Factorising is expanding backwards. Share the terms into equal rows: 6x + 9 is 3 rows of 2x + 3, so 3(2x + 3). Take out the highest common factor, then check by expanding.
6 x-tiles and 9 ones. Can you share them into equal rows, with nothing left over? The number of rows goes outside the bracket.
In Expanding brackets you turned 3(2x+4) into 6x+12. Factorising goes the other way: from 6x+12 back to a bracket.
In this lesson you share algebra tiles into equal rows. The number of rows goes outside the bracket, and one row goes inside.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Fully factorised
Think about it: which number goes into both 6 and 9?
TryTry it on the board and compare the result with your prediction.
Share into rows
k is the number of rows. When the tiles share equally, the board writes the bracket: k outside, one row inside.
Try it yourselfYour edits are saved in this browser.
The board has 6x+9. Change k, the number of rows, until the tiles share into equal rows with more than one row. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
The highest common factor
Here is how to factorise, then check.
Worked exampleBefore each step, say what you think comes next.
12x−8Find a number that goes into every term.
HCF of 12 and 8=4Take out the biggest: the highest common factor.
12x÷4=3xDivide every term by 4.
−8÷4=−2The sign comes too.
4(3x−2)The HCF goes outside; what is left goes inside.
4×3x−4×2=12x−8Check by expanding.
Try one: factorise 10x+15.
Read Ava's working one line at a time. Is her answer wrong, or just not finished?
Factorise 10x+15 fully.
Ava factorises 12x−8. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Which numbers go into both 10 and 15? Try the biggest.
No board from here: divide each term, then check by expanding.
Try it yourselfYour edits are saved in this browser.
The board has 10x+15. Share it into the most rows that work.
Use the board. This ticks when you get there.
Factorise 9x−6 fully.
Three terms: factorise 4x+8y−12 fully.
A letter can be a common factor too. Factorise 2x2+4x fully.
PractiseA few on your own, to check you have it.
Factorise and check
Factorise 15x+20 fully.
Factorise 8x−12 fully.
Factorise 18x+12 fully. You may take out a factor in steps. Write one step per line.
Start from 18x+12
Review: expand 3(2x−5).
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Four rows
Find an expression sx+t that factorises fully as 4(…). Then find another.
Stretch: why can't s and t both be multiples of 8?
Try it yourselfYour edits are saved in this browser.
Need a hint?
Both numbers must be in the 4 times table.
Another hint
But the rows must not share further: 8 and 16 have HCF 8, not 4.
Set s and t so that sx+t is fully factorised with 4 outside the bracket, then set k to 4.
Use the board. This ticks when you get there.
For teachers
Students share algebra tiles into equal rows to factorise sx + t, learn to take out the highest common factor (numbers, then letters) and check by expanding (AC9M8A01, node 2). About 45 minutes.
I can factorise an expression by taking out a common factor.
I can take out the highest common factor, including a letter, so the expression is fully factorised.
I can check a factorisation by expanding it.
Before class
Do the Run board and the Make task yourself first. Students need Expanding brackets (the lesson before) and highest common factors from Year 7 Number.
The one idea: factorising divides every term by a common factor, and the factor goes outside the bracket. The common mistake is taking out a common factor that is not the highest (12x − 8 = 2(6x − 4)); it is planted after the method, in Ava's working. Her answer is equal to 12x − 8, so it is not wrong, just not finished: that is the conversation to have.
Answers are marked by form: an equal answer that could still be factorised further is told so ("could take out more") rather than marked right.
Running the lesson
Predict: take a show of hands. Students who chose 6(x + 9) divided only the first term; those who chose 2(3x + 4.5) found a factor of 6 but not of 9. Do not correct it yet.
Run: ask why 2 rows do not work (9 ones do not share into 2) and why 3 does.
Investigate: model the worked example live, including the check by expanding. Then ask: is Ava's answer wrong?
Modify: before the letter task, ask what goes into both 2x² and 4x. Students often find the 2 and miss the x.
Make: collect students' expressions on the board and ask which ones are fully factorised with 4 outside.
When students get stuck
A student divides only the first term (6x + 9 = 3(2x + 9)). Ask: Expand your answer. Do you get back to where you started? Fix: Expanding is the check: 3 × 9 = 27, not 9.
A student stops at a common factor that is not the highest (Ava's mistake). Ask: Look inside the bracket. Do those terms share anything? Fix: Take out what they share as well: 2(6x − 4) = 2 × 2(3x − 2) = 4(3x − 2).
A student cannot find a common factor. Ask: List the factors of each number. Which appear in both lists? Fix: Use the board: try k = 2, 3, 4… and see which numbers of rows work.
Stretch and support
Stretch: Factorise 6x²y − 9xy² fully. (3xy(2x − 3y).) Then: write three different expressions whose full factorisation has 6 outside.
Support: Use the tiles board for every two-term question with positive numbers: try each k and read off the bracket.
Where it leads
Next: rearranging formulas (node 3), which divides every term by a number in the same way.
Where it leads: factorising quadratics in Years 9 and 10, which the same tiles show as a rectangle.
What each step is for
fully factorised: Committing to a factorisation of 6x + 9 before using the board tests whether they divide every term, and whether they take out the biggest number.
share into rows: Trying k = 1, 2, 3 on the tiles shows which numbers of rows work (the ones that divide both totals) and what each row holds.
the highest common factor: Factorising divides every term by a common factor; taking out the highest common factor leaves nothing more to share, and expanding checks it. Ava's mistake takes out a common factor that is not the highest.
one change at a time: One change at a time: different numbers on the board, a minus sign, three terms, a letter as the common factor, then the edge case of nothing to take out.
factorise and check: Factorising with each sign and full working, mixed; stopping at a common factor that is not the highest is the planted slip, and the review keeps expanding fresh.
four rows: Choosing tiles that factorise fully with exactly 4 outside reverses the lesson: from the factor back to the expression.
Now you can
Factorise an expression by taking out a common factor.
Take out the highest common factor, including a letter, so the expression is fully factorised.