Solving one-variable linear equations: 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 equation and solution cards, think-of-a-number 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: equation and solution cards (Day 3) and think-of-a-number cards (Day 5).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

An equation says that two sides have the same value, like a balance scale sitting level. Solving it means finding the value of the letter that makes it true. Whatever you do to one side, you do to the other, so the scale stays level. To free x in 2x + 3 = 11, undo the operations in reverse order: subtract 3 from both sides (2x = 8), then divide both sides by 2 (x = 4). Substituting back, 2 × 4 + 3 = 11, proves the answer. Every equation this week has a natural-number solution.

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 set out a solution

Write the equation, then one line per step, keeping the equals signs under each other. Say what was done to both sides beside each line if it helps. Finish with a check line that substitutes the answer into the original equation, and for a word problem, a sentence that answers the question. 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.

How the week builds

Day 1 reads an equation as a balance; Day 2 solves one-step equations with inverse operations; Day 3 undoes two steps in reverse order; Day 4 makes checking a habit and adds guess, check and improve; and Day 5 writes equations from words. It builds on formulas and expressions (AC9M7A01 and AC9M7A02), and it prepares the equations with fractions, negatives and letters on both sides of Year 8.

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 equation cards and 12 solution cards (Day 3).
• Cut-out sheet 2, one per pair: 8 think-of-a-number cards (Day 5).
• Optional: a real balance or a coat hanger with two cups, for Day 1.

Words for the word wall

equation, equals, balance, both sides, solve, solution, unknown, inverse operation, undo, isolate, substitute, check, verify, true, false, natural number, trial, improve, expression

Dear families

This week in maths, Year 7 is solving equations such as 2x + 3 = 11: finding the number that makes both sides equal, by undoing the operations one step at a time, and then checking the answer by putting it back in.

Try this at home

• Play think of a number: “I think of a number, double it and add 5, and get 17.” What was it? (6.)
• Ask your child to explain why you subtract 5 before you halve.
• Ask: is x = 3 a solution of 4x − 2 = 10? How can you tell without solving it? (Put 3 in: 12 − 2 = 10, yes.)
• Look for a real equation: a bill, a fare or a recipe where one amount is missing.

Equations this week

Fill one row a day with an equation, its solution and the check.

Printed from the free seegongsik Solving one-variable linear equations teaching pack · seegongsik.com/au/y7/algebra/AC9M7A03/pack

What an equation says

An equation says two sides have the same value, like a balance scale sitting level. Students decide whether equations are true, tell equations from expressions, and solve simple equations in their heads, checking each by substituting.

We are learning to

• explain what the equals sign means in an equation,
• decide whether a value makes an equation true,
• solve simple equations by thinking, and check them.

Success criteria

• I can say whether x = 6 makes 4x − 5 = 19 true, and why.
• I can solve x + 8 = 15 in my head and check it.

You need

The worksheet. A real balance, or a coat hanger with two cups, makes a good prop.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after true or false; start the second with Part A.

Watch for these ideas

• Reading = as “the answer is”, so 3 × 4 = 2 × 6 seems wrong. Both sides are 12.
• Trying to solve an expression. 3x + 2 has no equals sign.
• Guessing without checking. Substitute to make sure.

Answers

• Part A: 1 true, 2 true, 3 false, 4 true, 5 false, 6 true.
• Part B: 5n + 2 expression; 5n + 2 = 17 equation; y = 4 equation; 3(a + 1) expression; 12 = 2k equation; m − 6 expression.
• Part C: n = 11, k = 8, m = 7, y = 17, p = 10, t = 45.
• Part D: for example x + 4 = 10, 2x = 12 or 3x − 8 = 10. No: x + 3 is always 2 less than x + 5, so they are never equal.

What an equation says

A. True or false?

B. Equation or expression?

C. Solve by thinking, then check

D. Stretch

Write two different equations that both have the solution x = 6. ________________________________

Is there a value of x that makes x + 3 = x + 5 true? Explain. ________________________________

One-step equations

Every operation has an inverse: adding and subtracting undo each other, and so do multiplying and dividing. To solve x + 7 = 12, subtract 7 from both sides; to solve 3x = 18, divide both sides by 3. The check is substituting back.

We are learning to

• choose the inverse operation that undoes the one on x,
• do the same to both sides and write each step,
• check a solution by substituting it back.

Success criteria

• I can solve x − 9 = 14 and 6x = 54, showing the step for each.
• I can check my answer by substituting.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Doing the same operation instead of its inverse: x + 15 = 32 does not give x = 47.
• Changing one side only. Both sides change together.
• Stopping at the step without the check.

Answers

• Part A: x + 12 = 30: subtract 12; x − 4 = 11: add 4; 5x = 45: divide by 5; x ÷ 3 = 7: multiply by 3.
• Part B: x = 17, x = 27, x = 12, x = 30, x = 7, x = 13, x = 42, x = 14.
• Part C: x − 14 = 27, x = 41; 5x = 85, x = 17.
• Part D: x = 18; x = 10.

One-step equations

A. What do you do to both sides?

B. Solve, then check

C. Write an equation, then solve it

1. After spending $14, Kai has $27 left. How much did Kai start with?
2. Five equal boxes hold 85 apples altogether. How many apples are in each box?

D. Stretch

Solve 0.5x = 9. ________    Solve x ÷ 4 = 2.5. ________

Two-step equations

When x has been through two operations, undo them in reverse order. In 2x + 3 = 11, x was doubled and then 3 was added, so subtract 3 first and then divide by 2, each time on both sides.

We are learning to

• say which operation to undo first, and why,
• solve two-step equations, one line per step,
• check each solution by substituting.

Success criteria

• I can solve 4x + 5 = 33 in two lines and check it.
• I can explain why I undo the + 5 before the × 4.

You need

The equation and solution cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Undoing in the order the operations were done: dividing 33 by 4 first in 4x + 5 = 33.
• Adding when the equation needs subtracting, as in 2x + 6 = 20 becoming 2x = 26.
• Dividing only part of a side: in 4x = 28, all of 28 is divided by 4.

Answers

• Part A: 3x + 4 = 19: subtract 4, then divide by 3; x ÷ 4 − 2 = 5: add 2, then multiply by 4; 6x − 7 = 29: add 7, then divide by 6.
• Part B: 5x = 35, x = 7; 2x = 16, x = 8; 4x = 44, x = 11; x ÷ 2 = 6, x = 12; 6x = 24, x = 4; x ÷ 3 = 3, x = 9; 7x = 42, x = 6; 8x = 80, x = 10.
• Part C: 6 was added instead of subtracted; 2x = 14, x = 7. Check: 2 × 7 + 6 = 20.
• Part D: x + 2 = 9, so x = 7. 4x = 32, so x = 8.
• Cards: x + 6 = 7 → 1, 2x + 1 = 5 → 2, 4x − 5 = 7 → 3, x ÷ 2 + 1 = 3 → 4, 3x − 8 = 7 → 5, 5x + 2 = 32 → 6, x − 3 = 4 → 7, 2x − 9 = 7 → 8, x ÷ 3 + 5 = 8 → 9, 6x − 15 = 45 → 10, 4x + 6 = 50 → 11, x ÷ 4 − 1 = 2 → 12.

Two-step equations

A. Which step comes first?

B. Solve, one line per step, then check

C. Find the mistake

Ari solved 2x + 6 = 20 like this: 2x = 26, so x = 13. Find the mistake, fix it, and check your answer. ________________________________

D. Stretch

Solve 3(x + 2) = 27. ________    Solve 20 = 4x − 12. ________

Checking and closing in

Substituting a solution back into the original equation proves it. When the steps are not clear, guess, check and improve also works: try a value, see whether the left side is too low or too high, and adjust.

We are learning to

• verify a solution by substituting it into the original equation,
• decide whether a given value is a solution,
• solve an equation by guess, check and improve.

Success criteria

• I can show that x = 9 solves 4x − 7 = 29 and that x = 8 does not.
• I can close in on the solution of 6x + 9 = 81 with a table of tries.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Checking in a later line of working, where a slip has already happened.
• Thinking a wrong try is wasted. Each try says which way to go.
• Stopping at close enough. The solution makes both sides exactly equal.

Answers

• Part A: 1 yes (38), 2 no (11; x = 6), 3 no (29; x = 9), 4 yes (10), 5 yes (31), 6 no (38; x = 9).
• Part B: x = 5 gives 39, too low; the solution is x = 10, giving 74.
• Part C: 3x − 7 = 20, x = 9; 9x + 2 = 74, x = 8.
• Part D: for example 2x + 5 = 29 or x ÷ 3 + 1 = 5. 15x − 11 = 199 has x = 14.

Checking and closing in

A. Is the value a solution? Substitute to decide.

B. Guess, check and improve

Solve 7x + 4 = 74. The first try is done for you.

C. Which value is the solution?

1. 3x − 7 = 20: x = 7, x = 8, x = 9? ________
2. 9x + 2 = 74: x = 8, x = 9, x = 10? ________

D. Stretch

Make up an equation with two operations whose solution is 12. Swap with a partner and check each other’s. ________________

Solve 15x − 11 = 199 by guess, check and improve. ________________

Equations from words

A word problem becomes an equation: choose a letter for the unknown, write what the words say, solve, check, and answer in a sentence. Students warm up with think-of-a-number cards.

We are learning to

• write an equation for a word problem, saying what the letter stands for,
• solve it and check it in the story,
• answer the question in a sentence, with units.

Success criteria

• I can write “double a number and add 11 to get 35” as 2n + 11 = 35, and solve it.
• I can give the answer as a sentence that fits the story.

You need

The think-of-a-number cards from cut-out sheet 2 (one set per pair), the worksheet, the self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Writing the operations in the wrong order: double then add 11 is 2n + 11, not 2(n + 11).
• Ending with x = 12 and no sentence. Say what 12 means in the story.
• Forgetting that a perimeter counts every side.

Answers with working

1. 4n + 7 = 43; 4n = 36, n = 9; the number is 9.
2. 2k + 5 = 29; 2k = 24, k = 12; the trip was 12 km.
3. 4s = 52; s = 13; each side is 13 cm.
4. b ÷ 3 = 24; b = 72; the bill was $72.
5. 8w = 64; w = 8; it is 8 cm wide.
6. 3n + 8 = 92; 3n = 84, n = 28; there are 28 students.
7. n + (n + 1) + (n + 2) = 75, so 3n + 3 = 75, 3n = 72, n = 24: 24, 25 and 26.
8. 40 + 12w = 20w, so 40 = 8w and w = 5: after 5 weeks both cost $100.

Cards: 1. n + 9 = 25, n = 16; 2. 3n = 51, n = 17; 3. 2n − 7 = 23, n = 15; 4. n ÷ 4 + 6 = 11, n = 20; 5. 5n + 12 = 72, n = 12; 6. 3(n − 4) = 27, n = 13; 7. n ÷ 2 − 3 = 8, n = 22; 8. 7n − 5 = 58, n = 9.

Equations from words

For each problem: choose a letter, write an equation, solve it, check it, and answer in a sentence.

Problems (7 and 8 are the stretch)

1. I think of a number, multiply it by 4 and add 7. The answer is 43. What is my number?
2. A taxi charges $5 plus $2 a kilometre. A trip cost $29. How long was the trip?
3. The perimeter of a square is 52 cm. How long is each side?
4. Three friends share a bill equally, and each pays $24. What was the bill?
5. A rectangle is 3 times as long as it is wide, and its perimeter is 64 cm. How wide is it?
6. Each student in a class gets 3 pencils from a box of 92, and 8 pencils are left over. How many students are there?
7. Stretch. Three consecutive whole numbers add up to 75. What are they?
8. Stretch. One gym charges $40 to join plus $12 a week. Another charges $20 a week with no joining fee. After how many weeks do the two cost the same?

Equation and solution cards

Cut out the 24 cards. Day 3: solve each equation and pair it with its solution card. Every solution card is used once.

Equations

Solutions

Think-of-a-number cards

Cut out the 8 cards. Day 5: take turns to read a card aloud. Your partner writes an equation for it, solves it and checks it, then you swap.

What we know: Solving equations

Work on your own, without a calculator. Show each step and check each answer.

1. Is x = 6 a solution of 3x − 4 = 14? Show how you know. ________________
2. Solve x + 9 = 23. ________________
3. Solve 7x = 56. ________________
4. Solve x ÷ 4 = 9. ________________
5. Solve 3x + 5 = 26. ________________
6. Solve 5x − 8 = 37. ________________
7. I think of a number, double it and add 13. The answer is 45. Write an equation and find the number. ________________
8. A taxi charges $6 plus $3 a kilometre. A trip cost $36. Write an equation, find the distance, and check. ________________

Answers and marking guide

Answers with working

1. 3 × 6 − 4 = 14, so yes.
2. x = 14, x = 14; check 23 = 23.
3. x = 8; check 56 = 56.
4. x = 36; check 9 = 9.
5. 3x = 21, x = 7; check 26 = 26.
6. 5x = 45, x = 9; check 37 = 37.
7. 2n + 13 = 45; 2n = 32, n = 16; the number is 16.
8. 3k + 6 = 36; 3k = 30, k = 10; the trip was 10 km; check 6 + 3 × 10 = 36.

A quick three-level guide

Eight questions, four ideas. A student at standard solves most equations with one line per step and checks each answer in the original equation.

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: equations, one step, two steps, checking, and equations from words.