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.
