Tag Archives: Commutative Property
Using arrays has become much more prominent in the classroom. At first glance arrays seem very straightforward and simple for students. But what are the connections that are essential for students to build understanding of the concept of multiplication through arrays?
Arrays are a model of multiplication. Just because your students can build an array and write a corresponding multiplication fact doesn’t mean they have a deep understanding of the concept of multiplication.
Connecting the Counting Model to Arrays
The Counting Model (Teaching Multiplication, Part One), which I wrote about in my previous post, uses students’ knowledge of counting and skip counting to help them solve multiplication problems. This is similar to when students use counting to solve Change Plus/Change Minus Situations in Addition and Subtraction. A simple way to build on students’ prior knowledge of counting is to count to solve a multiplication problem and then use those objects to build an array. Click here to download a blank array work mat. Ask your students, “How are the two models the same and different?”
It is important for students to build experiences in looking at objects through a different lens. This allows students to have more analogies, which aid in deeper understanding.
Important Academic Language for Arrays
Now that students have built an array, lets talk about the language they need to use. The K-5 Operations and Algebraic Thinking Progression mentions that using the words “rows and columns” is difficult for students. Instead, the K-5 Operations and Algebraic Thinking Progression recommends saying “rows” and “in each row.” This ties in nicely with “groups” and “in each group.” Eventually you can challenge your students to use the words “rows” and “columns.”
Arrays Connection to Area
Another concept arrays are connected to is area. An array’s connection to area is significant because it allows students to use two lengths, even fractional lengths, to find the area. I’ll blog more about this at another time. What is important is that as you teach the array model, you lay a solid foundation for future learning and for supporting the concept of multiplication.
Arrays support the Commutative Property
Arrays also provide an important opportunity to talk about commutative property of multiplication, which is seen when you take an array and rotate it by 90 degrees. Rotation flips the factors, but the product stays the same. This property is best articulated with arrays because arrays are very visual for students. The rectangle or product doesn’t change; it simply rotates. The factors change, but if your students are fluent at articulating rows and numbers in each row, seeing the commutative property will be simple.
If you have made it to the end of the post, congrats! I know this one is a little longer. This is because arrays can be connected to quite a few big ideas. How have you taught arrays? Are you confident that most of your students are making the important connections to other mathematical concepts? How have you helped them make those connections?