Linear expressions: a week of ready-to-teach maths
Five lessons for Year 8 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 algebra tiles to cut out, equivalent-expression 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, and cut them before Day 2: the algebra tiles are for Days 2 to 4, the match cards for 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 linear expression such as 3x + 5 is a recipe for a number: multiply the unknown by 3, then add 5. Its parts have names: 3 is the coefficient, x the variable, 5 the constant. Expressions can be tidied without changing their value. Terms can swap places and regroup (the commutative and associative properties), so like terms collect: 2x + 5 + 3x + 1 = 5x + 6. A bracket multiplies every term inside it (the distributive property): 3(x + 4) = 3x + 12, which is the area of a 3 by (x + 4) rectangle. Factorising runs that backwards: 6x + 9 = 3(2x + 3). Adding 0 or multiplying by 1 changes nothing (identity), and x + (−x) = 0 (inverse). Every step can be checked by substituting a number.
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 expressions
We write the coefficient before the letter (3x, not x3), leave out a coefficient of 1 (x, not 1x), and use a true minus sign. 3x means 3 × x. A negative multiplier sits outside the bracket, as in −2(x + 6). The tiles model each step: an x tile is a long strip, a unit tile a small square, and an outlined tile with a minus is negative; a tile and its negative make a zero pair.
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 | Anatomy of an expression | Name terms, coefficients and constants, write expressions for word descriptions, and substitute values | Anatomy of a linear expression |
| 2 | Collecting like terms | Tell like from unlike terms, rearrange and collect with negatives, and write perimeters as expressions | Collecting like terms; algebra tiles |
| 3 | Expanding brackets | Expand k(x + c) with an area model and with tiles, including negative multipliers, then expand and simplify | Expanding with an area model; match cards |
| 4 | Factorising | Take out the greatest common factor, check by expanding, and decide whether an expression is fully factorised | Factorising by a common factor |
| 5 | The five properties | Name the property behind each step, justify a simplification line by line, rearrange for mental arithmetic, and play the quiz | The five properties; self-check quiz |
How the week builds
Day 1 reads and writes expressions; Day 2 tidies them by collecting like terms; Day 3 opens brackets; Day 4 closes them again by factorising; and Day 5 names the properties that made every step legal. It builds on Year 7, where students wrote and evaluated simple algebraic expressions. It prepares solving linear equations and graphing linear relations later in Year 8, and expanding pairs of brackets in Year 9.
Materials for the week (one trip)
- From the classroom: scissors, pencils, an envelope per pair for the tiles, this pack printed.
- Cut-out sheet 1, one per pair: algebra tiles, 12 x tiles and 20 unit tiles, positive and negative (Days 2 to 4).
- Cut-out sheet 2, one per pair: 24 equivalent-expression match cards (Day 3).
- No calculators needed. No other equipment to buy.
Words for the word wall
expression, term, like terms, coefficient, variable, constant, substitute, evaluate, simplify, collect, expand, bracket, factorise, common factor, highest common factor, equivalent, associative, commutative, identity, distributive, inverse
Dear families
This week in maths, Year 8 is working with algebraic expressions such as 3x + 5. Students learn to tidy expressions by collecting like terms (2x + 3x = 5x), to multiply out brackets (3(x + 4) = 3x + 12) and to put brackets back in by factorising (6x + 9 = 3(2x + 3)), and to name the rules that make each step correct.
Try this at home
- Play “think of a number”: add 5, multiply by 3, take away 15, divide by 3. Everyone ends with their starting number. Why? (3(n + 5) − 15 = 3n.)
- Ask: simplify a + a + a + b + b. (3a + 2b.)
- Ask: what is 4(x + 5) without brackets? (4x + 20.)
- Ask: is 2x + 3 the same as 5x? Try x = 2. (No: 7 against 10.)
My expressions this week
Fill one row a day. Write an expression, a simpler equal expression, and check both with x = 2.
| Day | My expression | Simpler and equal | Check with x = 2 |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Linear expressions teaching pack · seegongsik.com/au/y8/algebra/AC9M8A01/pack
Anatomy of a linear expression
An expression is a recipe for a number. Students name its parts, write expressions for word descriptions, and substitute values, including negatives, to see the recipe at work.
We are learning to
- name the terms, the coefficient of x and the constant in a linear expression,
- write a linear expression for a word description or a simple situation,
- substitute positive and negative values into an expression and evaluate it.
Success criteria
- I can say that in 7 − 2x the coefficient of x is −2 and the constant is 7.
- I can work out 5x − 2 when x = −1 and get −7.
You need
The worksheet, one per student. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Name the parts Show the board (see the box). Then evaluate 5x − 2 at x = 3 and at x = −1: 13 and −7. Ask: “The recipe stays the same. What changes when x changes?” |
| 15 min | Words into algebra Pairs write expressions for three phrases on the board, then swap and decode each other’s. Discuss “10 less than n” (n − 10, not 10 − n). Ask: ““Less than” turns the order around. Why?” |
| 35 min | Read, write, substitute Parts A to C; early finishers do Part D. Share Part C with the substitution written in brackets. Ask: “Why write 3 × (−3), with brackets, when x = −3?” |
Two shorter sessions instead? Split after the words; start the second with Part A.
Watch for these ideas
- Reading the coefficient without its sign: in 7 − 2x it is −2.
- Writing 3x as 3 + x, or reading 3x with x = 4 as 34.
- Forgetting brackets when substituting a negative: 3 × (−3) = −9.
Answers
- Part A: 4x + 9: 2 terms, coefficient 4, constant 9; 7 − 2x: 2 terms, coefficient −2, constant 7; x + 1: 2 terms, coefficient 1, constant 1; −5x: 1 term, coefficient −5, constant 0; 12 + 3x − 4: 3 terms, coefficient 3, constant 8; x/2 + 6: 2 terms, coefficient 1/2, constant 6.
- Part B: 3n + 5; 2n − 7; 4p + 3; n − 10; 4s.
- Part C: 3x + 5: x = 2 gives 11, x = −3 gives −4; −2x + 7: x = 2 gives 3, x = −3 gives 13; 4x − 1: x = 2 gives 7, x = −3 gives −13.
- Part D: 2x + 2 and 3x are equal only at x = 2 (both 6); at x = 0 they give 2 and 0, so they are not equivalent.
Anatomy of a linear expression
A. Name the parts
| Expression | Number of terms | Coefficient of x | Constant |
|---|---|---|---|
| 4x + 9 | |||
| 7 − 2x | |||
| x + 1 | |||
| −5x | |||
| 12 + 3x − 4 | |||
| x/2 + 6 |
B. Words into algebra
| 1. 5 more than 3 times a number | |
| 2. a number doubled, then 7 taken away | |
| 3. the cost in dollars of p pies at $4 each and one $3 drink | |
| 4. 10 less than a number | |
| 5. the perimeter of a square with side s |
C. Substitute
| Expression | x = 2 | x = −3 |
|---|---|---|
| 3x + 5 | ||
| −2x + 7 | ||
| 4x − 1 |
D. Stretch: equal once, or always?
Work out 2x + 2 and 3x for x = 0, 1, 2 and 3. Are they equivalent expressions? Explain using your table.
Collecting like terms
Like terms have the same letter part, and only like terms combine. Students model expressions with tiles, rearrange terms (commutative, associative), collect them, cancel zero pairs, and write perimeters as expressions.
We are learning to
- tell like terms from unlike terms,
- rearrange and collect like terms, including negative terms and zero pairs,
- write and simplify an expression for the perimeter of a shape.
Success criteria
- I can simplify 4x + 1 + 2x + 5 to 6x + 6 and show it with tiles.
- I can explain why 4x + 4 cannot be written as 8x.
You need
Algebra tiles (cut-out sheet 1), one envelope per pair; the worksheet.
Lesson flow (about 60 minutes)
| 10 min | Group, then add Show the board (see the box). Pairs lay out 4x + 1 + 2x + 5 in tiles, slide like tiles together and read off 6x + 6. Ask: “We moved tiles around. Why does that not change the total?” |
| 15 min | Zero pairs Lay out 3x − 2 + (−x) + 5 with negative tiles: an x and a −x cancel, as do 1 and −1, leaving 2x + 3. Ask: “Which property says x + (−x) = 0?” |
| 35 min | Collect and apply Parts A to C; early finishers do Part D. Share Part A item 4, where the x terms vanish. Ask: “An expression simplified to just 6. Where did the x go?” |
Two shorter sessions instead? Split after the zero pairs; start the second with Part A.
Watch for these ideas
- Combining unlike terms: 4x + 4 is not 8x.
- Losing the sign with its term when rearranging: −2x moves with its minus.
- Thinking x and x² are like terms.
Answers
- Part A: 3x + 4 + 5x + 2 = 8x + 6; 7x − 3 − 2x + 8 = 5x + 5; 6 − x + 4x − 9 = 3x − 3; 2x + 5 − 2x + 1 = 6; −4x − 1 − 3x − 6 = −7x − 7; x + x + x + 10 = 3x + 10.
- Part B: 3x and 5x: like; −2x and x: like; 4x and 4: unlike; x and y: unlike; 7 and −3: like; 2x and x²: unlike.
- Part C: rectangle: 6x + 6; triangle: 6x + 3; square: 12x − 8.
- Part D: middle row 3x + 1 and 3x + 2; top 6x + 3.
Collecting like terms
A. Collect like terms
| 1. 3x + 4 + 5x + 2 | |
| 2. 7x − 3 − 2x + 8 | |
| 3. 6 − x + 4x − 9 | |
| 4. 2x + 5 − 2x + 1 | |
| 5. −4x − 1 − 3x − 6 | |
| 6. x + x + x + 10 |
B. Like or unlike?
Circle like or unlike for each pair of terms.
C. Perimeters
Write the perimeter as an expression, then simplify it.
| 1. A rectangle with sides 2x + 3 and x | |
| 2. A triangle with sides x + 4, 2x and 3x − 1 | |
| 3. A square with side 3x − 2 |
D. Stretch: an algebra pyramid
Each brick is the sum of the two bricks under it. Fill the pyramid and simplify each brick.
Expanding brackets
A number outside a bracket multiplies every term inside it: the distributive property. Students expand with an area model and with tiles, handle negative multipliers, then expand and simplify.
We are learning to
- expand k(x + c) using an area model and the distributive property,
- expand brackets with a negative number outside, minding every sign,
- expand two brackets and collect like terms into one simplified expression.
Success criteria
- I can draw the area model for 3(x + 4) and write 3x + 12.
- I can expand −2(x − 5) and explain why the last term is +10.
You need
Algebra tiles and the match cards (cut-out sheets 1 and 2), one set per pair; the worksheet.
Lesson flow (about 60 minutes)
| 10 min | Rows of tiles Show the board (see the box). Pairs make 3 rows of (x + 4) with tiles and count: 3x + 12. Ask: “Why is 3(x + 4) not 3x + 4? What did we forget to multiply?” |
| 20 min | Match the pairs Pairs match the 12 tangled expressions to their 12 tidy forms, expanding or collecting on scrap paper; check any match by substituting x = 2. Ask: “Two cards look different but simplify the same way. How can you be sure?” |
| 30 min | Expand and simplify Parts A to C; early finishers do Part D. Share Part A items with a negative outside. Ask: “Minus times minus: what happens to the last term in −(3x − 7)?” |
Two shorter sessions instead? Split after the match cards; start the second with Part A.
Watch for these ideas
- Multiplying only the first term: 4(x + 3) is 4x + 12, not 4x + 3.
- Losing a sign: −2(x + 6) = −2x − 12.
- Forgetting the hidden −1 in −(x + 4).
Answers
- Part A: 4(x + 3) = 4x + 12; 5(2x − 1) = 10x − 5; −2(x + 6) = −2x − 12; 3(4x + 5) = 12x + 15; −(3x − 7) = −3x + 7; 6(−x + 2) = −6x + 12.
- Part B: 3(x + 5): areas 3x and 15, so 3x + 15; 7(x + 2): areas 7x and 14, so 7x + 14.
- Part C: 2(x + 3) + 4(x − 1) = 6x + 2; 3(2x + 1) − (x + 4) = 5x − 1; 5(x − 2) + 2(−2x + 3) = x − 4.
- Part D: n, n + 3, 2(n + 3) = 2n + 6, 2n + 6 − 6 = 2n, 2n ÷ 2 = n: you always get back your number.
Expanding brackets
A. Expand
B. Area models
Write the area of each part inside it, then write the expansion.
C. Expand and simplify
| 2(x + 3) + 4(x − 1) | |
| 3(2x + 1) − (x + 4) | |
| 5(x − 2) + 2(−2x + 3) |
D. Stretch: think of a number
Think of a number n. Add 3, double the answer, take away 6, then halve it. Write an expression for each step and expand to explain the result.
Factorising by a common factor
Factorising is expanding in reverse: find what every term shares, take it outside a bracket, and check by expanding. Students find the highest common factor and decide whether an expression is fully factorised.
We are learning to
- find the highest common factor of the terms of a linear expression,
- factorise by taking out the highest common factor, including a negative one,
- check a factorisation by expanding, and tell whether it is complete.
Success criteria
- I can factorise 10x + 15 as 5(2x + 3) and check by expanding.
- I can explain why 2(6x + 9) is not fully factorised.
You need
Algebra tiles (cut-out sheet 1), one set per pair; the worksheet.
Lesson flow (about 60 minutes)
| 10 min | What do the terms share? Show the board (see the box). Then 10x + 15: the factors of 10 and 15 share 5, so 10x + 15 = 5(2x + 3). Expand to check. Ask: “The common factor 1 always works. Why is it not the answer we want?” |
| 15 min | Tiles into rectangles Pairs make 6x + 9 with tiles and arrange them into equal rows: 3 rows of (2x + 3). The number of rows is the factor outside. Ask: “Could 6x + 9 make 2 equal rows? Why not?” |
| 35 min | Factorise and check Parts A to C; early finishers do Part D. Share Part B: the factor left inside still had a common factor. Ask: “How can you tell, by looking inside the bracket, that you have finished?” |
Two shorter sessions instead? Split after the tiles; start the second with Part A.
Watch for these ideas
- Stopping at a common factor that is not the highest: 2(6x + 9).
- Dividing only one term: 8x + 12 = 4(2x + 12) is wrong.
- Getting the signs inside wrong when the factor is negative.
Answers
- Part A: 8x + 12 = 4(2x + 3); 15x − 10 = 5(3x − 2); 10x + 25 = 5(2x + 5); 14x + 21 = 7(2x + 3); −4x − 20 = −4(x + 5); 9x − 3 = 3(3x − 1).
- Part B: 2(6x + 9): not fully; 6(2x + 3); 5(x + 3): fully factorised; 4(2x + 4): not fully; 8(x + 2); 3(4x − 1): fully factorised; 10(x + 2): fully factorised; 2(10x − 15): not fully; 10(2x − 3).
- Part C: check each Part A answer by expanding back to the starting expression.
- Part D: 1 by 12x + 18, 2 by 6x + 9, 3 by 4x + 6, 6 by 2x + 3; 6(2x + 3) is the full factorisation, with the largest whole-number side.
Factorising by a common factor
A. Factorise fully
B. Fully factorised?
Tick the ones that are fully factorised. Finish the others.
| 2(6x + 9) | |
| 5(x + 3) | |
| 4(2x + 4) | |
| 3(4x − 1) | |
| 10(x + 2) | |
| 2(10x − 15) |
C. Check by expanding
Choose three of your Part A answers. Expand each one and show that you get back the starting expression.
D. Stretch: rectangles with area 12x + 18
A rectangle has area 12x + 18 square units. One side is a whole number. List every possible pair of side lengths, and say which one is the full factorisation.
The five properties
Every step of the week was allowed by one of five properties. Students name the property behind each statement, justify a simplification line by line, use the properties for mental arithmetic, and play the quiz.
We are learning to
- name the associative, commutative, identity, distributive and inverse properties from examples,
- justify each step of a simplification with the property it uses,
- rearrange a calculation with the properties to make it easy to do mentally.
Success criteria
- I can say which property turns 2(x + 5) into 2x + 10, and which turns x + 3 into 3 + x.
- I can work out 4 × 17 × 25 in my head and name the property I used.
You need
The worksheet, one per student; the self-check quiz on the unit page for the game.
Lesson flow (about 60 minutes)
| 10 min | Five names, five pictures Show the board (see the box). Students write each property’s name with their own example as it appears. Ask: “Which property did we use most this week without naming it?” |
| 30 min | Name, justify, rearrange Parts A to C; early finishers do Part D. Share Part B line by line, a student naming each property. Ask: “Subtraction is not commutative: 7 − 2 ≠ 2 − 7. How do we still rearrange 7 − 2x?” |
| 20 min | Self-check quiz as a class game Teams of four hold up one, two or three fingers for their option, then reveal and read the explanation aloud. A team that was wrong names the property it mixed up. Ask: “Before we reveal it: which property is this question about?” |
Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.
Watch for these ideas
- Mixing up associative (regroup) and commutative (swap order).
- Thinking subtraction and division are commutative.
- Forgetting that identity has two forms: + 0 and × 1.
Answers
- Part A: 1. Commutative; 2. Distributive; 3. Identity; 4. Associative; 5. Inverse; 6. Identity; 7. Commutative.
- Part B: 3(x + 4) + 2x = 3x + 12 + 2x: Distributive; = 3x + 2x + 12: Commutative; = (3x + 2x) + 12: Associative; = 5x + 12: collecting like terms.
- Part C: 25 × 7 × 4 = 700 (25 × 4 = 100 first); 38 + 59 + 62 = 159 (38 + 62 = 100 first); 5 × 13 × 2 = 130 (5 × 2 = 10 first); 6 × 99 = 594 (6 × (100 − 1)).
- Part D: 3(x + 4) − 12 = 3x + 12 − 12 (distributive) = 3x + (12 + (−12)) (associative) = 3x + 0 (inverse) = 3x (identity).
The five properties
Properties: Associative, Commutative, Identity, Distributive, Inverse.
A. Name the property
| 1. x + 7 = 7 + x | |
| 2. 4(x + 2) = 4x + 8 | |
| 3. 3x + 0 = 3x | |
| 4. (x + 2) + 5 = x + (2 + 5) | |
| 5. 6x + (−6x) = 0 | |
| 6. 1 × (x + 3) = x + 3 | |
| 7. 5 × x × 2 = 5 × 2 × x |
B. Justify every step
| 3(x + 4) + 2x = 3x + 12 + 2x | |
| = 3x + 2x + 12 | |
| = (3x + 2x) + 12 | |
| = 5x + 12 |
C. Make it easy
Rearrange each calculation so you can do it in your head. Write the rearranged form and the answer.
| 25 × 7 × 4 | |
| 38 + 59 + 62 | |
| 5 × 13 × 2 | |
| 6 × 99 |
D. Stretch: prove it step by step
Show that 3(x + 4) − 12 = 3x, writing one step per line and naming the property each step uses.
Algebra tiles
For Days 2 to 4, one set per pair (keep them in an envelope)
A long tile is x, a small square is 1. An outlined tile marked with a minus is negative: a tile and its negative make a zero pair. Cut along the edges of each tile.
Teacher note: the largest tile task in the week is 3x − 2 + (−x) + 5 on Day 2 and 3 rows of (2x + 3) on Day 4 (six x tiles, nine unit tiles).
Equivalent-expression match cards
For Day 3, one set per pair: twelve expressions to tidy
Twelve tidy forms
Teacher note: 3(x + 4) = 3x + 12; 2(5x − 1) = 10x − 2; −4(x + 2) = −4x − 8; x + x + x + 2 = 3x + 2; 6(2x + 3) = 12x + 18; 5x + 7 − 2x = 3x + 7; 2(x + 6) − x = x + 12; −(3x − 5) = −3x + 5; 4(3x) + 1 = 12x + 1; 7 + 2x − 7 = 2x; 3(x + 2) + 3(x − 2) = 6x; 10x − 10 − 9x = x − 10.
What we know: Linear expressions
Work on your own, without a calculator. Show your working.
- In −3x + 8, what is the coefficient of x, and what is the constant? ______________________
- Work out 4x − 5 when x = −2. ______________________
- Simplify 6x − 2 − 4x + 9. ______________________
- Expand 5(2x − 3). ______________________
- Expand −3(x − 4). ______________________
- Factorise 18x + 12 fully, then check by expanding. ______________________
- Expand and simplify 4(x + 2) + 3(2x − 1). ______________________
- Which property makes x + 9 = 9 + x true? Which makes 7x + 0 = 7x true? ______________________
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
- Coefficient −3, constant 8.
- 4 × (−2) − 5 = −13.
- (6x − 4x) + (−2 + 9) = 2x + 7.
- 10x − 15.
- −3x + 12: −3 × (−4) = +12.
- 6(3x + 2); check: 6(3x + 2) = 18x + 12.
- 4x + 8 + 6x − 3 = 10x + 5.
- Commutative; identity (adding 0).
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Reading and substituting (Q1, Q2) | drops the sign of a coefficient, or forgets brackets when substituting | names the parts and evaluates with negatives | writes expressions for situations |
| Collecting (Q3, Q7) | combines unlike terms | collects like terms with negatives | explains collecting with the commutative and associative properties |
| Expanding and factorising (Q4, Q5, Q6) | multiplies only the first term, or takes out a small factor | expands with negatives and factorises fully, checking by expanding | explains both as the distributive property |
| Properties (Q8) | cannot name the property used | names the property for each step | justifies a whole simplification |
Eight questions, four ideas. A student at standard answers most questions with working and checks by substituting a number.
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 | Parts and values | Like terms | Expanding | Factorising | Properties |
|---|---|---|---|---|---|
The five columns are the five days: parts and values, like terms, expanding, factorising and the properties.