Expanding and factorising: a week of ready-to-teach maths
Five lessons for Year 9 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, factor match cards, sign sort 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 and 4, the factor match cards for Day 4 and the sign sort cards for Day 5.
- No calculators are needed this week.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
The idea in plain words
Like terms have exactly the same letters with the same exponents, so 3x² and −x² collect but x² and x do not. Expanding a binomial product multiplies every term of one bracket by every term of the other: an area grid shows the four products, and the two middle ones usually collect. Two products are worth knowing by sight: a sum times a difference, (x + 5)(x − 5) = x² − 25, where the middle terms cancel, and a perfect square, (x + 4)² = x² + 8x + 16, where they double. Factorising runs the expansion backwards. For a monic quadratic x² + bx + c it is a search for two numbers that multiply to c and add to b, and the signs of c and b say which pairs to try.
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 these expressions
Answers are written with the highest power first, as in x² + 7x + 12, and the letters in alphabetical order, as in 7xy. A factorised answer lists its brackets in either order, as (x + 2)(x + 3) or (x + 3)(x + 2). In this pack a monic quadratic has x² with no number in front. Every factorisation is checked by expanding it again.
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 | Like terms with powers | Tell x² from x and xy from x, collect like terms, expand single brackets and collect, and write perimeters and areas as expressions | Collecting like terms |
| 2 | Expanding binomial products | Build (x + 3)(x + 2) from algebra tiles, fill area grids, expand products with negatives and with a coefficient in front of x, and find the pattern x² + (p + q)x + pq | Area model of a binomial product |
| 3 | Special products | Expand a sum times a difference and a perfect square, see why the middle terms cancel or double, and use both for mental arithmetic | Difference of squares; Perfect square trap |
| 4 | Factorising as a number search | Search the factor pairs of c for the pair that adds to b, factorise with negatives, check by expanding, and play factor match | Factorising as a number search |
| 5 | Signs, common factors and special forms | Predict the signs from c and b, take out a common factor first, factorise squares and differences of squares, sort quadratics by sign, and play the quiz as a class game | Sign rules from c then b; self-check quiz as a class game |
How the week builds
Day 1 sorts like from unlike terms once powers appear; Days 2 and 3 multiply two brackets, first with an area grid and then by sight for the special products; and Days 4 and 5 run the expansion backwards to factorise. It builds on Year 8, where students collected like terms, expanded a single bracket and took out a common factor, and on the index laws earlier this year. It prepares solving quadratic equations and sketching parabolas later in Year 9, where a factorised quadratic shows where the graph meets the x-axis.
Materials for the week (one trip)
- From the classroom: scissors, envelopes for the cards, pencils, this pack printed.
- Cut-out sheet 1, one per pair: algebra tiles, two x² tiles, ten x tiles and fifteen unit tiles (Days 2 and 4).
- Cut-out sheet 2, one per pair: 16 factor match cards (Day 4) and 12 sign sort cards (Day 5). No other equipment to buy.
Words for the word wall
term, like terms, coefficient, constant, expression, simplify, expand, binomial, binomial product, area model, difference of squares, perfect square, factorise, factor pair, monic quadratic, sum, product, common factor, check by expanding
Dear families
This week in maths, Year 9 is learning to expand and factorise. Expanding turns a product such as (x + 3)(x + 4) into a sum, x² + 7x + 12; factorising turns it back. Most of the factorising is a number puzzle: find two numbers that multiply to one number and add to another.
Try this at home
- Ask: which two numbers multiply to 12 and add to 7? (3 and 4. So x² + 7x + 12 = (x + 3)(x + 4).)
- Ask: what is 21 × 19 in your head? (20² − 1 = 399, from (20 + 1)(20 − 1).)
- Ask: is (x + 4)² the same as x² + 16? (No: it is x² + 8x + 16. The 8x is the part people forget.)
- Ask: can 3x and 2x² be added into one term? (No: they are unlike terms.)
My expand-and-factorise week
Fill one row a day. Write an expression from class, what you did to it, and how you checked.
| Day | Expression | Simplified, expanded or factorised | How I checked |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Expanding and factorising teaching pack · seegongsik.com/au/y9/algebra/AC9M9A02/pack
Like terms with powers
Like terms have exactly the same letters with the same exponents. Once squares and products of letters appear, students sort before they collect: 3x² and −x² collect, x² and x do not, and neither do xy and x.
We are learning to
- decide whether two terms are like terms by comparing letters and exponents,
- collect like terms in expressions with squares and products of letters,
- expand single brackets and collect, including a negative in front of a bracket.
Success criteria
- I can simplify m² − 4m + 3m² − m + 2 and say why m² and m stay apart.
- I can expand and simplify 3(x² + 2x) − x(x − 4) without losing a sign.
You need
The worksheet. Highlighters in two or three colours help with Part A. No calculators.
Lesson flow (about 60 minutes)
| 12 min | Same letters, same powers Show the board (see the box). Then collect 4x² + 3x − x² + 5x = 3x² + 8x and 2ab + 3a − ab = ab + 3a, colouring each kind of term before adding. Ask: “Why can 4x² and −x² be added, but not x² and 3x?” |
| 13 min | Brackets, then collect Model the first item of Part B: multiply each term inside by the term in front, sign included, then collect. A minus in front changes the sign of every term inside. Ask: “−x(x − 4): what is −x × −4?” |
| 35 min | Collect, expand, measure Parts A to C; early finishers do Part D. Share Part C by checking at x = 2: a rectangle 5 by 4 has perimeter 18 and area 20. Ask: “Why is the perimeter of the rectangle linear but its area quadratic?” |
Two shorter sessions instead? Split after the bracket model; start the second with the two board examples.
Watch for these ideas
- Adding unlike terms: 3x + 2x² is already simplest; it is not 5x³ or 5x².
- Changing the exponent when collecting: x² + x² = 2x², not x⁴.
- Sign slips after a minus: −2(m² − 3m) = −2m² + 6m.
Answers
- Part A: 5x + 3 − 2x + 4 = 3x + 7; 7a² + 2a − 3a² + a = 4a² + 3a; 6xy − 2x + xy + 5x = 7xy + 3x; m² − 4m + 3m² − m + 2 = 4m² − 5m + 2; 3p²q − pq² + 2p²q = 5p²q − pq²; −2y² + 8 − y² − 3 = −3y² + 5.
- Part B: 3(x² + 2x) − x(x − 4) = 2x² + 10x; 2a(a + 5) + 3(a − 1) = 2a² + 13a − 3; 5 − 2(m² − 3m) = −2m² + 6m + 5; x(x + y) − y(x − y) = x² + y².
- Part C: rectangle perimeter 2(x + 3) + 2(2x) = 6x + 6, area 2x(x + 3) = 2x² + 6x; triangle perimeter 3x² + 2x + 5.
- Part D: at x = 3, x² + x² = 18 but x⁴ = 81, and x² × x² = 81 but 2x² = 18. Adding two of the same term doubles it (2x²); multiplying adds the exponents (x⁴). Ana and Kai swapped the two.
Like terms with powers
A. Collect like terms
Colour each kind of term, then collect. Write the answer with the highest power first.
| 5x + 3 − 2x + 4 | 7a² + 2a − 3a² + a |
| 6xy − 2x + xy + 5x | m² − 4m + 3m² − m + 2 |
| 3p²q − pq² + 2p²q | −2y² + 8 − y² − 3 |
B. Expand, then collect
Multiply every term inside each bracket by the term in front, sign included. Then collect like terms.
| 3(x² + 2x) − x(x − 4) | |
| 2a(a + 5) + 3(a − 1) | |
| 5 − 2(m² − 3m) | |
| x(x + y) − y(x − y) |
C. Perimeter and area
A rectangle is x + 3 long and 2x wide. Write and simplify its perimeter and its area. A triangle has sides x² + 1, 2x² − x, 3x + 4. Write and simplify its perimeter.
D. Stretch: two mix-ups
Ana says x² + x² = x⁴. Kai says x² × x² = 2x². Test both at x = 3, then explain what each one mixed up.
Expanding binomial products
Every term of one bracket multiplies every term of the other. An area grid lays out the four products, and the two middle ones usually collect into one term.
We are learning to
- expand a binomial product with an area grid and with algebra tiles,
- expand products with negative terms and with a number in front of x,
- use the pattern (x + p)(x + q) = x² + (p + q)x + pq.
Success criteria
- I can expand (y − 4)(y + 9) and show the four products before collecting.
- I can say what (x + p)(x + q) gives for any p and q, and why.
You need
The worksheet; the algebra tiles (cut-out sheet 1), one set per pair. No calculators.
Lesson flow (about 60 minutes)
| 12 min | Tiles into a rectangle Show the board (see the box). Pairs build a rectangle (x + 3) long and (x + 2) wide with tiles and read off its area: one x², five x tiles, six units. Then expand (x + 2)(x + 5) = x² + 7x + 10 in a grid. Ask: “Where are the 3x and the 2x in your rectangle? Why do they collect?” |
| 13 min | Negatives and a number in front Grid (x − 3)(x + 6) = x² + 3x − 18 (a negative cell) and (2x + 1)(x + 4) = 2x² + 9x + 4 (2x times x gives 2x²). Ask: “Why is the cell for −3 times 6 negative, and what happens to the middle terms?” |
| 35 min | Grids, then without Parts A to C; early finishers do Part D. Students who are sure may drop the grid in Part B, but write the four products. Ask: “In (x + p)(x + q), which cells give the x term, and which gives the number?” |
Two shorter sessions instead? Split after the second model; start the second with the tile rectangle again.
Watch for these ideas
- Multiplying only first with first and last with last: (x + 3)(x + 7) is not x² + 21; the middle products matter.
- Losing a sign: in (x + 4)(x − 2), 4 × (−2) = −8.
- Writing 3x × x as 3x: it is 3x².
Answers
- Part A: (x + 3)(x + 7): cells x², 7x, 3x, 21, so x² + 10x + 21; (x + 4)(x − 2): cells x², −2x, 4x, −8, so x² + 2x − 8; (x − 5)(x − 1): cells x², −x, −5x, 5, so x² − 6x + 5; (a + 6)(a + 1): cells a², a, 6a, 6, so a² + 7a + 6.
- Part B: (x + 8)(x + 2) = x² + 10x + 16; (y − 4)(y + 9) = y² + 5y − 36; (m − 6)(m − 3) = m² − 9m + 18; (2x + 3)(x + 5) = 2x² + 13x + 15; (3a − 1)(a + 2) = 3a² + 5a − 2; (x + 2y)(x + 3y) = x² + 5xy + 6y².
- Part C: p = 2, q = 9: x² + 11x + 18; p = -3, q = 10: x² + 7x − 30; p = -4, q = -6: x² − 10x + 24; p = 7, q = -7: x² − 49. The x term is p + q and the number is pq.
- Part D: (x + 1)(x + 2)(x + 3) = x³ + 6x² + 11x + 6.
Expanding binomial products
A. Area grids
Fill each cell with the product of its row and column, then add the cells and collect.
| × | x | +7 |
|---|---|---|
| x | ||
| +3 |
| × | x | −2 |
|---|---|---|
| x | ||
| +4 |
| × | x | −1 |
|---|---|---|
| x | ||
| −5 |
| × | a | +1 |
|---|---|---|
| a | ||
| +6 |
B. Expand and simplify
Write the four products, then collect.
| (x + 8)(x + 2) = | (y − 4)(y + 9) = |
| (m − 6)(m − 3) = | (2x + 3)(x + 5) = |
| (3a − 1)(a + 2) = | (x + 2y)(x + 3y) = |
C. The pattern
Expand (x + p)(x + q) for each pair. What are the x term and the number each time, in terms of p and q?
| p | q | (x + p)(x + q) expanded |
|---|---|---|
| 2 | 9 | |
| −3 | 10 | |
| −4 | −6 | |
| 7 | −7 |
D. Stretch: three brackets
Expand (x + 1)(x + 2)(x + 3). Expand two brackets first, then multiply the answer by the third.
Special products
Two products are worth knowing by sight. In a sum times a difference the middle terms cancel: (a + b)(a − b) = a² − b². In a perfect square they double: (a + b)² = a² + 2ab + b².
We are learning to
- expand a sum times a difference and explain why the middle terms cancel,
- expand a perfect square and explain where the doubled middle term comes from,
- use both patterns for mental arithmetic.
Success criteria
- I can write (2y + 3)(2y − 3) and (3m + 1)² straight down, and show the grid if asked.
- I can work out 52 × 48 in my head with a difference of squares.
You need
The worksheet. No calculators: Part C is for mental arithmetic.
Lesson flow (about 60 minutes)
| 12 min | Cancel or double Show the first board picture (see the box). Then (x + 6)(x − 6) = x² − 36: the cells +6x and −6x cancel. Show the second picture, then (x − 3)² = x² − 6x + 9: two cells of −3x make −6x. Ask: “Why is the last term of (x − 3)² positive?” |
| 13 min | Spot the pattern Write five products on the board; pairs name each one (sum times difference, perfect square or neither) and expand the special ones by sight. Ask: “(x + 4)(x − 5) looks like a difference of squares. Is it?” |
| 35 min | By sight, then in your head Parts A to C; early finishers do Part D. Share Part C: each mental product written as its special product first. Ask: “How does 99² become easy?” |
Two shorter sessions instead? Split after the pattern spotting; start the second with Part C.
Watch for these ideas
- Squaring each term only: (x + 5)² is not x² + 25; the middle term 10x is there twice over.
- Forgetting to square the number with the letter: (2y + 3)(2y − 3) = 4y² − 9, not 2y² − 9.
- Calling any product with a minus a difference of squares: the two brackets must hold the same two terms.
Answers
- Part A: (x + 7)(x − 7) = x² − 49; (a − 9)(a + 9) = a² − 81; (2y + 3)(2y − 3) = 4y² − 9; (m − 12)(m + 12) = m² − 144.
- Part B: (x + 5)² = x² + 10x + 25; (a − 4)² = a² − 8a + 16; (3m + 1)² = 9m² + 6m + 1; (y − 10)² = y² − 20y + 100; (x + 2y)² = x² + 4xy + 4y².
- Part C: 21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399; 52 × 48 = (50 + 2)(50 − 2) = 2500 − 4 = 2496; 31² = (30 + 1)² = 900 + 60 + 1 = 961; 99² = (100 − 1)² = 10000 − 200 + 1 = 9801.
- Part D: the a² and b² squares leave two ab rectangles, so 2ab. The box is 14 or −14: (x + 7)² or (x − 7)².
Special products
A. A sum times a difference
Expand. Show the two middle products cancelling in one of them.
| (x + 7)(x − 7) = | (a − 9)(a + 9) = |
| (2y + 3)(2y − 3) = | (m − 12)(m + 12) = |
B. Perfect squares
Expand. Remember the middle term: twice the product of the two terms.
| (x + 5)² = | |
| (a − 4)² = | |
| (3m + 1)² = | |
| (y − 10)² = | |
| (x + 2y)² = |
C. In your head
Write each as a special product first, then work it out without a calculator.
| 21 × 19 = | 52 × 48 = |
| 31² = | 99² = |
D. Stretch: the missing term
Draw a square of side a + b split into four parts, and use it to explain (a + b)² = a² + 2ab + b². Then find every number that can go in the box to make x² + □x + 49 a perfect square.
Factorising as a number search
Factorising runs an expansion backwards. Since (x + p)(x + q) = x² + (p + q)x + pq, factorising x² + bx + c is a search for two numbers that multiply to c and add to b.
We are learning to
- list the factor pairs of c and pick the pair that adds to b,
- factorise monic quadratics with negative b or c,
- check a factorisation by expanding it again.
Success criteria
- I can factorise x² + 11x + 24 by listing the factor pairs of 24.
- I can show that x² + 6x + 8 is not (x + 1)(x + 8) by expanding.
You need
The worksheet; the algebra tiles (cut-out sheet 1) and the 16 factor match cards (cut-out sheet 2), one set per pair.
Lesson flow (about 60 minutes)
| 12 min | The search Show the board (see the box). Pairs make x² + 7x + 10 into a rectangle with tiles: the sides read (x + 2)(x + 5). Then x² + 8x + 12 by search: 1 × 12 (sum 13), 2 × 6 (sum 8), 3 × 4 (sum 7), so (x + 2)(x + 6). Ask: “Which pairs multiply to 12? Which one adds to 8?” |
| 13 min | Factor match Pairs lay the 16 cards face down and turn two at a time, keeping a quadratic and its factorised form only after expanding the brackets aloud. Ask: “How do you know (x − 2)(x − 4) is not x² + 6x + 8?” |
| 35 min | Search and check Parts A to C; early finishers do Part D. Every answer is expanded once as a check. Ask: “In x² − 7x + 12 the product is positive but the sum is negative. What does that tell you?” |
Two shorter sessions instead? Split after factor match; start the second with the tile rectangle again.
Watch for these ideas
- Matching the product only: 1 × 8 = 8, but 1 + 8 is not 6, so x² + 6x + 8 is (x + 2)(x + 4).
- Signs: a positive c with a negative b means both numbers are negative.
- Not checking: one expansion catches almost every slip.
Answers
- Part A: x² + 9x + 14: pairs 1, 14; 2, 7; (x + 2)(x + 7). x² + 11x + 24: pairs 1, 24; 2, 12; 3, 8; 4, 6; (x + 3)(x + 8). a² + 10a + 21: pairs 1, 21; 3, 7; (a + 3)(a + 7). x² + 13x + 36: pairs 1, 36; 2, 18; 3, 12; 4, 9; 6, 6; (x + 4)(x + 9).
- Part B: x² − 7x + 12 = (x − 3)(x − 4); x² + 2x − 15 = (x − 3)(x + 5); m² − 4m − 21 = (m + 3)(m − 7); x² − 10x + 16 = (x − 2)(x − 8).
- Part C: 1 right; 2 wrong: it expands to x² + 9x + 8; (x + 2)(x + 4); 3 wrong: it expands to x² − 11x + 18; (x − 3)(x − 6); 4 right.
- Part D: b = −25, −14, −11, −10, 10, 11, 14, 25, one for each factor pair of 24.
Factorising as a number search
A. List the pairs, then choose
List the factor pairs of the number term, circle the pair that adds to the x coefficient, and write the brackets.
| Quadratic | Factor pairs (circle one) | Factorised |
|---|---|---|
| x² + 9x + 14 | ||
| x² + 11x + 24 | ||
| a² + 10a + 21 | ||
| x² + 13x + 36 |
B. With negatives
Decide the signs first, then search. Check one by expanding.
| x² − 7x + 12 = | x² + 2x − 15 = |
| m² − 4m − 21 = | x² − 10x + 16 = |
C. Right or wrong?
Expand each claimed answer. If it is wrong, write the right factorisation.
| Claim | Expanded, and the fix if needed |
|---|---|
| x² + 8x + 15 = (x + 3)(x + 5) | |
| x² + 6x + 8 = (x + 1)(x + 8) | |
| x² − 9x + 18 = (x − 2)(x − 9) | |
| y² − y − 20 = (y + 4)(y − 5) |
D. Stretch: how many b?
Find every integer b for which x² + bx + 24 can be factorised with integers. How do you know you have them all?
Signs, common factors and special forms
The sign of c says whether the two numbers share a sign; the sign of b then says which. A common factor comes out first, and squares and differences of squares factorise by sight.
We are learning to
- predict the signs in the brackets from the signs of c and b,
- take out a common factor before factorising the monic quadratic left inside,
- factorise perfect squares and differences of squares by sight, and recognise one that will not factorise.
Success criteria
- I can say before searching that x² + 3x − 28 has one positive and one negative number.
- I can factorise 2x² + 14x + 24 fully.
You need
The worksheet; the 12 sign sort cards (cut-out sheet 2), one set per pair; the unit’s self-check quiz for the game.
Lesson flow (about 60 minutes)
| 12 min | Read c, then b Show the board (see the box). Then x² + 3x − 10: c is negative, so opposite signs, and b is positive, so the larger number is positive: (x − 2)(x + 5). Then 3x² + 12x + 9 = 3(x² + 4x + 3) = 3(x + 1)(x + 3). Ask: “Why take out the 3 before searching?” |
| 33 min | Sort, then factorise Pairs sort the 12 sign cards into both positive, both negative and opposite signs, by reading c and b only. Then Parts A to C; early finishers do Part D. Ask: “Which card was hardest to sort, and why?” |
| 15 min | Self-check quiz as a class game Run the unit’s self-check quiz on the board. Teams vote on each answer, then the reason is read aloud. |
Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.
Watch for these ideas
- Stopping before the common factor is out: (2x + 6)(x + 4) is right but not fully factorised; 2(x + 3)(x + 4) is.
- Thinking x² + 9 factorises like x² − 9: a sum of two squares does not, with integers.
- Writing a perfect square as two different brackets: x² + 12x + 36 = (x + 6)².
Answers
- Part A: x² − 8x + 15: both −, (x − 3)(x − 5); x² + 3x − 28: opposite, (x − 4)(x + 7); x² + 10x + 9: both +, (x + 1)(x + 9); p² − 5p − 36: opposite, (p + 4)(p − 9); x² − 13x + 40: both −, (x − 5)(x − 8).
- Part B: 2x² + 14x + 24 = 2(x² + 7x + 12) = 2(x + 3)(x + 4); 5a² − 5a − 30 = 5(a² − a − 6) = 5(a + 2)(a − 3); 3x² − 24x + 48 = 3(x² − 8x + 16) = 3(x − 4)².
- Part C: x² − 64 = (x + 8)(x − 8); m² − 1 = (m + 1)(m − 1); x² + 12x + 36 = (x + 6)²; y² − 18y + 81 = (y − 9)².
- Part D: x² + 5x + 7 and x² + 9 do not factorise with integers (no pair has the right sum and product); x² − 9 = (x + 3)(x − 3) and x² − 2x − 35 = (x + 5)(x − 7).
Signs, common factors and special forms
A. Predict the signs, then factorise
Circle the signs the two numbers must have before you search.
| Quadratic | Circle one | Factorised |
|---|---|---|
| x² − 8x + 15 | both + · both − · opposite | |
| x² + 3x − 28 | both + · both − · opposite | |
| x² + 10x + 9 | both + · both − · opposite | |
| p² − 5p − 36 | both + · both − · opposite | |
| x² − 13x + 40 | both + · both − · opposite |
B. Common factor first
Take out the common factor, then factorise what is left. Check by expanding.
| 2x² + 14x + 24 | |
| 5a² − 5a − 30 | |
| 3x² − 24x + 48 |
C. Squares and differences of squares
Factorise by sight, then check one by expanding.
| x² − 64 = | m² − 1 = |
| x² + 12x + 36 = | y² − 18y + 81 = |
D. Stretch: will it factorise?
Which of these can be factorised with integers? x² + 5x + 7; x² + 9; x² − 9; x² − 2x − 35. Factorise the ones that can, and explain how you know the others cannot.
Algebra tiles
For Days 2 and 4, one set per pair: 2 x² tiles, 10 x tiles and 15 unit tiles. Cut along the dashed lines. The long side of an x tile is the same as the side of an x² tile; the short side is 1.
Teacher note: to build (x + 3)(x + 2), lay one x² tile, three x tiles down one side and two along the other, then fill the corner with unit tiles. The x side is drawn a little more than three units long so that it is never read as 3.
Factor match and sign sort cards
Factor match (Day 4): 16 cards
Lay the cards face down. Turn two over; keep them if one is a quadratic and the other is its factorised form.
Sign sort (Day 5): 12 cards
Sort by the signs of the two numbers in the brackets: both positive, both negative, or opposite signs. Read c and b only.
Teacher note: the keys are with the mini-check answers, so this sheet can go to pairs whole.
What we know: Expanding and factorising
Work on your own. No calculator. Show your working.
- Simplify 4a² − 3a + 2a² + 7a. ______________________
- Expand and simplify 2(x² − 3x) − x(x + 1). ______________________
- Expand (x + 6)(x − 2). ______________________
- Expand (2y − 5)(y + 3). ______________________
- Expand (x − 7)² and (m + 9)(m − 9). ______________________
- Factorise x² + 11x + 28. ______________________
- Factorise x² − 3x − 40, then check by expanding. ______________________
- Factorise 3x² + 15x + 18 fully. ______________________
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
- 4a² − 3a + 2a² + 7a = 6a² + 4a (a² terms and a terms collected apart).
- 2(x² − 3x) − x(x + 1) = 2x² − 6x − x² − x = x² − 7x.
- (x + 6)(x − 2) = x² + 4x − 12.
- (2y − 5)(y + 3) = 2y² + y − 15.
- (x − 7)² = x² − 14x + 49; (m + 9)(m − 9) = m² − 81.
- Pairs of 28: 1, 28; 2, 14; 4, 7. Only one pair adds to 11, so (x + 4)(x + 7).
- Opposite signs (c is negative), the larger number negative (b is negative): (x + 5)(x − 8); check: x² − 8x + 5x − 40 = x² − 3x − 40.
- 3x² + 15x + 18 = 3(x² + 5x + 6) = 3(x + 2)(x + 3).
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Like terms | Adds unlike terms, or changes an exponent when collecting | Collects only terms with the same letters and exponents | Expands and collects with negatives in front of brackets without a slip |
| Expanding | Multiplies first with first and last with last only | Writes the four products and collects the middle terms | Uses the special products by sight and explains them with a grid |
| Factorising | Matches the product but not the sum, or loses a sign | Searches the factor pairs and checks by expanding | Predicts the signs first and takes out a common factor |
| Explaining | States steps without reasons | Explains a step with the grid or the sum and product | Says why an expression will not factorise |
Eight questions, four ideas. A student at standard answers most questions with working, and checks each factorisation by expanding.
Card keys
Factor match (Day 4): x² + 6x + 5 = (x + 1)(x + 5); x² + 9x + 18 = (x + 3)(x + 6); x² − 6x + 8 = (x − 2)(x − 4); x² + 4x − 12 = (x − 2)(x + 6); x² − x − 30 = (x + 5)(x − 6); x² + 14x + 49 = (x + 7)²; x² − 25 = (x + 5)(x − 5); x² − 11x + 28 = (x − 4)(x − 7). Sign sort (Day 5): both +: (x + 1)(x + 6), (x + 2)(x + 10), (x + 1)(x + 8); both −: (x − 4)(x − 5), (x − 5)(x − 7), (x − 2)(x − 6), (x − 6)(x − 9); opposite: (x + 2)(x − 4), (x − 6)(x + 7), (x − 2)(x + 7), (x + 3)(x − 6), (x − 3)(x + 7).
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 | Like terms | Expanding | Special | Factorising | Signs |
|---|---|---|---|---|---|
The five columns are the five days: like terms, expanding binomial products, special products, factorising, and signs with common factors.