Bodmas.

It is essential when using multiple mathematical operations involving addition, subtraction, multiplication or division to know the order in which the operations are to be performed. The default order is always to do the multiplications and/or divisions before the additions and/or subtractions. When this order is not intended, brackets (usually parentheses) should be used to highlight that the operations within the brackets should be performed first. The mnemonic BODMAS is used to remind students of the order in which a multi-operational calculation should be attacked. Here B stands for brackets, O for of, D for divide, M for multiply, A for add, and S for subtract.

The use of BODMAS enables calculations such as

7 + 6 x 5

to have a unique answer. Students should be told that the spreadsheet calculations in Excel use BODMAS, and therefore the answer given will be 37 since the multiplication is to be performed first of all. When the intention is to do the addition first, then brackets have to be used producing

(7 + 6) x 5

and now the answer is 65. Note that brackets could also be used as

7 + (6 x 5)

but are unnecessary because of BODMAS. However the overuse of brackets should not be discouraged as this frequently helps students in many complicated operational problems.

Here are some fun and creative exercises that you could use with your students to help them develop competence with BODMAS.

Exercise 1

Use exactly two twos (2) and any BODMAS operations to make as many different numerals as possible.

Solution

0 = 2 - 2
1 = 2 / 2
4 = 2 + 2 = 2 x 2
22 = 22

Only four different answers are possible and BODMAS did not have to be used because there were no multiple operations.

Exercise 2

Use exactly three threes (3) and BODMAS operations to make as many numerals as possible.

Solution

0 = (3 - 3) x 3 = (3 - 3) / 3
2 = (3 + 3) / 3 = 3 - (3 / 3)
3 = 3 x 3 / 3 = 3 + 3 - 3
4 = 3 + (3 / 3)
6 = 3 x 3 - 3
9 = 3 + 3 + 3
11 = 33 / 3
12 = 3 x 3 + 3
18 = (3 + 3) x 3
27 = 3 x 3 x 3
30 = 33 - 3
36 = 33 + 3
99 = 33 x 3
333 = 333

Here brackets were used in four of the fourteen different answers. Perhaps you could simply tell your students that there are fourteen different possible answers and see how many they can find.

Note that 1 cannot be obtained unless powers are allowed as in

1 = [(3 / 3).sup.3] = [3.sup.(3-3)]

As powers or indices are not part of BODMAS and the latter expression is probably beyond the mathematical knowledge of many junior high school students, they are not included here.

Exercise 3

Use four fours (4) and BODMAS to obtain as many numerals as possible.

Solution

This problem is rich in BODMAS examples. Ask your students to find solutions for all numerals from 0 to 10. Try to find them yourself before looking at the answers below.

0 = 4 - 4 + 4 - 4 = (4 - 4) x 44 = (4 / 4) - (4 / 4)
1 = 44 / 44 = (4 + 4) / (4 + 4) = (4 + 4 - 4) / 4
2 = (4 / 4) + (4 / 4) = 4 - (4 + 4) / 4
3 = (4 + 4 + 4) / 4 = (4 x 4 - 4) / 4
4 = 4 + (4 - 4) x 4
5 = (4 x 4 + 4) / 4
6 = 4 + (4 + 4) / 4
7 = 4 + 4 - (4 / 4) = 44 / 4 - 4
8 = 4 + 4 + 4 - 4 = 4 x 4 - 4 - 4 = (4 + 4) x (4 /4)
9 = 4 + 4 + (4 / 4)
10 = (44 - 4) / 4

For the numerals 11 to 20 only the following have been found:

12 = (4 - (4 / 4)) x 4 = (44 + 4) / 4
15 = 44/ 4 + 4 = 4 x 4 - (4 / 4)
16 = 4 + 4 + 4 + 4 = 4 x 4 x 4 / 4
17 = 4 x 4 + (4 / 4)
20 = (4 + (4 / 4)) x 4

The numerals 11, 13, 14, 18 can be obtained if we allow the use of [square root of 4 = 2], but 19 has not yet been found by your author. Can any reader help?

Other numerals past 20 are possible and your author has found another twenty-six. See how many your students can obtain. You could suggest that they try a systematic approach since there are four operations (+, - , x , +) and the four fours can be grouped into the five sets:

(4, 4, 4, 4),
(4, 4, 44),
(44, 44),
(4, 444) and
(4444).

Further exercises can be generated using five fives, six sixes, up to nine nines, but the number of possibilities grows rapidly.

Since four fours can produce more than forty different numerals, your students could also investigate four Zs, where Z is any digit from 1 to 9. Not only does this help with BODMAS but it can also assist in developing their algebraic manipulations. Happy discoveries!

with Neville de Mestre