Math Detours · Grade 5 · Sharpen · 13

Brackets say what goes first

2 + 3 × 4 and (2 + 3) × 4 are different sums. Brackets are how you tell a reader which one you mean.

Name ____________________Date ____________

Tonight's idea

Look at \(2 + 3 \times 4\). Add first and you get \(5 \times 4 = 20\). Multiply first and you get \(2 + 12 = 14\). One sum should not have two answers.

So people everywhere agreed on a rule. With no brackets, multiply and divide before you add and subtract. That makes \(2 + 3 \times 4\) equal to 14.

What if you want the adding done first? Then you say so, with brackets. Whatever is inside brackets gets done first: \((2 + 3) \times 4 = 5 \times 4 = 20\).

23 rows of 4 2 + 3 × 4 = 14 5 rows of 4 (2 + 3) × 4 = 20
The same three numbers, two different pictures. The brackets tell you which picture is meant.

Brackets first. Then × and ÷. Then + and −, working from left to right.

Brackets are a message from the writer to the reader. They cost nothing, so use them whenever a sum could be read two ways. Extra brackets do no harm: \(2 + (3 \times 4)\) is still 14.

Watch one: tickets and drinks

Three friends go to the cinema. Each buys a ticket for $6 and a drink for $2. Write one sum for the total.

Each friend pays \(6 + 2\). There are three friends, so the whole of \(6 + 2\) is multiplied by 3:

\[(6 + 2) \times 3 = 8 \times 3 = 24.\]

Leave the brackets out and you get \(6 + 2 \times 3 = 6 + 6 = 12\). That is one ticket and three drinks. It is a different story.

Your turn

  1. Work out: (a) \(5 + 2 \times 6\)   (b) \((5 + 2) \times 6\)   (c) \(20 - 8 \div 4\)   (d) \((20 - 8) \div 4\)
  2. Put in one pair of brackets to make each one true.
    (a) \(3 + 4 \times 5 = 35\)   (b) \(24 \div 4 + 2 = 4\)   (c) \(10 - 6 - 3 = 7\)   (d) \(2 \times 3 + 4 \times 5 = 70\)
  3. A box holds 4 red pens and 2 blue pens. A teacher buys 5 boxes, then gives away 3 pens. (a) Write one sum, with brackets, for the number of pens left. Work it out. (b) Sam forgets the brackets. What answer does Sam get?
  4. Start with \(8 - 2 \times 3 + 1\). (a) Work it out as it stands. (b) Put in one pair of brackets to make 19. (c) Move the brackets to make 0. (d) Move them again to make 1.

Stretch: 1, 2, 3, 4

Keep 1, 2, 3, 4 in that order. Put + or × in each gap, and add any brackets you like. (a) Make 21. (b) What is the biggest answer you can make?

Answers

  1. (a) \(5 + 12 = 17\). (b) \(7 \times 6 = 42\). (c) \(20 - 2 = 18\). (d) \(12 \div 4 = 3\).
  2. (a) \((3 + 4) \times 5\). (b) \(24 \div (4 + 2)\). (c) \(10 - (6 - 3)\). (d) \(2 \times (3 + 4) \times 5\), which is \(2 \times 7 \times 5\).
  3. (a) \((4 + 2) \times 5 - 3 = 30 - 3 = 27\) pens. (b) \(4 + 2 \times 5 - 3 = 4 + 10 - 3 = 11\).
  4. (a) \(8 - 6 + 1 = 3\). (b) \((8 - 2) \times 3 + 1 = 19\). (c) \(8 - 2 \times (3 + 1) = 8 - 8 = 0\). (d) \(8 - (2 \times 3 + 1) = 8 - 7 = 1\).

Stretch. (a) \((1 + 2) \times (3 + 4) = 3 \times 7 = 21\). (b) \((1 + 2) \times 3 \times 4 = 36\). Multiplying by 1 does nothing, so it is better to add the 1 to the 2 first.

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