Division with fractions can be perplexing - for adults as well as children. So often, we rely on rules such as "invert and multiply" or "keep, change, flip" that help us perform a procedure but are disconnected to broader mathematical ideas. While done with the best of intentions, creating a model or drawing to help can also interfere with our understanding of division involving fractions. This was evident when tutoring my neighbor this morning. We haven't worked together for a few weeks - last time she was working on addition and subtraction with fractions. Today, we worked on a problem involving making milkshakes. There were 3 cups of chocolate syrup and different recipes: 1/8 cup of chocolate syrup for a chocolate milkshake 2/8 cup of chocolate syrup for a double chocolate milkshake 3/8 cup of chocolate syrup for a triple chocolate milkshake Seems like a simple pattern, right? Once you figure out the number of milkshakes made with one of the recipes, you can use that information to help you figure out the others. My neighbor didn't see the pattern, though. She also wasn't sure why it was a division situation rather than multiplication. When I asked her to draw a picture of her thinking about the problem, she made this: She explained she had seen her teacher draw this in class and it must be 1/8. She was adamant and I realized that the picture caused some interference in her thinking - looking at the picture as a "whole" rather than seeing that there was 1/8 of a cup being used in each row. (If each row represents one cup of chocolate syrup - so the 3 rows would represent the 3 cups of chocolate syrup.) . She was having difficulty distinguishing what was the "whole" - the 3 cups of chocolate syrup versus 1/8 of 1 cup of chocolate syrup. In an effort to help her visualize the repeated removal of 1/8 cup, we tried measuring sugar into bowls. (Unfortunately I didn't have the ingredients for chocolate milkshakes on hand - that would have been more delicious!) After measuring out the sugar, we went back to write equations to match our work. While she seemed a little more confident, she still left our session today still feeling frustrated by fractions. We'll continue to work on multiplication and division with fractions. I'm thinking she might benefit from using common denominators for division to help her see the connection to division with whole numbers. I'm looking forward to our next session to explore these ideas further.
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