B44 — Connecting Multiplication to Division using the Open Number Line
Problem Strings for Fluency and Beyond · Connecting Multiplication to Division using the Open Number Line
Division is the inverse of multiplication. The strings in this section juxtapose the operations together to help students understand how the operations are connected (for example 32 ÷ 8, 8 x 4 , and 4 x 8). As you begin to explore division with your students, you will notice that the three models which have been developed thus far (the open number line, the ratio table, and the array) are used again to support progressive development to eventually arrive at fluency with multiplication and division to 100. As the division strings are presented within this section, we will refer to one model at a time in a developmental progression from using the open number line first for skip counting, repeated addition (or repeated subtraction) and regrouping groups. The ratio table and the array are used later for scaling (and simplifying) and partial quotients and partial products. In the string below, use the open number line for representation. Present one expression at a time, continuing to listen to students’ strategies and representing them as they are described.
Notice how the number problems are written in pairs or triplets until the last problem? The second problem is the inverse of the first. They are paired to help generate a conversation about the relationship between multiplication and division. The numbers within this string were chosen to encourage students to use the multiplication facts they know to solve division. The last problem is a standalone to challenge students in finding their own helper multiplication fact to arrive at the quotient for the problem.
In addition, for the second problem (the first division problem), students might think of the problem quotatively asking, "How many 4s are there in 40?" To represent this strategy, show on the number line 10 groups of 4, repeated addition of fours, or skip counting by fours depending on what students say:
Some students will think of it partitively and ask themselves, "How can I divide 40 into 4 equal groups?" To represent this strategy, show the length of 40 cut into 4 tens:
Both representations are correct for the expression 40 ÷ 4. Representing them on the number line will elicit a conversation about how to think about division problems in relation to multiplication. The first representation helps students show 10 groups of 4. The second shows 4 groups of 10—the commutative property.
- 10 x 4
- 40 Ă· 4
- 5 x 4
- 20 Ă· 4
- 3 x 4
- 12 Ă· 4
- 12 Ă· 3
- 18 Ă· 3
