Variables and substituting into formulas: a week of ready-to-teach maths
Five lessons for Year 7 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 formula machine cards, formula match cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets, one copy per pair: formula machine cards (Day 1) and formula match cards (Day 3).
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
A variable is a letter that stands for a quantity that can change, and a formula is a rule written with variables. P = 4s says the perimeter of any square is four times its side, so one short line replaces every separate calculation. Substitution puts a number in place of each letter and works out the result: with s = 7, P = 4 × 7 = 28. The unknown can also be an input: if P = 20, then 20 = 4s, and undoing the multiplication gives s = 5.
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 formulas
A number next to a letter means multiply: 4s is 4 × s, and 2(a + b) is 2 × (a + b). The letter on its own on the left, the subject, is what the formula finds. Write each substitution on a new line, under the one before, so every step can be checked. Units go with the answer: P = 28 cm. Calculators are not needed this week.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Variables and formulas | Play formula machines, say what each letter in a formula stands for, and write formulas for everyday rules | A formula is a rule that processes inputs |
| 2 | Substitution | Replace each letter with its value, one line per step, following the order of operations | Substitution replaces letters with known values |
| 3 | Everyday formulas | Match stories to formulas, use taxi, hire and travel formulas, and compare two formulas with a table | Everyday formulas, all from variables |
| 4 | Working backwards | Substitute a known output and undo the operations to find an unknown input, then check by substituting | Finding an unknown that is an input |
| 5 | Two-step formulas | Convert temperatures, find areas and solve mixed formula problems, forwards and backwards | A two-step everyday formula; self-check quiz as a class game |
How the week builds
Day 1 meets variables and writes formulas; Day 2 substitutes into them; Day 3 uses everyday formulas and compares two of them; Day 4 runs a formula backwards to find an input; and Day 5 handles two-step formulas in both directions. It builds on the Year 6 work on number patterns and rules, and it leads straight into writing expressions and solving equations later this year.
Materials for the week (one trip)
- From the classroom: pencils, scissors, small envelopes for the cards, this pack printed.
- Cut-out sheet 1, one per pair: 12 formula machine cards (Day 1).
- Cut-out sheet 2, one per pair: 8 story cards and 8 formula cards (Day 3).
- No calculators and no other equipment to buy.
Words for the word wall
variable, constant, formula, rule, input, output, subject, substitute, value, unknown, perimeter, area, flagfall, per hour, order of operations, inverse operation, undo, check, Celsius, Fahrenheit
Dear families
This week in maths, Year 7 is starting algebra: using letters to stand for numbers that can change, writing everyday rules as formulas, and putting numbers into formulas to find an answer.
Try this at home
- Play formula machine: one person thinks of a rule such as “double it, then add 1”; the other gives numbers and guesses the rule.
- A taxi charges $3 plus $2 a kilometre, so C = 3 + 2k. What does a 6 km trip cost? ($15.)
- Ask: if a square has a perimeter of 24 cm, how long is each side? (6 cm.)
- Look for a formula on a bill, a sign or a recipe, and say what each part stands for.
Formulas this week
Fill one row a day with a rule you met, the formula for it, and one value worked out.
| Day | The rule in words | The formula | One value |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Variables and substituting into formulas teaching pack · seegongsik.com/au/y7/algebra/AC9M7A01/pack
Variables and formulas
A variable is a letter that stands for a quantity that can change, and a formula is a rule written with variables. Students play formula machines, then write formulas for everyday rules and say what each letter stands for.
We are learning to
- explain what each variable in a formula stands for,
- write a formula for an everyday rule,
- find the rule of a formula machine from its inputs and outputs.
Success criteria
- I can say what P and s stand for in P = 4s, with their units.
- I can write a formula for the cost of n tickets at $9 each.
You need
The formula machine cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 15 min | Formula machines Show the board (see the box). Then pairs play: one draws a secret rule card and is the machine; the partner calls inputs, guesses the rule and writes it as a formula, such as y = 2x + 1. Ask: “How many inputs did you need before you were sure of the rule?” |
| 10 min | Rules into letters Write formulas together: the cost C of n tickets at $9 is C = 9n; the legs L on b birds is L = 2b. Ask what each letter stands for, and its unit. Ask: “Why is one formula better than a list of every case?” |
| 35 min | Write the formulas Students complete Parts A to C; early finishers do Part D. |
Two shorter sessions instead? Split after the machines; start the second with the tickets.
Watch for these ideas
- Thinking the letter hides one secret number. A variable can take many values.
- Reading 4s as a two-digit number. 4s means 4 × s.
- Letting c stand for cows rather than the number of cows.
Answers
- Part A: A = l × w: A is the area, l the length and w the width; C = 3n: C is the cost in dollars and n the number of pens; L = 4c: L is the number of legs and c the number of cows; D = 7w: D is the number of days and w the number of weeks.
- Part B: C = 2s, L = 8p, c = 100m, b = 12k.
- Part C: y = 5x, so 10 gives 50; y = 3x + 2, so 10 gives 32.
- Part D: no; s = 1 gives 4 and 5, s = 2 gives 8 and 6, s = 5 gives 20 and 9. W = 2b + 4c.
- Machine cards: y = 3x, y = x + 7, y = 2x + 1, y = 5x, y = x − 4, y = 4x − 3, y = 10x, y = x + 12, y = 3x + 2, y = x ÷ 2, y = 6x − 1, y = 2x + 5.
Variables and formulas
A. What does each letter stand for?
| Formula | It works out | What each letter stands for |
|---|---|---|
| A = l × w | the area of a rectangle | |
| C = 3n | the cost of pens at $3 each | |
| L = 4c | the legs on a herd of cows | |
| D = 7w | the days in some weeks |
B. Write a formula
- Each sticker costs $2. Write a formula for the cost C of s stickers. ________________
- A spider has 8 legs. Write a formula for the number of legs L on p spiders. ________________
- There are 100 centimetres in a metre. Write a formula for the centimetres c in m metres. ________________
- A packet holds 12 biscuits. Write a formula for the biscuits b in k packets. ________________
C. Find the machine’s rule
Fill the last output, then write the rule as a formula: y = ...
| Input x | 1 | 2 | 4 | 7 | 10 |
| Output y | 5 | 10 | 20 | 35 |
| Input x | 1 | 2 | 5 | 8 | 10 |
| Output y | 5 | 8 | 17 | 26 |
D. Stretch
Is 4s the same as s + 4? Try s = 1, s = 2, s = 5 and explain. ________________________________
Write a formula for the number of wheels W on b bikes and c cars. ________________
Substitution
Substitution puts a number in place of each letter, then calculates. Writing one line per step, under the line before, keeps the order of operations clear and shows exactly where a slip happened.
We are learning to
- substitute values into a formula with one or more variables,
- set out the working one line per step,
- follow the order of operations after substituting.
Success criteria
- I can find P = 2l + 2w when l = 7 and w = 3, in three lines.
- I can explain why 3k + 4 and 3(k + 4) give different values.
You need
The worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Line by line Show the board (see the box). Then model P = 2l + 2w with l = 7 and w = 3: P = 2 × 7 + 2 × 3, P = 14 + 6, P = 20. Ask: “Why write 2 × 7 and not 27?” |
| 15 min | Order matters Substitute k = 2 into 3k + 4 and into 3(k + 4): 10 and 18. Pairs make up their own pair of formulas with the same letters and different values. Ask: “Where do the brackets change the order of the steps?” |
| 35 min | Substitute Students complete Parts A to C; early finishers do Part D. |
Two shorter sessions instead? Split after the order of operations; start the second with Part A.
Watch for these ideas
- Writing 2l as 28 when l = 8. 2l means 2 × 8.
- Adding first: 5 + 3n with n = 7 is 5 + 21, not 8 × 7.
- Chaining equals signs that are not equal. Start a new line for each step.
Answers
- Part A: A = 45, P = 28, V = 24, C = 26, y = 21, T = 30.
- Part B: 3k + 4 = 19, 3(k + 4) = 27, m ÷ 2 + 6 = 8, (m + 6) ÷ 2 = 5.
- Part C: 6, 12, 18, 30, 60.
- Part D: A = 9 × 6 ÷ 2 = 27; P = 4 × 2.5 = 10; any of 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6 (and decimals such as 4.5 and 8).
Substitution
A. Substitute, one line per step
B. Same letters, different order
C. A table of values
Rectangles are 6 cm long. Their area is A = 6w. Complete the table.
| w (cm) | 1 | 2 | 3 | 5 | 10 |
| A (cm²) |
D. Stretch
- The area of a triangle is A = b × h ÷ 2. Find A when b = 9 and h = 6. ________
- Find P = 4s when s = 2.5. ________
- A rectangle has A = l × w = 36. Give three different pairs of values for l and w. ________________
Everyday formulas
Everyday formulas often mix a fixed part with a part that grows: a taxi fare has a flagfall plus a charge for each kilometre. Students match stories to formulas, use formulas with units, and compare two formulas with a table.
We are learning to
- match a story to its formula,
- use a formula, with units, to answer a real question,
- compare two formulas with a table of values.
Success criteria
- I can say which part of C = 50 + 12p is fixed and which part grows.
- I can say which of two taxis is cheaper for a given trip, and show why.
You need
The formula match cards from cut-out sheet 2 (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Fixed and growing Show the board (see the box). Then a party costs C = 50 + 12p: $50 for the room and $12 a person. For 10 people, C = 50 + 120 = 170. Ask: “What would change in the formula if the room cost $80?” |
| 15 min | Formula match Pairs match the 8 story cards to the 8 formula cards, then explain a pair that nearly fooled them. Ask: “How are C = 3 + 2k and C = 3t + 2 different stories?” |
| 35 min | Use the formulas Students complete Parts A to C; early finishers do Part D. |
Two shorter sessions instead? Split after the formula match; start the second with Part A.
Watch for these ideas
- Swapping the parts: $3 to start and $2 a kilometre is 3 + 2k, not 3k + 2.
- Dropping units. D = 90t gives a distance in kilometres.
- Deciding from one trip that one taxi is always cheaper.
Answers
- Part A: $6, $10, $14, $24, $28.
- Part B: 1. C = 15h, $45; 2. C = 80 + 70h, $220; 3. D = 90t, 360 km.
- Part C: 5 km: A $14, B $17.50, A is cheaper; 20 km: A $44, B $40, B is cheaper.
- Part D: 12 km, where both cost $28. C = 20 + 8p gives $60 for p = 5.
- Cards: 1 C = 3 + 2k, 2 C = 3t + 2, 3 P = 4s, 4 D = 60t, 5 C = 12p + 5, 6 C = 25h, 7 P = 2l + 2w, 8 S = n ÷ 4.
Everyday formulas
A. A taxi fare
A taxi charges $4 to start and $2 a kilometre: C = 4 + 2k. Complete the table.
| k (km) | 1 | 3 | 5 | 10 | 12 |
| C ($) |
B. Write the formula, then use it
- Bike hire costs $15 an hour. Write a formula for the cost C of h hours, then find the cost of 3 hours.Formula: ______________ Answer: ______________
- A plumber charges $80 to come out, plus $70 an hour. Write a formula for the cost C of h hours, then find the cost of 2 hours.Formula: ______________ Answer: ______________
- A train travels at 90 km/h. Write a formula for the distance D in t hours, then find the distance in 4 hours.Formula: ______________ Answer: ______________
C. Two taxis
Taxi A: C = 4 + 2k. Taxi B: C = 10 + 1.5k. Which is cheaper for a 5 km trip? For a 20 km trip? Show the costs. ________________________________
D. Stretch
For what length of trip do the two taxis cost the same? ________________
Write a story that C = 20 + 8p could describe, and find C when p = 5. ________________________________
Working backwards
The unknown in a formula can be an input. Substitute what is known, then undo the operations in reverse order: for 30 = 4 + 2k, take away 4, then divide by 2. Substituting the answer back in checks it.
We are learning to
- substitute a known output into a formula,
- undo the operations in reverse order to find an unknown input,
- check an answer by substituting it back.
Success criteria
- I can find s when P = 4s and P = 36.
- I can find k when 19 = 5 + 2k, and check it.
You need
The worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Work back Show the board (see the box). Then P = 36: 36 = 4s, and dividing by 4 undoes the multiplying, so s = 9. Ask: “What undoes times 4? What undoes plus 5?” |
| 15 min | Two steps back 19 = 5 + 2k: take away 5 to get 14 = 2k, then divide by 2, so k = 7. Check: 5 + 2 × 7 = 19. Like socks and shoes, the last thing done is the first undone. Ask: “Why do we take away 5 before dividing by 2?” |
| 35 min | Find the unknown Students complete Parts A to C; early finishers do Part D. |
Two shorter sessions instead? Split after two steps back; start the second with Part A.
Watch for these ideas
- Doing the operation instead of undoing it: 20 = 4s does not give s = 80.
- Undoing in the wrong order, such as halving 19 first in 19 = 5 + 2k.
- Skipping the check. Substituting back catches most slips.
Answers
- Part A: s = 11, t = 2.5, w = 6, n = 9.
- Part B: k = 13, x = 7, l = 11, h = 4.
- Part C: Ali is wrong: 11 gives 26, not 18; the answer is 7; Mel is right (it gives 26).
- Part D: 2 × 3w + 2w = 8w = 40, so w = 5 cm and l = 15 cm. p = (260 − 50) ÷ 12 = 17.5, so 17 people (a half person cannot come).
Working backwards
A. One step back
B. Two steps back
C. Check by substituting
- Ali used C = 4 + 2k with C = 18, and got k = 11. Is Ali right? If not, fix it. ________________
- Mel used P = 2l + 2w with P = 26 and w = 5, and got l = 8. Is Mel right? If not, fix it. ________________
D. Stretch
A rectangle has a perimeter of 40 cm, and its length is 3 times its width. Find the length and the width. ________________
A party costs C = 50 + 12p dollars for p people. The budget is $260. How many people can come? ________________
Two-step formulas
Many real formulas take two steps, such as F = 9C ÷ 5 + 32. Students substitute and follow the order of operations, run formulas backwards, and write a formula of their own.
We are learning to
- substitute into formulas with two or more operations,
- work a two-step formula backwards to find an input,
- write and use a formula for a new situation.
Success criteria
- I can change 15 °C to Fahrenheit and show both steps.
- I can find the number of visits a budget buys from a cost formula.
You need
The worksheet and the unit’s self-check quiz on the board. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Two steps On the unit page, show “A two-step everyday formula” and press “20 °C” and “100 °C”. Then change 15 °C: 9 × 15 ÷ 5 = 27, and 27 + 32 = 59 °F. Ask: “Why multiply and divide before adding the 32?” |
| 30 min | Problems Students work through problems 1 to 6; 7 and 8 are the stretch. |
| 20 min | Class quiz game Run the self-check quiz as a class game (see the box), then wrap up. |
Two shorter sessions instead? Split after problem 3; start the second with problem 4.
Watch for these ideas
- Adding 32 first, so 15 °C becomes 9 × 47 ÷ 5.
- Undoing a two-step formula in the same order it was built. Undo the last step first.
- Writing 16 + 3n for $16 tickets and a $3 fee. The fee is paid once.
Answers with working
- F = 9 × 30 ÷ 5 + 32 = 54 + 32 = 86 °F.
- F = 9 × (−5) ÷ 5 + 32 = −9 + 32 = 23 °F.
- T = 40 × 2.5 + 20 = 100 + 20 = 120 minutes, which is 2 hours.
- A = 12 × 7 ÷ 2 = 42 cm².
- 180 = 60 + 15v, so 120 = 15v and v = 8 visits.
- C = 16n + 3; C = 16 × 5 + 3 = $83.
- 77 − 32 = 45, × 5 = 225, ÷ 9 = 25 °C.
- yes, −40: 9 × (−40) ÷ 5 + 32 = −72 + 32 = −40.
Two-step formulas
Write the formula, substitute on a new line, then work one step per line.
Problems (7 and 8 are the stretch)
- Change 30 °C to Fahrenheit using F = 9C ÷ 5 + 32.
- Change −5 °C to Fahrenheit.
- A recipe says a roast cooks for T = 40m + 20 minutes, where m is its mass in kilograms. How long does a 2.5 kg roast take? Give the answer in hours and minutes.
- The area of a triangle is A = b × h ÷ 2. Find the area when b = 12 cm and h = 7 cm.
- A gym charges C = 60 + 15v dollars for v visits. How many visits can you make for $180?
- Movie tickets cost $16 each, plus a $3 booking fee. Write a formula for the cost C of n tickets, then find the cost of 5 tickets.
- Stretch. What temperature in Celsius is 77 °F? Work backwards through F = 9C ÷ 5 + 32.
- Stretch. Is there a temperature where the Celsius and Fahrenheit numbers are the same? Try some values, or set F = C.
Formula machine cards
Cut out the 12 cards. Day 1: the machine draws a card and keeps it secret. The partner calls out inputs, the machine answers with outputs, and the partner guesses the rule and writes it as a formula: y = ...
Extension: write a rule of your own on a scrap of paper for your partner to find.
Formula match cards
Cut out the 16 cards. Day 3: match each story to its formula. Two pairs are close, so read carefully.
Stories
Formulas
What we know: Variables and formulas
Work on your own, without a calculator. Show your working, one line per step.
- In C = 6n, the cost of n notebooks, what do C and n stand for? ________________________________
- Write a formula for the number of legs L on h horses. ________________
- Find A = l × w when l = 11 and w = 4. ________
- Find y = 5x − 2 when x = 3. ________
- A plumber charges C = 90 + 60h dollars for h hours. Find the cost of 3 hours. ________
- P = 4s and P = 52. Find s. ________
- C = 6 + 3k and C = 33. Find k, then check by substituting. ________________
- Change 35 °C to Fahrenheit using F = 9C ÷ 5 + 32. ________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- C is the cost in dollars; n is the number of notebooks.
- L = 4h.
- A = 11 × 4 = 44.
- y = 5 × 3 − 2 = 15 − 2 = 13.
- C = 90 + 60 × 3 = 90 + 180 = $270.
- 52 = 4s, so s = 52 ÷ 4 = 13.
- 33 − 6 = 27 = 3k, so k = 9; check: 6 + 3 × 9 = 33.
- F = 9 × 35 ÷ 5 + 32 = 63 + 32 = 95 °F.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Variables and formulas (Q1, Q2) | treats a letter as a label for an object | says what each variable stands for and writes a formula for a rule | explains why a formula covers every case |
| Substitution (Q3 to Q5) | joins digits, so 2l with l = 8 becomes 28 | substitutes line by line and follows the order of operations | substitutes decimals and checks with units |
| Working backwards (Q6, Q7) | does the operation instead of undoing it | undoes the steps in reverse order and checks by substituting | explains why the last step is undone first |
| Two-step formulas (Q8) | adds before multiplying | substitutes into a two-step formula in the right order | runs a two-step formula backwards |
Eight questions, four ideas. A student at standard answers most questions with working set out one line per step, and checks an unknown by substituting it back.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Variables | Substituting | Everyday formulas | Working back | Two-step |
|---|---|---|---|---|---|
The five columns are the five days: variables, substituting, everyday formulas, working backwards, and two-step formulas.