BIDMAS through number puzzles

BIDMAS is the convention that tells us how to read an expression containing several operations. Brackets come first, then indices, then division and multiplication, then addition and subtraction. Division and multiplication have equal priority and are read from left to right. The same is true of addition and subtraction. That detail matters, and the acronym sometimes hides it.

A standard worksheet gives a child an expression and asks for its value. For example:

$$3 + 3 \times 3 = 12$$

Change the structure and the same symbols produce something different:

$$(3 + 3) \times 3 = 18$$

We prefer to reverse the task. Give the child a target, a small collection of identical numbers, and a set of permitted operations. Now they must build the expression. In Make 6 with Three Identical Numbers, the operations are the tools and 6 is the goal. BIDMAS is no longer a slogan to remember. It determines whether the construction actually works.

Many children can recite BIDMAS and still have a weak sense of structure. They move down a worksheet, apply the rule they were shown, and get most answers right. Then a less familiar expression appears and the knowledge falls apart.

Construction problems expose that gap quickly. A child who is trying to make 6 from three 3s has to ask better questions: should I create a large number and reduce it, make two smaller parts, or use a bracket to change what happens first? Every attempt gives immediate information. Moving one bracket may change the value completely.

This also changes the emotional character of the work. A wrong attempt is not simply a red cross. It is a result to inspect. The child can change one decision and see what that decision controlled.

At Primary level we normally keep the toolkit to addition, subtraction, multiplication, division, and brackets. At Junior level we can add powers, square roots, and factorials. Symbols such as $\sqrt{\phantom{x}}$ and $!$ are useful here because students meet them as purposeful operations: $\sqrt{9}$ turns 9 into 3, while $3!$ turns 3 into 6. The notation earns its place by solving a problem.

We begin with a tight set of rules. Which numbers must be used? Must each appear exactly once? Are joined digits, powers, roots, or factorials allowed? Clear constraints create the mathematics.

Students make a few targets and compare expressions. We ask them to predict the effect of a bracket before calculating, then explain why the same symbols can have different values. When an answer fails, we keep it on the board long enough to diagnose the structure rather than merely replace it.

The Problems.cc libraries provide the practice sequences: Make 6 is a natural Primary starting point, while Four Fours, Five Threes, Five Fives, Four Sevens, and Five Twos extend the search. This Exact Science page has a different job. It explains to parents and teachers why we use those tasks and what we listen for in a lesson.

We still use short calculation exercises when a convention needs checking. We just do not confuse successful worksheet completion with understanding. The stronger test is whether a child can choose the structure themselves, defend it, and adapt it when the target changes.