Modelling change: a week of ready-to-teach maths

Five lessons for Year 9 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 situation sort cards, model match cards, 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, and cut them before Day 1: the situation sort cards are for Days 1 and 5, the model match cards for Day 2.
4. Calculators are needed every day; graphing software helps on Days 3 and 4 if you have it.
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 an equation that stands in for a real situation. Modelling goes round a cycle: formulate the equation, solve it, interpret the answer in the words of the situation, evaluate how well the model fits, and report what was done and found. When something changes by the same amount each step, such as a fee per week or a burn rate per hour, the model is linear: a start value plus a constant rate. When a quantity is a product of two changing lengths, as the area of a fenced rectangle is, or when gravity bends a path, the model is quadratic, and its vertex gives the greatest or least value. Every model rests on assumptions, and its answers are only as good as they are.

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 models

Each model names its letters and units, as in C = 45d + 25, where C is the cost in dollars and d the number of days. Money is rounded to the cent; heights and lengths to a sensible place. A heights model uses 5 for half the pull of gravity, a rounded 4.9, so its answers are good to about a tenth of a second. Every answer ends with a sentence in the words of the situation.

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

Days 1 and 2 build and use linear models, many of them about money; Days 3 and 4 build quadratic models of area and of motion and read their vertices; and Day 5 chooses between the kinds from data, tests a model’s assumptions and writes up a report. It builds on Year 8, where students modelled with linear relations, and on solving quadratic equations earlier in Year 9. It prepares transforming graphs later this year and modelling growth and decay in Year 10.

Materials for the week (one trip)

• From the classroom: calculators, rulers, pencils, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 situation sort cards (Days 1 and 5).
• Cut-out sheet 2, one per pair: 12 model match cards (Day 2).
• If you have them: devices with free graphing software for Days 3 and 4. No other equipment to buy.

Words for the word wall

model, formulate, solve, interpret, evaluate, report, assumption, variable, rate of change, constant rate, start value, linear, quadratic, vertex, maximum, minimum, break-even, domain, financial, depreciation, savings goal

Dear families

This week in maths, Year 9 is using equations as models of real situations: phone plans, savings goals, garden fences and the flight of a thrown ball. We choose between a straight-line model, for things that change at a steady rate, and a curved (quadratic) one, and we check whether each model’s assumptions hold.

Try this at home

• Look at a phone, power or streaming bill: is there a fixed charge plus a rate? Write it as an equation.
• Ask: saving $25 a week from $50, how long until $300? (300 − 50 = 250, and 250 ÷ 25 = 10 weeks.)
• Ask: with 20 m of fencing, which rectangle has the most area? (A 5 m by 5 m square, 25 m².)
• Ask: what could make a savings plan go wrong? (A missed week, a big purchase: the model assumes neither.)

My modelling week

Fill one row a day. Write a situation from class, its model, and one assumption the model makes.

Printed from the free seegongsik Modelling change teaching pack · seegongsik.com/au/y9/algebra/AC9M9A05/pack

A constant rate is linear

A model is an equation that stands in for a situation, and modelling goes round five stages. A quantity that changes by the same amount each step has a linear model: start value plus rate times steps.

We are learning to

• name the five stages of the modelling cycle and what each does,
• write a linear model from a start value and a constant rate, naming the letters and units,
• tell a linear situation from a quadratic one or neither, with a reason.

Success criteria

• I can write C = 45d + 25 for car hire and say what 45 and 25 mean.
• I can use a model forwards (days to cost) and backwards (cost to days).

You need

The worksheet; calculators; the 12 situation sort cards (cut-out sheet 1), one set per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with the bike-hire model again.

Watch for these ideas

• Swapping rate and start: C = 25d + 45 would charge $25 a day.
• Missing a negative rate: the candle gets shorter, so its rate is −1.5.
• Calling every growing quantity linear: compound interest grows by more each year.

Answers

• Part A: C = 45d + 25 (d days); L = 24 − 1.5t (t hours); V = 120m + 600 (m minutes); D = 5t (t hours).
• Part B: C = $25, $70, $115, $160, $205, $250; 7 days cost $340; $430 buys 9 days ($430 − $25 = $405, ÷ 45).
• Part C: 1 linear (the same $20 is added every year); 2 quadratic (A = πr²: the radius is squared); 3 neither (each year adds 4% of a bigger amount); 4 linear (each step adds the same height); 5 quadratic (n(n − 1) ÷ 2 handshakes).
• Part D: C = 0.1t + 20 and C = 35; equal when 0.1t = 15, t = 150 texts. Above 150 texts a month the flat plan is cheaper.

A constant rate is linear

A. Formulate

Write a model for each situation. Say what each letter stands for, with units.

B. Forwards and backwards

Use your car-hire model. Complete the table, then answer the two questions in words.

How much is 7 days’ hire? ______________ For how many days can you hire the car with $430? ______________

C. Linear, quadratic or neither?

Decide, and give the reason in a few words.

D. Stretch: two phone plans

Plan A costs $20 a month plus 10c per text. Plan B costs $35 a month with free texts. Write a model for each. For how many texts a month do they cost the same, and which plan suits someone who sends a lot?

Money models forwards and backwards

One linear model answers two kinds of question: forwards, from a time to an amount, and backwards, from an amount to a time. Two models side by side show where one deal overtakes the other.

We are learning to

• use a linear model forwards by substituting and backwards by solving an equation,
• compare two linear models with a table and an equation to find where they are equal,
• interpret each answer in the situation and say where a model stops making sense.

Success criteria

• I can find when a savings plan reaches its goal by solving an equation.
• I can say which gym is cheaper for a given number of weeks, and explain why.

You need

The worksheet; calculators; the 12 model match cards (cut-out sheet 2), one set per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the match; start the second with the car model again.

Watch for these ideas

• Forgetting the start: $750 ÷ 35 ignores the $120 already saved.
• Rounding a week down: if the goal needs 17.4 weeks, it is first met in week 18.
• Trusting a model past its limits: a tank cannot hold a negative volume.

Answers

• Part A: after 12 weeks, 35 × 12 + 120 = $540; goal: 35w + 120 = 750, so 35w = 630 and w = 18: Jo first has $750 in week 18.
• Part B: Gym 1 $20, $65, $110, $155, $200; Gym 2 $65, $95, $125, $155, $185. Equal when 15w + 20 = 10w + 65: w = 9, both $155. Gym 1 is cheaper before week 9, Gym 2 after.
• Part C: t = 10: 1200 L; t = 40: 0 L; t = 50: −400 L. The tank is empty at t = 40; the model only works for 0 to 40 minutes, so t = 50 makes no sense.
• Part D: for 26 weeks (half a year), Gym 1 costs $410 and Gym 2 $325: Gym 2, by $85.

Money models forwards and backwards

A. A savings plan

Jo has $120 saved and adds $35 every week. The model is A = 35w + 120, where A is the amount in dollars after w weeks. How much is saved after 12 weeks? In which week does Jo first reach $750?

B. Which gym?

Gym 1 costs $20 to join and $15 a week. Gym 2 costs $65 to join and $10 a week. Write a model for each, complete the table, and find when they cost the same. Which gym would you choose, and when?

C. Where the model stops

A tank holds 1600 L and drains at 40 L a minute, so V = 1600 − 40t. Find V when t = 10, 40, 50. When is the tank empty? Which of your answers does not make sense, and why?

D. Stretch: half a year

A new member plans to go for 26 weeks. Which gym should they join, and how much do they save?

Quadratic models of area

With a fixed length of fencing, a rectangle’s area is a product of two lengths that change together, so the model is quadratic. Its graph opens downward and its vertex gives the width with the most area.

We are learning to

• formulate an area model from a fixed perimeter, A = x(half the fence − x),
• find the greatest area from a table and from the vertex, and interpret it,
• solve the model for a given area and explain why there are two widths.

Success criteria

• I can write A = x(16 − x) for 32 m of fencing and explain each part.
• I can find the width that gives the most area and say what the shape is.

You need

The worksheet; calculators; for the opener, a 24 cm loop of string or 24 matchsticks per pair (optional).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the string loop again.

Watch for these ideas

• Using the whole fence for one length: the two widths and two lengths share the fence, so length = half the fence − width.
• Reading the vertex as the answer to every question: it gives the most area, not a given area.
• Keeping both answers without thinking: two widths for 48 m² describe the same rectangle turned around.

Answers

• Part A: A = x(16 − x); A = 0, 28, 48, 60, 64, 60, 48, 28, 0; most area 64 m² at x = 8: an 8 m square.
• Part B: A = x(30 − 2x) = −2x² + 30x; vertex at x = 30 ÷ 4 = 7.5, so 7.5 m out from the wall and 15 m along it, 112.5 m².
• Part C: x(16 − x) = 48, so x² − 16x + 48 = 0 and x = 4 or x = 12: a 4 m by 12 m rectangle either way round.
• Part D: for perimeter P the area is x(P/2 − x), whose vertex is at x = P/4, the side of a square: of all rectangles with the same perimeter, the square holds the most.

Quadratic models of area

A. 32 m of fencing

A rectangular pen uses all 32 m of fencing. Its width is x m. Write the length and the area A in terms of x, then complete the table. Which width gives the most area?

Length = ____________   A = ____________   Most area: ______ m² when x = ______

B. Against a wall

A farmer has 30 m of fencing to make a rectangular pen against a long wall, so only three sides need fencing. Each side out from the wall is x m. Show that A = x(30 − 2x), find the vertex, and give the dimensions of the pen with the most area.

C. A given area

With the 32 m of fencing from Part A, which widths give an area of exactly 48 m²? Solve the equation, then explain why there are two answers.

D. Stretch: why a square?

For any perimeter P, show that the rectangle with the most area is a square. (Hint: where is the vertex of A = x(P/2 − x)?)

Quadratic models of motion

Something thrown straight up slows, stops and falls back: gravity bends its height into a parabola that opens downward. The vertex is the greatest height, and the roots are when it is at ground level.

We are learning to

• read a height model, h = −5t² + vt + c, and say what each part means,
• find the greatest height at the vertex and when the object lands,
• solve for a given height and reject a solution that does not fit the situation.

Success criteria

• I can find when a ball thrown up at 30 m/s is highest, and how high that is.
• I can explain why a ball is at 40 m twice, and why a time of −1 s is rejected.

You need

The worksheet; calculators; graphing software if you have it.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the board flight again.

Watch for these ideas

• Squaring the coefficient: −5 × 3² = −45, not (−15)² = 225.
• Thinking the ball is at a height once: on the way up and on the way down.
• Keeping a negative time: the model was not true before the throw.

Answers

• Part A: h = 0, 25, 40, 45, 40, 25, 0; peak at t = 3 s, 45 m; lands at t = 6 s.
• Part B: −5t² + 30t = 40, so t² − 6t + 8 = 0 and t = 2 or t = 4: at 2 s on the way up and 4 s on the way down.
• Part C: h = −5t² + 5t + 10; h = 0 gives t² − t − 2 = 0, t = −1 or t = 2. Reject −1: the diver enters the water after 2 s. Highest at t = 0.5 s, 11.25 m.
• Part D: double the launch speed to 60 m/s: the peak is at t = 6 s and 180 m, four times as high, because the height grows with the square of the speed.

Quadratic models of motion

A. A ball thrown up

A ball is thrown straight up at 30 m/s from the ground. Its height in metres after t seconds is h = −5t² + 30t. Complete the table, then find the greatest height and when the ball lands.

Greatest height ______ m at t = ______ s.   Lands at t = ______ s.

B. When is it at 40 m?

Solve −5t² + 30t = 40. Divide by −5 to make a monic equation, factorise and solve. Explain what each answer means.

C. A diver

A diver leaves a 10 m platform, rising a little first. The height above the water after t seconds is h = −5t² + 5t + 10. When does the diver reach the water? Which solution do you reject, and why? How high does the diver go?

D. Stretch: twice as fast

If the ball in Part A were thrown at 60 m/s instead, how high would it go? How many times higher is that, and why?

Choose, evaluate, report

The last stages of the cycle: choose the kind of model the data supports, test its assumptions against what really happens, and report the method, the answer and its limits.

We are learning to

• choose a linear or a quadratic model from the differences in a table,
• test a model’s assumptions and measure what breaking one costs,
• write a short modelling report: what was done, what was found and how far to trust it.

Success criteria

• I can say from its differences which table is linear, which is quadratic and which is neither.
• I can name an assumption behind a model and say what happens if it fails.

You need

The worksheet; calculators; the situation sort cards from Day 1; the unit’s self-check quiz for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Choosing a model from two points: any two points fit a line; it takes the differences to decide.
• Reporting a number with no sentence: a report says what it means and how far to trust it.
• Thinking evaluation means the model was wrong: it means knowing where it works.

Answers

• Part A: Table 1: first differences 15, 15, 15, 15: linear (y = 15x + 50); Table 2: first differences 9, 7, 5, 3, second −2, −2, −2: quadratic (y = −x² + 10x); Table 3: first differences −20, −16, −12.8, −10.24: neither (each value is 0.8 times the one before).
• Part B: the plan, A = 60w + 300, gives $900 after 10 weeks. Missing 2 weeks gives $780: $120 short, 2 weeks of saving. The model assumes a deposit every week.
• Part C: a good report names the model and its letters, shows the working, answers in a sentence, and names an assumption and a limit.
• Part D: Table 3 multiplies by 0.8 each step: a fixed percentage decrease, as a car’s value or a cooling drink, which needs an exponential model (Year 10).

Choose, evaluate, report

A. Which model fits?

Find the first differences (and the second if you need them). Decide linear, quadratic or neither, and write the rule if you can.

B. Test the assumption

Ari has $300 and plans to save $60 every week, so A = 60w + 300. What does the model predict after 10 weeks? In fact Ari misses 2 of those weeks. How much is really saved, and how far short is it? What assumption broke?

C. Write a report

Choose one model from this week. Write a short report under four headings: the model (with its letters and units), the working, the answer in a sentence, and one assumption with what happens if it fails.

D. Stretch: neither

Look again at Table 3. How does each value come from the one before? Name a real situation that behaves like it.

Situation sort cards

For Days 1 and 5, one set per pair: 12 cards

Sort the situations by the model that fits: linear (a constant rate), quadratic, or neither. Be ready to say why.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Model match cards

For Day 2, one set per pair: 12 cards

Match each situation to its model. Then point to the rate and the start in each model.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Modelling change

Work on your own. Use a calculator. Show your working and answer in words.

1. A plumber charges $80 to come out and $60 an hour. Write a model for the cost and find the cost of a 3.5-hour job. ______________________
2. Using your model, how long was a job that cost $350? ______________________
3. Streaming service A costs $25 plus $8 a month; service B costs $5 plus $12 a month. After how many months do they cost the same? ______________________
4. A rectangle uses 36 m of fencing. Write its area A in terms of its width x, and find the greatest area. ______________________
5. A ball’s height is h = −5t² + 25t. Find its greatest height and when it lands. ______________________
6. When is the ball in Question 5 at 20 m? ______________________
7. Is this table linear, quadratic or neither? x: 0, 1, 2, 3, 4; y: 3, 7, 13, 21, 31. Show the differences. ______________________
8. A model says a puppy gains 0.5 kg a week, so at 5 years old it will weigh over 130 kg. What is wrong with the model? ______________________

Answers and marking guide

Answers with working

1. C = 60h + 80; 60 × 3.5 + 80 = $290.
2. 60h + 80 = 350, so 60h = 270 and h = 4.5: a 4.5-hour job.
3. 25 + 8n = 5 + 12n, so 20 = 4n and n = 5: both cost $65after 5 months; A is cheaper after that.
4. A = x(18 − x); vertex at x = 9, so a 9 m square of 81 m².
5. Vertex at t = 25 ÷ 10 = 2.5 s, height 31.25 m; h = 0 at t = 0 and 5, so it lands after 5 s.
6. −5t² + 25t = 20 gives t² − 5t + 4 = 0, t = 1 or 4: 1 s going up, 4 s coming down.
7. First differences 4, 6, 8, 10; second differences 2, 2, 2: quadratic.
8. It assumes a constant rate for ever. Puppies grow fast and then stop; the linear model only fits the first months.

A quick three-level guide

Eight questions, four stages of the cycle. A student at standard answers most questions with working and a sentence in the words of the situation.

Card keys

Situation sort (Days 1 and 5): linear: hiring a trailer: $40 plus $15 an hour; a 30 cm candle burning down 2 cm an hour; a rain gauge filling 3 mm every hour of steady rain; a phone plan: $25 a month plus 10c per text; distance walked at a steady 6 km/h. quadratic: the area of a square photo of side x cm; the height of a ball thrown straight up, t seconds later; the area of a rectangle with perimeter 40 m and width x m; a car’s braking distance at speed v. neither: a $1000 deposit earning 5% a year, compounded; the temperature of a cup of tea left to cool; the time to drive 100 km at speed v.

Model match (Day 2): A gym charges $30 to join and $12 a week: C = 12w + 30; A $24 000 car loses $2400 of value each year: V = 24 000 − 2400y; Pocket money: $150 saved now, plus $40 a week: A = 40w + 150; A 900 L tank drains at 30 L a minute: V = 900 − 30t; Concert tickets are $55 each plus a $10 booking fee: C = 55n + 10; A tree 1.2 m tall grows 0.4 m a year: H = 0.4y + 1.2.

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: linear models, money models, area, motion, and evaluating and reporting.