We started the week of by exploring the commutative nature of multiplication. Students created arrays for various equations in which the factors were reversed, making the connection that although the arrangement of rows and columns were different in both arrays, the total number represented in the arrays was the same. 

We spent the past week looking at the relationship between multiplication and division.

Multiplication and division are  inverse operations. When we divide, we look to separate into equal groups, while multiplication involves joining equal groups. Students use the language of multiplication as they understand what factors are and differentiate between the size of groups and the number of groups within a given context. 

We also worked on solving division word problems where  students had to understand that the unknown, or the quotient, can represent something different based on the context of the problems. For example, in the problem on the left, students are solving for the number on each group (the size of each group) whereas in the problem on the right, students are solving for the number of groups. 

Vocabulary: quotient, dividend,  divisor, size of the group, number of groups, factor, product, array, commutative property

 

 

We have officially introduced the students to multiplication and division!

Students recognize multiplication as a means to determine the total number of objects when there are a specific number of groups with the same number of objects in each group. Multiplication requires students to think in terms of groups of things rather than individual things. Students learn that the multiplication symbol ‘x’ means “groups of” or “rows of” and problems such as 3 x 4 refer to 3 groups of 4 or 3 rows of 4.

To further develop this understanding, students interpret a problem situation requiring multiplication using pictures, objects, words, numbers, and equations.

In regards to division, students are introduced to the two models of division: 

Partition models focus on the question, “How many in each group?” A context for partition models would be: There are 12 cookies on the counter. If you are sharing the cookies equally among three bags, how many cookies will go in each bag? 

Measurement (repeated subtraction) models focus on the question, “How many groups can you make?” An example would be: There are 12 cookies on the counter. If you put 3 cookies in each bag, how many bags will you fill? 

To develop this understanding, students interpret division as the partitioning into groups.

This past week, we created arrays with counters and other manipulatives and related them to repeated addition equations. We also separated arrays by groups and related them to repeated subtraction equations. In addition, we looked at the two ways to interpret an array: with the rows serving as the groups and with the columns serving as groups. This will set the stage as we formally introduce multiplication and division this week. 

We have started subtraction of 3-digit numbers with regrouping. Rather than starting with the standard algorithm, we focus on strategies that reinforce the understanding of place value and build number sense. The video below shows how to subtract using drawings for base-10 blocks. 

We are also working with open number lines as a subtraction strategy. The video below shows the strategy. 

We have also introduced students to the ‘counting on’ strategy which is a strategy they are used to with smaller numbers.