3.1: Common Core and Multiplication

The Common Core State Standards (CCSS) define multiplication as an operation that involves repeated addition of equal groups, as well as an operation that can be understood through various models such as arrays, number lines, and area models. It emphasizes understanding the concepts behind multiplication, rather than just memorizing multiplication facts.

Note: The definition of multiplication does NOT include that A x B = B x A (this is called the commutative property and we will read more about it in the next chapter). The definition means A groups of B objects in a group, that is A x B = B + B +… + B where we are adding B together A times. This does not mean that  this sum equals A + A + … + A where we are adding A together B times. It turns out that THIS IS true! We will see why in the next chapter.

Key Aspects of Multiplication in the Common Core:

1. Repeated Addition:
   • Multiplication is often described as repeated addition of the same number. For example, $4\times 3$ can be interpreted as adding 3 four times:
     $$3+3+3+3=12.$$

   This helps students understand that multiplication is an operation that combines groups of equal size. Notice that 4 x 3 is the number in four groups with 3 objects in each group. It is not assumed that 4 x 3 = 3 x 4 (that is, 3 groups with 4 objects in each group).

2. Arrays and Grouping:
   • Multiplication can also be represented using arrays (arranging objects into rows and columns) and equal groups. For example, $4\times 3$ can be visualized as 4 rows of 3 objects.
   • Example: An array for $4\times 3$ is:

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3. Commutative Property (you will read about this more in the next chapters)

• One important concept in the Common Core approach is the commutative property of multiplication, which means that the order of the factors does not change the product. For example:
  $$3\times 4=4\times 3=12.$$

This property is fundamental for students to understand that multiplication is flexible and can be approached in multiple ways. Think about why this property is true. Consider using an array to visualize 3 x 4 and 4 x 3.

4. Understanding the Role of Factors:

In multiplication, the two numbers being multiplied are called factors, and the result is called the product. For instance, in $3\times 4=12$, the factors are 3 and 4, and the product is 12.

5. Connection to Division:

• • • Multiplication is closely related to division, and students are encouraged to understand this relationship. For example, $4\times 3=12$ can also be interpreted as dividing 12 into 4 equal groups, each containing 3 objects.
    • This relationship is vital as students move toward understanding division and later concepts such as fractions and ratios.

6. Use of Models:

• • • • Students are taught to represent multiplication in various ways, such as:
        • Number lines: Visualizing multiplication as “jumps” or steps along a number line. For example, $3\times 4$ involves making 3 jumps of size 4.
      • Area Models: Multiplication is seen as finding the area of a rectangle. For example, if the length is 4 units and the width is 3 units, the area (product) is $4\times 3=12$. 

         

7. Distributive Property: (you will see this more in the next chapters)

• The distributive property of multiplication over addition is a key concept introduced early on in Common Core. This property states that:

• For example:

This helps students break down more complex multiplication problems into simpler parts.

8. Scaling:

• Scaling is another way to understand multiplication, especially when students move to larger numbers. For example, multiplying a number by 2 means doubling it, and multiplying by 3 means tripling it. This approach helps students grasp the concept of enlarging quantities by a specific factor.

Common Core Standards for Multiplication (K-5):

• Kindergarten and 1st Grade: Introduce the concept of multiplication as repeated addition of equal groups. Students understand that multiplication involves equal groups of objects and that the order in which the groups are combined doesn’t matter (commutative property).
• 2nd Grade: By this grade, students should be able to multiply numbers up to 100 and begin understanding and using multiplication tables.
• 3rd Grade: Students develop fluency with multiplication facts up to 10 × 10 and learn to apply the distributive property to break up larger numbers for easier computation. They also work with arrays and number lines.
• 4th and 5th Grade: These grades expand on multiplication with multi-digit numbers, using algorithms and the distributive property. Students also explore fractions and decimals and how multiplication applies to those.

Summary:

In Common Core, multiplication is defined not just as an operation for finding a product, but as a process that involves understanding and applying properties such as repeated addition, grouping, arrays, and scaling. The focus is on conceptual understanding—ensuring that students grasp the meaning of multiplication before relying on formal algorithms. This approach helps build a solid foundation for more advanced mathematical concepts, such as division, fractions, and algebra.

Attributions

For more, see the Common Core Mathematics Standards here: https://www.thecorestandards.org/Math/.