Fraction division – Educational Aspirations

Fraction Division Strategies

My classes have been recently exploring fraction division. Students completed word problems involving dividing fractional pieces and they were finding the idea challenging. In order to gain clarity, I worked with students in small groups to determine where the trouble spots seemed to developed. I started to notice a couple things: 1) students were relying on a fraction division algorithm without context 2) students were not sure how to determine the dividend, which made creating a number model problematic.

Relying on the traditional fraction division shortcut ended up causing problems for more than a few of my students. Students were not able to explain their reasoning for flipping the second fraction. This become even more apparent when students attempted fraction division word problems. Because you have to “flip” the second fraction students were not sure how to identify the dividend. This caused confusion. I planned out a small fraction bootcamp for students to explore fraction division through visual models. Students started out with problems like 2 ÷ 1/4 and progressed to where a fraction is in the divisor and dividend. Students were making progress and relying less on the shortcut method, although some used that to check their work.

After our mini camp, students were given prompts to show their understanding of fraction division.

1.) Juliane has 12 bags of confetti to spread on 16 tables. She wants to put the same amount of confetti on each table. How much of one bag of confetti should she put on each table?

This was the first problem and achieved the highest accuracy. Students drew out the 12 bags and spread it on 16 tables, finding the answer to be 12/16. Some showed a number model of 12 ÷ 16 = 12/16 and others drew a picture.

2.) Write a number story that can be modeled by 4 ÷ 5 = 4/5

This was more challenging. The number stories indicated whether a students could determine what was being shared and in how many pieces. It was interesting to read the responses and revealed an understanding of what is being split equally. Here are a few response:

There were 4 candy bars and 5 children. How much of the candy bars will each child get?

I have 4 boxes of apples and I wanted to put them in 5 bags and all the bags have the same amount of apples. How much of the box of apple go into the bags?

Tyler has 4 rats and 5 carrots for his rats to each get equally fed how much will each rat get?

There were 4 oranges jamal and his four friends wanted to spilt the oranges to a even amount how much of and orange does each person get?

3. Explain using words and the process you would use to complete the problem 5 ÷ 1/3. Give the reason why you completed each step.

This problem caused a few student headaches – but in a good way. Students that relied on the shortcut were confused in how to explain the reasoning for flipping the second fractions. Out of all of the problems, this one highlighted the conceptual understanding of fraction division the most. Some students sent in pictures with written explanations while others created number models. Here are a few of the responses:

First I would do 5 ÷ 1/3 This works, because it is the same question just written in a different way. Next I would see how many 1/3 can fit in 5. To do this I would do 5*3. This works, because there is 3 1/3’s I one. And there is 5 ones in 5*3 = 15. So the answer is 15. (appreciate the thorough thinking behind this response!)

1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3+1/3

First I switched 5 to 5/1 and then 5/1 to 15/3. Why I did this is to make the denominators the same same number. Then I divided across numerators and denominators to get 15/1 then I simplified 15/1 to get 15. Why I divided across numerators and denominators is to get the answer. Why I simplified to make the number a whole number.

I think the answer is 15 because you can think about how many 1/3 are in 5 and that answer is the answer to your problem.

First I converted 5 to 5/1 then I did 5/1 divided by 1/3 to get 5/1/3 then I did 5/1/3 X 3/3 to get 15/1 which I simplified into 15

I was pleasantly surprised to see the improvement in being able to navigate fraction division. Being able to conceptually understand fraction multiplication/division can sometimes be a roadblock for students. I am hoping to break that and looking forward to discussing and highlighting a few student examples with the class next week.

Fraction Division – Models and Strategies

My fourth grade students have been exploring fractions. They’ve become familiar with how to add, subtract, and multiply fractions. They just started to divide fractions earlier in the week. Whenever I introduce fraction division I tend to have one or two kids that raise their hand quickly. Their quickly raised hand tends to cause me to slow down and prepare. They comment that there’s a “fast” way to divide fractions that they learned at Kumon or from someone at home. Sadly, that trick is infamous number 1 on the NCTM’s Tricks that Expire! These students can explain what to do, (change the numerator and denominator of the second fraction and multiply) but struggle when pushed to explain why it works. I feel like at times these particular students inadvertently or purposefly convince others in the class that this method is the quickest. Some agreed, but introducing this idea at the begging caused unneeded confusion.

I shifted the discussion to the meaning of the fraction bar. One of the students mentioned that the fraction 1/2 is the same as 1 divided by 2. Another student said that is the same as 0.5. This conversation was productive and moved the discussion back on course. Students started to build upon each response and were able to start thinking more about their own understanding of fractions. I then introduced the idea of fraction as division. This resonated well with students and I could tell that they were really thinking about how they view fractions. I then put this problem on the board.

Screen Shot 2018-03-14 at 8.25.39 PM

Students thought for a little while and then decided to split up each fraction into three pieces. They then counted up the pieces to find 9.

Screen Shot 2018-03-14 at 8.29.06 PM

I then introduced students to a common numerator and denominator model. Students thought about this problem and then started making a few guesses.

Screen Shot 2018-03-14 at 8.31.45 PM.png

One thing that seemed to shift this thinking was to look at fraction as division. In my years of teaching this seems to make quite a few connections Many students know that a half of a half is a quarter, but are a confused when it comes to dividing a half. One student mentioned that they both have common denominators and that might be useful when dividing. Another student said that a fraction is division, so you could divide the numerators and denominators.

Screen Shot 2018-03-14 at 8.34.07 PM.png

The class agreed that this will work as long as the denominators are the same. They also concluded that if the denominators aren’t the same, we can find an equivalent fraction to create ones that are. This conversation lasted for about five minutes. It was productive and not once was there mention of a “fast” method to divide fractions. I’m hoping that students hold on to visual models and using a variety of strategies when dividing fractions in the future. Next week, we’ll be investigating how to divide mixed numbers. That’ll most likely happen after our week long PARCC adventure.

Fraction Division Models

Screen Shot 2017-10-14 at 10.06.30 AM — Dividing Fractions

One of my classes started a unit on fractions around two weeks ago. We explored the lowest common multiple and greatest common factor during the first week. Moving on, the class investigated the different ways to multiply fractions. Most students had an idea of how to find the products by multiplying the numerators and denominators. They struggled when using visual models and also when converting fractions to/from a mixed-number form. We spent a couple days reinforcing how to visualize fraction multiplication and created multiple models to show how to multiply a fraction of a fraction. We used the folding paper method as well as free-hand drawing of different models. On Thursday, the class moved to the next topic: fraction division.

I’d say that this is a topic that’s a bit confusing every year. Many students, and I mean almost a third of my class tend to come into the class with an understanding that when you divide, the quotient will always be less. This is one of the tricks that expire. So, as I started to plan out what I was going to do during the introduction, I had to keep in mind that this topic has the potential to be a misconception minefield.

On Friday, that class started to study the topic of fraction division. I ended up using a Brian’s amazing resource to put together a Nearpod activity involving fraction division. Without discussing the topic too much, I asked students this question:

Show 3 ÷ 1/2. Write on the picture to show your model.

Screen Shot 2017-10-14 at 5.35.13 AM.png

Students worked in pairs with one device to find a solution. Some groups immediately started splitting up the fractions. The confidence from these groups seemed to be high. Other groups were discussing what was meant by the visual model. Here is one of the initial responses:

Screen Shot 2017-10-14 at 5.41.44 AM — Does this match the number model?

It was interesting as some students wanted to split the entire three sections in half. This had me wondering if the students understood that each block was one whole. I also had some students that were able to find a solution, but it had to do with using the trick and not the visual model.

Screen Shot 2017-10-14 at 5.42.09 AM — “I used the trick” – visual model wasn’t touched

When pushed to explain their thinking the students weren’t able to move past the process of finding the reciprocal of the second number and multiplying. The class then moved to the second question.

Show 3 ÷ 1/3. Write on the picture to show your model.

Screen Shot 2017-10-14 at 5.48.57 AM.png

Students seemed to be more comfortable with the problem. They were also a bit more careful when splitting up the shapes. I reinforced with the class that idea of dividing fractions. The responses showed more detail this time around.

All of the responses used some type of a visual model, which was a positive as this didn’t happen as much the first time. When asked to explain their reasoning, students were able to tell me with confidence why the model made sense. Also, some of the students started to find that that they could check their answer with multiplication – 3 ÷ 1/3 = x , then x * 1/3 = 3. There were still students that went to a default of using the trick to find the quotient. The third problem was designed to see if students could stretch their understanding.

Show 3 ÷ 2/3. Write on the picture to show your model.

Screen Shot 2017-10-14 at 5.58.32 AM

Students took more time with this model. Most of the groups started to divide each whole into thirds. That’s when trouble started to brew. Students counted up the wholes, but noticed that parts were still missing. There were a lot of questions here. I asked students to take a risk and put their mathematical thinking out there so we can all analyze the responses. The students submitted their ideas and I noticed that the answers were all over the place.

The class noticed that all of the groups split up each whole into thirds. The groups also shaded in or indicated two thirds of the whole. Students also noticed that some of the third shapes weren’t included in the quotient. There was a class debate on whether those thirds should be included. Most of the students agreed and said that they’re part of the whole so they should be included. This last question took up the most time and I think it was one of the better moments of the lesson.

There was only around ten minutes of class left, so students went back to their seats to work on a few more fraction division application problems. The struggle and perseverance to understand was evident and I’d like to find ways to incorporate this type of instructional routine more often.

Overall, I thought this lesson went well. I recorded the entire lesson for my NBCT video, so it’ll be interesting to see how it turns out. While watching the videos that I record I tend to remember instances where I could’ve done things differently. In some of the cases, I could’ve asked better questions or model a bit less and have students make the connections themselves. It’s a balance though. I’d say that watching a video of myself teaching is a humbling experience. It’s humbling, but the personal reflections that come out of those experiences are worthwhile. I think I could write an entire blogpost on that reflection process. Maybe next time.

Fraction Multiplication and Division

Screen Shot 2016-02-26 at 5.47.33 PM.png

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.

Screen Shot 2016-02-26 at 5.26.13 PM

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.

Screen Shot 2016-02-26 at 5.37.12 PM — 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.

Screen Shot 2016-02-26 at 5.35.39 PM — Showing number models

Screen Shot 2016-02-26 at 5.45.30 PM

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.