So, back on the fractional multiplying kick.
Disclaimor: Although the title says multiplying, many students do not see it as such. To me, it’s all about pattern recognition and application.
This approach to fractions and “multiplication” (more number sense/arithmetic sequences) came about this year as we were working on Linear functions and X/Y tables. Students really got into finding patterns in the table and were able to talk about how changes in the X relates to changes in the Y. Of course, I wanted to challenge them a bit so since I knew they feel they stink at fractions- it was the perfect example.
This table took my class less than 30 seconds to fill out. Of course I made a big deal of posting it on the SMARTBoard and having students come up and fill in a missing value. While I had students doing this task, one student asked “What happens when X is 1.5 or 1 1/2?” This took me off-guard for a second, but became that teachable moment.
me: What would happen?
s1: I don’t know, it’s not even on the table.
s2: but it goes between 1 and 2.
me: OK, let me adjust the table a bit…
me: So, now what?
s2: Well, the pattern is going up by 1/2.
s1: No, it’s going up by 1.
s3: The X was going up by one, and the Y is going up by 1/2. Mr. A changed it however and now X goes up by 1/2 in one spot.
s1: He can’t do that, it’s against the rules.
me: What rules?
s1: The rules of X/Y tables, X has to go up the same amount.
me: Where did you learn this rule?
s1: I don’t remember, some other teacher.
This is a problem that has always concerned me, partial information that students remember. My students have checked in and out of class so much that they have a huge hodge-podge of rules and information floating in their heads that they can’t rationalize when and where to use it. Many of them get frustrated with me when I don’t immediately tell them whatever method they choose is correct, they hate that I ask them why they chose to use it. My “Rules vs. Understanding” letters sometimes go missing….
Once we get the mis-concept of X values cleared up, we go back to the table and start looking at the value of the missing Y.
me: So did we find an answer for Y?
s4: It’s 3/4. (they are normally not the top student, but one who thinks critically)
me: Why did you choose 3/4? (notice I say choose, not guess- at this point there is a big difference)
s4: Y is going up by 1/2. If you only go half of that, you go by 1/4. So 1/4 more than 1/2 is 3/4.
me: Does anyone agree? Can you tell me if this is true? (I try to not immediately confirm claims, I still want students to analyze and justify)
<pause while students think>
s3: Yea, she is right. You shifted the pattern from 1/2 to 1/4.
This is when I throw up some more fractional examples for students to complete. I give them the same initial table, usually starting with some fraction that has an even denominator base- I want them to really understand this before I throw them a curveball. Then we edit the tables for increments of 1/2 in X instead of 1.
When students really seem to have this down, I write: “What is 1 1/2 X 1/2?” on the board. Depending on the students, some start furiously multiplying, some go for the calculator, and some just sit and wait for another student to get the answer (these students I address in other ways- that’s a different blog, but I still wanted to acknowledge that they existed in my classroom). Students will answer 3/4.
me: Huh, you said you got 3/4?
s1: Yes, Mr. A
me: Is that correct?
s2: I got that on my calculator as well, it’s right.
me: Interesting, did we need to do any calculations or calculator work to actually find that answer?
s4: No! It’s on our work for today.
me: What do you mean?
s4: We did this problem earlier, look at our first X/Y table for today.
me: How does that help?
s4: Multiplying 1/2 has the same pattern as the table, going up 1/2 each time. So multiplying 1 1/2 would be the same as our table, 1/2 and half of 1/2, or 1/4. The answer is 3/4.
me: So can you tell me 3 1/2 x 1/2?
s2: 1 3/4. (They finally caught on and understand what is happening, and like that fact)
me: What about 4 1/2 x 1/2? (I want them to look at the table on a whole number, not only the mixed one)
s2: Nice try Mr. A, it’s going up 1/4 so it’s 2 1/4.
At this point I ask the students to come up with 5 multiplication questions from their own tables, have the students ask them of their peers and moderate the answer/justification process. It really surprises me how quickly they catch on.
The one thing I don’t like about this is that it can become just another multiplication table in their notebooks, and I have had some students construct it. The BEST thing I like about this is that it gets students thinking about the patterns of fractions, they start doing mental math with fractions, and begin to believe that fractions are not as hard as they believe.
Did I give them any rule? No. Did I show them the standard algorithm? Never crossed my mind. Will I ask later if students notice a different way to multiply fractions? You bet I will. We’ll talk about those and refine thinking even more. And when there comes a time they tell me to convert to improper fractions, multiply numerators and denominators- I’ll know they understand that process a whole lot more than just giving them that rule.