Brackets under the microscope

So far you have followed the order of operations. In this layer you will experiment with it. What happens if you move a bracket? How many different answers can the same four numbers make? When do brackets change nothing at all? Is a rule you notice always true, or only sometimes?

That is how mathematicians work: predict, try, compare, and test. You will make guesses (called conjectures), try to break them with a single example that does not fit (a counterexample), and keep only the ideas that survive.

Why exactly five? Every fully bracketed expression has a last operation, the one done at the very end. Here there are three signs, so there are three choices for the last one:

• Last is +: the left side is just 2, and the right side is 3 × 4 − 1, which can be bracketed in 2 ways.
• Last is ×: left is 2 + 3 (1 way), right is 4 − 1 (1 way): 1 × 1 = 1 way.
• Last is −: left is 2 + 3 × 4 (2 ways), right is 1: 2 ways.

Total: 2 + 1 + 2 = 5. With five numbers the same thinking gives 14 ways, with six numbers 42 ways, then 132, 429… These are called the Catalan numbers, and they turn up all over mathematics (you will meet them again in the Extend layer).

Brackets do not always change the value. Test these:

• (2 + 3) + 4 = 9 and 2 + (3 + 4) = 9.
• (2 × 3) × 4 = 24 and 2 × (3 × 4) = 24.
• 2 + (3 × 4) = 14, the same as 2 + 3 × 4.

In the first two, the expression uses only + or only ×. Adding and multiplying are associative: how you group them does not matter. In the third, the brackets just surround a part that would have been done first anyway.

Now test subtraction and division:

• (20 − 8) − 2 = 12 − 2 = 10, but 20 − (8 − 2) = 20 − 6 = 14.
• (48 ÷ 4) ÷ 2 = 12 ÷ 2 = 6, but 48 ÷ (4 ÷ 2) = 48 ÷ 2 = 24.

Subtraction and division are not associative. That is precisely why the order of operations needs the left-to-right rule for them.

Here is a real investigation. Compare 20 − (8 + 2) with 20 − 8 + 2.

• 20 − (8 + 2) = 20 − 10 = 10
• 20 − 8 + 2 = 12 + 2 = 14

They differ. Now try 20 − 8 − 2 = 12 − 2 = 10. That matches!

Conjecture: when you remove a bracket that has a − in front of it, the signs inside flip: + becomes − and − becomes +.

• 20 − (8 + 2) = 20 − 8 − 2 ✓ (both 10)
• 20 − (8 − 2) = 20 − 8 + 2? Left: 20 − 6 = 14. Right: 12 + 2 = 14 ✓

It makes sense: taking away (8 − 2) means taking away 8 but then giving back the 2, because you took away 2 too many.

Mathematicians love statements that are always true. But many statements are only sometimes true, and some are never true. Deciding which, and backing it up, is a key skill.

• To show sometimes, give one example where it works and one where it fails.
• To show never or always, you need a reason that covers every possible number, not just a few tests.

Try the sort game below. For each statement, test it with a few numbers (include 0 and 1) before you decide.

Here is a different kind of puzzle. The numbers are fixed, but the signs are hidden: 6 ▢ 3 ▢ 2. Each box can be +, −, × or ÷. How many different values can you get?

There are 4 × 4 = 16 ways to fill the boxes. Before you look at the table, predict: which filling gives the biggest value? Which gives 0? Are any two fillings equal?

Take the digits 1, 2, 3, 4. Use each once, with any of + and ×, and any brackets. What is the biggest value?

You might guess 1 × 2 × 3 × 4 = 24. But try (1 + 2) × 3 × 4 = 36. Adding the 1 to something first is better than multiplying by 1, because multiplying by 1 changes nothing.

This leads to a useful observation: for numbers bigger than 2, multiplying beats adding; but 1s should be added, not multiplied. (2 and 2 are a tie: 2 + 2 = 2 × 2 = 4.)

In Understand you met two kinds of calculator. Now test them like a scientist. For each key sequence below, predict what a basic (step-by-step) calculator shows and what a scientific calculator shows, then check if you can.

The basic calculator works like reading a sentence: it finishes each operation before starting the next, and it has no brackets. The scientific calculator waits for = and follows the order of operations.

Now combine everything: signs, brackets and the order of operations. A target puzzle gives you numbers and a target; you must hit the target exactly.

Example: use 3, 5, 7, 2 in this order with any signs and brackets to make 30.

• 3 × 5 + 7 × 2 = 15 + 14 = 29. Close!
• (3 + 5 + 7) × 2 = 15 × 2 = 30. ✓
• 3 × (5 + 7) − 2 × 3? Not allowed: that uses 3 twice.

A good strategy is to work backwards: 30 = 15 × 2, so can the first three numbers make 15? 3 + 5 + 7 = 15. Done.

When you test a rule, always include 0 and 1. They behave in special ways, and they are where wrong rules most often seem to work.

• Multiplying by 1 changes nothing: 7 + 1 × 5 = 7 + 5 = 12, the same as if the × were not there.
• Multiplying by 0 wipes everything out: 7 + 0 × 5 = 7 + 0 = 7, but (7 + 0) × 5 = 35.
• Adding or subtracting 0 changes nothing: 9 × 4 − 0 = 36.
• Dividing by 1 changes nothing: 20 ÷ 1 × 3 = 60.
• Dividing by 0 is not allowed at all: 5 ÷ (3 − 3) has no value.

A classic trick question is 9 − 9 × 0 + 1. People who go left to right say (0) × 0 + 1 = 1. The rule says 9 − 9 × 0 + 1 = 9 − 0 + 1 = 9 + 1 = 10.

A batter's strike rate is the runs scored per 100 balls: runs × 100 ÷ balls. If Shafali scores 45 runs off 30 balls, her strike rate is 45 × 100 ÷ 30 = 4,500 ÷ 30 = 150.

Here is an investigation. The formula could also be written runs ÷ balls × 100 or 100 ÷ balls × runs. Do these always give the same value?

• 45 ÷ 30 × 100: 45 ÷ 30 = 1½, then 1½ × 100 = 150.
• 100 ÷ 30 × 45: 100 ÷ 30 = 3⅓, then 3⅓ × 45 = 150.

Same value! Each ÷ stays attached to its own number (30), so moving the × and ÷ around does not change the answer. But look at the path: only the first order kept every step a whole number. When you work by hand, multiply first and divide last to avoid fractions along the way.

Four friends eat at a dhaba. The bill is ₹640 for food, plus ₹80 for drinks that only two of them had. How much should each person pay?

This is a real investigation, because there are several fair answers, and each is a different expression:

• Everyone shares everything: (640 + 80) ÷ 4 = (640 + 80) ÷ 4 = 720 ÷ 4 = 180 each.
• Food shared by all, drinks by the two drinkers: non-drinkers pay 640 ÷ 4 = 160; drinkers pay 640 ÷ 4 + 80 ÷ 2 = 640 ÷ 4 + 80 ÷ 2 = 160 + 80 ÷ 2 = 160 + 40 = 200.
• Check the second plan adds up: 2 × 160 + 2 × 200 = 2 × 160 + 2 × 200 = 320 + 2 × 200 = 320 + 400 = 720 = 640 + 80 ✓.

The common wrong expression is 640 + 80 ÷ 4 = 640 + 20 = 660 per person, which charges everyone almost the whole bill! The ÷ grabbed only the 80.

A spreadsheet is a perfect laboratory for the order of operations: type a formula, press Enter, and the machine shows exactly how it reads it. Try this experiment (or predict the results if you have no computer to hand). Put 10 in A1, 4 in A2 and 2 in A3. Then type each formula into a new cell and predict before you press Enter.