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If you were to stop people on the street and ask them the answer to

3 + 2 × 4 = ?

most would give you 20, and a small minority would say 11.

I teach the convention of sequence of operations early in maths courses above level 1 (GCSE Intermediate and Access courses) as a lead into algebra. I use two activities - traditional worksheet and open ended group work. I then have a discussion about the different cognitive styles involved in the two contrasting activities. You'd be surprised how many people think that mathematics is just doing worksheets that get *really* hard. They have no idea of the creative side.

Just in case you have not heard the mnemonic for remembering the sequence of operations...

**B**rackets**O**peration (some say 'of' meaning multiply - the vowel is here to make a word. You can use the letter i for index, but I don't have the money to keep on registering domain names :- )**D**ivide**M**ultiply**A**dd**S**ubtract

Whole class interactive 'exposition' on the whiteboard / Interactive Whiteboard / flip chart.

- Put 3 + 2 × 4 or similar on board
- Write = 20 and ask for votes - most will agree
- Write = 11 and ask for votes - a few hands will go up and some people will remember and change their vote
- Relish the moment: how can a
*mathematical*calculation have two answers? - What are we to do?
- Explain the
*convention*that we do multiplication first, addition second. I usually mention some other conventions in Maths like the place notation system (the 3 in 34 means thirty as a result of a convention), and in everyday life (we drive on the left in the UK because someone somewhere decided to make a rule) - Ask how can I make 3 + 2 * 4 = 20 true? Someone will remember
*brackets*and usually where to put them. - Introduce the mnemonic

I usually go through several examples of increasing difficulty and encourage students to make up examples before moving into individual work on the worksheet. I differentiate nominated questions - can be tricky early in the year, so I keep the questions moving and refer back to the mnemonic often, ticking off the stages.

- Use a worksheet with about 40 questions covering all the types including 4(10-7) = 12 and 15 / 5 + 12 / 6 = 5 and worked examples at the top and answers at the bottom
- Explain the layout of the worksheet carefully pointing to all the sections (this lesson is about
*conventions*right? Maths books have conventions that are very strange until you get used to them) - Run through the worked examples on the worksheet one by one always relating them to the BODMAS mnemonic
- Invite students to work through the worksheet and I always point out the large table square poster we have in the classroom
- Move about making sure people are checking off against the BODMAS rules and re-explaining where necessary. This is a key part of the lesson as it where I learn what people know and what skills they can bring to the worksheet.

Next, I call the group together and write something like

= 7

on the whiteboard and ask for a calculation to put on the left hand side. 3 + 4 is a popular one, so then I ask for another that involves subtraction. I get 10 -3 or something so I add in 1 000 000 - 999 993 which gets people going a bit. Then I start asking for calculations that need two operations (5 × 2 - 3 or similar) and then one with brackets. People need practice chaining the calculations together and I challenge the group as a whole to check the rules...

Then I introduce the 4s activity with constraint....

4 - 4 + 4 - 4 = 0

Can you find other bodmas expressions that use the number 4 exactly 4 times and make 1, 2, 3, 4, 5, 6, 7, 8, 9??

This needs a bit of preparation (still whole class). I ask about what you can make with two 4s and build a table on the side of the whiteboard. Then...

- Invite students to work in groups of three or four. People will ask for clarification of the task
- Circulate prompting the groups to
*play*with combinations of the two fours, or to just 'take some scrap paper and write 4 down four times and stick operations between them and see what happens' then suggest changing an operation sign and see how that changes things. - Avoid people sitting there staring at a list of the numbers 0 to 9...
- After about 15 to 20 minutes most groups will be getting there - 6 and 8 are difficult to find. Many groups suss that 7 and 9 are related and 3 and 5 are related. I sometimes remind people that 2 is 1 + 1
- Wrap up when all the groups have a number of correct examples - some people need the rules re-explaining
- Feedback onto a prepared flip chart or board display. Ask who has one, and so on, going through the expressions and inviting the whole group to check. Welcome different calculations for each digit.
- Congratulate groups on completing a difficult task and invite views on the strategy they used and their roles in the group

- Ask people how the activities were different?
- Which one was more fun?
- Could they have done the group work without the skills base provided by the first activity?
- Which activity was more interesting? Any ideas why?

The answers enable me to check people's cognitive styles which is useful for future activities. I also recap the worksheet and suggest some exercises from the workbooks we use for homework. Recap in the next session usually reveals a good recall of the sequence of operations but also improved tables, and calculation.

Keith Burnett, Last update: Sun Sep 04 2011