B51 — Connecting Multiplication to Division using the Ratio Table
Problem Strings for Fluency and Beyond · Connecting Multiplication to Division using the Ratio Table
The strings in this next section use the ratio table to show the relationship between multiplication and division. Continue to encourage students to use the multiplication facts they know to solve the division problems. Presenting the problems in context will help students realize the meaning behind what they are doing. Just present the ratio table as shown in Appendix A on p. 161, like the table below but with the unknowns missing. (Do not show the arrows and x4 and x10 below — those are for you only, shown as ways to represent what students might say. See Appendix A.) Start by saying this is a ratio table that a car dealership uses when ordering tires. One car has 4 wheels. Write in the 4 under "wheels." Then write in the 40 (also under wheels) and ask, "How many cars?" Some students may use skip counting or repeated addition. If so, represent this strategy on the open number line showing the skip counting, or write to the side the repeated addition of ten fours: 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4. Then write the first two problems in the string and put 10 on the ratio table under "cars". Other students may scale x 10, saying, "We had 4 and I knew 40 was 10 x 4." Draw arrows on the sides of the table to show the scaling.
As you work through the string, solving for 5 cars and then 2 cars, note that these partial products (5x4 + 2x4) can be used for 28 wheels. Partial quotients can also be written in the string (28÷4 = 7 because 20÷4 + 8÷4 = 28÷4). Keep adding related expressions and equations into the string as they come up. The goal is to fill in the chart and to generate several equations. Some equations that will likely be generated: 10 x 4 = 40; 40 ÷ 4 = 10; 5 x 4 = 40 ÷ 2 = 20; 2 x 4 = 8; 7 x 4 = (5 x 4) + (2 x 4); 28 ÷ 4 = (20 ÷ 4) + (8 ÷ 4) = 7; 24 ÷ 4 = (28 ÷ 4) – (4 ÷ 4) = 6; 4 x 3 = 12; 12 ÷ 4 = 1/2 of (24 ÷ 4); 16 ÷ 4 = 8 ÷ 2 = 4.
Notice the NUMBERS CHOSEN and WHERE THEY ARE PLACED on the ratio table for consideration. They will likely generate scaling and simplifying strategies, as well as partial products and/or partial quotients. This will be the case throughout this section where the ratio table is being used. 5 is half of 10 and thus the number of wheels for 5 cars (20) will be half of the number for 10 cars (40). 40/10 = 20/5. This is an example of simplifying. The numbers of wheels for 5 cars and 2 cars are very helpful for determining the wheels for 7 cars. The strategies that evolve here are partial products, 7x4 = (5x4)+(2x4), and /or partial quotients, 28 ÷ 4 = (20÷4) + (8÷4).
Inside One Classroom: A Portion of the B51 Minilesson
Problem Strings for Fluency and Beyond · Grade 3
Julia (the teacher): Here is another problem (It's the third problem in the string.) We have 5 cars. How many tires will there be? (When most thumbs are up, Julia starts discussion.)
Christina: 20.
Julia: Did anyone else think it would be 20 tires? (All heads nod.) Great, how did you get that?
Christina: I knew if each car has 4 tires, then I knew 5 x 4 is 20.
Julia: So, which earlier problem in the string did you use?
Christina: I guess I used the first problem. Now that I think about it, I multiplied both of the numbers in the first problem by 5. I didn't see that until just now.
Julia: Nice. Let me write an equation for your first idea. (Julia writes "5x4=20" on the right to start a string of equations and then she draws arrows on the side of the table to represent Christina's second idea.)
Julia: Does that represent what you are thinking? (Christina nods her head.) Great, did anyone else think of this problem a different way? Nick?
Nick: I also got 20, but I used the problem with 10 cars. I knew that 10 cars had 40 tires, so since 5 is half of 10 I knew it had to have half as many tires, or 20.
Julia: That's some great thinking. Hmmm… so are you saying that because 5 is half of 10, the number of wheels for 5 cars (20) will be half of the number for 10 cars (40)? Let me show that on the ratio table too. (Julia records arrows on the sides to represent the halving that Nick is referring to and then writes more equations in the string, 1/2 of 10 = 5 and 1/2 of 40 = 20.) Hmmm… so could I also write this equation: 40/10 = 20/5? This question needs a lot of thinking time. Turn to a partner and talk about this question.
Nick: What's the bar mean?
Julia: Oh sorry, that's just like the division sign. They mean the same thing. I could write it this way also: 40÷10 = 20÷5. 1÷4 = 1/4, right? The bar just means division.
Nick: Oh, then they must be equal because 40 divided by 10 equals 4 and 20 divided 5 equals 4. Same answers.
| Number of Cars | Number of Wheels |
|---|---|
| 1 | 4 |
| 40 | |
| 5 | |
| 2 | |
| 28 | |
| 24 | |
| 12 | |
| 16 |
Appendix A – Ratio Table for String B51
