Number Foundations

Rules nobody discovered

Nobody ever discovered that a 3 in the tens column is worth thirty. Nobody proved that multiplication happens before addition. These are not facts about numbers. They are AGREEMENTS about how to read them. And the whole point of an agreement is that everyone gets the same answer. Without one, 2 + 3 × 4 would be 14 to some people and 20 to others, and no calculation could ever be checked by anyone else. ⚠️ That is worth knowing because it turns three separate things you have to remember into one idea you can work out. Where a digit sits tells you what it is worth. Where a number sits on the line tells you which is smaller. Which operation goes first is settled in advance so a written calculation means one thing only. Every rule below is one of those agreements, and the last one on the list is the one examiners actually reward: write your working so a reader can follow what you did.

Words for reading a number

Five terms. Every one of them names an agreement rather than a fact.

Match each digit to what it is worth

Why minus seven is the smaller number

Ask which is smaller, -7 or -3, and most people say -7. Ask why, and most people say it is because seven is bigger than three, which is the right answer for the wrong reason and stops working almost immediately. Put them on the line instead. -7 sits seven steps to the LEFT of zero. -3 sits three steps to the left. -7 is further left, so -7 is the smaller number. Written down: -7 < -3. ⚠️ Now watch the wrong reason break. Which is smaller, -0.8 or -0.08? If you compare how big the digits look, 8 against 08, you will get it wrong. On the line, -0.8 is further left, so -0.8 is the smaller number. A thermometer is the version of this you already believe. Minus seven degrees is colder than minus three degrees. Nobody hesitates over that, because on a thermometer you can SEE the positions. ⚠️ So the rule is not "ignore the minus sign" and it is not "reverse it". The rule is: WHICHEVER IS FURTHER LEFT IS THE SMALLER NUMBER. That one sentence handles positives, negatives and decimals without a special case for any of them.

Put these in order, smallest first

Five numbers. Furthest left on the number line goes first.

• -7
• -3
• -0.5
• 0.05
• 0.5

Which one is further left?

Five comparisons, three lives. Picture the line before you answer.

What goes first, and what waits

Three bands, worked all the way down. Anything in a later band waits for everything in an earlier one.

Two plus three times four

Two students work out 2 + 3 × 4 and get different answers. Only one of them followed the agreed order. Which value is the correct one?

• 14
• 20
• 24
• 9

Writing a calculation someone can follow

Your specification is blunt about this: a correct final answer with no working can still lose marks if it turns out to be wrong. That sounds unfair until you see what working actually is. ⚠️ Working is not proof that you did the sum. It is the calculation written so that a reader can redo it. One line per step. Take out exactly one thing at a time and write what is left. (12 - 4) × 3 + 5 should lose the bracket first, then the multiplication, then the addition, on three separate lines. Never do two things on one line. That is where mistakes hide, from you as much as from anyone marking it. Work outwards from the innermost bracket when there is more than one. ⚠️ And here is why it pays even when you are wrong. If your four lines are right and the fifth has a slip in it, a reader can SEE that everything up to the slip was correct, and you are credited for it. If there is only a final answer and it is wrong, there is nothing to credit. A right answer alone earns full marks; a wrong answer alone earns nothing at all. Working is the only thing standing between those two outcomes.

Work it out with the brackets

Work out (12 - 4) × 3 + 5. Take one step per line: the bracket, then the multiplication, then the addition.

Name the value of the five

In the number 3.457, the 5 is worth a certain amount on its own. Write that amount as a decimal.

Spot the true place-value facts

Tap the TWO statements that are true.

• The 6 in 0.06 is worth six hundredths
• Minus nine is less than minus two
• 0.25 is greater than 0.3
• The 4 in 40 and the 4 in 400 are worth the same

True about reading numbers

Select the TWO statements that are true.

• Brackets can change the answer to a calculation without changing any of the numbers in it
• Zero is greater than every negative number
• In 5.5, the left-hand 5 is worth one hundred times the right-hand 5
• A decimal with more digits after the point is always the larger number

Complete the number-reading rules

The agreement that a _____ is worth different amounts depending on where it sits is called _____. To decide which of two numbers is smaller, compare their positions on a _____, and the one further left is the smaller. The agreed sequence for working through a calculation is the _____.

Three number decisions

Three pieces of reasoning to judge. Each answer has to carry the reason, not just the verdict.

• A student says `-6` must be greater than `-2`, because 6 is greater than 2. What has gone wrong?
• A student writes only a final answer to a five-step calculation and it turns out to be wrong. They argue that the method was in their head, so it should still count. Why does it not?
• Someone wants to add 4 and 6 and then double the result. They write `4 + 6 × 2` and get 16. What should they have written, and why?

Explain why the rules are agreements

A friend has learned all three rules and still loses marks, because to them they are three unrelated things to memorise. Write them the explanation that ties the three together.

• Explain what place value means, using a digit that appears twice in the same number
• Explain how to decide which of two negative numbers is smaller, and why comparing the digits fails
• Explain why an order of operations has to be agreed in advance, using a calculation that would otherwise have two answers
• Say what brackets are for, given that the order is already fixed
• Finish with why showing working can earn marks even when the final answer turns out to be wrong