B2.5 Represent multiplication as repeated equal groups, including groups of one half and one fourth, and solve related problems, using various tools and drawings.
Skill: Representing and Solving Problems Involving Multiplication as Repeated Addition
Modelling and Counting
Initially, students usually need to model the facts using their fingers or objects. In the case of multiplication, students model these operations with objects, tallies, or drawings to represent objects organized in groups and then count these objects to find an answer. Students use these models to find a solution to problems such as "Three bowls contain apples. Each bowl contains 5 apples. How many apples are there in total?". Students can draw the bowls and count the apples drawn to determine the total number of apples.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 6e année, p.
12.
To address multiplication, students need to understand other mathematical concepts. In particular, students need to know that multiplication can be interpreted as repeated addition. Students also need to be able to create groups or sets of equal size.
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 6e année, p. 80.
Effect of Operations
Each operation has an effect on the quantities involved. Depending on the operation, certain quantities increase or decrease. They may increase or decrease by a lot or a little. Tracking the effect of operations on numbers allows students to make connections between operations and to anticipate the outcome of an operation.
The effect of addition can be compared to that of multiplication. Compared to multiplication, addition increases a number by a small amount. For example, when the number 2 is multiplied by 8, you get 16, whereas if you add 8 to it, you get only 10. People with good operational sense recognize the effect of operations on whole numbers, but learning students are often impressed and awed by the effect of, for example, multiplication. One caveat is that care must be taken when generalizing, as operations on decimal numbers or fractions may have different effects than the effects on whole numbers. In some cases, the effect may even be the opposite. For example, if you multiply one whole number by another whole number, the product is larger than both factors (for example, if you multiply 3 by 6, the product 18 is larger than both 6 and 3), whereas if you multiply a proper fraction by a whole number, the product is smaller than either factor (for example, if you multiply \(\frac{1}{2}\) by 6, the product 3 is smaller than 6).
Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 90-91.
Initially, multiplication represents the addition of the same quantity repeated a certain number of times. In an abstract way, multiplication is composed of two factors that give a product.
To understand multiplication, students need to understand that the two factors do not play the same role. Frequently, a number sentence such as 3 × 2 is read as "three times two." In this interpretation, the factor 3 represents three groups, while the factor 2 represents two items in each group.
The same number sentence 3 × 2 can also be interpreted using the words "multiplied by". Then 3 × 2 reads "three multiplied by two", which creates the image of groups of three, twice.
Both ways of interpreting the number sentence are correct. The context from which the number sentence comes from would help to specify the representation that corresponds to it.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 80.
Types of Problems Related to Multiplication
Students develop an understanding of multiplication as well as number relationships by solving written problems. The types of problems presented below, with examples, can help students see the basic multiplication facts in a variety of ways, using equal group problems. Using problems to present basic number facts forces students to reason their way to solutions, which helps them develop a better sense of operations.
The sample written problems below contain one-digit numbers. The structures of the 3 types of written problems also lend themselves to multi-digit numbers. The problems are represented using Cuisenaire rods.
Equal-Group Problems
- Equal Groups: Product Unknown (multiplication)
Julie bought 5 books for her classmates. Each book cost her $2. How much did Julie spend on all these books?
- Equal groups: Number of Groups Unknown (Equal Distribution)
Julie has bought 10 books for her classmates and is making gift bags. She puts 2 books in each bag. How many gift bags did Julie use?
- Equal Groups: Group Size Unknown (Sharing)
Julie has 10 books. She wants to give them to 5 of her classmates. How many books will each classmate receive?
Source : Guide d’enseignement efficace des mathématiques de la maternelle à la 6e année, p. 10-11.
Knowledge: Equal-Group Problems
With equal-group problems, a group of a given size is repeated a certain number of times to create a total. Sometimes the size of each group is unknown, sometimes the number of groups is unknown, and sometimes the total is unknown.