Always, sometimes or never?

So far you have been told the properties. In this layer you become the mathematician. You will meet claims, some true and some false, and your job is to predict, test and decide:

• Always true: true for every number you could try. To be sure, you need a reason, not just examples.
• Sometimes true: true for some numbers, false for others. Find one of each.
• Never true: false for every number. Again, you need a reason.

Keep a notebook (or the back of an old calendar) beside you. Every chapter has a prediction to commit to before you read on, and most have a lab where you can test your ideas quickly. Being wrong in a prediction is not failure; it is the moment the idea gets stored in your memory.

You know that subtraction is not commutative in general. But is a − b = b − a never true, or only sometimes false? Test pairs. With whole numbers, a − b only makes sense when a ≥ b, and b − a only when b ≥ a. Both make sense together only if a = b, and then both are 0. So:

a − b = b − a is sometimes true: exactly when a = b.

Division is similar. 6 ÷ 6 = 1 either way. But 6 ÷ 3 = 2 while 3 ÷ 6 = ½. And 0 ÷ 0 is undefined, so the equal pair must not be 0. a ÷ b = b ÷ a is sometimes true: exactly when a = b and neither is 0.

Subtraction is not associative: (a − b) − c and a − (b − c) can differ. But by how much? Rather than just saying "different", let's measure the gap. Work out both groupings for several triples and subtract.

Some multipliers are just one away from a round number: 9 = 10 − 1, 11 = 10 + 1, 99 = 100 − 1, 101 = 100 + 1, 999 = 1,000 − 1. The distributive property turns each into a shortcut. Before reading the table, try to predict each answer using the round number.

The × 11 column hides a famous trick. For a two-digit number, write the two digits apart and put their sum in the middle:

• 23 × 11: 2 _ 3, middle 2 + 3 = 5 → 253.
• 54 × 11: 5 _ 4, middle 9 → 594.
• 86 × 11: 8 _ 6, middle 14 → carry the 1 → 946.

Why? 86 × 11 = 860 + 86. Line them up and the tens digit of 86 lands in the same column as the ones digit of 860's tens place, so the middle column adds the two digits. It is the distributive property plus place value.

We cannot divide by 0, but we can divide by numbers closer and closer to 0 and watch what happens. This is exactly what mathematicians did to understand the problem. Predict the pattern before reading the table.

Consecutive numbers follow one another: 7, 8, 9. Here are three claims to investigate. Test each with a few examples, then look for a reason.

1. The sum of two consecutive numbers is always odd.
2. The sum of three consecutive numbers is always a multiple of 3.
3. The product of two consecutive numbers is always even.

All three claims survive, and each has a neat reason:

1. Of two consecutive numbers, one is even and one is odd. Even + odd = odd. Always.
2. Three consecutive numbers are (middle − 1), middle, (middle + 1). The −1 and +1 cancel, so the sum is 3 × middle. Always a multiple of 3. (Look at the table: 12 + 13 + 14 = 39 = 3 × 13.)
3. One of the two is even, and anything × even is even. Always.

Closure is not only about whole numbers. You can ask it about any family of numbers. Take the multiples of 3: 0, 3, 6, 9, 12, … Add two of them: 12 + 21 = 33, a multiple of 3. The reason: 3 × a + 3 × b = 3 × (a + b), by the distributive property. So multiples of 3 are closed under addition. The same argument works for multiples of any number.

Now try square numbers (1, 4, 9, 16, 25, …). Multiply two: 4 × 9 = 36 = 6 × 6, a square. Add two: 1 + 4 = 5, not a square. So squares are closed under multiplication but not under addition. Before the lab, make your own predictions about the families below.

Some number patterns look like magic. Almost always, the magic is a property of numbers in disguise. Work out a few rows yourself, predict the next row, then read the explanation.

Why do 111, 222, 333 … appear? Because 37 × 3 = 111. Then 37 × 6 = 37 × (3 × 2) = (37 × 3) × 2 = 111 × 2 = 222, by the associative property. In the same way 37 × 27 = 111 × 9 = 999.

And the square pattern of ones: 11 × 11 = 121, 111 × 111 = 12,321, 1,111 × 1,111 = 1,234,321. Split 111 × 111 by the distributive property into 111 × 100 + 111 × 10 + 111 × 1 = 11,100 + 1,110 + 111. Stack them and the columns add up to 1, 2, 3, 2, 1. The pattern survives until 111,111,111 × 111,111,111 = 12,345,678,987,654,321, then carrying spoils it.

Here is an experiment you can do with a friend and a stopwatch. Each of you does the same calculation, one the "straight" way and one with a property shortcut. Swap roles and repeat. Most people find the shortcuts win by a lot, but only once they have practised spotting them.

Arranging chairs. The school hall needs 36 chairs in a rectangle. How many different arrangements are there? The rows × columns pairs are: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6, 9 × 4, 12 × 3, 18 × 2, 36 × 1. That is 9 arrangements if you count 4 rows of 9 and 9 rows of 4 as different (the hall is longer one way than the other). If you only care about the shape, turn-arounds are the same, leaving 5 shapes: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The square 6 × 6 is its own turn-around, which is why the count is odd. (Only square numbers have an odd number of arrangements, because only a square number has an arrangement that is its own turn-around.)

Shopping totals. The shopkeeper does not add the bill from top to bottom. He spots partners: atta + dal = 245 + 155 = 400, sugar + tea = 90 + 110 = 200, soap + biscuits = 45 + 55 = 100. Total ₹700. Commutative and associative properties together promise that the total is the same in any order.

Cricket. A batter's score is the same however the runs arrive: 4 + 6 + 1 or 1 + 6 + 4. But the order of overs matters for a run chase: needing 20 off the last over is different from needing 20 off the first. Real life often mixes things where order matters with things where it doesn't. Spotting which is which is the skill.