B30 — 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 next section of strings focuses primarily on the development of the open array. Graph paper is no longer used unless needed as a fallback. The next several strings also introduce doubling and halving and eventually this strategy is generalized to the associative property.

Inside One Classroom: A Portion of the B30 Minilesson

Problem Strings for Fluency and Beyond · Grade 3

Julia (the teacher): Here is our first problem. (She writes "10x6=" and thumbs go up pretty quickly, so Julia starts discussion.)

Maya: 60. I just knew it.

Julia: Today I am going to use an open array to represent our work instead of graph paper. Let’s understand how we should label the array. (She shows the array.) Carlos?

An open array labeled 10 on the left side and 6 on top, with 60 written in the middle.

Carlos: The 10 is the number of rows, and the 6 is the number of columns.

Julia: Great and as Maya said, the product is 60 so I’m going to write that in the middle of the rectangular area. Do we all agree? (All heads nod.) Let’s go on to the next problem. (She writes "5x6=".)

James: That is half of 10 x 6 so 30.

Julia: That was a quick way of thinking about that. See if we can think about how to represent what James said. What do I do to the array? James, do you want to try?

James: Well, I think maybe we need to cut the 10 rows in half?

Julia: Like this?

The same array split into two halves: a top half labeled 5/6 with 30 in the middle, and a bottom half labeled 5.

Rosa: Yes. We could keep the top half because that is 5 x 6 and remove the bottom half.

Julia: Nice. Let’s see if we can use these ideas for the next problem. (She writes "5x12=")

James: Hey cool! I can use my strategy again. Now the 6 has been doubled. We could just move back the bottom part that we threw away, and put it on the side of the top.

Julia: Wow! So, can I write this equation? (Julia writes "10x6 = 5x12".)

An array 10 rows by 6+6 columns, with the second 6-column half shaded and an arrow showing it moved beside the first half to form 5 rows by 12 columns.

Author's notes

The array provides students with a representation for multiplication. Now that she has introduced the open array, Julia will use it, transforming it into representations of the relationships that come up in the discussion.

Inviting opportunity for paraphrasing of ideas allows opportunity for student reflection and understanding. It is also a good assessment opportunity for the teacher.

The string

10 x 6
5 x 6
5 x 12
4 x 12
8 x 6
6 x 4
3 x 8