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BODMAS as a first lesson
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...
- Operation (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 :- )
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.
Activity 1: worksheet
- 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.
Activity 2: Open ended
Next, I call the group together and write something like
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
Payoff (my word for 'meta-cognitive gain')
- 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