Multiplication and Division is the next step up from addition and subtraction, and should be taught with a graduated spiral approach throughout the elementary school years. Both concepts can and should be introduced together, as early as second grade.
Before building up fluency in multiplication tables, students should first understand the concepts of multiplication and division. To this end, we require students to master four concepts, expressed in the following “I can” statements:
- I can use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
- I can share a given number of concrete objects, explain if it can be done equally, and find the number in each group. E.g. Find the number of apples in each group when 12 apples are split into 3 groups.
- I can find the number of groups, given the number of concrete objects and number in each group. E.g. Find the number of groups when 12 apples are split into groups of 3.
- I can determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
Let’s look at these objectives in details:
1. Use addition to find total number of objects in rectangular arrays
First, we want students to be comfortable with pictorial notations of what we’re trying to accomplish, i.e. arrange items into groups for easy counting. Be consistent in convention, e.g. “3 groups of 4” is expressed as 3×4, and 3×4 is illustrated as 3 rows of 4.
Once we can “interpret” what the product of 3 and 4 means, we’ll introduce the simplest strategy to find the total number, i.e. the total number for 3 groups of 4 is the 4 + 4 + 4 (equal addends).
It is not critical for students to know the final answer (i.e. 12), getting the concept is what we want to achieve.
2. Division Concepts
Once students understand grouping items to count them, the next logical step is to learn about the complementary task of division – here, we have a total and we want to break it up into groups. For the same division equation, there can be two different interpretation.
1 – Number of items in each group
2 – Number of groups
We see that the same equation 12 ÷3 mean two different things. It is important at this stage to understand the difference between these two scenarios as this will happen again later on when they’re learning about fractions.
Similar to multiplication, it is more important to see how arranging the objects works for both scenarios than to get the correct answer “4” at this time.
3. Odd or Even by pairing
Here, we want to introduce a related concept – seeing that a group of objects is even if the objects can be arranged in two equal groups.
Try to avoid rules such as “numbers that end with an even number or zero is even”. Instead focus on understanding that if a group has an even number of items, it can be divided equally into two groups.
Once objects are divided into two groups, we can easily see that the total is made up of the sum of two equal numbers, in this case , 12 = 6 + 6.
Video explanation and lesson plan (member’s resource)
Common Core Standards
- C3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
- C4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Suggested Workbook Series
- Math in Focus workbook (2A) Chapter 5 – Multiplication and Division (pages 107 to 126)
- Primary Mathematics workbook (Common Core Edition) (2A) Chapter 4 – Multiplication and Division (pages 112 to 134)
For more related resources, please refer to our Multiplication and Division page.
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