So last year was my first year teaching fraction division, and we did a rushed job of it. It was the last (mini) unit of 6th grade math before the state exam, and I basically taught it as the inverse of multiplication, with some ideas of scaling using ratio tables (from Kara Imm’s number talks on paint cans on her website, numberstrings.com). Many of my students already knew the algorithm from their fifth grade classes, so we didn’t do a great job at teaching that topic. And it was sufficient, but not enough for me. This year, I rearranged the order of our units with my colleague to make the fraction/decimal operations unit come right after our second/third unit, so that after our students have done integers (first), coordinate plane (second), and factors/multiples (third), we would go into fraction/decimal operations. This will allow us to reinforce those calculations within the units on expressions/equations and ratios/proportional reasoning (though, I’m honestly thinking for next year, we might want to do ratios and proportional reasoning FIRST and then fraction and decimal operations – whoops!).
Anyway, since I’ve never taught this topic to the level of depth I wanted to, I decided not to reinvent the wheel, and instead, to look at known resources. While I used CMP3 to launch this unit on fractions and decimals, they go more in-depth into addition and subtraction than we wanted to spend. Plus, their division problems in problem 3 didn’t go into detail about the two different ways of thinking about division (quotative and partitive) and that was something I wanted to try out with my students this year.
I really have been enjoying the lessons from Illustrative Math’s open up resources. I do find that I don’t have enough time to go into quite the level of depth that they do (there’s 17 lessons on just developing the ideas about division of fractions! each of which are anywhere between 40 and 50 minutes depending on optional activities). I’ve had to cut out some of the activities to make it work, but I REALLY really like it. I think next year, I might go straight into this investigation, and not do the other lessons from CMP3 first (it’ll depend on how we rearrange the units and whether we need those ideas of equivalent fractions to be developed in this unit or if they’ll have already been developed in a ratios and proportional reasoning unit first).
On the Friday before break, I had my students do a set of problems with fraction strips. Some of the students used the diagrams, some of the students wrote equations and used the algorithm, and some of the students thought about it in other ways (that I didn’t highlight). What came up was that some students said they were using an algorithm they didn’t really understand and I asked if they could connect it to what the person who used the diagram had done. We ultimately decided that the reciprocal represents the number of the divisor in one whole (which is basically the definition of reciprocal), and we multiply it by “how many wholes we have” (because we were doing whole number divided by a fraction first). I had never realized that interpretation before of fraction division – and I was impressed! I know this is tricky stuff (Liping Ma’s book, the Knowing and Teaching of Elementary Mathematics made me realize how tricky fraction division is – and it’s part of what inspired me to become a math teacher!). But our discussion around these problems was what led them to this conclusion!
This week, I started with trying to get a bit too much accomplished… I wanted to squish ideas from 4 different lessons into just today. In my double period, I managed to go through three handouts and do an exit ticket – and the amount of struggle they had with the exit ticket made me realize I really rushed it too quickly. I’m struggling with the balance of the kids who “already know the algorithm” and the kids who have limited prior experience with division of fractions, and I’m not sure yet how to balance that.
I am also struggling with how to go over some of the topics/skills in the work from Illustrative Math. I’m doing a lot of partner talk (which I normally do), but not as much whole class conversation – because I’m feeling like some of it is tricky to hash out in whole class conversation without looking at their diagrams – but I don’t always find them to be using the diagrams in a way that actually makes sense to solve the problems. So I wonder if I should’ve had more direct-instruction about it? But I think that defeats some of the purpose. I think it went a bit better in the second class I taught today because I knew which things to point out to them in the warm-up (where the diagrams were already half drawn for them) and how to help them connect it to the diagrams in the second handout.
I also made some strategic choices about which items to focus on and which items to cut. I used 4.6 warm-up and created a handout where they were drawing the tape diagrams. I pulled some other questions from the IM curriculum (in the practice sections, etc) and made a two-sided handout. The students got a chance to create the diagrams for a number of problems, some referring explicitly back to the warm-up of “How many groups of 2 1/2 in ten?” and “How many groups of 2 in 7?”
I had the students discuss in partnerships the questions of how they began to draw their diagrams and how they saw the answer in the diagrams (how many groups). I decided not to go over each diagram individually as a whole class, but rather to put up my answer key, share that theirs might not be identical, but they could edit/revise or ask questions about mine. I saw some pencils/erasers moving as I put up my answer key, but no one had any questions they wanted to share with the whole class. I struggle some times – I’m not sure that it was the best way to go over these problems, but no student’s work had stood out to me sufficiently during the work time as being one I wanted to highlight under the document camera, and sometimes, I struggle with what to have students do… Now I wonder if I could’ve had them do a “notice/wonder” discussion with a partner about my answer key instead! (Darn! Next time!)
I also had the students share out their answers to the following question (also from IM, though I’m not sure if it was in the same lesson or not). “Diego said that the answer to the question ‘How many groups of 5/6 are in 1?’ is 6/5 or 1 1/5. Do you agree with his statement? Explain or show reasoning.”
I chose this question because one of my students who was “done early” had this question incorrectly answered with “No, he’s wrong. It should be 1 1/6 because he has 1/6 remaining.” This student didn’t share her thinking with the class and I didn’t want to call her out, so I just tried to play dumb/confused when the student responded that it should be 1 1/5. I asked the class, “Wait, I’m not really sure what she means by it should be 1/5. I thought there was 1/6 left over? Talk with your partner about which one it should be.” I had a few students share out and rephrase/restate each other’s reasoning. Then that first class (with only a single) ended, and I left them to think about making sense of the “remainders.”
My second class did this same problem, and we had a much richer discussion about it. I shared with them “One student in my other class thought it was 1 1/6. What might they have been thinking?” and they were able to articulate that they just took the extra 1/6 and put it on the answer. Then I asked them to think about about what they would say to that student to help them understand why it’s 1/5, and a student used the equation 1/6 x 5 = 5/6 therefore 1/6 / 5/6 = 1/5 (because you want to know “how much of the 5/6 the 1/6 is”). I felt more satisfied with their reasoning, but I’m not sure that every student was there yet, despite the fact that most students gave a thumb vote agreeing with Diego. I struggle sometimes to get whole-class QUICK formative assessment that doesn’t require me to grade their work, but also gives me a view into their thinking. I’m not sure it’s possible – either I need to look at their work, or I don’t get a view into their actual thinking.
Anyway, in that second class, we then went on to 4.7’s fractional batches of ice cream. I skipped over the ropes section (even though I included it in the handout) due to timing, and also because I wanted to stick with the same kind of diagrams and I felt like ice cream would be something my students understood well. I was quite happy with the way I did this section of the lesson, and I think I’ll repeat it similarly with tomorrow’s class.
I gave them the handout and some independent think time. Then we transitioned into “table talk” time where students can continue working alone OR working with partners. When most students had finished Monday, Tuesday, Thursday and Friday (only one table hadn’t), I asked students to turn to their elbow partners and compare their diagrams – discuss how they saw the numbers in the diagrams and how they could use the diagrams to answer each question. The one group where no one had finished all four, I gave my answer key, and I encouraged them to compare their answers with that one – it wasn’t ideal to me, but I didn’t know how else to manage the fact that that group was SO FAR behind everyone else. I don’t think it was an understanding issue, but maybe more of an organization issue? Or a visual-spatial issue of making sense of the diagrams? I’m not sure.
Anyway, I then showed the four diagrams from IM’s “answer key” and I asked students to share a division equation for each day. We got all four equations up on the board (though we almost had one mishap – one student said 9 / 6 = 2/3, but students saw that he had reversed the divisor and dividend, and we corrected it – while that mistake can be valuable, it wasn’t the focus I wanted to go with, so I quickly dealt with it, rather than exploring it more in-depth).
Then I used the alike/different routine and I asked students to discuss how Monday/Tuesday (both had quotients greater than 1) were different from Thursday/Friday (both had quotients less than 1) and how they were alike. We discussed that they all involved dividing by 9 (which made sense to them because 9 cups of milk was required for each batch, and we were really asking the question “how many times does 9 go in?” My students also noticed that none of the dividends were divisible by 9). We also discussed that they were different because of the quotients being greater or less than 1 – and then I asked them how they could predict whether the quotient would be greater or less than 1 without doing the calculation – and they told their partners and then we shared out that they could compare the dividend the the divisor – if the divisor was greater, the quotient would be less than 1, whereas if the dividend was greater, then the quotient would be greater than 1.
At this point, I should’ve made a different instructional decision than I did. I was feeling rushed because I’d only had the students work on 2 of the 4 handouts I’d prepared for our double today, and I was feeling the pressure of time. Unfortunately, I decided to forge ahead with another handout, but it was really pointless to give out. Students barely had any time to finish reading the instructions before I was asking them to share with their partners. Luckily, it was a matching handout, so they still could discuss something, even if they hadn’t finished – but I don’t think it was as productive as it could’ve been. I then showed the answer key, and saw a bunch of kids asking questions – which I didn’t give us time to answer. 😦 Tomorrow, I will go back to this handout, revisit the front, and ask the students to actually work on the back – and then make sense of it together!
The reason I rushed that handout was because I wanted to give an exit ticket. Unfortunately, I don’t think the students were ready for this exit ticket – it made some leaps above where we were in class, and went to the finish line. It’s the cool down from 4.9 (and we had barely even skimmed the surface of 4.8 in class!):
Noah fills a soap dispenser from a big bottle that contains 2 1/3 liters of liquid soap. That amount of soap will fill 3 1/2 dispensers. How many liters of soap fit into one dispenser?
Use the diagram below to answer the question. Label all relevant parts of the diagram.
I added in the question of asking my students to write a multiplication and division equation that matched the situation, because I wanted to know how they were thinking about it. I definitely rushed this (there were only two minutes remaining in class when I handed it out) – luckily, I had homeroom next period, and my students were “patiently” waiting in the hallway while my sixth graders finished the exit ticket. I had to encourage some of them to write to me about their confusion if they felt stuck because it became clear that not all of them were ready to answer this question yet.
After reading the exit tickets, I feel a bit of despair and frustration with myself. I don’t think my students are clear AT ALL about how to use these tape diagrams to reason. Out of 32 students, 11 got it correct by clearly using the division algorithm (i.e. they converted both mixed numbers to improper fractions and then multiplied and simplified).
5 students got the right answer, but I’m unclear from their work how – there’s little to no writing on the diagram and not enough reasoning shown elsewhere for me to interpret. They may have computed mentally (I have students with strong working memories!). The way these students labeled the diagrams makes varying degrees of sense – some labeled it mostly correctly, but didn’t use it to answer the question, while others labeled it in an incorrect way that makes sense to me (i.e. labeling three wholes 3, 3, 3, instead of 1, 2, 3 makes sense, but labeling the whole thing 2/3 instead of 2 1/3 doesn’t make sense to me). One of these four students may have actually labeled the diagram BASED on her answer using the algorithm, because she shows the same algorithm as above, but then she labels the diagram 3, 3, 3, 1/2 on the bottom, and 2 1/3 on the top. When it comes to each individual section, she labels the first three WHOLES as 2/3, but then writes a ? in the half container and writes “I’m confused.” for that section – she clearly hasn’t connected the extra 1/3 liters with the extra 1/2 container.
Three students who were sitting at the same table (the table that didn’t finish drawing the diagrams earlier in class!) wrote 7/9 in each of the three sections that are whole (and two kids wrote 7/18 in the half section and one wrote 3.5/9). I’m not sure how they got the 7/9, though, to be honest – any ideas?
The other mistake I saw on two papers (one is a normally strong student, also) was 5/6 – on the other paper I saw it on, I see them write 5/6, 10/6, 15/6, and then later show 2/3 – buy I feel very confused about where those numbers are coming from.
Two students wrote 2 1/3 / 3 1/2 = 1 1/6 or 1 2/12, which I think is because they divided the whole numbers and fraction parts separately? But they didn’t understand how to label the diagram at all!
I had three more students who could write the problem as a division situation, one of whom was also able to label the diagram correctly, but they were confused about how to solve 2 1/3 / 3 1/2 and either didn’t attempt it or got it incorrect (1 1/6 made an appearance again here).
I had one student label 2/3, 2/3, 2/3, 1/3, and even showed it adding up 2/3 >> 1 1/3 >> 2 >> 2 1/3. But then somehow he wrote 3/4 of a liter as his answer. I can only assume that it’s a “typo” because there’s literally nothing else on his paper to suggest 3/4!
I had five students (not including the previous one) write 2/3 in each of the three rectangles and 1/3 in the half rectangle. They were all able to write the equivalent division and multiplication equations as well.
So out of that whole class, only five students seem to have understood the diagrams well enough to actually use them to solve a problem. And ultimately, my goal isn’t even to get them to use diagrams – it’s to make sense of the problems and solve them appropriately (which they can mostly already do using an algorithm!). Now I’m left feeling unsure about tomorrow’s lesson for them as well as what to do with my other class.
I know I should go back tot he handout with the matching (for the other three diagrams we didn’t discuss), and I think I should do the handout I made but didn’t use at all about how much in each group with the goal to answer “how many in 1?” after I do the number talk from numberstrings.com about the paint cans (I think the ratio table is MUCH easier to see than the tape diagrams for this!). I can also do some direct instruction off the board to lead into this skill – I think the diagrams for 10 /5 and 6 / 4 are too easy relative to the “Two cups make two-thirds of a batch. How many cups make 1 batch?” (and it also goes right along with the cups of cheese per pizza that’s in Pamela Weber Harris’ book). I also think we might need to discuss more about how to make sense of the diagrams. I wonder if looking at student work for tomorrow’s warm-up (about today’s exit ticket!) might be helpful (especially if it’s a correct one?) or if there’s something different I should have the students do. It’s clear to me that they’re not USING the diagrams to solve the problems – they’re viewing the diagrams as a separate entity. I don’t know whether this is a case for direct instruction or if there’s a way to get a kid to say something helpful, or if we just need more analysis of the drawings? Comment here with suggestions or hit me up on twitter at @MrKitMath.