Manipulating formulas with several variables: 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 change and effect cards, spreadsheet 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: change and effect cards (Day 2) and spreadsheet cards (Day 4).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Book devices with a spreadsheet for Day 4 if you can. Every task also works by hand or with a calculator.

The idea in plain words

A formula such as P = 2(l + w) links several variables in one rule: give each input a value and the formula gives the output. To understand a formula, change one input at a time while the others stay fixed, and watch the output: in A = l × w, doubling l doubles A; in P = 2(l + w), each extra 1 in l adds 2 to P. A spreadsheet does the repeated working, so the pattern shows at a glance. When two inputs change together, their effects combine: double both sides of a rectangle and its area is four times as big.

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 letter stands for one variable, and the formula says what it means: h is a height in one formula and a number of hours in another. Write the signs in full when substituting: 2(7 + 3) = 2 × 10 = 20. Work brackets first, then multiplying and dividing, then adding and subtracting. In a spreadsheet, * means multiply, / means divide, and a cell name such as A2 stands for the number in that cell.

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

Day 1 reads a formula and substitutes every input; Day 2 changes one input at a time and describes the effect; Day 3 adds brackets; Day 4 hands the repeated working to a spreadsheet and changes two inputs together; and Day 5 takes formulas beyond shapes, to costs, trips and fuel. It builds on substituting into formulas (AC9M7A01) and on writing expressions with brackets (AC9M7A02), and it leads into modelling with linear relations in Year 8 (AC9M8A03).

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 8 change cards and 8 effect cards (Day 2).
• Cut-out sheet 2, one per pair: 8 formula cards and 8 spreadsheet entry cards (Day 4).
• Devices with a spreadsheet, one per pair, for Day 4 if you have them; calculators help on Days 4 and 5.
• Nothing to buy: every task also works by hand.

Words for the word wall

formula, variable, input, output, substitute, unit, bracket, order of operations, hold fixed, vary, systematic, effect, doubles, halves, goes up by, spreadsheet, cell, entry, fill down, scale factor

Dear families

This week in maths, Year 7 is working with formulas that link several quantities, such as area, perimeter, distance and cost, and finding out what happens to the answer when one quantity changes.

Try this at home

• Pizzas cost $14 each plus $6 delivery. Use a phone calculator to find the cost of 1 to 5 pizzas. What does each extra pizza add?
• Look at a recipe for 4 people. Which amounts change if you cook for 8, and how?
• On a trip, ask: if we drove at the same speed for twice as long, how far would we go? (Twice as far.)
• Measure a rug or a table. What would its area be if both sides were twice as long? (Four times as big.)

Formulas this week

Fill one row a day with a formula from class, what changed, and what happened to the answer.

Printed from the free seegongsik Manipulating formulas with several variables teaching pack · seegongsik.com/au/y7/algebra/AC9M7A06/pack

Formulas with several variables

A formula can link three or more quantities. Students name every variable and its unit, substitute a value for each input, work in the right order, and check that the answer and its unit make sense.

We are learning to

• say what each letter in a formula stands for, with its unit,
• substitute a value for every input and work out the output,
• check the units before substituting.

Success criteria

• I can name the inputs and the output of a formula.
• I can work out a formula with three inputs, showing one line of working.

You need

The worksheet and the unit open on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the formula together; start the second with Part A.

Watch for these ideas

• Leaving out an input. Every letter on the right needs a value.
• Mixing units: with a speed in km/h, 30 minutes must go in as 0.5 h.
• Working left to right through a bracket: 2(6 + 5) is 2 × 11.

Answers

• Part A: P = 2(l + w): P perimeter (cm), l length (cm), w width (cm), 3 variables; D = s × t: D distance (km), s speed (km/h), t time (h), 3 variables; C = 20 + 15h: C cost ($), h number of hours, 2 variables; V = l × w × h: V volume (cm³), l length, w width, h height (cm), 4 variables.
• Part B: 2(9 + 4) = 2 × 13 = 26 cm; 70 × 3 = 210 km; 20 + 15 × 4 = 20 + 60 = $80; 5 × 4 × 3 = 60 cm³; 10 × 6 ÷ 2 = 60 ÷ 2 = 30 cm²; 4 × 9 + 2 × 1 = 36 + 2 = 38 points.
• Part C: Sam doubled only the 6; the bracket comes first, so P = 2 × 11 = 22, not 17. Ari used minutes with a speed per hour; 30 minutes is 0.5 h, so D = 90 × 0.5 = 45 km.
• Part D: C = 14p + 3d + 6; 14 × 3 + 3 × 4 + 6 = $60.

Formulas with several variables

A. Read the formula

Write what each letter stands for, with its unit, and how many variables the formula has.

B. Every input gets a value

C. Find the slip

1. Sam used P = 2(l + w) with l = 6 and w = 5, and wrote 2 × 6 + 5 = 17. What went wrong? What is P?
2. Ari used D = s × t with s = 90 km/h and t = 30 minutes, and wrote D = 90 × 30 = 2700 km. What went wrong? How far is the trip?

D. Stretch

An order has p pizzas at $14 each and d drinks at $3 each, plus $6 delivery. Write a formula for the cost C. Use it for 3 pizzas and 4 drinks.

One variable at a time

To see what one input does, hold the others fixed and step that input through its values. Students fill tables, describe each effect with numbers, and find that doubling an input does not always double the output.

We are learning to

• change one variable at a time and hold the others fixed,
• record the results in a table,
• describe the effect in words, with numbers.

Success criteria

• I can say which variable I held and which I changed.
• I can describe an effect with numbers: “each extra 1 adds 2” or “it doubles”.

You need

Change and effect cards from cut-out sheet 1 (one set per pair) and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the cards; start the second with Part A.

Watch for these ideas

• Changing two inputs at once, so neither effect can be seen.
• Expecting every doubling to double the output. In P = 2(l + w), doubling l does not double P.
• Describing without numbers. Say by how much, or by what factor.

Answers

• Part A: P = 10, 12, 14, 16, 18; each extra 1 in l adds 2; doubling l from 2 to 4 takes P from 12 to 16, not double, because the 2 × 4 part stays.
• Part B: D = 90, 135, 180, 225, 270; each extra 15 km/h adds 45 km; doubling s doubles D (30 to 60: 90 to 180).
• Part C: 26, 30, 34, 38; 32, 34, 36, 38; a win adds 4 and a draw adds 2, so a win adds 2 more.
• Part D: P goes up by 4; V is multiplied by 8 (6 to 48).
• Cards: 1 C, 2 F, 3 H, 4 A, 5 G, 6 D, 7 B, 8 E.

One variable at a time

A. Hold the width

In P = 2(l + w), keep w = 4 and change only l.

B. Hold the time

In D = s × t, keep t = 3 h and change only the speed s.

C. A win or a draw?

AFL premiership points: T = 4w + 2d, for w wins and d draws.

D. Stretch

1. In P = 2(l + w), l and w both go up by 1. What happens to P? Test two rectangles.
2. In V = l × w × h, l, w and h all double. What happens to V? Test a box 1 by 2 by 3.

Brackets in formulas

A bracket in a formula sets the order: work inside it first. Students work formulas with brackets, find slips in working, and see how a change to one input inside a bracket ripples through to the output.

We are learning to

• work the bracket first, then multiply or divide,
• find and fix a slip in the order of working,
• describe the effect of changing an input inside a bracket.

Success criteria

• I can work out a formula with a bracket, showing the bracket first.
• I can explain why 4(12 + 5) is not 4 × 12 + 5.

You need

The worksheet and the unit open on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the slips; start the second with Part A.

Watch for these ideas

• Working left to right: 60 + 90 ÷ 2 halves only the 90.
• Multiplying only the first number in the bracket: 4(12 + 5) is 4 × 17.
• Forgetting what multiplies the bracket: in C = n(t + f), $1 more on t adds $n.

Answers

• Part A: 2(11 + 4) = 2 × 15 = 30; (68 + 84) ÷ 2 = 152 ÷ 2 = 76; 6(15 + 9) = 6 × 24 = 144; 12 × 7 ÷ 2 = 84 ÷ 2 = 42.
• Part B: Jo halved only the 90, getting 105; M = (60 + 90) ÷ 2 = 75. Ash multiplied only the 12, getting 53; C = 4 × 17 = 68.
• Part C: $24, $48, $72, $96, $120: each extra person adds $24, the cost of one ticket and food; $76, $80, $84, $88: each extra $1 on the ticket adds $4, one for each person.
• Part D: M goes up by 5; a up by 6 adds 3, both up by 4 adds 4, so both up raises M more.

Brackets in formulas

A. Bracket first

1. P = 2(l + w) (perimeter)   l = 11, w = 4
2. M = (a + b) ÷ 2 (mean of two marks)   a = 68, b = 84
3. C = n(t + f) (cost for a group)   n = 6, t = 15, f = 9
4. A = b × h ÷ 2 (area of a triangle)   b = 12, h = 7

B. Spot the slip

1. For M = (a + b) ÷ 2 with a = 60 and b = 90, Jo wrote 60 + 90 ÷ 2 = 105. What went wrong? What is M?
2. For C = n(t + f) with n = 4, t = 12 and f = 5, Ash wrote 4 × 12 + 5 = 53. What went wrong? What is C?

C. A change inside the bracket

A group of n friends goes to the movies: C = n(t + f), with a ticket of $15 and food of $9 each. Keep f = 9 all the way.

D. Stretch

1. In M = (a + b) ÷ 2, a goes up by 10 and b stays the same. By how much does M go up?
2. Which raises M more: a going up by 6 on its own, or a and b both going up by 4? Show your working.

A spreadsheet does the work

A spreadsheet works a formula for a whole column at once. Students turn formulas into entries, fill a column, change a fixed value and watch every result respond, then scale both sides of a rectangle.

We are learning to

• write a formula as a spreadsheet entry, with * and / and brackets,
• use a filled column to describe the effect of one variable,
• describe what happens when two variables change together.

Success criteria

• I can write a spreadsheet entry for a formula with two inputs.
• I can explain why doubling both sides makes the area four times as big.

You need

Spreadsheet cards from cut-out sheet 2 (one set per pair), the worksheet, and devices with a spreadsheet (one per pair) if you have them; without devices, students fill the printed sheet by hand or with a calculator.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the cards; start the second with Part A.

Watch for these ideas

• Typing x for multiply. A spreadsheet needs *.
• Leaving out a bracket: =A2+B2/2 halves only B2.
• Doubling the area when both sides double. Each side doubles, so the area is 2 × 2 = 4 times as big.

Answers

• Part A: 7, 14, 21, 28, 35, 42; each extra 1 of length adds 7.
• Part B: 14, 28, 42, 56, 70, 84; every area doubles.
• Part C: lengths 5, 10, 15, 20; widths 2, 4, 6, 8; areas 10, 40, 90, 160; multiplied by 4, 9, 16, the square of k.
• Part D: double the length, 200 m³; double the width and halve the depth, 100 m³; add 1 m to the depth, 150 m³; double the length gives the biggest pool. =A2*B2*C2.
• Cards: 1 G, 2 E, 3 B, 4 H, 5 D, 6 A, 7 F, 8 C.

A spreadsheet does the work

A. Fill a column

Rectangles 7 cm wide. Type the sheet below, then copy the entry in C2 down column C. No device? Work each cell by hand.

B. Change the fixed value

Change every width to 14. Write the new areas. What happened to every area?

C. Scale both sides

A 5 by 2 rectangle is scaled by k. B2 holds =5*A2 and C2 holds =2*A2; copy all three entries down.

D. Stretch

1. A pool is 10 m long, 5 m wide and 2 m deep, so V = l × w × d = 100 m³. Which change gives the biggest pool: double the length, or double the width and halve the depth, or add 1 m to the depth?
2. Write a spreadsheet entry for V with l in A2, w in B2 and d in C2.   ____________________

Formulas beyond geometry

Formulas for costs, trips and fuel work just like the formulas for shapes. Students substitute, compare the effects of different changes, and work a formula backwards to find a time, then play the quiz.

We are learning to

• use formulas for costs, speed and fuel,
• compare the effects of two different changes,
• use a formula backwards to find an input.

Success criteria

• I can say which of two changes has the bigger effect, with numbers.
• I can find a time from a distance and a speed.

You need

The worksheet, a calculator if you like, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 4; start the second with problem 5.

Watch for these ideas

• Reading 0.5w as 0.5 plus w. It means 0.5 × w, so 6 minutes cost $3.
• Working backwards by guess: t = D ÷ s, not s ÷ D. Check that the time makes sense.
• Comparing changes by their size alone. Compare what each change costs, in dollars.

Answers with working

1. $30, $32, $34, $36, $38; each extra gigabyte adds $2.
2. 4 + 2 × 8 + 0.5 × 6 = 4 + 16 + 3 = $23.
3. $14, $24, $34; each 5 km adds $10.
4. 2 km cost $4 and 6 minutes cost $3, so the kilometres cost $1 more.
5. 24 km, 48 km, 72 km, 96 km; when t doubles, D doubles.
6. 180 ÷ 60 = 3 h; 180 ÷ 45 = 4 h.
7. 250 × 8 ÷ 100 = 20 L; 250 × 6 ÷ 100 = 15 L; 5 L saved.
8. any formula with three variables, its letters named, and a test with numbers showing the effect of doubling one input.

Formulas beyond geometry

Problems (7 and 8 are the stretch)

1. A phone plan costs C = 30 + 2g dollars a month, where g is the number of extra gigabytes. Fill the table. What does each extra gigabyte add?

   g	0	1	2	3	4
   C ($)

2. A taxi fare is F = 4 + 2d + 0.5w dollars, for d kilometres and w minutes of waiting. Find the fare for d = 8 and w = 6.
3. For the taxi, keep w = 0 and work out F for d = 5, 10, 15. What does each 5 km add?

   d (km)	5	10	15
   F ($)

4. Which adds more to a taxi fare: 2 more kilometres, or 6 more minutes of waiting? By how much?
5. A cyclist rides at 24 km/h. Use D = s × t for each time. What happens to D when t doubles?

   t (h)	1	2	3	4
   D (km)

6. Two riders each cover 180 km, one at 60 km/h and one at 45 km/h. Use t = D ÷ s to find each time.
7. Stretch. Fuel used is F = d × r ÷ 100 litres, for d kilometres in a car that uses r litres per 100 km. Find F for 250 km at r = 8. How much fuel is saved over the same trip at r = 6?
8. Stretch. Make up a formula with three variables for something at school or at home. Say what each letter stands for, and what happens to the result when one input doubles.

Change and effect cards

Cut out the 16 cards. Day 2: match each change card to the effect it has on the formula’s result. Try numbers to check.

Changes

Effects on the result

Spreadsheet cards

Cut out the 16 cards. Day 4: x is in cell A2 and y is in cell B2. Match each formula to the spreadsheet entry that works it out. In a spreadsheet, * means multiply and / means divide.

Formulas

Spreadsheet entries

What we know: Formulas with several variables

Work on your own. Show one line of working for each answer.

1. C = n × d × p gives the cost of a camp for n students and d days at $p per student per day. Find C for n = 25, d = 3 and p = 40. ____________
2. A ferry travels at 36 km/h for 40 minutes. Use D = s × t to find how far it goes. ____________
3. C = n(p + 4) gives the cost of n tickets at $p each, with a $4 booking fee on each ticket. Find C for n = 3 and p = 16. ____________
4. Write a spreadsheet entry for C = n(p + 4), with n in cell A2 and p in cell B2. ____________
5. In A = l × w, w is held at 6 and l steps through 2, 3, 4, 5. Fill the table and describe the effect.

   l	2	3	4	5
   A
6. In the camp formula, d doubles and n and p stay the same. What happens to C? ____________
7. Both the length and the width of a rectangle are halved. What happens to its area? Explain. ________________________
8. Use D = s × t to find how long it takes to ride 45 km at 15 km/h. ____________

Answers and marking guide

Answers with working

1. 25 × 3 × 40 = $3000.
2. 40 minutes is 40/60 of an hour, so D = 36 × 40 ÷ 60 = 24 km.
3. 3(16 + 4) = 3 × 20 = $60.
4. =A2*(B2+4); without the bracket, =A2*B2+4 adds the fee only once, not on each ticket.
5. 12, 18, 24, 30; each extra 1 in l adds 6 to A.
6. C doubles: $3000 becomes $6000 for the camp in question 1.
7. The area is a quarter of what it was: each side is half, and half of a half is a quarter.
8. t = D ÷ s = 45 ÷ 15 = 3 h.

A quick three-level guide

Eight questions, four ideas. A student at standard substitutes every input with matching units, works brackets first, and describes the effect of changing one input with numbers.

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: several variables, one at a time, brackets, spreadsheets, and formulas beyond shapes.