# The order of operations

*Keywords:*

AO elaboration and other teaching resources

The purpose of this unit is to develop the students’ understanding of how number operations behave, to recognise that there is a need for rules to guide us in the order in which we carry out these operations, and to interpret and apply these rules in problem solving situations.

- Recognise that the order of operations matters in practical real life situations.
- Understand and explain the relationship between addition and subtraction, and between multiplication and division.
- Recognise the ambiguity of expressions and equations that include more than one operation.
- Recognise the order of operations in spoken number problems.
- Understand and explain the rules for the order of operations, including explaining the acronym, BEDMAS.
- Apply the order of operations to solve problems.

When students have come to understand and correctly use the common symbols for relationships (=, ≠, <, >) and for number operations (+ - x ÷), they can express simple mathematical ideas and problem situations using these symbols, and can and interpret (and solve) familiar mathematical equations and expressions. However, when they encounter problems that involve more than one operation, they may be perplexed by the ambiguity in some expressions and equations. This can lead to differing interpretations and results. For example, 4 + 2 x 5 = ☐. Is this equal to 30 or to 14?

In oral and written language we take for granted the very important punctuation that we use. The controversial statement, ‘A woman without her man is nothing,’ for example, has quite a different meaning when punctuation is inserted thus, ‘A woman: without her, man is nothing.’ So too with a simple mathematical problem such as, ‘Four plus two, times five, is equal to what?’ or ‘Four, plus two times five, is equal to what?’

In the absence of this punctuation, rules have been established in which parenthesis (brackets) and a stated order for carrying out the number operations, remove any ambiguity. Whilst scientific calculators produce the correct answers to multistep equations, students should understand the need for the agreed order, know the correct conventions and be able to apply these to solve problems.

The activities suggested in this series of lessons can form the basis of independent practice tasks.

**Links to the Number Framework **

Multiplicative

Advanced multiplicative

**Session 1**

SLOs:

- Recognise that the order of operations matters in practical real life situations.
- Understand and explain the relationship between addition and subtraction, and between multiplication and division.
- Explore conjectures and proof.
- Recognise the ambiguity of expressions and equations that include more than one operation.
- Recognise the order of operations in spoken number problems.

__Activity 1__

- Print onto card the sets of action statements in Attachment 1.

Begin by distributing one to each student in the class. As a whole group, have students read their statements aloud and listen closely to each other’s statements. Students then form groups of related statements and agree on the logical order of their action statements.

Have groups present their action statements in order to the whole class. For example:

1. Get out of bed. 2. Have a shower. 3. Get dressed. 4. Leave the house. 5. Arrive at school. - Have some groups then read them
*out of order*to highlight the fact the nonsense of this, and that there is a*sensible order*to their actions and that*some actions must be performed before others*.

__Activity 2__

- Ask a student to record on the class chart the
(use this term). Have students explain each of the operation symbols,**number operations****+ - x ÷**. - Have students discuss in pairs the
*relationship between the operations*of addition and subtraction, and between multiplication and division.

Highlight thein each pair of operations. Have students explain this with equation and word examples, such as:**inverse relationship**

30 + 4 = 34 and 34 – 4 = 30

‘Pip has in her wallet, thirty dollars in notes and four dollars in coins. This is thirty-four dollars altogether. She pays four dollars for a coffee which leaves her with thirty dollars.’

Subtraction undid addition. It is the inverse operation.

16 x 4 = 64 and 64 ÷ 4 = 16

Four friends earned sixteen dollars each. Altogether the employer paid out sixty-four dollars. Each person took their share of sixteen dollars so that was sixty-four divided by four.

Division undid multiplication. It is the inverse operation.

__Activity 3__

- Refer to the ordering actions task in Activity 1, Step 1 (above). The order was important.

Pose:*‘***In mathematics**the order that we carry out number operations does not change the outcome/result.’

Have students discuss this and indicate their agreement/disagreement and.**their reasons for their position**

Record and discuss their ideas. Building upon this: - Write this equation on the class chart.

12 + 4 x 5 – 4 = ☐

Have student pairs solve the equation and explain their solution(s).

Have three students.**read the equation aloud in three different ways**

Write these, using words, emphasising the different**punctuation**.*Twelve plus four, times five, minus four, is equal to seventy-six.*

Twelve, plus four times five, minus four, is equal to twenty-eight.

Twelve plus four, times five minus four, is equal to sixteen. - Have students work in pairs to write each of these three interpretations as equations,
*inventing their own ‘mathematical punctuation’*. Have them share these, writing these on the class chart for others to see. Accept and explore all suggestions. - Agree that
**brackets**are helpful, and record these. Explain that brackets are also known as**parenthesis**.

(12 + 4) x 5 – 4 = 76

12 + (4 x 5) – 4 = 28

(12 + 4) x (5 – 4) = 16 - Have students discuss and suggest possible
*practical*scenarios for*each**representation*of the problem. For example:

(12 + 4) x 5 – 4 = 76*Lewis earns twelve dollars per hour packing boxes, but gets a bonus of four dollars each hour if he exceeds the total of boxes set as a per-hour target. He works for five hours, getting a bonus each hour. Once he’s spent four dollars on a coffee he has seventy-six dollars left.*

__Activity 4__

- Make chart paper and coloured pens available to the students.

Have student pairs together write on charts at least three of their own*multistep*equations, showing each in*words with punctuation*, and in*symbols*.

Have them write real life scenarios for at least one ‘version, of each of their problems. - Have students pair share these.

NB. The charts will be added to in Session 2 when BEDMAS is introduced.

__Activity 5__

Conclude the session by reviewing the conjecture posed in Activity 3, Step 1 (above).*‘The order that we carry out number operations makes no difference to the outcome.’ **Disagree* with the statement. *Agree* that the order in which number operations are carried out **does** make a difference to the result, and that using brackets helps us to understand the order of operations.

**Session 2**

SLOs:

- Understand and explain the rules for order of operations, including explaining the acronym, BEDMAS.
- Apply the order of operations to solving problems.

__Activity 1__

- Review the ordering of actions undertaken at the beginning of Session 1 and the conclusions from that session.
- Explain to students that mathematicians have agreed on an
or mathematical actions. Write BEDMAS, explaining that this is an acronym for the agreed convention.**order of operations**

Ask, ‘What is an?**acronym**

Who can suggest another well know acronym?

What is a?*convention*

What could**BEDMAS**stand for?

Have students discuss each of these questions, then record their ideas beside the acronym letters on the class chart. (E for exponents may not be known).

B :brackets. E: exponents, D: division, M: multiplication, A: addition, S: subtraction.

Explain that the order of operations is sometimes known as*operation precedence*. It is a rule that we use to clarify what operation we do first. (to*precede*means to come (or go) before or first.)

If not well understood, explain*exponentiation*, demonstrating that it is repeated*multiplication*: ie 4^{2}= 4 x 4 = 16, or 10^{3}= 10 x 10 x 10 = 1000

__Activity 2__

- Distribute Attachment 2. Explain that the students should investigate how BEDMAS works by applying it to the equations. Remind them that it may be useful to first read each problem aloud, making sense of the equation by ‘hearing’ it.

Have students work through the problems in pairs.They should also discuss and note any questions they have as they do so.**They should record which operation they complete first, and explain in words why.** - Have students pair share their solutions to each of the equations, explaining the order of operations they applied in each example.
*Make the following***teaching points**as they do so:- Division and multiplication are undertaken in the order in which they appear, working left to right.
- Addition and subtraction are undertaken in the order in which they appear, working left to right.
- Note in examples, 3, 4, 5, 6, 13, and 15, the effect of using the inverse operations.
- When solving what is in brackets, the order of operations applies.

__Activity 3__

- Gather as a class. Have students read several of the equations aloud, punctuating the reading with commas (pauses) where expressions are in brackets, for example:

9. (25 – 5) ÷ 5 = ☐*Twenty five minus five, divided by five, is equal to four.*

and punctuating the reading with commas (pauses) to recognise the order of operations, for example:

13. 25 + 5 x 5 ÷ 5^{2}= ☐*Twenty five plus, five times five divided by five squared, is equal to twenty six.* - Conclude that
*the order that we carry out number operations certainly*.**does change**the outcome/result

__Activity 4__

Have student pairs add to the charts begun in Session 1. *Explaining the order of operations in their own words*, showing the meaning of the acronym BEDMAS and highlighting the ‘punctuation effect’ of the brackets.

**Session 3**

SLO:

- Apply the order of operations to solving problems.

__Activity 1__

- Pose:
*Having the BEDMAS convention means that we will all agree on the value of an unknown amount in an equation.*

Have students discuss this in pairs, then share with the class,*giving their reasons for agreeing or disagreeing*with the statement. - In the discussion clarify what is the unknown in an equation such as (25 – 5) ÷ 5 = ☐. Also clarify that sometimes
**an unknown**(amount) is shown with a**letter symbol**, such as**‘n’**.

__Activity 2__

- Make paper and pencils available to each student.

Pose this problem. “I’m thinking of a number. We’ll call this number ‘n’. I add six to it. I double it. This is equal to twenty eight.”

Ask:*Can you write this problem as an equation?*[Equation: (n + 5) x 2 = 28] - Ask for some students to record their equation on the class chart. Discuss the students’ ideas, highlighting the importance of using the convention of brackets to show the order of operations. The focus is on correctly
*recording*the equation at this point, not finding the value of n.

Pose two more problems, for example:

“I’m thinking of a number. I subtract 10 from it. I divide this by nine. This is equal to ten.” [Equation: (n – 10) ÷ 9 = 10 ]

“I’m thinking of a number. I subtract two. I square it and I get twenty five.” [Equation: (n – 2)^{2}= 25]

and have the students.**record each equation, share and discuss what they have written** - Now look at each of the equations in turn.

(n + 5) x 2 = 28

(n – 10) ÷ 9 = 10

(n – 2)^{2}= 25

Ask students for suggestion of how they might work out what number you were thinking of in each example (n).

Highlight the points made in Session 1, Activity 2 that there is an inverse relationship between addition and subtraction, and between multiplication and division.*The inverse operation ‘undoes’ the other operation.*

Work backwards through the first example, applying the inverse operation: 28 ÷ 2 = 14, 14 – 5 = 9 and check that n = 9, by substituting it for n in the original equation. (9 + 5) x 2 = 28 - Have students discuss the other two examples in pairs and share their solutions.

__Activity 3__

- Make 2 small pieces of paper, and pencils, available to each student. Have each student write five of their own, “What number am I thinking of?” problems. They should write the words of each of the problems on one piece of paper, and on the second piece, their own solutions to the problems, with the number they are thinking of in the place of ‘n’.
- Have students swap their problems with a partner. Have students record their solutions to their partner’s problems separately from the problem page. These problems can then be exchanged with another pair of students and discussed.

__Activity 4__

Have students compare the way in which simple calculators and scientific calculators give *different results* for their problems.

They should enter the solutions to their equations but omit the total. For example, for the original problem (n + 5) x 2 = 28, enter instead: **9** + 5 x 2 = ☐

Scientific calculators have been programmed to follow conventions, whilst simple classroom calculators have not.

Have students investigate their problems and discuss the results from both kinds of calculator.

__Activity 5__

- Conclude the session, by having students record one of their “I’m thinking of a number” word problems, with the equation, on the bottom of their poster from Sessions 1 and 2. Display the posters and reflect on learning.
- Reflect on the symbols, expressions and equations that we use to express mathematical problems and to help us to think these through. Highlight how important it is that we know
**how to correctly read and understand the symbols and expressions that we use.**

Dear Parents and Whānau,

In algebra this week the students have been exploring number operations and the order in which we carry these out in multistep problems. BEDMAS is a convention that we use to guide us in a problem such as 3 + 4 x 5 = ☐. Is the result 23 or 35?

Have your child explain the solution to you, then together create some number puzzle problems: “What number am I thinking of?” Your child would like to show you how to create these problems.

Enjoy your discussions.

Thank you.

Attachment | Size |
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OrderOfOperationscm1.pdf | 30.7 KB |

OrderOfOperationscm2.pdf | 29.06 KB |