B2.5 Represent and solve equal-group problems where the total number of elements is no more than 10, including problems in which each group is a half, using tools and drawings.
Skill: Representing and Solving Equal-Group Problems
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 will be able to draw the bowls of apples 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. 8.
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.
We can compare the effect produced by addition with that produced by 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, you get only 10. People with good operations 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 operations 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 6 and 3), whereas if you multiply a proper fraction by a whole number, the product is smaller than one of the two factors (for example, if you multiply one half 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.
Written Problems Related to Multiplication and Division
Students develop an understanding of multiplication and division and the relationships between numbers by solving written problems. The types of equal-group problems presented below with examples can help students see basic number facts about multiplication and division in a variety of ways. Using problems to introduce basic number facts forces students to reason their way to solutions and thus helps develop a better sense of operations.
The sample written problems below contain one-digit numbers. The structures of the three types of written problems also lend themselves to multi-digit numbers. The problems are represented using Cuisenaire rods. In Grade 1, we only deal with problems where the whole is unknown or the number of groups is unknown. We always give the size of the groups.
Equal-Group Problems
- Equal Groups: Total Unknown (Multiplication)
Julie bought 5 books for her classmates. Each book cost her $2. How much money 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?
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.
Note: In Grade 1, students are always given the size of the equal groups and they determine either the number of equal groups or the total needed (not to exceed 10).