |
| 4. Find b in a quadratic fit | −1. 5 − 6. |
| 5. Predict with an exponential model | 121.5. 54 × 2.25. |
| 6. Residual of a model | −4. 27 − 31. |
| 7. Sum of squared residuals | 18. 4 + 1 + 9 + 4. |
| 8. Choosing the family before predicting | B. A uses a first difference; C fits a quadratic without constant second differences; D is one step short. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Looks only at the first differences, uses ratios, or says none. | Write the second row of differences and ask whether it is constant. |
| 50, 72 or 220 in question 2 | Forgets the intercept, starts from the wrong x, or treats y as proportional. | Write the model as y = 7 + 5x and substitute. |
| 10 or 20 in question 3 | Gives the second difference, or multiplies by 2. | Ask: for y = ax squared, what is the difference pattern of 0, a, 4a? |
| 1, 3 or 5 in question 4 | Gives a difference, forgets c, or gives y. | Substitute x = 1 into the whole model. |
| 56, 81 or 182.25 in question 5 | Uses a difference, stops a step early, or goes a step too far. | Write the values at x = 4 and x = 5 with arrows of times 1.5. |
| 4, 31 or 58 in question 6 | Reverses the subtraction, gives the model value, or adds. | Ask: is the data above or below the model, so is the residual positive? |
| 2, 8 or 33 in question 7 | Adds the residuals, ignores signs, or takes the larger sum. | Make a column of squares for each model. |
| A, C or D in question 8 | Uses a first difference, fits a quadratic though the second differences change, or stops a step short. | Write the differences, the second differences and the ratios, and ask which row is constant. |
Questions worth asking
- Why is a constant second difference a quadratic? A quadratic's first differences change by a fixed amount, 2a, for each step.
- Why do we square the residuals? Positive and negative residuals cancel in a plain sum; squares are never negative.
- Can two different models fit the same data equally well? Yes, inside the data; they may disagree strongly outside it.
- What would make you doubt a GDC fit? A pattern in the residuals, or a prediction outside the data that is impossible.
Differentiation
A way in for anyone stuck
- Give a pre-drawn table with rows for first differences, second differences and ratios, blank.
- Provide the form y = ax2 + bx + c and the step: a, then c, then b by substituting x = 1.
- Let students check each hand-fitted model by substituting one point that was not used in the fit.
Stretch
- A quadratic y = ax2 + bx + c has first differences 7, 11, 15 for x = 0, 1, 2, 3. Find it, given that y = 1 at x = 0.
- Why would an exponential model of a population be sensible for 5 years and not for 500?
Students learning in English as an additional language
“Differences” here means subtracting each value from the next one. “Ratio” means dividing each value by the one before. “Fit” means choosing a function whose graph is close to the data. “Residual” is what is left over: data minus model.
| Word | Plain meaning | Example |
|---|---|---|
| first difference | a value minus the one before it | the gaps in a column of y-values |
| second difference | a difference of the first differences | constant for a quadratic |
| ratio | a value divided by the one before it | constant for an exponential |
| regression | finding the best-fitting function for data, usually with technology | a GDC fit |
| residual | the data value minus the model value | above the line is positive |
| extrapolation | using a model outside the range of the data | predicting far ahead |
Sentence starters for explaining aloud or in writing:
- The first / second differences are …, so the model is ….
- The residual at x = … is … because the data is … the model.
- This prediction is / is not reliable because ….
Two things not to say
“The GDC's answer is the answer.” It is the best fit of the family you chose.
“A model that fits well can be used anywhere.” It may fail outside the 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.
- DifferencesFind the first differences of 5, 8, 11, 14. Which family?
3, 3, 3: linear. - RatiosFind the ratios of 6, 18, 54. Which family?
3, 3: exponential.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA farm's yields over 4 seasons are 20, 23, 28, 35. Which family fits, and what is the next value?
Second differences are 2, 2: quadratic; next first difference 9 so 44. - Earlier work (Lesson 3)y = 6 × 2t. First whole t with y more than 100.
5. 6 × 32 = 192; 6 × 16 = 96.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningResiduals of a line are 3, 1, −1, −3, −1, 1, 3. What do they suggest?
The residuals form a curve (a U shape), so a quadratic is likely better. - Working backwardsA line y = 2x + 1 has residuals 1, −1, 2 at x = 1, 2, 3. Find the data values.
4, 4, 9.
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.
No device? Give each pair a data table on paper and a ruler to draw lines and measure the vertical gaps.