Digging Deeper into Division with Rationals

I have been studying how to get more kids involved in our math lessons. I feel like sometimes they get bogged down by the number of words on the page, all the different instructions, and all the different materials and resources they are expected, or allowed, to use.

In the past, I would use the lesson 3.3.2 in CPM’s CC2 book as is.  While there’s some good word problems in there, I found that kids got confused by the number of different steps and they shifted their focus to checking off that the problem was done rather than trying to understand the lesson target.

Recently, my students have been working on dividing positive integers by fractions and then by decimals.  So I just asked the class how would you take 15 and divide by .2 and then invited different strategies to be put on the board. The classroom exploded with ideas.  We found some good misconceptions: one girl who made the ratio table below originally set 15 and 1 in the same row and scaled it down to .2. She was confused, but we shared it with the class and we realized that she was actually finding the answer to a different question: what is 15 x .2?

My students shared some really inventive strategies like the incomplete circle with the lines on it.  That team knew that there were five .2s in each 1. So they counted by fives with the lines off of the circle (Note: they realized they were a few lines short). We connected it to the 15 by 5 area model and the linear model where students were repeatedly adding by .2s. 

One team remembered what we learned the previous day and turned .2 into 2/10. Then they used the “keep-change-flip” method. 

Another team used scaling up and proportional reasoning as shown below. 

Scaling up led us to The Giant One. That led us into the traditional method…and a picture of Deadpool for some reason. 

I found that my kids were much more engaged with understanding decimal division at a deeper level than in the past when I would just go by the book.  I was able to facilitate many conversations about the connections different students made between strategies.  

By simplifying the problem, asking for multiple ways to solve it, and then searching the room for novel ideas, I was able to get the students to develop connections and make sense of the math.

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