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
- 24 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 24
Visual introductionPicture this
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.
In a nutshell
Model situations or procedures by translating them into algebraic expressions or formulae
What you already know
Part 2 of 24
Before we start
What you already know
You should already be able to form and solve linear equations and substitute values into an expression.
Key words
Part 3 of 24
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.
Reset break
Part 4 of 24
Pause
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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.
Learning cycle
Part 5 of 24
Learning cycle 1 of 2
Turning words into algebra
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 6 of 24
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.
Explore the idea
Part 7 of 24
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.
Reset break
Part 8 of 24
Pause
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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.
Quick check
Part 9 of 24
Quick check
Part 10 of 24
Learning cycle
Part 11 of 24
Learning cycle 2 of 2
Using and testing a model
A short piece of teaching, then a check to make sure it has landed.
Reset break
Part 12 of 24
Pause
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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.
Explore the idea
Part 13 of 24
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.
Explore the idea
Part 14 of 24
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.
Quick check
Part 15 of 24
Reset break
Part 16 of 24
Pause
That's sitting 4 of 6 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.
Quick check
Part 17 of 24
Common mix-ups
Part 18 of 24
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.
Challenge round
Part 19 of 24
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
Reset break
Part 20 of 24
Pause
That's sitting 5 of 6 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.
Challenge round
Part 21 of 24
Game · Recall cards
A gym charges £20 to join and £15 a month. The cost C after m months is...
Card 1 of 4
Exit quiz
Part 22 of 24
Exit quiz
Show what you've learned
6 questions, marked together at the end. Nothing is timed.
1. A taxi charges £3 then £1.20 per mile. The fare F for m miles is...
2. The first step in modelling should be to...
3. Make r the subject of A = 2rh.
4. In C = 50 + 0.2n, the cost when n = 0 is...
5. Which is an assumption in a phone-cost model?
6. A model gives a fare of −£4 for 1 mile. This shows the model is...
Mastery quiz
Part 23 of 24
Marked quiz
End of lesson quiz: Modelling with expressions and formulae
4 questions, marked with the reasoning shown. No timer.
1. A gym charges £20 to join and £15 a month. The cost C after m months is...
2. In C = 12 + 3h, the 3 represents...
3. Make x the subject of y = 4x − 7.
4. A good model should always state...
Lesson round-up
Part 24 of 24
Lesson round-up
Ready when you are
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Ask LunaPart 1 of 24 · Watch & discover