Anyhow - I had a pretty decent response from my session on dividing fractions - mainly about the materials and people requesting them - so even though I posted them on the website for the conference, I decided I should post them here too so it's easier for people to find.
Now - I made a number of points about my intent and how *I* think these things should be used in the talk, and those things are not obvious by just downloading my stuff... So I will attempt to summarize my big points as well. If you don't give a crap what I have to say and you just wanna steal my stuff, well I will be an enabler and give you those links at the bottom. I just REALLY hope you don't run off copies then stand at the front of the class and go, "Ok, everyone, watch me do number 1 so you can copy that process for the rest..." ;-)
My goal:
Give teachers usable ready-to-go materials that actually help develop conceptual understanding of division, of fractions, and of division of fractions, and that lead sequentially towards noticing the pattern that division is just multiplying by the reciprocal.
Required Background Knowledge:
Many of my activities assume that students had the types of experiences expected in the 3-5 grade common core standards dealing with fractions, and in the way(s) that have been presented in the progressions document. This means that if students do not have those experiences (understanding fractions with various models, building fractions from unit fractions then using this understanding to make sense of multiplication of unit fractions times whole numbers, division represented in many ways with different models, thinking about partitive vs. quotative division, etc. - the list goes on), they will struggle with what the heck they are supposed to do. It doesn't mean not to do this stuff... It just means you may have to give them some of those experiences first in order for things to make sense to them. If you want some example slides of what I am talking about in terms of background knowledge, check out slides 5-9 in the full presentation. (Yeah - I know it would've been nice if I pulled those slides and linked them separately here, but seriously - I still have a whole period of tests to grade before tomorrow - I have faith that you can figure it out).
Some of My Philosophy for Activities:
- When I introduce a new concept in math, I think it's important to attempt to include context - mainly because context aids in comprehension. For this reason, I often start with a "situation" or a word problem. If you want to be really cool, you could actually put the students themselves into the situation and have them do it for real (give string, have them measure and cut, etc).
- For some reason, students often do not use comprehension strategies in word problems. I see a lot of kiddos who just circle the numbers and start doing random operations with no regard to situation. We do "Notice and Wonder" a LOT with word problems. I have also started having students justify which operation they are using based on the context of the word problem (embedded in first "assignment").
- Models are awesome for supporting understanding of concept, and the types of problems you should focus on when you are trying to support conceptual understanding should be transparent - simpler numbers, relatively easy to make work with the model. Students need enough time with these simpler versions and the models so that they can start to notice the patterns that lead to a more efficient strategy (the algorithm). Our goal is NOT that students solve every problem with the model! Our goal is to give students a tool that models how math works so that they have the foundation of understanding necessary to uncover the pathway to where we REALLY want them to wind up - the algorithm.
- Avoid "teaching" algorithms at all costs... BUT - carefully select problems that expose patterns, and embed activities where students practice noticing those patterns so that they can see the "rules" for themselves. There may still be those kids who stick to the longer and more unruly methods because it makes sense, but they usually need that anyway. And the kids who figure out the "short-cut" tend not to forget how it works as easily as if I had just given it all away.
- Embed everything possible. In many of the documents, students are expected to prove the answers to their division problems by writing the inverse problem (like if I said 8÷2=4, my proof by inverse would be that 4x2=8). This isn't "the standard" for dividing fractions per se, but it supports future algebraic thinking, it reinforces the idea that you can check yourself in math to see if you are making sense, and it can start to expose the idea of multiplying by the reciprocal in certain circumstances if students are paying attention. If you see other ways to embed more cool stuff - by all means, please do!
What is IN this packet of stuff? Why?
- "Cutting Things in Pieces" - I start with division with common denominators. While I have many reasons, two are the most important to me. First, they just make sense because it is VERY similar to working with whole numbers. We want students to connect fraction operations to operations with whole numbers and to understand that all of the same "properties" of numbers and operations still apply to fractions JUST like whole numbers. Often students think everything has to go out the window and they are learning some new alien thing that isn't a number anymore but is called a "fraction" - I want to show that that is not the case. Secondly, there are some interesting patterns and ideas that start out easy to see, but also come back in ways that make things easier in the future. Also - a note about the word problems and reasoning: It seems stupid to have this here because (duh) all the problems are division. BUT, if this is something that is happening regularly in the classroom, it won't feel so weird. I like to mix up the operations of practice and then the reasoning feels more appropriate, but if we don't ever practice it, they get frustrated when the operations are mixed. I am imagining that students have practiced all of these things before, and could dive in to this without the teacher "showing" them anything. If that is NOT the case... perhaps go back and build up some of that background knowledge stuff first.
- "Noticing Patterns: Dividing Fractions with Common Denominators" - This is just an extension of the previous problems. Basically some practice as well as gathering enough "data" to analyze for patterns. Often, students will see the "pattern" ahead of time :) - Notice, the first 4 problems on the first "assignment" are part of this data collection as well. Once data is collected, students look for a pattern - have them make conjectures and test them out (usually, this one is easy to see, so it doesn't take long). Then have them explain...
- "Dan's Division Strategy" - This is an Illustrative Mathematics task that jumps into the rule that the students probably just devised. The information about how students might justify the rule is really nice, so you should check that out here. I am imagining that these first 6 pages would all be copied double sided (so 3 pages) and would be given in series. *A brief note: Students will need more time/practice/models/versions that this. They need to be challenged a little bit with things that DON'T work out perfectly (like 7/3 ÷ 2/3) etc. I did NOT make an entire unit - just some key activities that I think can support conceptual understanding.
- "How Many Servings" - This activity is kind of designed for students to struggle. That is pretty much the point. When I do things like this with my students. I tell them straight out that my goal is to see perseverance and creativity. I tell them that I am going to challenge them with something that I have never shown them how to do, but that they can use everything they do know as a tool to try to make sense of going forward. We do notice and wonder (embedded in the assignment). In classes where students tend to struggle more, we brainstorm a list of strategies. They each commit to trying one (I have them write it down). And I set a timer (start at 5 min, tell them I will check in). Usually what happens is that when I check in at 5 minutes, they complain that it wasn't long enough and I should give them more time ("Oh, so you WANT me to let you struggle to figure out that math problem longer than 5 minutes... I guess I can do that" *win*). Sometimes certain strategies are panning out better and I might have some students share ideas if kids are stuck - or they work in groups, etc. This is where I would do a "5 Practices" type discussion and focus on sequencing student work. If no one figured it out, I would STILL do this and discuss where our sticking points are and have students determine what it is they need in order to proceed.
- I show ONE POSSIBLE model in my powerpoint with animation and all that good stuff on slides 26-31. It is NOT the only way, and students who solve the problem may not have used this method. That is ok too! But I do think it is a valuable model for students to play with.
- This is a procedure that you may have to teach, but it is really much cooler if a kid comes up with this type of idea first :)
- "Dice Practice!" - This is just a fun way to practice. For some reason, when I give kids dice to determine numbers in a problem they like it more. Totally shameless, to be honest, but I thought someone else might like it too, so if you do, have at it :)
- "How many ___ and in...?" - Another IM task. Now, I wouldn't just do this as is. I placed it here in the order of things on purpose because it is a really nice set of problems for students to have to differentiate which model makes more sense to use. I would give them THAT as their goal ahead. We want them to think about the type of "tool" they select for the task. Have them practice doing that here, while also looking at patterns for dividing fractions :) *Side note: There is a blank page because it would be printed double sided and go with the following two pages which have spaces for students to draw a diagram and justify answers by doing the inverse operation.
- "Multiplicative Inverse and Identity Investigation" - Basically, this is just dividing 1 by a bunch of different fractions. BUT, there are a LOT of mathematical properties that are really visibly when doing that. I would tell students that they will be investigating the Multiplicative Inverse and the Multiplicative Identity, and to see if they can figure out what those are and what they mean. Again - at the end would be a nice place to have a discussion with students about what they notice, any patterns, etc. This leads REALLY nicely into the whole multiply by the reciprocal idea, and also the idea that WHEN you multiply a number times its reciprocal, the product is always 1.
- "Developing a Rule for Dividing Fractions" - So again, this is basically a series of problems that is designed to lead students to seeing the pattern of the algorithm by collecting data about what problems they started with and what answer they got. I PURPOSELY chose problems where the numerator in both fractions was 1 FIRST, so students might first come up with a conjecture that the denominator of the second fraction just goes on top. Then, when one numerator is no longer one, they still have that idea but realize you are then multiplying across after you flip things over. I am hoping to COMPLETELY AVOID the idea of "cross multiply" - and here is why: "Cross multiply" has no meaning in terms of mathematical properties - how do you know what thing goes on top after you multiply? You are just memorizing some procedure. If students make the connection that the MEANING of division is the inverse of multiplication (meaning - multiply by the multiplicative inverse which is the reciprocal), this actually has important mathematical implications for the future. It helps make sense of things like negative exponents (positive and negative are inverses, so it makes sense that if positive exponents mean repeated multiplication, that negative exponents should mean the inverse - repeated division), and is used throughout high school mathematics. I could go on... but I think you get the idea.
OK! So, that's it. A crazy long post. Dunno if anyone will even read it. If you did - gold star for you ;)
Let me know what you like/hate about this stuff. If you use it, let me know how it goes! If you change things, let me know what and why and if it worked better for you that way.
And if you have questions, let me know. I hope it helps someone out there!
Here are links for my PowerPoint presentation and the Student Activities Documents
*Please realize that I uploaded editable versions, which means formatting is probably not correct. If things look ridiculous, just shoot me a message and I will email you the documents directly.
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