B2.1 Use the properties and order of operations, and the relationships between operations, to solve problems involving rational numbers, ratios, rates, and percents, including those requiring multiple steps or multiple operations.
Activity 1: Take a Position (Order of Operations)
Project a mathematical expression on the board that includes some or all of the following: addition/subtraction, multiplication/division, brackets, and exponents.
Recommendation: start with whole numbers and then increase the level of complexity.
Ask the students to solve the expression. Circulate and take note of the approach of two or three students who do not have the same answer. Take a picture of all the steps of each targeted student's work. Reconvene as a whole group and project the different approaches to solving. Give the students time to reflect individually, analyze the various solutions with their own and then take a position as to which approach they think is correct. Feedback and questioning will be necessary to establish relationships with the many concepts required to master this content. Be careful to value the different approaches to arrive at the correct answer, and help students to understand that the conventions we use around the order of operations have been developed by mathematicians to ensure consistency in approach so that if we follow the conventions, we will all end up with the same answer.
Examples of mathematical expression
From simple examples:
- \(3\; + \;4\; \times \;2\; - \;2\; \times \;1\; + \;4\)
- \(5\; \times \;7\; - \;2\; \times \;4\)
To more complex examples
- \(5\; \times \; - \frac{1}{4}\; + \;{\left( {2\; - \;\frac{1}{2}} \right)^ 2}\;-\;5\;\times\;0.5\)
- \(\left({ - \frac{1}{5}\; \times \;\sqrt {25} } \right)^2\; - \;6\; \times \;50 \% \)
Activity 2: The Carousel (Order of Operations)
Students should have frequent opportunities to practise the order of operations.
Write problems (depending on the number of stations). Photocopy and laminate.
Arrange the desks in work islands. Each island corresponds to a station.
Group students together (three or four per team). Manage the working time for each station.
Begin the Carousel. Throughout the activity, record comments, discussions and especially computational strategies.
Students can consult the Internet as needed and use a calculator. Take pictures of different student approaches (two per station) and use these pictures to provide feedback to the students.
Examples of problems at stations
Station 1:
Josianne's cell phone plan includes an initial cost of $150.00 and a monthly cost of $32.50 thereafter. Alberto's plan includes a monthly fee of $55.00 and no initial cost. Josianne and Alberto have signed a two-year agreement with their respective companies. At the end of their agreement, what is the monetary difference between the plans?
Station 2:
The municipal pool is 20 metres long and 15 metres wide. The manager has just ordered a solar cover at a cost of $32 plus tax per square metre. How much will the bill be?
Station 3:
During a trip to France, Mr. Foster purchased a beanie for 22.99 euros. At the time of the transaction, 1 euro = $1.33 CDN. When he arrived at Canadian customs, Mr. Foster had to pay an 18% duty on the hat and then the harmonized sales tax (HST). What is the total price, in Canadian dollars, for the hat?