My fourth grade students have been exploring fractions. They’ve become familiar with how to add, subtract, and multiply fractions. They just started to divide fractions earlier in the week. Whenever I introduce fraction division I tend to have one or two kids that raise their hand quickly. Their quickly raised hand tends to cause me to slow down and prepare. They comment that there’s a “fast” way to divide fractions that they learned at Kumon or from someone at home. Sadly, that trick is infamous number 1 on the NCTM’s Tricks that Expire! These students can explain what to do, (change the numerator and denominator of the second fraction and multiply) but struggle when pushed to explain why it works. I feel like at times these particular students inadvertently or purposefly convince others in the class that this method is the quickest. Some agreed, but introducing this idea at the begging caused unneeded confusion.
I shifted the discussion to the meaning of the fraction bar. One of the students mentioned that the fraction 1/2 is the same as 1 divided by 2. Another student said that is the same as 0.5. This conversation was productive and moved the discussion back on course. Students started to build upon each response and were able to start thinking more about their own understanding of fractions. I then introduced the idea of fraction as division. This resonated well with students and I could tell that they were really thinking about how they view fractions. I then put this problem on the board.
Students thought for a little while and then decided to split up each fraction into three pieces. They then counted up the pieces to find 9.
I then introduced students to a common numerator and denominator model. Students thought about this problem and then started making a few guesses.
One thing that seemed to shift this thinking was to look at fraction as division. In my years of teaching this seems to make quite a few connections Many students know that a half of a half is a quarter, but are a confused when it comes to dividing a half. One student mentioned that they both have common denominators and that might be useful when dividing. Another student said that a fraction is division, so you could divide the numerators and denominators.
The class agreed that this will work as long as the denominators are the same. They also concluded that if the denominators aren’t the same, we can find an equivalent fraction to create ones that are. This conversation lasted for about five minutes. It was productive and not once was there mention of a “fast” method to divide fractions. I’m hoping that students hold on to visual models and using a variety of strategies when dividing fractions in the future. Next week, we’ll be investigating how to divide mixed numbers. That’ll most likely happen after our week long PARCC adventure.