First, I had to go over what is an improper fraction, what it means, and how to model it different ways. This is key because student answers to multiplication of whole numbers by fraction problems may result in improper fractions. I also want them to be able to write answers in mixed form.
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| I didn't formally teach students they could divide the fraction to get the mixed number. Students know that the fraction bar is essentially a division sign, but I want them to understand the concept of decomposing and seeing their work in their model first. |
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| Again, I didn't formally teach the "multiply & add" trick to change mixed numbers to improper fractions for the same reason as above. However, I asked if students saw a pattern (for mixed -> improper & improper -> mixed). Then I taught it to those who mastered the concept of the visuals. I explained how the algorithm helps us with numbers that are more difficult or tedious to model, such as 39/4 or 17 and 3/5. |
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| Here are the notes we took together. I wanted to relate what multiplication means first by bringing it down to a very basic level students would all understand, such as 6 x 3. (For some reason, you throw in a fraction and they "forget" what operation signs mean.) I taught the bar model because it was easier for some to see conceptually at first. However, I eventually pushed them to the number line model only. |
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| Students who were having a difficult time with this concept were confused about 1) why we partitioned each whole by the unit size (denominator) or 2) how many "jumps" they needed to take. Even some of my other kids who knew what to do had a difficult time explaining their thought process to others. If I gave them an incorrect problem to critique and analyze, they were able to find mistakes, but couldn't do so as easily in their own work. So... I wanted to reinforce this explaining thing by having them create a problem with their partner, listing steps they took to solve, AND (the most important) stating the reasons WHY they did each step. |
One thing I tried to reinforce, but didn't work as well, was the use of the math vocabulary while they explained the reasons why each step was needed. However, this was really their first attempt at breaking down each step in addition to listing their purposes. So, I wanted to cut them some slack and see what they would give me first. Next time, I plan to put a list of vocabulary words up and then expect a minimum number of words used.
Some vocabulary words to use could be:
-number line
-improper
-mixed
-whole, fractional parts
-partition/iterate
-unit size
Overall, I am VERY impressed with students on their quizzes. I felt like a proud parents when one of my struggling students received 100% on her test! They all did a great job and we are ready to move on!!! And still review, of course. ;)
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