B2.1 Use the properties and order of operations, and the relationships between operations, to solve problems involving whole numbers, decimal numbers, fractions, ratios, rates, and percents, including those requiring multiple steps or multiple operations.
Activity 1: What a Great Challenge! (Order of Operations)
The following activity is a challenge for students. It challenges them to apply their knowledge of the order of operations and the effect of operations on numbers. Students must use each of the numbers in a given series only once, along with some of the 4 operations to get as close as possible to a target number. For example, the sequence of numbers is 2, 4, 5, 7, 10 and 25 and the target is 433.
Example 1
\(\begin{align}10\; \times \;25\; &= \;250\\5\; \times \;7\; &= \;35\\250\; - \;35\; &= \;215\\215\; \times \;2\; &= \;430\\430\; + \;4\; &= \;434\end{align}\)
Example 2
\(\begin{align}10\; \times \;7\; &= \;70\\70\; \times \;5\; &= \;350\\350\; + \;25\; &= \;375\\4\; + \;2\; &= \;6\\375\; + \;6\; &= \;381\end{align}\)
Example 3
\(\begin{align}5\; \times \;7\; &= \;35\\10\; + \;2\; &= \;12\\35\; \times \;12\; &= \;420\\420\; + \;25\; &= \;445\\445\; - \;4\; &= \;441\end{align}\)
The level of difficulty of the activity can vary according to certain modalities, namely:
- choice of numbers - it can be helpful to provide at least 4 or 5 numbers smaller than 10 and 2 or 3 numbers that make calculations easier (for example, 15, 25, 40, 50, 75, 100) ;
- the use of numbers - determine whether a number can be used once or repeatedly;
- choice of calculation strategy - on paper, mentally or with a calculator.
If the order of operations has been studied in class, students can summarize their solution with a number sentence. For example, they might summarize their solution as follows \((5\; \times \;7)\; \times \;(10\; + \;2)\; + \;25\; - \;4\; = \;441\) for Example 3.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 110-111.
Extensions
Do this same activity with rational numbers that include decimal numbers and positive and negative fractions.
Example
Sequence of numbers: -20, -10, -0.5, \(\frac{1}{3}\), \(\frac{1}{2}\), 2, 40
Target: 5
Example of a possible answer
\(\ ((40\; + \; {}^ - 20 )\; + \;(\frac{1}{2}\; \times \; {}^ - 10)\; + \;2\; \times \; {}^ - 0.5) \; \times \;\frac{1}{3}\)
Answer : \( - \frac{{14}}{3}\; = \;4\frac{2}{3}\)
Activity 2: Trios
The following activity exposes students to the use of brackets to specify the order of operations in a number sentence. The activity is designed for students who have not been previously exposed to the order of operations. If students are already familiar with the order of operations or know how to use brackets, the activity could be conducted differently.
As a class, tell students that it is possible to represent the number 0 in a number sentence using only the number 3 and one or more of the arithmetic operations. Give them an example by writing on the board \(0\; = \;3\; - \;3\).
Then introduce the number sentence \(3\; \times \;3\; - \;3\; \times \;3\) and show them that if you do the operations in the order they appear, you get 18:
\(\begin{align}3\; \times \;3\; - \;3\; \times \;3\; &= \;9\; - \;3\; \times \;3\\ &= \;6\; \times \;3\\ &= \;18\end{align}\)
However, if we perform the 2 multiplications first, we get 0 :
\(\begin{align}3\; \times \;3\; - \;3\; \times \;3\; &= \;9\; - \;9\\ &= \;0\end {align}\)
It is also possible to obtain 0 by first performing the subtraction :
\(\begin{align}3\; \times \;3\; - \;3\; \times \;3\; &= \;3 \times 0\; \times 3\\ &= \; 0 \times 3\\ &= \; 0\end{align}\)
Explain that by convention, brackets can be used to give priority to one or the other operation. For example, if one wants to give priority to the 2 multiplications, one can write \((3; \times \;3)\; - \;(3; \times \;3)\). However, if one wants to give the priority to the subtraction, one can write \(3\; \times \;(3\; - \times \;3)\).
To ensure that students have understood, ask them to represent the number 1 in number sentences using only the number 3 and one or more operations. Point out that they should use brackets to give priority to one or more operations, if necessary. Ask a few students to write a sentence on the board and ask the others to check its accuracy.
Here are some examples of possible answers:
\(1 \; = 3 \div 3\) or \(\frac{3}{3}\)
\(\begin{align} 1\; &= \;3\; - \;(3\; \div \;3)\; - \;(3 \div 3)\\ &= \; 3 \; -\; 1\; -\;1 \\ &= \; 2 \; -\; 1 \\ &= \; 1 \end{align}\)
\(\begin{align} 1 \; &= \frac{{(3\; + \;3)}}{{(3\; + \;3)}} \\ \; &= \frac{6 }{6} \\ \; &= \;1 \end{align}\)
Group students in pairs and ask them to present numbers from 2 to 10 in different number sentences using the number 3, the 4 operations and brackets.
Once the task is completed, ask a few students to take turns writing their number sentences on the board and grouping them according to the number in question.
Facilitate a gallery walk or other method to share their thinking and invite other students to observe the number sentences and check for accuracy.
Note: It is possible that some students use brackets within brackets. In such situations, it should be mentioned that priority is given to the brackets inside the others first.
Example
\(\begin{align} 3\; - \;((3\; + \;3) \div \; 3 ) &= \; 3 \; - \;(6 \div 3) \\ &= \; 3 \; - \; 2 \\ &= \; 1\end{align}\)
Here are some examples of possible answers:
\(2\; = \;(3\; + \;3)\; \div \;3\) or \((3\; \div \;3)\; + \;(3\; \div \;3)\)
\(3\; = \;(3\; \times \;3)\; - \;(3\; + \;3)\) or \((3\; + \;3)\; - \;3\)
\(4\; = \;3\; + \;(3\; \div \;3)\) or \((3\; + \;3)\; - \;((3\; + \;3)\; \div \;3)\)
\(5\; = \;3\; + \;3\; - \;(3\; \div \;3)\) or \((3\; \times \;3)\; - \; 3\; - \;(3\; \div \;3)\)
\(6\; = \;3\; + \;3\) or \((3\; \times \;3)\; - \;3\)
\(7\; = \;(3\; \times \;3)\; - \;\frac{{3\; + \;3}}{3}\) or \((3\; + \;3)\; + \;(\;3\; \div \;3)\)
\(8\; = \;3\; + \;3\; + \;3\; - \;(3\; \div \;3)\) or \(3\; + \;3\; + \;(3\; \div \;3)\; + \;(3\; \div \;3)\)
\(9\; = \;3\; + \;3\; + \;3\) or \(3\; \times \;3\)
\(10\; = \;3\; + \;3\; + \;3\; + \;(3\; \div \;3)\) or \((3\; \times \;3 )\; + \;(3\; \div \;3)\)
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 209-211.
Extensions
Do the same exercise using fractions or decimal numbers.
For example, ask students to present numbers from 1 to 10 in different number sentences using the fraction \(\frac{1}{2}\), the 4 operations and brackets.
Activity 3: Let's Play With Numbers (Distributive Property of Multiplication)
Write each of the following sets of related operations, one at a time, on the board or a large piece of flip chart paper and have students perform the operations in it.
| Set 1 | Set 2 | Set 3 | Set 4 |
|---|---|---|---|
|
\( 3 \times 6\) \(3 \times 40 \) \(3 \times 46 \) |
\(2 \times 13\) \(22 \times 10\) \(22 \times 3\) \( 20 \times 13\) \(\ 22\; \times \;13\) |
\(4 \times 2\) \( 4 \times 50\) \( 4 \times 25\) \(\ 4 \times \;77\) |
\(5 \times 5\) \(5 \times 30\) \(5 \times 100\) \(5 \times 95\) |
Once a set is completed, draw out the various mental math strategies by asking questions such as:
- How did you solve the last operation?
- To solve the last operation, did you use any elements of the previous operations?
- Did you solve the operations in order?
If necessary, before doing the same with the next set, present other examples of similar related operations.
If students have difficulty seeing and applying the distributive property of multiplication to solve the last operation in each set, lead them to do so by representing these operations as area models.
Example
Findings for each set
Set 1
This set allows to review the distributive property of multiplication over addition at its simplest. The idea is to recognize that to obtain the product of 3 × 46, it is possible to perform an operation on a sum of terms and obtain the same result as if the operation had been performed on each term, that is :
\(\begin{align} 3\; \times \;46\; &= \;3\; \times \;(40 \; + \; 6)\\ &= \;(3 \times 40) + \;(3 \times 6)\end{align}\)
Set 2
This set allows to see that the decomposition related to distributive property can be performed on the second term as well as on the first, that is :
\(22\; \times \;13\; = \;(20\; \times \;13)\; + \;(2\; \times \;13)\)
Or
\(22\; \times \;13\; = \;(22\; \times \;3)\; + \;(\;22\; \times \;10)\)
Set 3
This set shows that the decomposition of a number to apply the distributive property can be done in more than 2 parts. For example:
\(\begin{align}{l}4\; \times \;77\; &= \;4\; \times \;(50\; + \;25\; + \;2)\\ &= \;(4\; \times \;50)\; + \;(4\; \times \;25)\; + \;(4\; \times \;2)\end{align}\)
Set 4
This set allows to see that we can solve the last operation using the distributive property of multiplication over subtraction. For example:
\(\begin{align}{l}5\; \times \;95\; &= \;5\; \times \;(100\; - \;5)\\ &= \;(5\; \times \;100)\; - \;(5\; \times \;5)\end{align}\)
It also shows that this operation can be solved by using the distributive property of multiplication over addition and other properties. For example:
\(5\; \times \;95\; = \;3\; \times \;(5\; \times \;30)\; + (5\; \times \;5)\)
To go further : add a set with a fraction
| \(\frac{1}{4} \times 40\) |
| \(\frac{1}{4} \times 8\) |
| \(\frac{1}{4} \times 1\) |
| \(\frac{1}{4} \times 49\) |
\(\begin{align} \frac{1}{4}\; \times \;49\; &= \;\frac{1}{4}\;(40\; + \;8\; + \;1)\\ &= \;(\frac{1}{4}\; \times \;40)\; + \;(\frac{1}{4}\; \times \;8)\; + \;(\frac{1}{4}\; \times \;1)\\ &= \;10\; + \;2\; + \;\frac{1}{4}\\ &= \;12\frac{1}{4}\end{align}\)
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 112-115, p. 221-222.
Activity 4: Who is Telling the Truth? (Property of Rational Numbers)
Introduce the rules of the game which consists of presenting 3 statements, 2 of which will be false and the other true.
The student must justify to their partner (or other students in the class) why each statement is true or false.
The purpose of the activity is to determine which student is telling the truth.
Note: The student is not allowed to use a calculator.
Examples of statements
- Who is telling the truth?
Student 1: \(5\; \times \;3\; - \;3\; \times \;4\; = \;0\)
Student 2: \(5\; \times \;3\; - \;3\; \times \;4\; = \;3\)
Student 3: \(5\; \times \;3\; - \;3\; \times \;4\; = \;48\)
Justify.
- Who is telling the truth?
Student 1: \(25\; \times \;40\; = \;(10\; \times \;40)\; + \;(10\; \times \;40)\; + \;(5 \; \times \;40)\)
Student 2: \(25\; \times \;40\; = \;(20\; + \;40)\; \times \;(5\; + \;40)\)
Student 3: \(25\; \times \;40\; = \;(25\; \times \;4)\; + \;(25\; \times \;10)\)
Justify.
- Who is telling the truth?
Student 1 : \((30 - \;10)\; - \;5 = \;30 - \;(10 - \;5)\)
Student 2 : \((30\; - \;10)\; - \;5\; = \;30\; + \;({}^ - 10\; + \;5)\)
Student 3 : \((30 - \;10)\; - 5 = \;30 - \;(10 + \;5)\)
Justify.
- Who is telling the truth?
Student 1 : \((\frac{1}{2}\; \times \;0)\; + \;(0.5\; \times \;1)\; = \;0\)
Student 2 : \((\frac{1}{2}\; \times \;0)\; + \;(0.5\; \times \;1)\; = 0.5\)
Student 3 : \((\frac{1}{2}\; \times \;0)\; + \;(0.5\; \times \;1)\; = \;1\)
Justify.
- Who is telling the truth?
Student 1 : \((50\;\% \; \times \;80)\; + \;(20\;\% \; \times \;80)\; = \;70\;\% \; \times \;160\)
Student 2 : \((50\;\% \; \times \;80)\; + \;(20\;\% \; \times \;80)\; = \;70\;\% \; \times \;80\)
Student 3 : \((50\;\% \; \times \;80)\; + \;(20\;\% \; \times \;80)\; = \;70\;\% \; \times \;6 400\)
Note: Construct the decoys based on common student errors.