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The order of operations

Keywords:
Achievement Objectives:

Achievement Objective: NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
AO elaboration and other teaching resources

Purpose: 

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.

Specific Learning Outcomes: 
  • 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.
Description of mathematics: 

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

Required Resource Materials: 
Chart paper
Coloured (felt) pens
Calculators
Scientific calculators
Activity: 

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

  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.
  2. 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

  1. Ask a student to record on the class chart the number operations (use this term). Have students explain each of the operation symbols, + - x ÷.
  2. Have students discuss in pairs the relationship between the operations of addition and subtraction, and between multiplication and division.
    Highlight the inverse relationship in each pair of operations. Have students explain this with equation and word examples, such as:
    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

  1. 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:
  2. 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.
  3. 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.
  4. 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
  5. 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

  1. 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.
  2. 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

  1. Review the ordering of actions undertaken at the beginning of Session 1 and the conclusions from that session.
  2. Explain to students that mathematicians have agreed on an order of operations or mathematical actions. Write BEDMAS, explaining that this is an acronym for the agreed convention.
    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 42 = 4 x 4 = 16, or 103 = 10 x 10 x 10 = 1000

Activity 2

  1. 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 record which operation they complete first, and explain in words why. They should also discuss and note any questions they have as they do so.
  2. 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

  1. 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 ÷ 52 = ☐
    Twenty five plus, five times five divided by five squared, is equal to twenty six.
  2. 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

  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.
  2. 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

  1. 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]
  2. 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.
  3. 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
  4. Have students discuss the other two examples in pairs and share their solutions.

Activity 3

  1. 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’.
  2. 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

  1. 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.
  2. 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.
AttachmentSize
OrderOfOperationscm1.pdf30.7 KB
OrderOfOperationscm2.pdf29.06 KB