Year 11 Standard · Unit 5 Ready for the Diploma · Lesson 6 of 6

What would a piece of mathematics look like if you chose the question yourself?

An exploration starts with a question you care about, collects data, fits a model, tests it and says honestly where it stops working. This lesson walks through one, small enough to finish in a lesson.

FunctionsGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 4: the ratios of a table are all 1.4. Which family of function fits?
    AnswerExponential.
  2. From Lesson 1: in y = 80 × 0.5x, what is y when x = 2?
    Answer20. 80 × 0.25.
  3. From Year 9: a line goes up 3 for each 1 across and passes through (0, 4). What is y when x = 5?
    Answer19. 4 + 3 × 5.
Words used in this lesson
exploration
a piece of mathematics you choose, investigate and write up. For example: a study of how traffic builds up outside a school.
model
a function chosen to describe data. For example: a rule that matches the readings of a bus's distance against time.
residual
the data value minus the model value. For example: a small positive number.
limitation
a place where a model stops being reliable. For example: a prediction far outside the data.
evaluate
to judge how good something is, giving evidence. For example: the last section of a report.
excess
the amount above a fixed value. For example: degrees above room temperature.

A cooling cup of tea

In a kitchen in Kampala someone makes tea, leaves it on the table, and records how many degrees it is above the room temperature every 5 minutes for 20 minutes: 64, 45, 32, 22 and 16. She wants to know how long before it is only a few degrees above the room, and she wants to be able to say honestly how far her answer can be trusted.

Predict first: does the tea lose the same number of degrees every 5 minutes? What would you work out from her table to decide? And what would you be worried about if she used the model to say what happens after an hour?

50
0.6
833.07sum of the squared residuals
15largest residual (size)

Model 50 × 0.6 to the power (t ÷ 5): the residuals are 14, 15, 14, 11.2, 9.52. The sum of squares is 833.07 and the largest residual is 15.

Show me the steps
  1. a = 50, b = 0.6: every residual is positive (14, 15, 14, 11.2, 9.52), so the curve is below every reading. The squares add to 833.07.
  2. a = 64, b = 0.8: the start is right, so the first residual is 0, but the curve falls too slowly and is above the readings: the residuals are 0, −6.2, −8.96, −10.77, −10.21 and the squares add to 339.01.
  3. a = 64, b = 0.7: the residuals are 0, 0.2, 0.64, 0.05, 0.63, small and all on one side. The squares add to 0.85, which is the smallest this slider grid can give.

The stages of a short exploration.

  1. Question. Ask something you can measure and that matters to you, and say what you expect.
  2. Data. Collect it carefully: equal steps, units stated, a note of how it was measured.
  3. Model. Choose a family from differences or ratios, fit by hand, then check with a GDC.
  4. Test. Look at the residuals, and compare the model with a reading it was not fitted to.
  5. Limits and reflection. Say where the model fails and what you would do next.

Writing counts as much as calculating: define every symbol, state units, and give reasons.

0
64exponential model
64quadratic model

At 0 minutes the exponential gives 64 and the quadratic gives 64. Both models pass through a reading here.

Show me the steps
  1. At 0, 10 and 20 minutes both models give 64, 32 and 16, the readings. Between them they are close but not equal: at 5 minutes the exponential gives 45.3 and the quadratic 46.
  2. Slide to 25 minutes: the exponential gives 11.3 and the quadratic gives 14, the lowest it ever gets. The quadratic has stopped falling.
  3. Slide to 40 minutes: the exponential gives 4 and the quadratic gives 32. The quadratic says the drink is getting warmer, which a cooling drink does not do.

Fitting is not explaining. Two different models can pass through the same readings and disagree sharply afterwards. The data cannot choose between them; what you know about the situation can. A cooling drink falls towards the room temperature and does not warm up, so the family has to be one that behaves like that.

Worked examples

Readings: 64, 45, 32, 22, 16 at 0, 5, 10, 15, 20 minutes. Choose a model and write it down.

  1. The differences are 19, 13, 10, 6, which are not constant. The ratios are 45 ÷ 64 = 0.70, 32 ÷ 45 = 0.71, 22 ÷ 32 = 0.69 and 16 ÷ 22 = 0.73, all close to 0.7.
  2. That suggests an exponential: with t in minutes and E the degrees above the room, E = 64 × 0.7t/5.

A new reading at 25 minutes is 11. Test the model.

  1. The model gives 64 × 0.75 = 64 × 0.16807 = 10.76.
  2. The residual is 11 − 10.76 = 0.24, small compared with the readings, so the model predicts a new reading well. The limit is still there: the model has never been tested at an hour.

Your go. The first step is given. Use E = 64 × 0.7t/5 to find the degrees above the room at 30 minutes, to one decimal place.

  1. 30 minutes is 30 ÷ 5 = 6 steps of 5 minutes.
  2. Work out 0.76 and multiply by 64.
Check the answer

0.76 = 0.117649, and 64 × 0.117649 = 7.53, which is 7.5 degrees above the room. The model says the excess keeps falling but never reaches 0, which agrees with how a drink cools.

With a GDC (optional)

Use the list and regression screens as in Lesson 4: enter the times and the readings, choose the family you picked from the ratios, and read the parameters. Then store the model and use the table or the graph to compare it with the readings, and to look at what it predicts for a time outside your data.

Write the GDC's model in your report with the symbols defined and the units stated, and say what you did by hand to check it. Everything here can be done by hand.

Check before you go on

1. A quadratic is fitted to the height of a sunflower measured each week for 6 weeks, and it fits all six readings closely. It predicts that the plant will be shorter at week 20 than at week 12. What does that tell you?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A battery level falls over the hours as 100, 82, 67, 55, 45 (per cent). Each value is about 0.82 times the one before. What does that suggest for the next value, to 1 decimal place?

3. The model predicts 30.4 per cent at hour 6, and a new reading is 31.0 per cent. What is the residual, data minus model?

Level 2 · Building

4. The battery model is y = 100 × 0.82t, with t in hours. What does it predict at hour 10, to 1 decimal place?

5. Model A has residuals 0.2, −0.4, 0.3. Model B has residuals 0.5, 0.1, −0.1. What is the sum of squared residuals of the better model?

Level 3 · Stretch

6. A quadratic model for the excess temperature of a drink is E = 0.05t2 − 3t + 50, with t in minutes. A cooling drink never warms up. After how many minutes does this model stop falling?

7. The room is at 20 degrees and the excess is E = 64 × 0.7t/5. What is the drink's temperature at 15 minutes, in degrees to 1 decimal place?

Something to think about

A friend says her exploration is finished because her model has a perfect fit to all her readings. What questions would you ask her about the model, the readings and what she wants to say?

One way to think about it

How many readings are there and how many numbers does the model have? A model with as many parameters as readings can always fit perfectly. Does it behave sensibly between and beyond the readings? Would a new reading agree with it? And what does she want it to tell her? A fit is the start of the discussion, not the end.

Exit ticket

8. A model of a cooling drink was fitted to readings from 0 to 20 minutes. Its residuals are 0.3, −0.2, 0.4, −0.3 and 0.1, and it predicted a new reading at 25 minutes to within 0.3 degrees. Which conclusion does this evidence support?