Checking a model

Every model makes assumptions: a steady speed with no stops, a fare with no starting charge, an exchange rate with no fees. When a model's answer and real life differ, look for the assumption that does not hold, then refine the model, for example by adding a fixed starting amount.

A map app says the drive takes 3 hours. It takes 3 and a half. The maths was right; the model was too simple. Finding out why is the last step of modelling.

A model is a simplified version of a real situation. Every model makes assumptions: things it takes to be true to keep the maths simple.

Model	Assumption	What can go wrong
time = distance ÷ speed	a steady speed, no stops	traffic and breaks make it slower
water bill = rate × litres	no fixed charge	bills add a service charge
US dollars = 0.65 × Australian dollars	no fees, a fixed rate	banks charge fees; rates change daily

To review a model: compare its answer with real life, name the assumption that does not hold, and refine the model (add the stop, the starting charge, the fee). A simple model is fine when the difference does not matter for the decision.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

The taxi fare

A model says a 5 km taxi ride costs 5 × $2.10 = $10.50. The real fare is $14.70. Why?5 × 2.10 is not 10.50The model leaves out a starting chargeTaxi fares are randomThe taxi went faster than expected

TryTry it on the board and compare the result with your prediction.

The refined model

Try it yourselfYour edits are saved in this browser.

This model adds the $4.20 start. Set the ride to 5 km. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Assumptions

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$\text{model: water bill} = \$2.50 \times \text{kL}$ A simple model: 20 kL costs $50.
$\text{real bill: } \$80$ Compare with what happened.
$\text{assumption: no fixed charge}$ Name what the model took to be true.
$\$30 + \$2.50 \times 20 = \$80$ Refine: add the $30 service charge.

Read Sam's review. Is the arithmetic wrong, or the model?

A model says a 300 km trip takes 300 ÷ 100 = 3 hours. What does it assume?The road is straightThe car travels at exactly 100 km/h the whole way, with no stopsThe car uses no petrolNothing: it is exact

Sam's model and the real trip disagree. Click the line where his review goes wrong.

How could you refine the trip model?Divide 300 by 100 againUse 3.14 instead of 3Add the time for a stop, or use a lower average speedMake the trip shorter

ModifyChange the problem on purpose and see what happens.

Refine and judge

Need a hint?

Take off the $4.20 first.

Another hint

21.00 ÷ 2.10 = ?

Try it yourselfYour edits are saved in this browser.

With the refined model, how far can you ride for $25.20?

Use the board. This ticks when you get there.

A money model says 500 Australian dollars buys 500 × 0.65 = 325 US dollars. At the airport you get 305. What assumption failed?US dollars are worth more500 × 0.65 is not 325The model should divideNo fees: the exchange counter charges a fee or a worse rate

Refine the trip model: 300 km at 100 km/h, plus two 15-minute stops. How many hours does it take?

When is the simple trip model (300 ÷ 100 = 3 hours) good enough?For a rough plan, such as which day to travelFor a train timetable to the minuteNeverOnly if the answer is a whole number

PractiseA few on your own, to check you have it.

Checking models

A model for a phone plan is cost = $0.10 per MB used. The real bill was $20 more than the model said. Jo says: 'The model's sum must be wrong.' What is more likely?The model leaves out a fixed $20 monthly feeJo is right: the sum is wrongThe phone company made a mistakeThe price per MB changed

With the refined taxi model ($4.20 plus $2.10 per km), what does an 8 km ride cost, in dollars?

A refined phone model is cost = 20 + 0.10 × MB. What is the cost for 350 MB, in dollars?

Review: a car goes 270 km in 3 hours. What is its average speed in km/h?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A plumber's quote

Then name one assumption in the plumber's model, and say how a real bill might differ.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Take off the $80 call-out first.

Another hint

380 ÷ 95 = ?

A plumber charges $80 to come out, then $95 per hour. Find the job length that costs $460.

Use the board. This ticks when you get there.

For teachers

Students compare simple models with real outcomes (taxi fares, trip times, exchange rates), name the assumption that failed, and refine the model with a starting charge, stops or fees (AC9M8M07, node 2). About 35 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can name an assumption a model makes.
I can explain how an assumption makes a model's answer differ from real life.
I can refine a model and say when a simple one is good enough.

Before class

Do the Run board yourself first. Students need rate models (previous lesson).

The one idea: when a model and real life disagree, look for the assumption that did not hold, then refine. The common mistake is blaming correct arithmetic (Sam's mistake).

The board's starting charge (base) shows a refined model: a fixed amount plus a rate. The taxi fare, plumber's prices and exchange rate are round examples, not real tariffs.

Wider: compare a simple tax model (one rate for everyone) with the bracket model from Money models; constant versus changing exchange rates.

Running the lesson

Predict: ask whether 5 × 2.10 is right.
Run: ask what the fare is for 0 km.
Investigate: ask Sam to check 300 ÷ 100.
Modify: ask when the simple trip model is fine.
Make: ask for an assumption about the plumber's time.

The hard part (on purpose)

Separating the maths from the model. When an answer is 'wrong', ask students to check the arithmetic first, then ask what the model took to be true.

When students get stuck

A student says the maths is wrong. Ask: Is the calculation right? What did the model assume? Fix: Check the arithmetic, then name the assumption.
A student forgets the fixed charge when working backwards. Ask: What do you pay before the meter starts? Fix: Take the fixed part off first, then divide by the rate.
A student cannot name an assumption. Ask: What would have to be true for the answer to be exact? Fix: That is the assumption.

Stretch and support

Stretch: Find a real taxi or rideshare fare structure for your city and build its model. What does it still leave out (waiting time, surge pricing)?

Support: A frame: The model assumes … In real life … So the answer … A better model would …

Where it leads

This completes AC9M8M07.

Where it leads: linear models with a starting value (Linear relations: y = mx + c), and modelling in Years 9 and 10.

What each step is for

the taxi fare: Committing to why a simple fare model is wrong shows whether they look for a missing part of the model rather than an arithmetic slip.
the refined model: Moving along the refined fare line shows the fixed start and the rate working together.
assumptions: Name the assumption behind a model; a mismatch is usually the assumption, not the arithmetic (Sam blames the maths); refine by adding what was left out.
refine and judge: Using the refined fare model backwards, naming a hidden assumption, adding stops, and judging when a simple model is enough.
checking models: Assumptions, refined models and a unit-rate review, mixed; blaming the arithmetic for a model's mismatch is the planted slip.
a plumber's quote: Using a refined model (a call-out fee plus a rate) backwards, then naming its assumptions.

Now you can

Name an assumption a model makes.
Explain how an assumption makes a model's answer differ from real life.
Refine a model and say when a simple one is good enough.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.You reached the endBack to Number Foundations

Source: content/maths/number/checking-models/lesson.md