B1 — Multiplication within 100: Automatizing Basic Facts using the Open Number Line

Problem Strings for Fluency and Beyond · Multiplication within 100: Automatizing Basic Facts using the Open Number Line

Teacher notes

When students are first introduced to multiplication, they often need to count each object within each group. Once they begin to group and realize the groups are of the same size, repeated addition and skip counting strategies emerge. The strings in this section offer opportunities for students to use these strategies, but they also offer opportunities to regroup the groups into larger groups and to use partial products. Doing so allows the properties of the operation to also emerge. The numbers in these strings are limited to products of 100 as an opportunity to support automatizing of the facts. By focusing on the relationships between the facts, there are fewer facts to memorize. As the string progresses encourage children to use the relationships in the string. The open number line is used at this point in development because it is a helpful model to represent skip counting, repeated addition, and the regrouping of groups. As you represent children’s strategies, the equivalent relations can be shown on the top and bottom of the number line (see the dialogue box that follows as an example of this). Show one problem at a time. Record each child’s strategy by drawing leaps as lengths on an open number line.

Inside One Classroom: A Portion of the B1 Minilesson

Problem Strings for Fluency and Beyond · Grade 3

Julia (the teacher): Here is our first number problem. (Writes "2x5=") Ok, thumbs up when you are ready to share what the sum of these numbers will be and how you came up with your answer. (All thumbs go up quickly and Julia smiles.)

Dandre: 10. I just thought of it as 5 + 5.

Julia: Let me show that on the number line.

A number line with two arcs labeled 5 and 5, landing on 5 then 10.

Julia: Who else just knew that one? (All thumbs again go up quickly.) Great!? So, what about this one? (Writes "5x5") … Rosa?

Rosa: 25. I counted 5, 10, 15, 20, 25

A number line with five arcs labeled 5 each, landing on 5, 10, 15, 20, 25.

Julia: Here is a picture of what you said. You skip counted by 5’s. Great. It is helpful to skip count, isn’t it. Is there a different way? Carlos?

Carlos: I also got 25 but I used the first problem. I knew 2 x 5 was ten from the first problem, so then I thought of another 2 x 5 and then one more 5.

Julia: That’s interesting! I’m going to represent that on the same number line. (Using the number line already shown.)

The same number line now with larger arcs labeled 10 and 10 drawn above the five arcs labeled 5, showing 25 as (2x5)+(2x5)+(1x5).

Julia: Let’s turn and talk and try to understand why Carlos’ strategy works… (after some time) Kendra?

Kendra: We talked about knowing 2 groups of 5, so 5 x 5 is 5 groups of 5. So, 4 groups of 5 is another 2 groups of 5 then we only needed to count one more group of 5.

Julia: Mathematicians use parentheses when they want their audience to know which parts they did first, so let me do that (Julia writes "5x5 = (2x5) + (2x5) + (1x5)") Turn and talk about this representation. Is it what Kendra said? Do the parentheses help?

Author's notes

The representation on the number line allows a visual for equivalent relationships.

Providing time for students to discuss the big ideas in the string will encourage students to articulate the ideas and to begin using them.

Although students are not yet writing the equations, Julia notates for them so that they become aware and can reflect on it. Note how naturally she introduces parentheses.

The string

2 x 5
5 x 5
7 x 5
10 x 5
9 x 5
12 x 5
6 x 5
3 x 5