B24 — Multiplication within 100: Automatizing the Basic Facts Using the Array

Problem Strings for Fluency and Beyond · Multiplication within 100: Automatizing the Basic Facts Using the Array

Teacher notes

The array model is another helpful thinking tool for modeling the properties of the operation of multiplication, and it builds a bridge to an understanding of area. When introducing the array model, it is best to start by using graph paper. Although most adults understand and see the rows and columns in an array, most children initially do not. Research by Battista has provided evidence of the difficulties that third graders have coordinating the two dimensions and understanding that a square is in a row and a column simultaneously. It is also very important to standardize that the first factor represents the number of rows, and the second represents the number of squares in the row (as well as the number of columns). The next several strings are designed to help students understand how the rows and columns are related and to help them construct the commutative and distributive properties.

Inside One Classroom: A Portion of the B24 Minilesson

Problem Strings for Fluency and Beyond · Grade 3

Julia (the teacher): Here is our next problem (Writes, "10x3=". Thumbs go up quickly, so Julia starts discussion.)

Kendra: 30. I just knew it. I know when you multiply by a ten you just add a zero.

Julia: Hmmm… the first factor is 10. That means we have 10 rows of 3. I’ve cut that out for us to look at. Here is an array of that.

A vertical open array, 10 rows by 3 columns.

Julia: Why does the pattern of adding a zero to the right of 3 make 30 happen? We’ve noticed that a lot, right? But we aren’t really adding a 0 to 3. That would be 3, right? How did the 3 become 30? Everybody, find a partner and talk about this. (Julia moves around and listens into the conversations to determine her next move. She notes that Bella and Henry are discussing the rows and columns and how they are moving and recognizes that they are discussing commutativity.) Henry and Bella, you are having an interesting discussion about something turning. Would you share what you noticed? You are imagining that the array turned like this?

The same array turned: 3 rows by 10 columns.

Henry: Yes, now I can count 3 rows and there are ten boxes in each row.

Julia: Which is the row, and which is the column?

Bella: The rows go across and there are 10 squares in each row. 3 rows of 10. And there are 10 columns of 3. We think the rows have 3 tens and that is why the 3 moved over, because now there are 3 tens.

Julia: Wow! So, can I write this equation 10 x 3 = 3 x 10 because it is the same array; it just turned? (All heads nod.) Great, here’s our next one. (Writes "9 x 3")

Author's notes

The array provides students with an area representation for multiplication. Make sure that students can articulate where the rows and columns on the array are. This will also provide connections later on when students are working with division and the connections to multiplication.

Note how Julia listens to the students’ strategies and records their thinking. As you use the array, others in the class will begin to see relationships between the problems.

Inviting opportunity for reflection of new strategies and insights provides time for students to talk with their peers to facilitate understanding and help each other. These moves also help to develop a real sense of community—one that focuses on thinking, understanding, and sense-making.

The string

10 x 3
3 x 10
9 x 3
10 x 8
9 x 8
8 x 9
10 x 9
9 x 9