The order of operations

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Purpose

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

Achievement Objectives
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
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 in 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.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing support to students and by varying the task requirements. Ways to support students include:

  • providing students with smaller numbers to work with to enable the use of early additive strategies for number problems
  • supporting students to represent problems using diagrams of the numbers and operations involved
  • altering the complexity of the numbers that are used - particularly for multiplication and division (multiplication and division problems with two, four, five, and ten tend to be easier to solve than problems working with three, six, seven, eight and nine)
  • encouraging students to collaborate in small groups (mahi tahi relationship) and to share, and justify, their ideas and solutions.

The context for this unit can be adapted to suit the interests, experiences, and cultural makeup of your students. By varying the contexts which frame the different equations throughout the unit, you can make the mathematics more relevant and engaging for your students. Consider linking all of the problems to one shared context (e.g. number of people in Olympic sports teams, amount of people attending the marae visit, numbers of food items needed for the fiefia night). You may find it effective to introduce the equations without context, and then link to a context once students have a firm grasp on the calculations involved in using BEDMAS. 

The activities suggested in this series of lessons can form the basis of independent practice tasks. Understanding of BEDMAS is foundational to developing understanding of algebraic processes. Therefore, it may be beneficial for you to work with small groups of students whilst other students work on the problems independently. Students ready to work independently could explore representing the problems they have solved and made up with the use of digital tools, paper, materials, short videos, and movement.

Te reo Māori vocabulary terms such as raupapa paheko (order of operations, BEDMAS), tairinga korero (conjecture), and hapono (proof) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
  • Chart paper
  • Coloured (felt) pens
  • Calculators
  • Scientific calculators
  • Copymasters One and Two
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.
  • 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 Copymaster 1. Alternatively, you could create 4 sets of ordered action cards related to a topic you are currently exploring in another curriculum area (e.g. parts of a water cycle, how to perform the school haka).
    Distribute one card 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 out their statements out of order. Highlight the nonsensical order and point out that there is a sensible order to these actions, meaning 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. Ask  a student to record on the class chart the relationship symbols (use this term). Have students explain each of the common relationship symbols, =, ≠, <, >.
  3. 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
    Piripi has $30 in notes and $4 in coins, this is $34 altogether. He pays four dollars for a juice which leaves him with $30.
    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.
    Adapt these equations to suit the number knowledge and context of your class as necessary. If needed, provide students with more written equations to cement the idea of these inverse operations.

Activity 3

  1. Refer to the ordering actions task in Activity 1, Step 1 (ordering the actions). The order was important.
    Pose: In mathematics the order that we carry out number operations does not change the outcome/result.’ Introduce this statement as a conjecture (a statement we are proposing as true, but have not yet proved or disproved the answer to).
    Have students discuss this and indicate their agreement/disagreement and their reasons for their position. (their proof). Consider pairing up students with mixed levels of confidence and mathematical understanding to allow for greater tuakana-teina (peer sharing).
    Record and discuss students' ideas. 
  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’.  If necessary, work with a small group of students and generate a shared list of invented punctuation marks to be used. With the whole class, share the new equations. Write 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 works for pocket money on the whānau orchard, he 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 juice 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 multi-step 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.
    Encourage students to think of real life scenarios that are relevant to learning from another curriculum area (e.g. remember we have been learning about different Olympic sports, can you base your multi-step equations on the number of people in different sports teams?)
  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 (discussion of “in mathematics the order that we carry out number operations does not change the outcome/result.”).
Present two of the equations from activity 3. Tell the students that you agree with the statement. Ask students to prove to you, using the statements you have presented, that you should disagree with the statement. Look for them to explain to you that 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 idea that the ordering of actions is important in life and in maths. Encourage students to reflect on the activities from session 1 and briefly explain to a partner one situation in which the order of actions was important (i.e. it changed the outcome). 
  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.
  3. Ask students to share their prior knowledge of the term acronym. You may choose to distinguish between acronyms (which are read as words, for example, NASA, PIN, and ASAP) and initialisms (which are read out as the letters used, for example, ATM, FBI and OMG).
  4. Ask students to share their prior knowledge of the term convention. Explain that a convention is a way in which something is usually done. Make a list of conventions students follow in a relevant context (e.g. arriving at school, visiting a marae, sitting in assembly, losing a sports game).
  5. Discuss: What could BEDMAS stand for?
    Have students discuss this acronym and what they think each of the letters stand for. Record their ideas beside the acronym letters on the class chart. 
    Write the correct parts of the acronym on the board. 
    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.)
  6. If not well understood, explain exponents. The exponent indicates the number of times the base is used as a factor. For example, 104 = 10 x 10 x 10 x 10 = 10 000. 104 is read as "ten to the power of four".  If necessary, provide some exponent problems for students to work through with a pair (e.g. 33, 42).

Activity 2

  1. Distribute Copymaster 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. Remind students of the meaning of BEDMAS, and if necessary, how exponents work. You might model a question from Copymaster 2 for the whole class, to ensure they understand what to do when working in their small groups. Explain that the students should investigate how BEDMAS works by applying it to the equations.
    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. Emphasise 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 pauses where expressions are in brackets, for example:
    Question 9: (25 – 5) ÷ 5 = ☐
    Twenty five minus five, divided by five, is equal to four. 
    and punctuating the reading with pauses to recognise the order of operations, for example:
    Question 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 does change the outcome/result.

Activity 4

Have student pairs add to the charts begun in Session 1 (number operations). Ask them to explain 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’, or 'x'.

Activity 2

  1. Make paper and pencils available to each student. Allow students to work independently or in pairs.
    Pose this problem. “I’m thinking of a number. We’ll call this number ‘n’. Remember letters like ‘n’ can stand for an unknown number or amount. 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. If students want to try solving the equation, the value of n is 8 (28/2 = 14, 14 - 6 = 8).
    Pose two more problems, for example:
    “I’m thinking of a number. Let's call this number 'n'. 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 suggestions 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 (in step 3), 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’.
    Work with students who require additional support, and simplify the cognitive demands for writing their own problems. You could ask students to write a problem that includes a multiplication and an addition, and a problem that includes a subtraction and a division. Gradually increase the amount of information you expect students to include in their problems. Where possible, use materials to model problems with small numbers (e.g. (n + 3) x 2 = 10).
  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.
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