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SavannahYear 8
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MathsYear 9 Maths: standard form, modelling and Pythagoras30 min★★★ difficulty

Modelling with expressions and formulae

A phone tariff, a taxi fare and a growing plant all follow rules. Modelling is the skill of writing that rule as algebra, so one formula answers every version of the question.

Lesson overview

What you'll learn in this lesson

Model situations or procedures by translating them into algebraic expressions or formulae

Key learning points

  • • Naming the variables first
  • • Fixed parts and rates
  • • Changing the subject
  • • Assumptions and limits

This lesson at a glance

  • 30 minutes
  • 18 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

situationsprocedurestranslatingalgebraicexpressions

Scroll down — the lesson carries on below

Part 1 of 18 · Discover6%
1

Watch & discover

Part 1 of 18

Watch first

Modelling with expressions and formulae

An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.

In a nutshell

Model situations or procedures by translating them into algebraic expressions or formulae

2

What you already know

Part 2 of 18

Before we start

What you already know

You should already be able to form and solve linear equations and substitute values into an expression.

Recap: Solving equations that need rearranging
3

Key words

Part 3 of 18

Key words

Words you'll need today

Model
A formula that stands in for a real situation.
Variable
A quantity that can change, shown by a letter.
Subject
The letter on its own on one side of a formula.
Assumption
Something the model treats as true to keep it simple.
4

Learning cycle

Part 4 of 18

Learning cycle 1 of 2

Turning words into algebra

A short piece of teaching, then a check to make sure it has landed.

5

Explore the idea

Part 5 of 18

Learn

Naming the variables first

Start by writing down what changes and give each one a letter with a unit: let n be the number of minutes, C the cost in pounds. Half the mistakes in modelling come from letters that stand for something vague. Then write one sentence describing the rule in words, and translate it piece by piece. 'A fixed fee plus a rate per minute' becomes C = 8 + 0.05n.

6

Explore the idea

Part 6 of 18

Learn

Fixed parts and rates

Most linear models have two parts: something fixed that happens once, and something that grows steadily. The fixed part is the constant; the rate multiplies the variable. Recognising which is which lets you read someone else's formula quickly. In C = 8 + 0.05n, the 8 is the fee before any calls and the 0.05 is the cost of each extra minute.

7

Quick check

Part 7 of 18

Quick check

A gym charges £20 to join and £15 a month. The cost C after m months is...

8

Reset break

Part 8 of 18

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
9

Quick check

Part 9 of 18

Quick check

In C = 12 + 3h, the 3 represents...

10

Learning cycle

Part 10 of 18

Learning cycle 2 of 2

Using and testing a model

A short piece of teaching, then a check to make sure it has landed.

11

Explore the idea

Part 11 of 18

Learn

Changing the subject

A model written to find cost can be rearranged to find anything else in it. To make n the subject of C = 8 + 0.05n, subtract 8 then divide by 0.05, giving n = (C − 8) ÷ 0.05. The same inverse operations you use to solve equations work here, but the answer stays in letters. Rearranging is faster than solving the same equation repeatedly with different numbers.

12

Explore the idea

Part 12 of 18

Learn

Assumptions and limits

Every model simplifies. C = 8 + 0.05n assumes the rate never changes and there is no cap. A good answer states the assumption and the range over which the model works, for example 0 ≤ n ≤ 500. Testing a model with a value you already know is the fastest check: if n = 0 should cost £8, substitute it and see.

13

Quick check

Part 13 of 18

Quick check

Make x the subject of y = 4x − 7.

14

Quick check

Part 14 of 18

Quick check

A good model should always state...

15

Common mix-ups

Part 15 of 18

Common mix-ups

Things people often get wrong

  • People often use a letter without saying what it means. Always define the variable and its unit.
  • People often treat a model as exact. A model is a useful simplification, valid over a stated range.
16

Challenge round

Part 16 of 18

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

17

Mastery quiz

Part 17 of 18

Marked quiz

End of lesson quiz: Modelling with expressions and formulae

6 questions, marked with the reasoning shown. No timer.

  1. 1. A taxi charges £3 then £1.20 per mile. The fare F for m miles is...

  2. 2. The first step in modelling should be to...

  3. 3. Make r the subject of A = 2rh.

  4. 4. In C = 50 + 0.2n, the cost when n = 0 is...

  5. 5. Which is an assumption in a phone-cost model?

  6. 6. A model gives a fare of −£4 for 1 mile. This shows the model is...

18

Lesson round-up

Part 18 of 18

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 18 · Watch & discover

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