The number outside a bracket multiplies every term inside it. 3(2x + 4) is 3 rows of 2x + 4, so 6x + 12. Then collect any like terms.
Three friends each buy a drink and a $4 snack. Three drinks and three snacks: the 3 goes to both. That is all expanding a bracket is.
In Algebra Foundations you wrote expressions with brackets: 3(x+2) means "add 2 to x, then multiply by 3". Expanding writes the same expression without the bracket.
In this lesson you find out what the number outside the bracket does to the terms inside, using algebra tiles: a long tile for each x and a small square for each 1.
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.
3(x + 2)
Think of x+2 as one row of tiles: one x and two ones. What do 3 rows hold?
TryTry it on the board and compare the result with your prediction.
Build it with tiles
k is the number outside the bracket. a is the number of x inside, and b the number on its own. The board draws k rows of the bracket, then counts every tile.
Try it yourselfYour edits are saved in this browser.
Set the board to 3(x+2) and count the tiles. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Multiply every term
Here is how to expand a bracket and then collect like terms.
Worked exampleBefore each step, say what you think comes next.
3(2x+4)−xExpand the bracket first.
3×2x=6xThe 3 multiplies the first term.
3×4=12And the second term: every term in the bracket.
6x+12−xThe bracket is gone. The −x was outside it, so it stays as it is.
5x+12Collect like terms: 6x−x=5x. The 12 has no like term.
Try one: expand 2(5x−3).
Read Sam's working one line at a time. Which term did the 4 miss?
Write each step on its own line. The checker marks the first line that does not follow.
Expand 2(5x−3).
Sam expands 4(x+3). Click the line where his working goes wrong.
Expand and simplify 3(2x+4)−x. Write one step per line.
Start from 3(2x+4)−x
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing. The board shows the tiles; type the expansion.
Need a hint?
Set k, a and b, then count: k × a x-tiles and k × b ones.
Need a hint?
b is negative here: each row has 2 negative ones.
Multiplying two negatives, from Number Foundations.
k can be negative now: watch every tile change sign.
Need a hint?
Set k to −2. Every tile flips sign.
Another hint
−2 × x = −2x and −2 × (−5) = +10.
Need a hint?
Expand the bracket first: 2(x + 3) = 2x + 6.
Another hint
Then collect: 2x + 3x, and 6 − 1.
No board for this one: expand each bracket, then collect.
Try it yourselfYour edits are saved in this browser.
Expand 5(2x+3). Build it on the board to check.
Expand 4(3x−2).
Quick check before a negative outside: what is −2×(−5)?
Expand −2(x−5).
Expand and simplify 2(x+3)+3x−1. Use p and q for the terms after the bracket.
Edge case, two brackets: expand and simplify 2(x+3)+3(x−1).
PractiseA few on your own, to check you have it.
Expand and simplify
Expand 6(x+2).
Expand −3(2x−4).
Expand and simplify 4(2x−3)+5. Write one step per line.
Start from 4(2x−3)+5
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Brackets backwards
Find a bracket that expands to 6x+12. Then find a different one.
Stretch: which bracket with the biggest possible k expands to 12x+18?
Try it yourselfYour edits are saved in this browser.
Need a hint?
Which number goes into both 6 and 12?
Another hint
Try k = 3: then 3 × a = 6 and 3 × b = 12.
Set k, a and b so the bracket expands to 6x+12, with k bigger than 1.
Use the board. This ticks when you get there.
For teachers
Students build k(ax + b) as rows of algebra tiles, find that the number outside multiplies every term inside (signs included), expand, then collect like terms, writing their working one step per line (AC9M8A01, node 1). About 45 minutes.
I can expand a bracket by multiplying every term inside by the number outside.
I can expand when there are negative numbers inside or outside the bracket.
I can expand, then collect like terms, and show each step.
Before class
Do the Run board and the working check yourself first. Students need Algebra Foundations' brackets and expressions, and adding and subtracting negatives (Number Foundations). Multiplying two negatives (−2 × −5 = 10) is Year 8 Number: the tiles show it (a negative row flips every tile), but teach or recap it in class if it is new.
The one idea: the number outside multiplies every term inside. The common mistake is multiplying only the first term (4(x + 3) = 4x + 3); it is planted after the method, in Sam's working.
Answers are marked by form: an answer that is equal but still has brackets, or still has like terms to collect, is told what to do next rather than marked right. The working check marks each line against the one before, so students see which step went wrong.
Running the lesson
Predict: take a show of hands. Students who chose 3x + 2 are thinking the 3 only reaches the x. Do not correct it yet.
Run: ask students to read the count under the tiles aloud: '3 rows of x + 2: 3 x-tiles and 6 ones'.
Investigate: model the worked example live. Draw 3 rows of tiles for 2x + 4, write 3 × 2x and 3 × 4 at writing pace, then the − x, then collect. Reveal the prepared lines afterwards to compare.
Modify: before the negative-outside task, ask: what happens to a tile if you take it away twice? Then let the board show −2(x − 5).
Practise: the working check marks the first wrong line. Ask students who get a cross to say what that step did.
Make: collect the different brackets for 6x + 12 and ask which uses the biggest number outside. That is the lead into factorising.
When students get stuck
A student multiplies only the first term (Sam's mistake). Ask: How many rows are there? What is in each row? Fix: Build it on the board: every row has the ones as well as the x.
A student gets the sign wrong with a negative inside (4(3x − 2) = 12x + 8). Ask: What is in each row: add 2 or take away 2? Fix: 4 rows each take away 2, so 8 taken away: −8.
A student expands correctly but stops before collecting. Ask: Are there any terms that are the same kind? Can they be added? Fix: Underline the x terms and the number terms in different colours, then add each kind.
Stretch and support
Stretch: Expand and simplify 3(2x − 1) − 2(x − 4). (4x + 5.) Then: find the bracket with the biggest number outside that expands to 12x + 18.
Support: Use the tiles board for every expansion: set k, a and b, read the count, then write it. Physical algebra tiles help if you have them.
Where it leads
Next: factorising, which is expanding backwards (the Make task), then rearranging formulas (nodes 2 and 3).
Where it leads: binomial products (x + 2)(x + 3) in Year 9, which the same area model shows as a grid.
What each step is for
3(x + 2): Committing to an expansion of 3(x + 2) before building it tests whether they think the 3 reaches the 2.
build it with tiles: Building k rows of the bracket and counting the tiles shows the number outside multiplies every term inside, not just the first.
multiply every term: Expanding multiplies each term in the bracket by the number outside, signs included; collecting like terms then tidies up. Sam's mistake multiplies only the first term.
one change at a time: One change at a time: bigger numbers, subtraction in the bracket, a negative outside (after a quick check of multiplying negatives), extra terms to collect, then the edge case of two brackets.
expand and simplify: Expanding with each sign, collecting, and full working, mixed; the review keeps writing brackets from words fresh.
brackets backwards: Finding a bracket that expands to a given expression reverses the lesson, and shows more than one bracket can give the same result.
Now you can
Expand a bracket by multiplying every term inside by the number outside.
Expand when there are negative numbers inside or outside the bracket.
Expand, then collect like terms, and show each step.