Brackets keep a part together so it is worked out first. If a sentence says to do something to a whole result, such as "add 3, then double it", the expression needs brackets.
"Add 4, then double it" and "double it, then add 4" sound alike. They give different answers, and different expressions.
In the last lesson each sentence was written from left to right. Some sentences need more care: "add 3 to n, then double the result" doubles the whole of n+3. Brackets group that part so it is worked out first.
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.
Stickers
Try Dan with 4 stickers: 4 and 3 more is 7, doubled is 14. Which expression gives 14?
TryTry it on the board and compare the result with your prediction.
Check the words
Work out 2×4+3 and 2(4+3). Which one matches the board?
Try it yourselfYour edits are saved in this browser.
Set n=4. Dan has 4 stickers, gets 3 more and doubles the lot. How many does the board show?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Where do the brackets go?
Here is how to decide where the brackets go.
Worked exampleBefore each step, say what you think comes next.
add 3 to n, then double the resultFind the step done first, and what is done to its result.
n+3The first step: add 3 to n.
double the result"The result" is the whole of n+3, not just n.
2(n+3)Brackets keep n+3 together, so it is worked out first and then doubled.
n=4:2(4+3)=2×7=14Check: add 3 to 4 to get 7, then double it.
2n+3Compare: this doubles n only and then adds 3. For n=4 it gives 11, not 14.
Read Jo's working one line at a time. Which operation is done to the whole of x+2?
Say the steps, then decide whether a part needs brackets.
Write an expression for "add 4 to n, then multiply the result by 5".
Jo writes "add 2 to x, then multiply by 6". Click the line where she goes wrong.
ModifyChange the problem on purpose and see what happens.
New sentences
The board does what the words say. Write your expression, then check it with a number.
Try it yourselfYour edits are saved in this browser.
Write an expression for "subtract 2 from n, then multiply the result by 3".
Write an expression for "multiply n by 3, then add 2".
Write an expression for "add a and b, then multiply the result by 4".
Write an expression for "add 5 to n, then halve the result".
PractiseA few on your own, to check you have it.
Brackets or not?
Books cost 8 dollars each. Write an expression for the cost of n books after taking 5 dollars off the total.
A group of n adults and 3 children each pay 6 dollars. Write an expression for the total.
Review: find 4(n−1) when n=6.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Find the rule
Say the steps in words: what does the rule do first, and what next?
Stretch: write a rule of your own with brackets and ask a partner to find it from the board.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Try n = 1, 2 and 3. How much does T go up each time?
Another hint
At n = 1 the board shows 0. What step makes that zero?
The mystery board follows a rule in n. Move n and watch T. Write an expression for T, using brackets.
For teachers
Students decide where brackets belong when a sentence applies a later step to a whole result, and see that brackets change the value (AC9M7A02, node 2). About 45 minutes.
I can say which step is done first, and what is then done to its whole result.
I can put brackets around the part that is done first, and leave them out when they are not needed.
I can show with a number that 2(n + 3) and 2n + 3 are different.
Before class
Do the Run and Modify boards yourself first. They hide the formula: the board does what the words say, so students check their brackets against it.
The new move is brackets as "do this part first". The common error is leaving them out (Jo's mistake: 6x + 2), so the multiplier only reaches the x. Always check with one number: add 2 to 1 and multiply by 6 is 18, but 6 × 1 + 2 is 8.
Not every sentence needs brackets. The 'books' and 'multiply then add' questions are there on purpose: when a later step is add or subtract, no brackets are needed. Ask: "what is done to what?"
Running the lesson
Predict: take a show of hands. Ask students to try 4 stickers on each option.
Run: ask them to say which of 2 × 4 + 3 and 2(4 + 3) the board matches, and why.
Investigate: model the worked example live. Underline the part done first, then bracket it.
Modify: before the no-brackets task, ask 'is anything done to a whole result here?'
Make: ask students to say the steps in words before they check their expression.
When students get stuck
A student writes 6x + 2 for 'add 2, then multiply by 6' (Jo's mistake). Ask: Try x = 1 in the words and in your expression. Do they agree? Fix: To multiply the whole x + 2, put x + 2 in brackets: 6(x + 2).
A student puts brackets everywhere (8(n − 5) for 'take 5 off the total'). Ask: What is the 5 taken off: n, or the cost of all the books? Fix: Brackets are for a part that is done first and then has something done to it. Here 8n is made first, then 5 is taken off.
Stretch and support
Stretch: Write 'double n, then add 3 more than n' in two ways. Which words need brackets? Test both with n = 5.
Support: Give a two-step frame: Step 1 (done first): ___; Step 2 (done to the result): ___. Then write the expression.
Where it leads
Next: an expression for a whole situation, where you say what the letter stands for. Brackets come back when expressions are expanded and factorised (Year 8) and in Equations with brackets.
What each step is for
stickers: Committing to an expression exposes leaving the brackets out, so the doubling only applies to n.
check the words: The board follows the words exactly, so its number for n = 4 shows which expression keeps the order of the steps.
where do the brackets go?: Brackets go around the part that is done first, when something is then done to the whole result.
new sentences: One change at a time: subtract first, a sentence that needs no brackets, two letters, and dividing the whole result.
brackets or not?: Choosing when brackets are needed, mixed, with one question where they are not; the review is substituting.
find the rule: Finding an expression with brackets from a mystery board reverses the lesson: from numbers back to the order of steps.
Now you can
Say which step is done first, and what is then done to its whole result.
Put brackets around the part that is done first, and leave them out when they are not needed.
Show with a number that 2(n + 3) and 2n + 3 are different.