Modelling growth and decay: a week of ready-to-teach maths

Five lessons for Year 10 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus twelve situation cards to sort, twelve cards that match each stage of modelling to what was done, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair. Cut the 12 situation cards before Day 1 and the 12 stage cards before Day 5, and keep each set in an envelope.
4. Calculators are used every day. A spreadsheet on a laptop or tablet makes the money and data tables quick, but every task also works on paper.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A mathematical model is a piece of mathematics that stands in for a real situation: a rule for a taxi fare, a formula for a savings balance, a curve for the path of a ball. Modelling is a cycle. We formulate the problem, choosing a model that suits how the quantity changes (adding the same amount, linear; rising and turning, quadratic; multiplying by the same factor, exponential); we solve it; we interpret the answer in the situation; we evaluate it against reality and data; we modify the model when it falls short; and we report what we assumed, what we did and what we found. A model is never the real thing, so its assumptions matter as much as its answers.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these

Money is given to the cent, $1,338.23, and interest rates per year (p.a.). A model names its variables first: “let V be the value in dollars after n years”. Every answer ends with a sentence about the situation, and every model with at least one assumption.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 sets out the cycle and the three model shapes; Days 2 and 3 model growth and decay with money and value; Day 4 adds the quadratic models that rise and turn; and Day 5 runs the whole cycle on real data, through evaluation and modification to a report. It draws on the linear and quadratic modelling of Year 9 and the exponential relations of the Year 10 unit before, and it prepares the financial mathematics and modelling of senior mathematics.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, pencils, this pack printed.
• Cut-out sheet 1, one per pair: 12 situation cards (Day 1).
• Cut-out sheet 2, one per pair: 12 stage cards (Day 5).
• If you have them: laptops or tablets with a spreadsheet for Days 2, 3 and 5 (one per pair is plenty). No other equipment to buy.

Words for the word wall

model, variable, formulate, solve, interpret, evaluate, modify, report, assumption, linear, quadratic, exponential, simple interest, compound interest, principal, rate per annum, depreciation, decay, second difference

Dear families

This week in maths, Year 10 is building mathematical models of real things: savings and loans, a car losing value, a cup of tea cooling, and a video gathering views. Students choose a model, use it, test it against reality and say what they assumed.

Try this at home

• Ask: a bank pays 5% a year on $1000. How much interest in the first year? In the second, if the interest stays in? ($50, then $52.50.)
• Ask: a new car loses value every year. Does it lose the same number of dollars each year, or the same share? (Usually about the same share.)
• Ask: with 40 metres of fence, what shape of rectangle holds the most? (A square, 10 m by 10 m.)
• Ask: a video’s views double every day. Can that go on for a month? (No: there are not enough people.)

My models week

Fill one row a day. Write something that changes over time, the model that fits it, and one thing the model leaves out.

The modelling cycle

Modelling turns a real situation into mathematics and back again. Students put the six stages in order, run them on a taxi fare and a filling pool, and sort twelve situations by the model that suits them.

We are learning to

• name the stages of modelling and say what happens at each,
• run the cycle on a simple situation, ending with a sentence and an assumption,
• choose a linear, quadratic or exponential model from how a quantity changes.

Success criteria

• I can explain what “evaluate” means for the pool model.
• I can say why compound savings are exponential and a taxi fare is linear.

You need

Calculators; the situation cards (cut-out sheet 1), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; sort the cards and do Part D next time.

Watch for these ideas

• Skipping Interpret: “12.29” is not an answer; “about 12 km” is.
• Treating Evaluate as checking the arithmetic: it is checking the model against reality.
• Choosing a model by the size of the numbers: decide by how the quantity changes, adding, multiplying or turning.

Answers

• Part A: Formulate, Solve, Interpret, Evaluate, Modify, Report.
• Part B: t = 9000 ÷ 40 = 225 minutes, so 3 hours 45 minutes. It assumes the flow stays at 40 litres a minute and the pool starts empty; measure the hose with a bucket and a timer, or subtract any water already in.
• Part C: the card key is with the mini-check answers.
• Part D: for example, savings V = P × 1.04ⁿ, the car V = P × 0.85ⁿ, the rumour N = 3^t, the town P × 1.02ⁿ, the water’s gap G × 0.9^t.

The modelling cycle

A. The stages in order

Number the stages 1 to 6 in the order a modeller works through them.

B. Two models

A worked model: a taxi charges $4.20 plus $2.10 a kilometre. How far can you go for $30?

Your turn: a hose fills an empty 9,000-litre pool at 40 litres a minute. How long will it take?

C. Which model?

Sort the twelve situation cards by how the quantity changes, then write the letters.

D. Stretch: write the rules

Money that grows

Simple interest adds the same amount each year, a linear model; compound interest earns interest on the interest, an exponential model. Students tabulate both, use the formulas, and find when a higher simple rate stops being the better deal.

We are learning to

• find simple interest with I = Prn and a compound balance with A = P(1 + r)ⁿ,
• explain why compound interest is an exponential model and simple interest a linear one,
• compare two accounts over different times, and say which is better and why.

Success criteria

• I can find the balance of $5,000.00 at 4.5% p.a. compounded yearly for 10 years.
• I can explain why the better account depends on how long the money stays in.

You need

Calculators; a spreadsheet if you have one; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Using 4.5 for 4.5% in the formula: r = 0.045.
• Compounding the simple account: simple interest is always worked on the first amount.
• “The bigger rate always wins.” Over 2 years 5.2% simple beats 5% compound; over 10 years it does not.

Answers

• Part A: simple $1,000.00, $1,060.00, $1,120.00, $1,180.00, $1,240.00, $1,300.00; compound $1,000.00, $1,060.00, $1,123.60, $1,191.02, $1,262.48, $1,338.23.
• Part B: 1. A = $5,000.00 × 1.045¹⁰ = $7,764.85, so $2,764.85 interest. 2. I = $5,000.00 × 0.045 × 10 = $2,250.00, so $7,250.00: $514.85 less.
• Part C: 2 years: X $2,205.00, Y $2,208.00, Y better; 3 years: X $2,315.25, Y $2,312.00, X better; 10 years: X $3,257.79, Y $3,040.00, X better. Assumes the rates stay fixed and nothing is added, taken out or charged.
• Part D: $1,000.00 × (1 + 0.06/12)¹² = $1,061.68, which is $1.68 more than yearly compounding.

Money that grows

A. Simple and compound

$1,000.00 is invested at 6% p.a. Fill each balance to the cent. The first two rows are done for you.

B. The formulas

C. Which account?

Account X pays 5% p.a. compounded yearly; account Y pays 5.2% p.a. simple interest. Ana has $2,000.00 to invest.

D. Stretch: more often

Decay and depreciation

Decay is growth with a factor below 1. Students model a $32,000 car losing 15% of its value a year against straight-line depreciation of $4,000 a year, find where the two disagree, and model a cup of tea cooling towards room temperature.

We are learning to

• model exponential decay with V = P(1 − r)ⁿ,
• compare it with a linear decay model and say which suits the situation better,
• interpret and evaluate a decay model, including where it breaks down.

Success criteria

• I can find a car’s value after 5 years at 15% depreciation a year.
• I can explain why the straight-line model fails after 8 years.

You need

Calculators; a spreadsheet if you have one; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Taking 15% of the price every year: that is the straight-line model; decay takes 15% of the current value.
• Multiplying by 0.15 instead of 0.85: the car keeps 85% of its value each year.
• Cooling to 0 °C: the tea approaches the temperature of the room, not zero.

Answers

• Part A: V = 32000 × 0.85ⁿ: $32,000.00, $27,200.00, $23,120.00, $19,652.00, $16,704.20, $14,198.57.
• Part B: straight line $32,000.00, $28,000.00, $24,000.00, $20,000.00, $16,000.00, $12,000.00. The exponential value is lower up to year 3 and the straight-line value from year 4. The straight line reaches $0 after 8 years, but a car still has value then, so the exponential model fits better (assuming a steady 15% a year).
• Part C: the gap of 64 °C halves every 10 minutes: 85 °C, 53 °C, 37 °C, 29 °C. It never quite reaches 21 °C.
• Part D: half price is $16,000.00. After 4 years $16,704.20; after 5 years $14,198.57. So the car is first worth under half its price after 5 years.

Decay and depreciation

A and B. Two ways to lose value

A new car costs $32,000.00. Model 1: it loses 15% of its value each year. Model 2: it loses $4,000.00 each year. The first two rows are done for you.

C. A cooling cup

Tea at 85 °C sits in a room at 21 °C. The gap between the two halves every 10 minutes.

D. Stretch: half price

Models that turn

Some quantities rise and then fall, or grow with the square of a length: quadratic models. Students find the largest pen for 40 m of fencing, follow a thrown ball, and choose models from first and second differences.

We are learning to

• build a quadratic model from a situation and tabulate it,
• find and interpret the greatest value, and the two answers a quadratic can give,
• choose a linear, quadratic or exponential model from differences and ratios.

Success criteria

• I can find the width that gives the largest pen, and say how big the pen is.
• I can explain why the ball is at 15 m at two different times.

You need

Calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Taking the length as 40 − x: two widths use 2x of the fence, so the length is 20 − x.
• Giving one time for h = 15: the ball passes 15 m on the way up and again on the way down.
• Deciding from first differences alone: a quadratic has constant second differences.

Answers

• Part A: A = x(20 − x): 0, 36, 64, 84, 96, 100, 96, 84, 64, 36, 0. The largest is 100 m² at x = 10: a square, 10 m by 10 m.
• Part B: h = 20t − 5t²: 0, 15, 20, 15, 0. Highest 20 m at t = 2 s; lands at t = 4 s; h = 15 at t = 1 s (rising) and t = 3 s (falling). It assumes no air resistance.
• Part C: a linear (first differences 4, 4, 4, 4); b exponential (first differences 4, 12, 36, 108; ratio 3); c quadratic (first differences 5, 3, 1, −1; second differences −2, −2, −2).
• Part D: A = x(40 − 2x): largest at x = 10, giving 200 m² (10 m by 20 m, the long side along the wall), double the free-standing pen.

Models that turn

A. The biggest pen

A rectangular pen is made from 40 m of fencing. Its width is x metres. Fill the table, then graph the area. The first two columns are done for you.

B. A thrown ball

A ball is thrown straight up. Its height after t seconds is h = 20t − 5t² metres.

C. Which model?

Find the first differences, then the second differences or the ratios. Name the model.

D. Stretch: against a wall

Evaluate, modify, report

The whole cycle on one data set: a school video’s views over six days. Students test a linear and an exponential model against the data, improve the better one, decide how far ahead to trust it, and write a short report. The self-check quiz ends the week.

We are learning to

• test a model against data by comparing its predictions with the values,
• modify a model to fit better, and judge how far ahead it can be trusted,
• report the assumptions, the method and the findings.

Success criteria

• I can show that the exponential model fits the views far better than the linear one.
• I can name an assumption that fails if we use the model to predict a month ahead.

You need

Calculators; a spreadsheet if you have one; the stage cards (cut-out sheet 2), one set per pair; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; do Parts B to D and the quiz next time.

Watch for these ideas

• Judging a model by one or two points: compare it with every point.
• Trusting a model forever: doubling cannot last, because there are only so many people to watch.
• A report with no assumptions: every model leaves something out, so name at least two things.

Answers

• Part A: linear v = 120 + 796t gives 120, 916, 1,712, 2,508, 3,304, 4,100, off by up to 1,498; exponential v = 120 × 2^t gives 120, 240, 480, 960, 1,920, 3,840, never more than 7% out.
• Part B: (4,100 ÷ 120)^1/5 ≈ 2.03, so v = 120 × 2.03^t gives 120, 243, 493, 998, 2,023, 4,100. For day 7 it predicts about 16,800 views.
• Part C: the stage cards key is with the mini-check answers. A good report names the model, gives the day 7 prediction, and lists assumptions such as the doubling going on and every view counting once.
• Part D: 120 × 2³⁰ ≈ 129 billion views, more than 16 times the number of people on Earth. The model must break down long before then.

Evaluate, modify, report

A. Two models against the data

A school video’s views, t days after it was posted. Model 1 is the straight line through the first and last points; Model 2 doubles each day. The first row is done for you.

B. Modify

C. Report

Match each stage card to what was done at that stage, then write a short report: the model, the prediction, and two assumptions.

D. Stretch: a month ahead

Which model fits?

For Day 1, one set per pair. Sort the cards by how the quantity changes: by the same amount (linear), by rising and turning or with a square (quadratic), or by the same factor (exponential).

Teacher note: the key is with the mini-check answers.

Stages of the cycle, and what was done

For Day 5, one set per pair. Six cards name a stage of modelling; six say what was done at that stage for the video views. Match each stage to its action, then put the pairs in order.

Teacher note: the key is with the mini-check answers.

What we know: Modelling growth and decay

Work on your own. A calculator is allowed. Show your working, and answer in sentences where a situation asks.

1. Name the model that suits each: (a) a phone plan of $20 a month plus 10 cents a text; (b) bacteria that double every 20 minutes; (c) the height of a diver above the water after leaving the board. ______________________
2. $4,000.00 is invested at 5% p.a. compounded yearly. Find the balance after 6 years. ______________________
3. How much more is that than 5% p.a. simple interest over the same time? ______________________
4. A laptop bought for $1,800.00 loses 25% of its value each year. Find its value after 3 years. ______________________
5. A ball’s height is h = 30t − 5t² metres after t seconds. How high does it go, and when does it land? ______________________
6. The data 5, 10, 20, 40 is best modelled as, circle one:   linear  ·  quadratic  ·  exponential
7. A seedling 4 cm tall is modelled as doubling its height every week. What does the model predict after 10 weeks? Is the model reasonable? ______________________
8. Name two assumptions behind your answer to Question 2. ______________________

Answers and marking guide

Answers with working

1. (a) linear, (b) exponential, (c) quadratic.
2. A = $4,000.00 × 1.05⁶ = $5,360.38.
3. Simple: $4,000.00 + $4,000.00 × 0.05 × 6 = $5,200.00, so compound gives $160.38 more.
4. V = $1,800.00 × 0.75³ = $759.38.
5. Highest at t = 3 s (half of the landing time): h = 45 m. It lands when 30t − 5t² = 0, at t = 6 s.
6. exponential: each value doubles the one before.
7. 4 × 2¹⁰ = 4096 cm, about 41 m: not reasonable. Plants slow down, so the model only fits the first weeks.
8. For example: the rate stays at 5% for all 6 years; nothing is added or taken out; no fees or tax.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, gives money to the cent, and answers Questions 5, 7 and 8 in sentences.

Card keys

Situation cards (Day 1): linear B, E, H, K; quadratic C, G, J; exponential A, D, F, I, L. Stage cards (Day 5): Formulate S and M, Solve P and V, Interpret Q and X, Evaluate N and T, Modify W and R, Report U and O.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the modelling cycle, interest, decay and depreciation, quadratic models, and evaluating, modifying and reporting.