Just following orders
I have been reading a thread on The TES maths forum (www. tes.co.uk section staffroom) which seems to imply that BIDMAS, the order of operations, doesn't work. Should I be teaching my pupils this or not?
The thread you refer to, "Is there a case for using SAMDIB (BIDMAS in reverse) as a model for solving simple linear type equations?" was started by dydx. The next poster suggests that there is never a case for using BIDMAS. The discussion leads to examples where it looks like the rules cannot be followed to arrive at the correct answer. For the non-mathematician, BIDMAS is a mnemonic for the order in which mathematical operations should be carried out, rather than simply reading left to right: B (brackets), I (indices), D (division), M (multiplication), A (Addition) and S (subtraction). Thus the expression 3 + 4 X 2 equals 11, not 14. This is because the 4 X 2 is calculated first.
Take the example given: 5 - 3 + 2. Following BIDMAS, you might work out the addition first (3 + 2 = 5) and then the subtraction (5 - 5 = 0), which is incorrect.
Taking 5 - 3 + 2 depends on how you read it. I teach that the sign in front of a number is part of it. So I would read the addition with a - 3 not +3, getting -3 +2 = - 1. Then I get 5 - 1, and the correct answer of 4.
When my husband Hugh was programming the BIDMAS programme for www.mathagonyaunt.co.uk under interactive resources he had to be very careful to follow the rules of BIDMAS in the programming. Think of collecting terms in algebra: 5x - 3x + 2x. If I swap them around they become - 3x + 5x + 2x.
These are also the rules we use for solving equations. To read 5 - 3 + 2 as adding the three and two first, you would need a bracket: 5 - (3 + 2).
Emily Isobel suggests that if you read questions that have both addition and subtraction in the order in which they are presented then this prevents mistakes. If we want pupils to be able to manipulate algebraic expressions then we need to teach them the rules correctly and consistently. This made me consider an example involving multiplication and division, as you can see in the second diagram. I applied the same "rules" as before, moving the numbers with the sign in front into different positions. The only number I haven't moved is the 10, as this has a hidden "+" in front of it and so comes after both the division and the multiplication in BIDMAS.
In the examples I have followed the rule that division should take place before multiplication. In the first example, if we multiply first, we get 10 V 10 V 2 = 0.5, which is incorrect; in the second example, we get 10 V 2 V 10 = 0.5, which is also incorrect; and in the last example, we get 50 V 2 V 2 = 12.5, which is correct.
However, using multiplication first gives inconsistent answers, as you can see. We get the correct answer if we complete the calculations in the order in which they appear, as suggested on the forum, provided the mix is the inverse operation.
This provides the basis of a great investigation exploring BIDMAS and trying to "crack" the rule. Pupils love a challenge. Manipulating the numbers in this way will strengthen their confidence in the correct manipulation of algebraic sentences.
I was teaching a Year 6 boy who has been shown to record recurring decimals with a small "r" at the end. Is this notation correct?
Denoting a recurring decimal with "r" on the end could be confusing as this symbol is also used for remainder - Jfor example, I have seen pupils record an answer of 8r2 as 8.2! So I would say definitely "no" to writing a recurring decimal with an "r". Take 0.516r - Jis this 0.51666I or 0.5161616I or 0.516516I? The notation tells us nothing about the number. The notations I have coloured green are the most useful. Writing a dot above each recurring number in the sequence can be laborious. Usually it is a dot above the first and last number in the recurring sequence. I have also seen and used a line over the recurring sequence; on the forum it is suggested that this is common practice in the US.
* Find BIDMAS interactive activities at www.mathagonyaunt.co.uk
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
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