B2.1 Use the properties of operations, and the relationships between operations, to solve problems involving whole numbers and decimal numbers, including those requiring more than one operation, and check calculations.
Activity 1: Trios
Strategy: Commutative Property (Addition or Multiplication)
Materials
- a set of Trios addition cards (Appendix FR15 and Appendix FR16) or prepare addition cards according to grade level
- a set of Trios cards for multiplication (Appendix FR17 and Appendix FR18) or prepare multiplication cards according to grade level
This activity is done with the whole class. Each student receives 1 card from the Trios game. Once the cards have been dealt, the students look for the other members of their trio. They need to know the answer to the question, or consider the possible questions to which their card has the answer. For example, the trios could be \(3\; + \;5,\;5\; + \;3\;{\rm{and}}\;{\rm{8}}\); or \(3\; \times \;6,\;6\; \times \;3\;{\rm{and}}\;18\). When a trio is complete, the 3 students sit together. Once all the students are seated, each trio presents the cards to the class.
Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6e année, p. 78.
Activity 2: True or False?
To explore the properties of operations, facilitate a mini-lesson by grouping students together and writing these number sentences one at a time. For each, ask students to determine whether the sentence is true or false.
For example, for the distributive property, write these sentences one at a time.
\(47\; \times \;21\; = \;\left( {40\; \times \;20} \right)\; + \;\left( {40\; \times \;1} \right)\; + \;\left( {7\; \times \;20} \right)\; + \;\left( {7\; \times \;1} \right)\)
\(92\; \times \;34\; = \;\left( {90\; \times \;30} \right)\; + \;\left( {90\; \times \;4} \right)\; + \;\left( {2\; \times \;30} \right)\; + \;\left( {2\; \times \;4} \right)\)
\(\left( {40\; \times \;50} \right)\; + \;\left( {40\; \times \;0} \right)\; = \;2\;000\)
It is also important to trace students' reasoning either by crossing out numbers on each side of an equation or putting arrows between numbers. It is also good to show reasoning using a double open number line.
Activity 3: What Do You See?
Show students a picture of objects (for example, fruits) placed in an array.
Example
In pairs, ask students to observe the image to determine as many operations as possible that it could represent.
For example: \(15\; \times \;12\; = \;180\)
Pair up the pairs so that students can compare their operations.
Pool the results to identify properties and relationships.
Examples
Commutative Property
\(12\; \times \;15\; = \;180\)
\(15\; \times \;12\; = \;180\)
Distributive Property
\(12\; \times \;15\; = \;\left( {10\; \times \;15} \right)\; + \;\left( {2\; \times \;15} \right)\)
\(12\; \times \;15\; = \;\left( {10\; \times \;10} \right)\; + \;\left( {10\; \times \;5} \right)\; + \;\left( {2\; \times \;10} \right)\; + \;\left( {2\; \times \;5} \right)\)
Inverse Relationship of Division and Multiplication
\(12\; \times \;15\; = \;180\)
\(180\; \div \;15\; = \;12\)
\(15\; \times \;12\; = \;180\)
\(180\; \div \;12\; = \;15\)
Activity 4: A Beautiful Zero
Strategy: Identity principle and the Zero Property (in multiplication)
Materials
- Appendix FR20 (a roulette wheel Beautiful Zero [+ 0, - 0, + 1, × 0, × 1])
- Appendix FR20 (a number wheel according to the level of study)
- Appendix FR21 (blank game sheets Beautiful Zero) (1 per student)
- counters
Students are given a blank game sheet and write the numbers from 0 to 8 in the boxes, repeating them as they wish, until all the boxes are filled in. They then work in groups of 2, 3 or 4. The student who opens the game spins the 2 spinners and performs the operation indicated. If the box indicating the answer is still empty, the student places a counter on the number. It is then the next student's turn. If there is already a counter on the number, the next student plays. The game continues until a student has a counter in every box in one row of their game sheet.
Source: Guide d'enseignement efficace des mathématiques de la maternelle à la 6e année, p. 80.
Activity 5: Related Operations
Write each of the following sets of related operations on the board or on chart paper, one at a time, and have students perform the operations.
Once a set is completed, draw out the various mental math strategies by asking questions such as:
- How did you perform the last operation?
- To perform the last operation, which, if any, elements of the previous operations did you use?
- How would you describe the order in which you performed the operations?
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 using an area model.
Example
Findings for each set
Set 1
This set allows us to review the distributive property of multiplication over addition at its simplest. The idea is to recognize that to obtain the product of \(30 \times 46\), 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 :
\(30\; \times \;46\; = \;30\; \times \;\left( {40\; + \;6} \right)\)
\(30\; \times \;46\; = \;\left( {30\; \times \;40} \right) + \;\left( {30\; \times \;6} \right) \)
Set 2
This set allows us to see that the decomposition related to distributive property can be performed on the second term as well as on the first, that is :
\(\begin{array}{l}22\; \times \;13\; = \;\left( {20\; \times \;13} \right) + \;\left( {2\; \times \;13} \right)\;{\rm{or}}\\22\; \times \;13\; = \;\left( {22\; \times \;10} \right) + \;\left( {22\; \times \;3} \right)\end{array}\)
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:
\(40\; \times \;77\; = \;40\; \times \;\left( {50\; + \;25\; + \;2} \right)\)
\(40\; \times \;77\; = \;\left( {40\; \times \;50} \right)\; + \;\left( {40\; \times \;25} \right)\; + \;\left( {40 \times \;2} \right)\)
Set 4
This set allows us to see that we can solve the last operation using the distributive property of multiplication over subtraction. For example:
\(25\; \times \;95\; = \;25\; \times \;\left( {100\; - \;5} \right)\)
\(25\; \times \;95\; = \;\left( {25\; \times \;100} \right) - \;\left( {25\; \times \;5} \right) \)
Source: adapted from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 221-222.