Grouping symbols
The agreed order says multiply before you add. What do you write when the story needs the adding done first?
Where you are 5 in Order of operations
Why this exists
Dayo is buying paint at Jiaqi’s paint store. He picks 3 cans of blue and 4 cans of white, and every can costs 5 dollars. To find what he pays, you add the cans and then multiply by the price.
Try writing that as one line with no extra marks: 3 + 4 \times 5. The agreed order from Numerical expressions says multiply first, so that line names 3 + 20 = 23. But Dayo does not pay 23 dollars. He has 7 cans at 5 dollars each, and he pays 35 dollars.
The agreed order is a good one, and nobody wants to change it. What is missing is a way to say “do this part first” when a story needs it. That is what grouping symbols do:
The parentheses say: add 3 and 4 first, and treat the result as one number.
The lesson takes three days. Part 1 is parentheses: what they mean, how to work an expression that has them, and where to put them to reach a target value. Part 2 is brackets and braces, for a group inside a group, worked from the inside out. Part 3 is the fraction bar, which is a grouping symbol too, and parentheses that change nothing.
Definition
A grouping symbol marks a part of an expression that is to be worked out first and then treated as one number. There are four of them.
- Parentheses, (\;\;), are the most common: (3 + 4) \times 5.
- Brackets, [\;\;], and braces, \{\;\;\}, mean exactly what parentheses mean. They are used when one group sits inside another, so the eye can tell which closing mark goes with which opening mark: 2 \times [12 - (3 + 5)].
- The fraction bar is the line in \displaystyle \frac{12 + 8}{9 - 4}. Everything above the bar is one group, everything below it is another, and the bar means divide the top by the bottom.
The opening and closing marks always come in pairs. The part between a pair is called the inside of the group.
The full order, with grouping symbols, is this. Work out the inside of each group first. Inside a group, and outside it, use the agreed order: multiply and divide from left to right, then add and subtract from left to right. When the inside is done, the group is one number and the work goes on.
Intuition
Figure 1 shows Dayo’s cans. The 3 blue cans and the 4 white cans together make one row of 7, and every can in the row has the same price tag.
The price is not paid on the white cans alone. It is paid on each of the 7 cans. That is why the 3 + 4 is one piece of the story, and why it has to be one piece of the expression. The parentheses draw a fence around 3 + 4 and say “this is a single number”.
Without the fence, 3 + 4 \times 5 tells a different story: 3 loose dollars, plus 4 cans at 5 dollars each. Both stories are fine. They are just different stories, and grouping symbols let you write either one.
Results
Part 1: parentheses
Parentheses first. In an expression with parentheses, work out the inside first. Then the parentheses and what is in them are replaced by one number.
The inside follows the agreed order. If the inside has more than one operation, multiply and divide before you add and subtract, just as anywhere else:
Moving parentheses changes the value. The same numbers and signs, grouped in different ways, can name different numbers. So to reach a target value, you can try placing a pair of parentheses around different parts and work out each placement.
Part 2: brackets and braces
Inside out. When a group sits inside another group, work the innermost group first, then the group around it, and so on outward. Figure 2 in Example 4 shows this one step at a time.
Brackets and braces are only there to be read easily. Parentheses inside parentheses mean the same thing, 2 \times ((12 - 8) + 1), but 2 \times [(12 - 8) + 1] is easier on the eye. A common habit is parentheses on the inside, brackets around them, and braces around those.
Part 3: the fraction bar, and parentheses that change nothing
The fraction bar groups. A fraction means the top divided by the bottom. When the top or the bottom is an expression, the bar works like two pairs of parentheses:
So work out the whole top and the whole bottom, then divide.
Some parentheses change nothing. Parentheses change nothing when they go around a part the agreed order would work first anyway, as in (5 \times 6) + 4. They also change nothing when the only operation is adding, or the only operation is multiplying, because then any grouping gives the same value. Everywhere else they may change the value, and you have to work it out to know.
Why it is true
Grouping symbols are part of how expressions are written, like the plus sign. Nobody proves that parentheses go first; people agreed on it. What can be shown is why the agreement works the way it does.
Why the inside comes first. A group stands for one number. You cannot multiply (3 + 4) by 5 until you know which number (3 + 4) is. Finding it means working out the inside.
Why inside out. In 2 \times [12 - (3 + 5)] the bracket holds 12 - (3 + 5). That bracket cannot become one number until everything in it is a number, and (3 + 5) is not yet one. So the inner group has to be done first. The outer group always waits for the groups inside it.
Why the fraction bar groups. A fraction such as \displaystyle \frac{20}{5} also means 20 \div 5, one of the ideas in Fractions. A bar that runs under the whole of 12 + 8 is dividing the whole of 12 + 8, not just the 8. The length of the bar shows what it divides, the way the fence in Figure 1 shows what is added.
Why some parentheses change nothing. For adding, the associative property says (a + b) + c = a + (b + c): you may add either pair first. The same holds for multiplying, (a \times b) \times c = a \times (b \times c). So in an expression that only adds, or only multiplies, moving the parentheses never changes the value. Subtracting and dividing have no such property: (20 - 6) - 3 = 11, but 20 - (6 - 3) = 17. So parentheses after a minus sign or a division sign often matter.
Worked examples
Examples 1 to 3 go with Part 1, Examples 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.
What the parentheses change
Evaluate each expression. Then evaluate it again with the parentheses taken out, and compare.
(a) (2 + 8) \times 6; (b) 30 - (12 - 4); (c) (18 + 6) \div 3
Solution(a) The inside comes first: 2 + 8 = 10. Then 10 \times 6 = 60.
Without the parentheses, 2 + 8 \times 6 multiplies first: 2 + 48 = 50. A different number.
(b) The inside first: 12 - 4 = 8. Then 30 - 8 = 22.
Without them, 30 - 12 - 4 goes left to right: 18 - 4 = 14. With the parentheses only 8 is taken away. Without them, 12 and then 4 more are taken away.
(c) The inside first: 18 + 6 = 24. Then 24 \div 3 = 8.
Without them, 18 + 6 \div 3 divides first: 18 + 2 = 20.
The usual order inside the parentheses
Tuan buys 2 cans of paint at 9 dollars each and one brush for 6 dollars. He pays with a 50 dollar bill. Jiaqi works out his change as
Find his change. Then say what goes wrong without the parentheses.
SolutionThe parentheses hold what Tuan spends, so they come first. Inside them there is a multiplication and an addition, and the agreed order still holds there: multiply first, 2 \times 9 = 18, then add, 18 + 6 = 24. Now the group is one number:
Tuan gets 26 dollars back.
Without the parentheses, 50 - 2 \times 9 + 6 is 50 - 18 + 6 = 32 + 6 = 38. That takes away the cans and then adds the brush back, as if the store paid Tuan for it. The parentheses make sure the whole amount he spends is taken away.
Placing parentheses to reach a target
With no parentheses, 4 + 6 \times 8 - 2 has the value 4 + 48 - 2 = 50.
(a) Place one pair of parentheses to make the value 78.
(b) Place one pair to make the value 40.
(c) Can one pair make the value 60? Can two pairs?
SolutionA pair of parentheses goes around two or more numbers that sit next to each other, with the signs between them. Here are all the places one pair can go, each worked out.
| Parentheses around | Expression | Value |
|---|---|---|
| 4 + 6 | (4 + 6) \times 8 - 2 = 80 - 2 | 78 |
| 6 \times 8 | 4 + (6 \times 8) - 2 = 4 + 48 - 2 | 50 |
| 8 - 2 | 4 + 6 \times (8 - 2) = 4 + 36 | 40 |
| 4 + 6 \times 8 | (4 + 6 \times 8) - 2 = 52 - 2 | 50 |
| 6 \times 8 - 2 | 4 + (6 \times 8 - 2) = 4 + 46 | 50 |
| all of it | (4 + 6 \times 8 - 2) | 50 |
(a) (4 + 6) \times 8 - 2 = 78. Adding first gives 10 eights instead of 6 eights.
(b) 4 + 6 \times (8 - 2) = 40. Subtracting first gives 6 sixes instead of 6 eights.
(c) No single pair makes 60: the table shows every placement, and the only values are 50, 78 and 40. Two pairs do it. Group both the 4 + 6 and the 8 - 2:
Four of the six placements give 50, the same as no parentheses at all. Part 3 comes back to parentheses that change nothing.
Brackets around parentheses
Evaluate 2 \times [12 - (3 + 5)].
SolutionThere is a group inside a group. The parentheses are inside the brackets, so they are the innermost group and go first. Figure 2 shows each step.
The parentheses: 3 + 5 = 8. The bracket now reads [12 - 8], which is all numbers, so it can be worked out: 12 - 8 = 4. Last comes the multiplication outside:
Starting on the outside does not work. The 2 \times needs one number to multiply, and the bracket is not one number until its inside is done.
Braces, then what is outside
Evaluate 4 + 3 \times \{20 - [2 \times (4 + 3)]\}.
SolutionThree groups sit one inside the next: parentheses, then brackets, then braces. Work from the inside out.
The parentheses: 4 + 3 = 7. The brackets now hold 2 \times 7 = 14. The braces now hold 20 - 14 = 6.
Every group is gone, and what is left follows the agreed order. Multiply first, 3 \times 6 = 18, then add:
Notice that the 4 + 3 inside the parentheses was added first, but the 4 + 3 \times at the front was not. The same signs are treated differently because one pair is inside a group and the other is not.
The fraction bar as a grouping symbol
(a) Evaluate \dfrac{12 + 8}{9 - 4}, and write it on one line with a division sign.
(b) Evaluate \dfrac{6 \times 7 - 2}{2 \times 4}.
Solution(a) The bar groups the whole top and the whole bottom, as in Figure 3. Top: 12 + 8 = 20. Bottom: 9 - 4 = 5. Then divide the top by the bottom: 20 \div 5 = 4.
On one line there is no bar to show what is grouped, so both groups need parentheses:
Leave the parentheses out and 12 + 8 \div 9 - 4 divides only the 8 by only the 9. Its value is not even a whole number, because 8 \div 9 is the fraction \displaystyle \frac{8}{9}.
(b) Top first, with the agreed order inside it: 6 \times 7 = 42, then 42 - 2 = 40. Bottom: 2 \times 4 = 8. Then divide:
Parentheses that change nothing
For each expression, decide whether taking out the parentheses changes the value.
(a) (5 \times 6) + 4; (b) 18 - (6 + 2); (c) 5 + (10 + 20); (d) 24 \div (6 \div 2)
Solution(a) No change. With them, 30 + 4 = 34. Without them, 5 \times 6 + 4 multiplies first anyway, and gives 34. The parentheses go around a part the agreed order would do first.
(b) A change. With them, 18 - 8 = 10. Without them, 18 - 6 + 2 goes left to right: 12 + 2 = 14. With the parentheses the 2 is taken away. Without them it is added.
(c) No change. With them, 5 + 30 = 35. Without them, 5 + 10 + 20 = 15 + 20 = 35. Here the parentheses do change which sum is found first. The value stays the same anyway, because the only operation is adding, and by the associative property any pair may be added first.
(d) A change. With them, 24 \div 3 = 8. Without them, 24 \div 6 \div 2 goes left to right: 4 \div 2 = 2. Dividing has no associative property.
Parentheses that change nothing are still allowed. Writers sometimes add them so a reader sees the pieces at a glance.
Boundaries
Every group needs its partner. An opening mark with no closing mark, as in (3 + 4 \times 5, does not say where the group ends, so it names no number. Each ( needs a ), each [ a ], and each \{ a \}.
A group may not divide by zero. \displaystyle \frac{6}{3 - 3} asks for 6 \div 0, and no number does that. An expression whose bottom group comes out to 0 has no value.
A times sign is written here. In this lesson a number next to a group always has a sign, as in 4 \times (5 - 2). Later books often leave the sign out and write 4(5 - 2); the two mean the same thing.
Staying with whole numbers. Every group here comes out to a whole number. A group such as (9 - 12) asks for a number less than zero, which belongs to the order of operations with negative numbers, and a fraction bar whose top is not a multiple of its bottom gives a fraction.
No powers yet. Powers, written with a small raised number, have their own place in the order, between grouping and multiplying. They come in the order of operations with powers.
Where this goes next
With grouping symbols, any story with adding, subtracting, multiplying and dividing can be written in one line. The next step is turning words into those lines, and reading a line back as words without working it out, in writing and reading expressions from words. “Add 8 and 7, then multiply by 2” is 2 \times (8 + 7), and the parentheses are what make it say that.
The full order later adds powers: grouping first, then powers, then multiplying and dividing, then adding and subtracting. Grouping stays at the top.
The fraction bar keeps grouping in algebra. When a fraction with a sum on top is subtracted, the whole sum is taken away, which is why adding and subtracting fractions with letters needs the same care as Example 7 (b).
Exercises
Part 1: parentheses
Evaluate each expression.
- (9 + 6) \times 3
- 45 - (20 - 8)
- (36 - 12) \div 6
- 8 \times (11 - 4)
- (5 + 3 \times 9) \div 4
- 7 \times (10 - 6) + 3
- At Jiaqi’s paint store, each can of paint costs 12 dollars, plus 2 dollars to mix in the color. Dayo buys 5 cans. He writes his cost as 5 \times 12 + 2, and Jiaqi writes it as 5 \times (12 + 2). Who is right, and what does Dayo pay? What does the other expression name?
Part 1: placing parentheses
- With no parentheses, 3 + 9 \times 6 - 1 has the value 56. Place one pair of parentheses to make the value (a) 71; (b) 48.
- With no parentheses, 40 - 10 \div 2 + 3 has the value 38. (a) Place one pair of parentheses to make the value 32. (b) Place two pairs to make the value 6.
Part 2: brackets and braces
Evaluate each expression.
- 3 \times [20 - (4 + 6)]
- [18 - (2 \times 5)] \div 4
- 50 - [6 \times (11 - 7)]
- 2 \times \{30 - [4 \times (9 - 3)]\}
- \{[8 + (5 \times 4)] \div 7\} + 9
- Tuan buys 3 paint kits. Each kit holds one can of primer at 7 dollars and 2 cans of color at 6 dollars each. A coupon takes 10 dollars off the whole order, and he pays with 100 dollars. His change is 100 - [3 \times (7 + 2 \times 6) - 10]. Say what the parentheses and the brackets each stand for, and find his change.
- Rewrite 4 \times ((10 - 2) - 3) with brackets in place of the outer parentheses, and evaluate it.
Part 3: the fraction bar
Evaluate each expression.
- \dfrac{15 + 9}{10 - 4}
- \dfrac{7 \times 8}{2 + 5}
- \dfrac{45 - 3 \times 5}{6}
- Write \dfrac{30 + 18}{4 \times 3} on one line with a division sign, and evaluate it.
- For each, say whether taking out the parentheses changes the value. (a) (7 \times 4) - 3; (b) 30 - (10 - 4); (c) 2 \times (3 \times 5); (d) 48 \div (4 \times 2)
- Jiaqi mixes 14 liters of white paint with 10 liters of blue, then pours the mix equally into 6 cans. Dayo writes the amount in each can as \dfrac{14 + 10}{6}. How many liters go in each can? Why would 14 + 10 \div 6 be the wrong way to write it on one line?
Reasoning
- Dayo evaluates 5 \times [16 - (6 + 2)] by starting on the bracket: he does 16 - 6 first, then adds 2, then multiplies by 5. What does he get? What is the value, and why must the parentheses go first?
- Tuan says a fraction bar is just a division sign, so \dfrac{20 - 8}{4} is the same as 20 - 8 \div 4. Is he right? Explain, using the values of both.
- Jiaqi says that putting parentheses into an expression that only adds, such as 2 + 3 + 4 + 5, can never change its value. Dayo says the same holds for an expression that only subtracts, such as 30 - 9 - 5 - 2. Who is right? Explain.
Answers
- 45. Inside first: 9 + 6 = 15, then 15 \times 3 = 45.
- 33. Inside first: 20 - 8 = 12, then 45 - 12 = 33.
- 4. Inside first: 36 - 12 = 24, then 24 \div 6 = 4.
- 56. Inside first: 11 - 4 = 7, then 8 \times 7 = 56.
- 8. Inside, multiply first: 3 \times 9 = 27, then 5 + 27 = 32. Then 32 \div 4 = 8.
- 31. Inside first: 10 - 6 = 4. Then multiply before adding: 28 + 3 = 31.
- Jiaqi is right: each of the 5 cans costs 12 + 2 = 14 dollars, so Dayo pays 5 \times 14 = 70 dollars. Dayo’s expression names 60 + 2 = 62, which charges for mixing only once.
- (a) (3 + 9) \times 6 - 1 = 72 - 1 = 71. (b) 3 + 9 \times (6 - 1) = 3 + 45 = 48.
- (a) 40 - (10 \div 2 + 3) = 40 - 8 = 32. (b) (40 - 10) \div (2 + 3) = 30 \div 5 = 6.
- 30. Inner first: 4 + 6 = 10, then 20 - 10 = 10, then 3 \times 10 = 30.
- 2. Inner first: 2 \times 5 = 10, then 18 - 10 = 8, then 8 \div 4 = 2.
- 26. Inner first: 11 - 7 = 4, then 6 \times 4 = 24, then 50 - 24 = 26.
- 12. 9 - 3 = 6, then 4 \times 6 = 24, then 30 - 24 = 6, then 2 \times 6 = 12.
- 13. 5 \times 4 = 20, then 8 + 20 = 28, then 28 \div 7 = 4, then 4 + 9 = 13.
- The parentheses are the cost of one kit, 7 + 12 = 19 dollars. The brackets are the cost of the order after the coupon, 3 \times 19 - 10 = 57 - 10 = 47 dollars. His change is 100 - 47 = 53 dollars.
- 4 \times [(10 - 2) - 3] = 4 \times [8 - 3] = 4 \times 5 = 20.
- 4. Top: 15 + 9 = 24. Bottom: 10 - 4 = 6. Then 24 \div 6 = 4.
- 8. Top: 7 \times 8 = 56. Bottom: 2 + 5 = 7. Then 56 \div 7 = 8.
- 5. Top: 3 \times 5 = 15, then 45 - 15 = 30. Then 30 \div 6 = 5.
- (30 + 18) \div (4 \times 3) = 48 \div 12 = 4.
- (a) No change: both give 28 - 3 = 25. (b) A change: 30 - 6 = 24 with them, 30 - 10 - 4 = 16 without. (c) No change: both give 30, since only multiplying is involved. (d) A change: 48 \div 8 = 6 with them, 48 \div 4 \times 2 = 12 \times 2 = 24 without.
- 4 liters: the top is 14 + 10 = 24 liters, and 24 \div 6 = 4. On one line it must be (14 + 10) \div 6. Without the parentheses, 14 + 10 \div 6 divides only the blue paint.
- Dayo gets 5 \times 12 = 60, since 16 - 6 = 10 and 10 + 2 = 12. The value is 40: 6 + 2 = 8, then 16 - 8 = 8, then 5 \times 8 = 40. The parentheses say 6 + 2 is one number, so all 8 is taken from 16. Working the bracket first takes away 6 and adds 2, which is a different calculation.
- No. The bar divides the whole top, so \dfrac{20 - 8}{4} = 12 \div 4 = 3. But 20 - 8 \div 4 divides only the 8: 20 - 2 = 18. The bar is a division sign with parentheses built in: (20 - 8) \div 4.
- Jiaqi is right. By the associative property any pair may be added first, so 2 + 3 + 4 + 5 is 14 however it is grouped. Dayo is wrong: 30 - 9 - 5 - 2 = 14, but 30 - (9 - 5) - 2 = 30 - 4 - 2 = 24. Subtracting has no associative property.