The Math Primer: Number 5.3 Order of Operations

8/8/2019 The Math Primer: Number 5.3 Order of Operations

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Number 5.3 Order of Operations1. Why we need to perform operations in a certain order

Stumpy Pete, Ailing Jenny and Greasy Joan are at The Dead Mans Chest half-drunk and

doing speed math.

Whoever finds the value of the following expression first, wins:

3 + 2 4 1

They all solve the problem in a few seconds, but get completely different answers:

StumpyPete: 19

AilingJenny: 15

GreasyJoan: 10

The first thing they do is check their workings, but they still get the same answers.

The problem is that each pirate added, multiplied and subtracted in a different order:

Pete worked left right:

3 + 2 4 1

5

5 4 1

20 1= 19

20

Jenny did + and first:

3 + 2 4 1

5

5 4 1

Joan did first:

3 + 2 4 1

3

5 3 = 15

8

3 + 8 1

11

111= 10

Whoisright?

Being pirates, they argue about who is right and end up drawing cutlasses and pistols

and starting a fight.

Before anyone gets hurt, Captain Jack throws a bucket of water over the lot of them and

says, There are rules about the order in which you do things, and only one of you

followed the rules! So only one of you is right!

But what are the rules, and who is right?


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2. Two rules for order of operations, but not the only rules

There are four rules, called the order of operations:

Rule 3: First, do any and working from left right

Rule 4: Next, do any + and working from left

rightWhataboutrules1and2?Wellgettothat,youscurvymonkey!

Lets try some examples:

Problem 1

2 + 6 2

2 +12 = 14

2 + 6 2

Dothe

first

Problem 2

4 2 2

Problem 3

3 + 6 3

4 2 2

=0 =21

Problem 4

6 4 2

Problem 5

8 9 3

Problem 6

12 6 3

Problem 7

4 12 6

The rule is the same for division:

6 4 2

62=

4

After multiplication or division, you do addition and subtraction from left to right:

Problem 8

3 + 6 2 4

Problem 9

3 + 4 2 3

Problem 10

6 + 8 4 + 2

Problem 11

12 + 8 12 4

3 + 6 2 4

3 + 68

9 8 = 1

3 + 4 2 3

3 + 23

6 + 8 4 + 2

=5 =10 =2

=2 =10 =17


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2b. Two rules for order of operations, but not the only rules (continued)

If there is more than one or then you do each one from left to right:

Problem 1

6 3 2 1

2 2 1

4 1= 3

6 3 2 1

Problem 2

8 4 2 + 4

Problem 3

4 3 2 +1

Problem 4

16 4 2 1

8 4 2 + 4

2 2 + 4

=5

4 3 2 +1

=25 =1

Problem 5

8 2 3 3

=3

Problem 6

6 3 + 8 4

=20

Problem 7

5 5 + 6 3

=3

Problem 8

4 2 6 3

=6

So which of the pirates was right? Look at their work again and see:

Stumpy Pete

3 + 2 4 1

= 5 4 1

= 20 1

= 19

Ailing Jenny

3 + 2 4 1

= 5 4 1

= 5 3

= 15

Greasy Joan

3 + 2 4 1

= 3 + 8 1

= 111

= 10

GreasyJoanistheonlyonewhofollowedtheorderofoperationsrules.

Now lets do some practice.


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Problem 14

18 +15 3 2 7

3. Practice

Remember the rules weve learnt so far:

Rule 3: First, do any and working from left right

Rule 4: Next, do any + and working from left

right

Problem 1

52 13+17

=56

Problem 2

20 +12 15

=17

Problem 3

3314 10

=9

Problem 4

12 9 + 3 6

=0

Problem 5

49 7 + 8

=15

Problem 6

49 7 8

=56

Problem 7

49 + 7 8

=105

Problem 8

49 7 7

=0

Problem 9

12 12 6

=24

Problem 10

12 6 12

=24

Problem 11

6 12 12

=6

Problem 12

6 +12 12

=150

Problem 13

15 3 3 + 8 4

=8

Problem 15

49 7 2 3 4

=9 =2


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4. Describing whats happening using words and pictures

To be really good at this, it helps to be able to describe what were doing. For instance:

4 2 3

This means 4 groups of 2 joined together, then take away 3.

4 2 = 8 3 = 5

Division is a little bit harder to describe (see if you can fill in the blank).

6 2 1

This means take one group of 6 and split it into 2 even groups. Now looking at just

one of those new groups, subtract 1, and count how many are in the group now.

6 = 3 = 2 2 11

Division is a little bit harder to describe (see if you can fill in the blank).

3 3 2 4

Join 3 groups of 3 to make one big group, then take away 2 groups of 4 and see

what you have left.

3 3 = 1= 9=9

2 4

Example 1

Example 2

Example 3


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4. Describing whats happening using words and pictures (continued)

Now you try. Describe each expression using sketches and words (you can draw squares,

circles, faces, animals, whatever you like).

6 2 9Join 6 groups of 2 into one big group, then take away 9.

6 2 = 12 9

Share your descriptions with somebody else and see if they can understand you.

6 + 8 4 2

Take one group of 8 and divide it into 4 even groups. Now focus on just one of those

new, smaller groups. Add 6 to this group, then subtract 2.

8 = 24 +6=6

This last one is tricky. You have to find a way to describe it using words. There are many

ways to do this. One option is in the answer at the bottom.

4 2 6 3

=3

2

Problem 1

Problem 2

Problem 3

Possibledescription:Join4groupsof2tomake8.Nowlookatagroupof6andbreakitinto3evenpieces.Howmanyineachpiece?Thereare2.Takethismanyawayfromthefirstgrouptoleaveuswith6.


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=25

5. Using brackets

Back at The Dead Mans Chest, Captain Jack has stopped the pirates fighting about what

order they do their operations in. But being pirates, its only a few minutes before

somebody is arguing again.

Stumpy Pete says, I get the rules, but what if I dont want to follow them? What if Iwant to do something in a different order? For instance:

3 + 2 5

According to the rules this is 3, plus 2 groups of 5, to make 15...

+ =

3 2 5 = 15

But what if I want to say, combine 3 and 2 to make a new group of 5, and then multiply

that group by 5?

3 + 2

=+= 5

5 5= 25

5=

Greasy Joan, having just won the previous argument, shrieks, You cant do that! He

just told you the rules! and tries to hit Stumpy Pete with a mug.

But Captain Jack says, I didnt tell you all the rules.

Rule 2: First, do anything in (brackets)

This means you can change the way numbers are grouped by using brackets. InStumpy Petes example above, you would simply do this:

Example 1

(3 + 2) 5

= 5 5

= 25

2 (3 +1)Tryityourself.Dothebracketsfirst.

=8

Thisisthenormalway...

Butwhatifyoureallywantedtodothis?

Dothebracketsfirst.


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Problem 2

(6 3) 3

Problem 3

(3 + 6) 3

6. Brackets practice

So now we have three rules:

Rule 3: Next, do any and working from left

rightRule 4: Finally, do any + and working from left right

Wheresrule1?Yourenotreadyforrule1!Thatsforanothertime!Gogetsomebooty!But-Yaargh!

Rule 2: First, do anything in (brackets)

Lets try some examples:

Problem 1

(2 + 3) 4

=

20

= 5 4

Brackets

first

=1 =3

Problem 4

10 (5 3)

Problem 5

4 + (2 +1) 2

= 4 + 3 2

=

=

Problem 6

6 (5 1) 2

=

=

=

Problem 7

8 + (4 2) 2

=5 =4 =12=10

Problem 8

10 (6 3) 3

Problem 9

5 (2 + 2) 10

= 5 10

= 20 10

=

Problem 10

8 (5 4) 4

= 8 4

=

=

Problem11

(12 3) 4 15

=

=

==9 =32 =21=2

Problem 12

28 6 (2 + 4) 3

=16

Problem 13

18 + (28 7) 311

Problem 14

8 + (20+ 8 2) 4 (15 11)

=14 =10

Whats first?


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7. Spot the mistakes

Take this expression:

30 3 15 3 + (12 3) 5

Find the mistakes in each solution below:

Solution 1

30 3 15 3+ (12 3) 5

= 90 5 + 9 5

= 85 + 9 5

= 94 5

= 470

Mistakeisat905

Solution 2

30 3 15 3 + (12 3) 5

= 90 5 + 9 5

= 90 14 5

= 90 70

= 20

Mistakeisat5+9

Solution 3

30 3 15 3+ (12 3) 5

= 90 5 + 9 5

= 90 5 + 45

= 85 + 45

= 130

Mistakeisat...No,itscorrect!

Solution 4

30 3 15 3+ (12 3) 5

= 90 15 3+ 9 5

= 75 3+ 9 5

= 25 + 9 5

= 25 + 45

= 70Mistakeisat9015


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8. Practice

Problem 1

16 + 5 +11

Problem 2

46 + 68 37

Problem 3

3 7 5

Problem 4

81 9 3

=32 =105 =3=77

Problem 13

(24 12) (5 + 3)

Problem 14

(32 4) 7 3

Problem 15

71+ (24 8)10

Problem 16

15 (20 4 3)

=16 =231 =30=1

Problem 5

10 2 8

Problem 6

26 14 2

Problem 7

(18 2) 2

Problem 8

14 9 3

=40 =8 =42=19

Problem 9

14 (9 3)

Problem 10

48 5 3

Problem 11

95 5 3+ 64

Problem 12

(6 + 5) (15 7)

=42 =144 =88=33

Problem 17

(45 18) (32 8)

=108

Problem 18

(89 14) (20 3 5)

Problem 19

2 5 (5 4 8 4)

=15 =180

(Calculators ok)

Documents

The Math Primer: Number 5.3 Order of Operations