Anyway, division of fractions by whole numbers is a much more difficult concept, so I taught it after students grasped division of whole numbers by fractions, as detailed in my previous post here.
What's difficult about this is that students start with something less than one and then have to further divide it into smaller pieces. In doing this, they need to be able to determine equivalence in order to represent the answer in relation to the whole.
I will be using the same theme of the dog food example as I did with my other division of fractions notes. I plan to stick with it so after we learn all division of fraction types, students can compare/contrast the different ways the problems are worded, how you would write the equation, and how they will be solved based on the question asked.
I anticipate that students will continue to have misconceptions about the answer. In this example, they would think 3/15 of the bag is the answer. Therefore, students will really need to go back to the question, reread it, and then answer accordingly. I also want them to know that 3/15 will have a meaning, but the question is not asking how much the dogs eat altogether. If that was the question, there would be no need to find equivalence or partition the fifths into thirds since together, they'd eat 1/5 of the bag.
Once again, after they grasp the modeling and meaning of this concept, I'll be going back and asking if they see a pattern between the expression and the answer.
- How do we get from one to the other without modeling?
- Is it the same pattern for when we were doing division of whole numbers by decimals?
This way, when students see the pattern for the algorithm, I can explain why we invert and multiply. My high math students who tend to see patterns pretty quickly go straight to that first and then use their answer to check their model. Hopefully all this conceptual teaching will help ingrain in students the reason behind math, not just to complete the problems like robots.
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