B12 — Multiplication within 100: Automatizing Basic Facts Using the Ratio Table
Problem Strings for Fluency and Beyond · Multiplication within 100: Automatizing Basic Facts Using the Ratio Table
As students become comfortable with the representation of multiplication on a number line and with repeated addition and the use of partial products, you can begin to build a bridge to scaling on the ratio table by representing the repeated addition next to the ratio table. Working developmentally like this will continue to support the use of partial products and the regrouping of groups, but it also challenges students to think multiplicatively using proportional reasoning and scaling. When using the ratio table, develop a context to help the children understand the meaning behind what they are doing multiplicatively. Some contexts that can be used are cars and wheels, rows of stamps, 6-packs of juice or bottles of water, quarters to dollars, price of tickets, pennies to nickels, cooking time per pound, etc. Start by drawing a t-chart or ratio table. As students share strategies, record on the ratio table, and where helpful to bridge understanding add the repeated addition.
As with all problem strings, the problems have been scaffolded to encourage students to use facts they know to find answers to problems that are more challenging. At this point in development, students are still working on automatizing the facts, so these strings continue to help students learn the facts while at the same time supporting them to focus on developing number relationships and the properties of operations—the foundation for algebra.
Inside One Classroom: A Portion of the B12 Minilesson
Problem Strings for Fluency and Beyond · Grade 3
Julia (the teacher): Here is our first problem. (Writes "2x4=") Today when we are working on this string, think about a car and its 4 wheels. Each car has 4 wheels, so how many wheels would 2 cars have? Ok, thumbs up when you are ready to share. (When most thumbs are up, Julia starts discussion.)
Jack: 8. Because 4 + 4 is equal to 8.
Julia: I know that was an easy one. Today I am going to record that on a new model—a ratio table.
Julia: Here's our next one. How many wheels with 4 cars?
Henry: 16. If there are 8 wheels with 2 cars then there are 16 with 4 cars because 8 + 8 = 16. The cars doubled so the wheels doubled.
Julia: Wow! Let's see that on our ratio table:
Julia: I'm also going to record the addition because you said 8 + 8. (Julia records (4 + 4) + (4 + 4) adding parentheses to show the regrouping.) That was a really quick way to think about that problem! Did anyone else think of it that way? (Many hands go up. Julia goes on with the string and writes, 8 x 4. Most students double again producing 32. Julia also records the repeated addition and the regrouping.) Ok. This next one is a big challenge. What about 10 cars? Are there any numbers on the ratio table we could use for this one? Turn and talk with a partner about this. (Julia moves around and listens to conversations as students talk. This helps her decide how to begin the next discussion.)
| Cars | Wheels |
|---|---|
| 2 | 8 |
| 4 | 16 |
| 8 | 32 |
| 10 | 40 |
| 9 | 36 |
| 12 | 48 |
| 5 | 20 |
