How Grouping Symbols Change the Value of an Expression | Grade 5 Math | LevelKite

Here are two expressions:

$3 \times 4 + 2$
$3 \times (4 + 2)$

A numerical expression is a row of numbers joined by operation signs ( $+$ , $-$ , $\times$ or $\div$ ). The one number it stands for is its value, and to evaluate an expression is to work out that value.

Both expressions use $3$ , $4$ and $2$ , with $\times$ and then $+$ . The only change is a pair of parentheses, the curved marks around $4 + 2$ . Do the two expressions still have the same value?

Crayons can show what each expression means.

Three boxes of 4 crayons, plus 2 loose crayons. Multiply first to count the crayons in boxes, then add the loose ones. In symbols, that is $3 \times 4 + 2 = 12 + 2 = 14$ crayons.

Three boxes, each with 4 red crayons and 2 blue crayons. Add first to find that each box holds 6 crayons, then multiply by the 3 boxes. In symbols, that is $3 \times (4 + 2) = 3 \times 6 = 18$ crayons.

The parentheses make $4 + 2$ one group. So the $3$ multiplies the whole group, not just the $4$ . The numbers and signs are the same, but the values are $14$ and $18$ .

The same numbers and signs with parentheses in different places

Expression	Goes first	Rest of the work	Value
$8 + 4 \times 3 - 1$	$4 \times 3 = 12$	$8 + 12 - 1 = 20 - 1$	$19$
$(8 + 4) \times 3 - 1$	$8 + 4 = 12$	$12 \times 3 - 1 = 36 - 1$	$35$
$8 + 4 \times (3 - 1)$	$3 - 1 = 2$	$8 + 4 \times 2 = 8 + 8$	$16$
$8 + (4 \times 3) - 1$	$4 \times 3 = 12$	$8 + 12 - 1 = 20 - 1$	$19$

Every row of the table uses $8$ , $4$ , $3$ and $1$ with $+$ , $\times$ and $-$ in the same order. Only the parentheses change. Look at which part goes first in each row.

No parentheses: $\times$ goes first, so you multiply only the $4$ by $3$ . The value is $19$ .
Parentheses around $8 + 4$ : you add $8$ and $4$ first. Now the $3$ multiplies all of $12$ , so the value rises to $35$ .
Parentheses around $3 - 1$ : you subtract first and get $2$ . Now you multiply $4$ by $2$ instead of by $3$ , so the value falls to $16$ .
Parentheses around $4 \times 3$ : this part goes first anyway. Nothing changes, and the value is still $19$ .

Key concept

Grouping symbols choose what goes first

Grouping symbols tell you which part of an expression to work out first. When they make a different part go first, the value can change, even though the numbers and signs stay the same. When they go around a part that would go first anyway, the value stays the same. To be sure of a value, evaluate the expression in the agreed order.

Worked example

Compare two groupings

The expressions $(40 - 24) \div 2 + 4$ and $40 - 24 \div (2 + 4)$ use the same numbers and signs in the same order. Only the parentheses are in different places. Compare their values and explain the difference.

Start with the first expression. Its parentheses are around the subtraction, so subtract first. Taking 24 from 40 leaves 16.
$(40 - 24) \div 2 + 4 = 16 \div 2 + 4$
No parentheses are left. Division comes before addition, so divide 16 by 2 to get 8. Then add 4.
$16 \div 2 + 4 = 8 + 4 = 12$
Now the second expression. Its parentheses are around the addition, so add first. Adding 2 and 4 makes 6.
$40 - 24 \div (2 + 4) = 40 - 24 \div 6$
No parentheses are left here either. Division comes before subtraction, so divide 24 by 6 to get 4. Then subtract 4 from 40.
$40 - 24 \div 6 = 40 - 4 = 36$
Compare the values and say why they differ. In the first expression the subtraction goes first, so you divide the 16 that is left by 2. In the second the addition goes first, so you divide 24 by 6 and take only 4 away from 40.
$12 < 36$

Answer

$(40 - 24) \div 2 + 4$ has the value $12$ , and $40 - 24 \div (2 + 4)$ has the value $36$ . The values differ because the parentheses make a different part go first.

What goes first also matters when a sign is missing. An equation is a number sentence with an equals sign. It is true when both sides have the same value.

Sometimes a box hides one operation sign, as in $18 - 6 \;\square\; 3 = 16$ . A different sign in the box can change which part goes first, so test the signs one at a time:

Put one sign in the box. Use the same order every time: $+$ , then $-$ , then $\times$ , then $\div$ .
Evaluate the side with the box in the agreed order.
Compare with the other side. If the two values match, that sign makes the equation true.

Worked example

Find the missing sign

Which operation sign makes this equation true? $18 - 6 \;\square\; 3 = 16$

Test the plus sign. There are no grouping symbols, and with only addition and subtraction you work from left to right. So subtract 6 from 18 first, then add 3. The value is 15, not 16.
$18 - 6 + 3 = 12 + 3 = 15$
Test the minus sign. Work from left to right again: subtract 6, then subtract 3. The value is 9, not 16.
$18 - 6 - 3 = 12 - 3 = 9$
Test the times sign. With no grouping symbols, multiplication comes before subtraction, so multiply 6 by 3 first. Then subtract that 18 from 18. The value is 0, not 16.
$18 - 6 \times 3 = 18 - 18 = 0$
Test the division sign. Division also comes before subtraction, so divide 6 by 3 first. Then subtract that 2 from 18. The value is 16, so both sides match.
$18 - 6 \div 3 = 18 - 2 = 16$

Answer

The missing sign is $\div$ . The equation $18 - 6 \div 3 = 16$ is true, and the other three signs do not give $16$ .

Last, you choose where the grouping symbols go. You get an equation that is not true yet, and you insert grouping symbols, such as parentheses, to make it true. This plan helps:

Evaluate the expression as it is written. Now you know its value before any new grouping.
Find the part that goes first already. Parentheses around that part will not change the value, so skip it.
Put parentheses around a different part and evaluate again.
If the two sides still do not match, move the parentheses and test again. When they match, your evaluation shows that the equation is true.

Worked example

Place the parentheses

Insert one pair of parentheses to make this equation true: $6 + 12 \div 3 - 1 = 12$

Evaluate the left side as it is written. With no grouping symbols, division comes before addition and subtraction, so divide 12 by 3 first. Then add and subtract from left to right. The value is 9, not 12, so the equation is not true yet.
$6 + 12 \div 3 - 1 = 6 + 4 - 1 = 10 - 1 = 9$
The division already goes first, so parentheses around it would change nothing. Put them around the addition instead, so the addition goes first. Then divide 18 by 3, and then subtract 1. The value is 5, not 12.
$(6 + 12) \div 3 - 1 = 18 \div 3 - 1 = 6 - 1 = 5$
Move the parentheses to the subtraction, so the subtraction goes first. Then divide 12 by 2, because division comes before addition, and then add 6 and 6. The value is 12.
$6 + 12 \div (3 - 1) = 6 + 12 \div 2 = 6 + 6 = 12$
Check the equation. The left side has the value 12 and the right side is 12, so the two sides match.
$6 + 12 \div (3 - 1) = 12$

Answer

Put the parentheses around $3 - 1$ . The equation $6 + 12 \div (3 - 1) = 12$ is true because the left side evaluates to $12$ .

Takeaway: Grouping symbols choose which part of an expression you work out first. That choice can change the value, even when the numbers and signs stay the same. So when you compare two groupings, test a missing sign or place parentheses yourself, evaluate each time in the agreed order and let the value decide.