Fraction Multiplication and Division

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My fourth graders are deep into a unit on fractions.  They’ve been multiplying common fractions and tiptoeing into fraction division.  That was until Wednesday of this week.  On Wednesday students explored different ways to divide fractions.  Students used visual models to divide, but that didn’t seem to help students understanding them better.  They encountered abstract problems and used the “flip” method to find the quotient.  This still didn’t help improve much in the conceptual understanding department.  Students wanted me to show the exact process of what to do to solve fraction division problems.  I wasn’t thrilled.  It was evident that students needed more exposure and practice with fractions. So I took a step back and reviewed fraction multiplication.

The class reviewed fraction multiplication and scenarios that are needed to find products.  Students were aware of many different situations where they might need to multiply fractions.   They were able to show visual models and computation strategies to find solutions involving multiplication.  I had a few students also indicate that it’s important to simplify the product.  So the class was rolling in a  positive direction and I decided to bring the lesson back to division.  The break through moment occured when the class connected fact families to the current lesson.  Similar to addition and subtraction, multiplication and division fact families can also contain fractions.  This helped students make connections.  Students wrote out different fact families using unit fractions (1/2,1/4,1/3…).  Students then changed the fact families related to only multiplication and division.  The class was starting to wrap their heads around fraction division with a bit more ease.  I felt as though students were ready for the next activity which was related to food.

Students were placed in teams of three and given a blueberry muffin recipe.

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Students reviewed the sheet and wondered where this was going.  Each group then received a sheet related to the original recipe.  Each half-sheet asked students to modify the recipe based on the serving size.

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Recipe modification

 

Some students were asked to make 12 muffins, while others said 36, 24, 60, 96 or 72.  I felt as though some students were relieved when they were asked to half or double the recipe.  Other groups tackled the problem with some major perseverance.  Students were asked to show their number model and explain why their answers were reasonable.  Some students wrote number models that multiplied fractions by the recipe amount.

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Showing number models

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Groups also used fraction division to show a number model.  The majority of groups connected how multiplication and division of fractions can be part of a fact family.  This was especially apparent when students started to see that 1/4 * 4/1 = 1.  I feel like this is laying groundwork for next year’s class when we start pre-algebra equations. Having a solid understanding of how to “undo” operations is a great tool to have in the math toolbox. Once students found the fractional reduction or addition they changed each ingredient accordingly.  After showing their work, students took a picture of the whiteboard and recorded their voice.  Student groups explained how they found each answer and why it was a reasonable answer.  Some student groups were amazing when communicating their reasoning.  They actually explained that the ingredients needed to be increased by a factor of 4.  Other groups were very general with their reasoning in saying that the recipe increased because they were asked to make more muffins.  I can tell this is an area that’ll need strengthening throughout the year.

Overall, this activity seemed to help reinforce skills taught earlier in the year.  The most complicated part was where to start.  Students had trouble knowing what do do with the problem at first.  Students seemed comfortable with the number model and computation components.  Explaining their reasoning needed some tweaking, but that might also be an expectation that needs to be set more in the future.