B52 — Connecting Multiplication to Division using the Ratio Table
Problem Strings for Fluency and Beyond · Connecting Multiplication to Division using the Ratio Table
This string is like B51, however here a string is given that you can use, if you wish to do so, instead of generating a list of equations as was the intent of B51. You can also do both. Just be sure when you place a value on the table that you keep the function (x5) constant. Use Appendix B on p. 162 for the string so that the x5 and red arrow are not present. It is shown below for you only, to make sure you keep the function on the table consistent as you record.
Inside One Classroom: A Portion of the B52 Minilesson
Problem Strings for Fluency and Beyond · Grade 3
Julia (the teacher): Here is the next problem. (Julia writes "10÷ 5"; it’s the fourth problem in the string.) Think about the solution to the problem, but also where to place the numbers on the ratio table. (When most thumbs are up, Julia starts discussion.)
Carlos: It’s 2. I know that if we divide 10 into groups of 5 then it would be 2.
Julia: Why are you thinking of it as 10 divided into groups of 5?
Carlos: Because I know there are 5 fingers on a hand so there are 5 in the group.
Julia: Do others agree? (Heads nod.) Ok so where should I put these numbers on the ratio table?
Bella: You should put the 10 in the column for the Number of Fingers and the 2 in the column with the Number of Hands. When you have 2 hands, you have 10 fingers.
Julia: The problem was to divide by 5. Carlos said he divided 10 into groups of 5. Where should I put Ă·5? Maya?
Maya: It’s kind of there already with the other numbers… 5÷5 = 1 and we could also think of 1x5 = 5. It just depends on which way you are going, fingers to hands, or hands to fingers.
Julia: So, I could draw an arrow from hands to fingers and mark it x5 and if we want to go from fingers to hands the arrow goes the other way? That’s pretty cool. What equations can we write so far?
Fiona: 10x5 = 50 and 50÷5 = 10. And also… 5x5 = 25 and 10÷5 = 2.
Julia: But we already have those in the string, right? Can we add any more now that Maya has pointed out the direction of hands to fingers and fingers to hands?
Fiona: Maybe 25Ă·5 = 5?
Julia: And that is a new one, right? It is not in our string. Ok, let’s add it and see what we can do with 60÷5. Turn and talk about this one with a partner. (Julia moves around and listens to the many discussions going on in her community. This helps her figure out her next move.) Bella, share the conversation you were having with Maya.
Bella: We used two problems we knew already. We used 50Ă·5 = 10 because we knew 10x5 = 50. But then we needed 10 more fingers to make 60 so we used 10Ă·5 = 2. Then we added 10+2 and got 12.
Julia: Great thinking. I’m going to write that like this: 60÷ 5 = (50÷5) + (10÷5) = 12. Did anyone else do it a different way? Let’s see what other equations we can write for this.
- 10 x 5
- 50 Ă· 5
- 5 x 5
- 10 Ă· 5
- 60 Ă· 5
- 20 Ă· 5
- 40 Ă· 5
- 30 Ă· 5
Appendix B – Ratio Table for String B52
