B2.1 Use the properties of operations and the relationships among addition, subtraction, multiplication, and division to solve problems involving whole numbers, including problems requiring more than one operation, and check calculations.
Activity 1: Trios
Strategy: Commutative Property (Addition or Multiplication)
Material
- a set of Triplets addition cards (Appendix FR15 and Appendix FR16) or prepare addition cards according to grade level
- a set of Triplets multiplication cards (Appendix FR17 and Appendix FR18) or prepare multiplication cards according to grade level
This activity is done with the whole class. Each student receives one card from the Triplets game. Once the cards are dealt, students look for the other members of their trio. They need to know the answer to the question, or consider the possible questions if their card has an answer. For example, the triplets could be \(3 + 5\), \(5 + \;3\) and \(8\); or \(3\; \times 6\), \(6 \times \;3\) and \(8\). When a triplet is complete, all three students sit together. Once all the students are seated, each triplet presents their cards to the class.
Source: translated from Guide d'enseignement efficace des mathématiques de la maternelle à la 6e année, Fascicule 5, p. 78.
Activity 2: True or False?
To explore the properties of operations, lead a mini-lesson by grouping students together and writing these math statements one at a time. For each, ask students to determine whether the statement is true or false.
For example, for the distributive property, write these statements one at a time.
\[7 \times 215 = \left( {7 \times 200} \right) + \left( {7 \times 10} \right) + \left( {7 \times 5} \right)\]
\[9 \times 345 = \left( {9 \times 300} \right) + \left( {9 \times 40} \right) + \left( {9 \times 5} \right)\]
\[\left( {4 \times 500} \right) + \left( {4 \times 50} \right) + \left( {4 \times 5} \right) = 4 \times 555\]
It is also important to record 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 or an array.
Activity 3: What Do You See?
Show students a picture of objects (for example, pieces of fruit) placed in an array.
Example
Ask students to work in pairs to represent the array using as many operations as they can.
For example: \(2\; \times \;10\; = \;20\)
Pair up students so that they can compare their operations.
Take stock of all the results to identify properties and relationships.
Examples
- Commutative Property
\(\begin{array}{l}4 \times 5 = 20 \ and \ 5 \times 4 = 20\end{array}\)
- Distributive Property
\(\ 2 \times 10 = \left( {2 \times 5} \right) + \left( {2 \times 5} \right)\)
- Inverse Relationship of Division and Multiplication
\(\begin{array}{l}10 \times 2 = 20\\20 \div 2 = 10\\2 \times 10 = 20\\20 \div 10 = 2\end{array}\)
Etc.
Activity 4: A Beautiful Zero
Targeted Strategy: Rule for Zero (multiplication)
Material
- Appendix FR20 (Spinning for Zero spinner (+ 0, - 0, + 1, × 0, × 1) or a spinner of numbers according to the grade level)
- Appendix FR21 (Spinning for Zero Game Sheets) (1 per student)
- counters
Students are given a blank game sheet (FR21) and write the numbers 0 to 8 in the boxes, repeating them at will until all the boxes are filled. Students then work in groups of two, three or four. The student who starts the game spins the two wheels and performs the operation indicated. If the box indicating the answer is free on their game sheet, the student places a counter on the number. The next student then plays. If there is already a counter on the desired number, the next student plays. The game continues until a student has covered all the squares in a row on their game sheet.
Source: translated from Guide d'enseignement efficace des mathématiques de la maternelle à la 6e année, Fascicule 5, p. 80.
Activity 5: Related Operations
Write the following string on the board or on chart paper, one computation at a time, and ask students to calculate the answers.
| String 1 | String 2 | String 3 | String 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 string is completed, draw out the various mental computational strategies by asking questions such as:
- How did you solve the last computation?
- To solve the last computation, did you use any elements of the previous computations?
- Did you solve the computations in order?
If necessary, before doing the same with the next math string, present other examples of similar related computations.
If students have difficulty seeing and applying the distributive property of multiplication to solve the last computation in each string, lead them to do so by representing these computations as arrays.
Example
Findings for Each String
String 1
This string allows us to review the distributive property of multiplication over addition at its simplest. The point is to recognize that to obtain the product of \(\ 3\; \times \;46\), it is possible to perform a computation on the sum of terms and obtain the same result as if the computation had been performed on (or distributed across) each term. For example:
\(\begin{array}{l}3 \times 46 = 3 \times \left( {40 + 6} \right)\\3 \times 46 = \left( {3 \times 40} \right) + \left( {3 \times 6} \right)\end{array}\)
String 2
This string allows us to see that the decomposition related to the distributive property can be performed on the second term as well as on the first. For example:
\(\begin{array}{l}22 \times 13 = (20 \times 13) + (2 \times 13) \ or \\22 \times 13 = \left( {22 \times 3} \right) + \left( {22 \times 10} \right) \end{array}\)
String 3
This string shows that decomposing a number to apply the distributive property can be done in more than two parts. For example:
\(\begin{array}{l}4 \times 77 = 4 \times \left( {50 + 25 + 2} \right)\\4 \times 77 = \left( {4 \times 50} \right) + \left( {4 \times 25} \right) + \left( {4 \times 2} \right)\end{array}\)
String 4
This string allows us to see that we can solve the last computation using the distributive property of multiplication over subtraction. For example:
\(\begin{array}{l}5 \times 95 = 5 \times \left( {100 - 5} \right)\\5 \times 95 = \left( {5 \times 100} \right) - \left( {5 \times 5} \right)\end{array}\)
It also shows that we can solve this computation using the distributive property of multiplication over addition and other properties. For example:
\(\begin{array}{l}5 \times 95 = 3 \times \left( {5 \times 30} \right) + (5 \times 5) \end{array}\)
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 1, Nombres naturels, p. 221-222.