Multiplication Strategies

Young mathematicians’ first formal introduction to multiplication and division happens in 3rd Grade.

In our unit, Equal Groups, situations are presented in context. The situation usually requires students to identify the number of groups and the number of items within each group.

There are 5 tricycles. Each tricycle has three wheels. How many wheels are there in all?

Or, the situation may read,

Tricycles have three wheels. There are 5 tricycles. How many wheels are there in all?

Number of Groups: 5
Number of Items in Each Group: 3

Students are taught to represent the problem with both an addition equation and a multiplication equation to illustrate the connection, and use a variable for the missing piece of information.

Addition equation: 3+3+3+3+3=w
Multiplication equation: 5x3=w (read 5 groups of 3)

Students generally begin solving the multiplication situations with their prior knowledge of repeated addition.

Repeated Addition: 3+3+3=9 and 3+3=6, so 9+6=15.

Then, some students move into skip counting if they can easily skip count by that number. If the situation allows counting skip counting by 2’s, 5’s, and 10’s, students will almost always start with this strategy. If however, the situation has them skip counting by 8’s, then we’ll teach students to use 10 as their anchor. Skip count by 10 and go back 2.

Skip Counting: 3,6,9,12,15.

One tool that we teach students to use to organize their thinking when skip counting is a ratio table. Ratio tables help students keep track of the number of groups as they skip count.

During the unit, students are introduced to multiplication situations using arrays, too. An array is a rectangular arrangement with rows and columns.

To create a common language with our students, we have them give us the dimensions of the row first and then the column. The array above would be a 5x3.

There are many ways to solve the array. One of the things students do is skip count the array.
They can skip count the array be either the rows or columns.

Students naturally begin to see and explore the commutative property of multiplication. Though the situation is 5x3, they realize they can applying their number knowledge properties, and solve 3x5 instead (5,10,15 or 5+5+5=15).

Students’ next level of understanding develops when they recognize that they can decompose an array into smaller arrays to help them solve problems.

In this situation, if a student didn’t know the product of 5x3 quickly, they could decompose the array into (5x2) + (5x1) = 15.

This understanding is extremely important as students move into problems that are more difficult when they begin multiplication like 6x8. When in the early stages of developing automaticity, they may not know 6x8, but if they know (3x8), they can solve 2(3x8), or, if they know 6x4, they can solve 2(6x4).

Students’ ability to think flexibly with decomposing arrays in multiple ways, builds a strong foundation for fluency in multiplication. The skill allows students to attack any multiplication equation for which they don’t automatically have a product, and leads into being able to solve more difficult equations like 14 x 12.

After the unit, Equal Groups, students have a solid conceptual foundation and can think about multiplication flexibly. But, if we stop there, they may never become fully fluent. We continue to practice fact fluency with our combination club flash cards, by playing multiplication bingo, doing fluency clicker reviews, and doing a timed fluency snapshot several times a week. The fluency snapshots are presented by similar facts. (5’s and 10’s together) (2’s and 4’s and 8’s together) (3’s, 6’s, 9’s, 12’s together) (7’s) and (11’s). Presentation with similar facts promotes the conceptual understanding we build throughout this unit.

Our goal is for every student to leave third grade knowing each of their multiplication facts within three seconds. This foundational knowledge creates automaticity and will help them be successful in fourth grade as they embark on more complex multiplication problems like 49 x 58.

Arrays Make Math Easy to Visualize

As young mathematicians, we are exploring the use of arrays to help us with multiplication. It is imperative that we develop strong visual images of multiplication to develop conceptual strategies for solving multiplication problems. When we can clearly visualize how the numbers being multiplied are related, we can develop flexible, efficient, and accurate strategies for solving any multiplication problems.

One way we are learning to visualize these relationships is through the use of arrays. An array is a rectangular arrangement of an equal number of items in rows and columns. Arrays can be helpful when solving more “difficult” multiplication situations. Being able to visualize how to break multiplication problems into parts becomes even more important when we begin to solve multi-digit problems.

The following is an example of how we can split an array into smaller arrays making it easier to find the product.

Another way we can lay a strong foundation for multiplication is to practice skip counting by multiples of numbers 2-12. The goal is to skip count fluently (within 3 seconds) from one multiple to the next.This task can be practiced at home, between commercial breaks, or even in the car. Being able to skip count fluently will undoubtedly help us in our work with multiplication and division. Check out this website that helps with skip counting.

Comparing Arrays

Last week in Math Workshop, our young mathematicians spent time arranging different amounts of chairs into rows and columns (arrays). We looked first at ways to arrange 12 chairs. Then, during Work Session students paired up to explore with two other arrangements. Exploring with the numbers 9, 15-21, 23-25, 27, and 30 helped us to see how differently numbers can be arranged, and gave us the opportunity to discuss the similarities and differences among the varying arrays.  

For example, look at the arrangement of 16 and 17 chairs below:

What do you notice about these arrays? Do any of the arrays have a special or unique shape? What do you notice about the number of arrays that can be made with 16 chairs compared to 17 chairs?

Hopefully, you notice that 16 chairs can be arranged in several different arrays. That is because it has many factors: 1, 2, 4, 8, 16. A number that has more than two factors is called a composite number. You probably noticed that 17 only has two arrays. This is because 17 is a prime number. Any number that has only two factors, one and itself, is a prime number. The factors of 17 are 17 and 1.Also, you may have noticed that 16 can be arranged into a perfect square with 4 rows and 4 columns. Any number that has an array that is a square is a square number.  Our young mathematicians talked about square numbers when we analyzed the arrays for 9 (3x3) and 25 (5x5).

Students also realized that you could skip count by either the column or the row to get the product. Sixteen, for example, has an array that is a 2x8. You can count by 2's eight times... 2, 4, 6, 8, 10, 12, 14, 16 or you can count by 8's two times... 8, 16.

Stay tuned this week as our young mathematicians learn how to decompose an array to make a more difficult multiplication equation simpler.