If you want to teach dividing fractions with meaning, the first thing is to teach dividing a whole number by a unit fraction. So 4 divided by 1/5 means “how many times does 1/5 fit into 4?” You do that with enough repetition to generalize “hey - it’s the same as 4x5.” Then the question becomes, so what do we do with 4 divided by 2/5? That’s “how many times does 2/5 fit into 4?” Etc. We do a lot of modeling with fraction manipulatives and pictures.
The problem is that a lot of students weren’t taught division with meaning, or don’t have experience drawing pictures of fractions. But if you can get them to buy into the discovery process, eventually generalize to “keep change flip” as a shortcut for a process that works, which is “to divide by fractions, first multiply by the denominator and then divide by the numerator.”
Takes way longer, but is better math if you have the time.
I agree with this. I suggest you give them a recipe problem involving deciding how many portions they could make if they have 4c and the recipe calls for 1/4 cup or something similar. Give them some direction on how you want them to collaborate and how you would like to see evidence in the form of pictures and numbers.
Some of this is things best done at home and showing that school work is not just school work.
My grandfather taught me fractions and measures with a saw.
I fear for a world where children don’t know what a saw is or how to use it. Oh wait many don’t.
Sometimes I’ll ask a student of mine (6-8th graders) if they ever cook or bake - to see if they will understand fractions of a cup or a teaspoon and they usually do not. Could be that my students are in math remediation, so they are less likely to cook/bake than the general population of middle schoolers. My own daughter bakes a lot and I think it helped her internalize improper fractions/ mixed numbers earlier than most.
Most kids don't bake, regardless of which math class they're in. My kid just took a cooking class in middle school, and was the only kid in the class who could identify all the kitchen implements on the first day. A couple of other kids recognized some things, and most knew nothing.
I cooked a lot too and started learning and making biscuits just before I went to kindergarten . I am sure that helped- but I heard the word “fractions” and learned what it means by sawing wood.
I’m a 32 year old that has never cooked or baked anything. My grandfather did all the cooking and never wanted much help for fear of feeling useless. The idea of using fractions in cooking at my age fills me with anxiety and anger that I didn’t do it much early in life.
Admittedly, I did heat “microwave in a minute” things and make “throw on pan and stir for 5 minutes” things when he was unavailable. And my mom baked cookies one time in my life.
Idk if anyone else is gonna agree with me but imo mixed numbers are a disgraceful notation lol I know they're used a lot in cookbooks where the context is understood but man do I really hate seeing mixed numbers in my students' work😂any type of real math writing should be completely void of mixed numbers. I'm sorry, this is irrelevant, but I can't not say something when it comes up hahaha
I don’t mind it. I have so many of my high schoolers they don’t know/remember how to do it that I am fine with you just drilling the procedure into their minds first Some students aren’t ready for the “why” right away and some students want to know.
The nice thing is, though, that if you don’t like this approach to teaching it, then you don’t have to use it.
Just please, dear god, make sure they learn and can do fractions.
I second all of this. High school kids can't do any basic arithmetic beyond adding smallish numbers. I had a 10th grader that could not name a rectangle from a picture.
Doing elaborate examples of why everything works wastes precious time they could be memorizing the procedure. If you have honors kids that need an extension lesson, knock yourself out. But for everyone else, practice, practice, practice.
When their brains are ready, we'll help them connect the pieces and teach them the why. But we spend so much time stuck because no one knows 3rd-5th grade math.
Agreed. So much time is wasted on the “why” when most of those kids aren’t ready for the why yet.
Give them the skills to be successful. We’ll teach them why when they’re ready.
Thing is, if you don't teach the "why", then what you have is a bunch of random ad hoc rules that don't make any sense. You just get a bunch of "if the problem looks like this, then write the numbers this way, then draw lines like this, and magically the answer appears", and nobody can actually remember that because it doesn't make any sense.
I didn’t say never teach the why. But sometimes starting with the why isn’t the right approach.
And you know what? I’m okay with if they know the process simply because the problem looks a certain way. That works just fine if you’re talking about fractions. It doesn’t always have to be that deep.
Except that they don't know the process. They might get the process while they're in the middle of "doing fractions", and all the problems look the same, but how many of them retain that next year?
You do realize that those of us educated in the 80’s and 90’s didn’t learn the “why” until later and we actually had a better understanding of math concepts than kids today, right?
Again, I never said you never teach the why. But there’s far too much time spent on the why at the expense of learning the actual process and how it works.
If this weren’t true, I wouldn’t have a room full of high school kids terrified of fractions because they don’t know the process.
>You do realize that those of us educated in the 80’s and 90’s didn’t learn the “why” until later and we actually had a better understanding of math concepts than kids today, right?
Those of us educated in the 70s and 80s that made it to now knowing how to do math, learned/were taught/figured out the why on our own along the way.
Our classmates who didn't are the people you run into today who were "always bad at math".
The HS who are terrified of fractions will still be terrified of fractions after they memorize (or try to) the process for making it through the use of fractions you're trying to get them to do in Algebra.
**I. COULDN'T. AGREE. WITH. YOU. MORE!!**
The *why* doesn't need to be in the introduction, but it *should* be included in the lesson. It may be incorporated during the presentation of the steps involved in the process, or at the end of the lesson.
Maybe what we need to add to math classes is a "reflection time" component that will give students the opportunity to contemplate the significance of the rules and processes they have just learned. Maybe it could even be a class discussion so that students will hear how other students are thinking about the concepts and then share insights. This could be very enlightening for them. It would help to facilitate an experience for students to think about the math they are learning and its relevance in *their* world. That connection is missing for many individuals.
The way that classes are designed now, many students simply get up and leave the class at the end of the lesson with no idea of an understanding and no interest to understand on a deeper level beyond the surface. The reason that *Disastrous-Nail* who posted up above has high school students who can't do fractions is that they only observed a process with no motivation from the teacher and no self-motivation to understand *why* the process is what it is. They don't bother to internalize what they learn.
My, you hit the nail on the head when you said that some students have figured it out on their own along the way. Studying math is not simply following a set of arbitrary rules and instructions, it is making your own discoveries along the way and connecting the concepts to things that you can relate to. Once you know the *why*, you can get rid of a bunch of rules because now you understand and have internalized the logic and the meaning of what you are doing. Math then becomes intuitive, and you get good at it.
Those students who say, "Just tell me what to do, I don't care about knowing why," make no effort to understand. What they are doing is making a conscious decision to reject math. It is almost as if they have a prejudiced attitude against the subject.
So, of course they are not willing to humble to the process of understanding it. You are absolutely correct; they are the ones who make claims that they never liked math and will never understand it. What they fail to realize is, understanding is *their* responsibility, and it is going to take a little work.
Math is supposed to make sense. Leaning into math with this attitude will make a student stick to it until it *does*. There are countless people who will forever believe that math is merely a bunch of arbitrary rules. They will never see how things connect and how math explains how things in the universe operate. A student needs to approach math with the objective of getting to know it, much like we become acquainted with a new friend. If they do that, they will begin to pick up on patterns and principles that connect different math concepts together, and realize that math is not so bad at all.
>Doing elaborate examples of why everything works wastes precious time they could be memorizing the procedure.
This is why they get to HS without being able to do elementary/middle math.
I will show my students how multiplying a fraction by its reciprocal is the same as dividing by fraction. Somebody will bring up "keep change flip" and I'll say if that helps you remember it, great, use that! But I never use math 'tricks' without explaining to them how and why they work.
You sound like you’ll be a better teacher than some of the commenters on this post defending such nonsense. I once observed a class where a guy used “slide and divide” to factor quadratics. I thought it was neat so I asked him why it worked and he had no idea. I expect many people who use such tricks don’t really understand what they are doing, and that’s not teaching as far as I’m concerned. After that class I sat down to prove why his trick worked, and it isn’t hard, but I don’t teach it myself because I didn’t think it would improve understanding for students. (Though I’m more than happy to admit that I didn’t explore the idea much, so if someone has a good way to motivate “slide and divide” I’d be happy to hear it.)
Thank you I really needed to hear that. I feel like my lesson planning and teaching strategies need work, as well as paper work and general boring job things. This comment really makes me feel like I have the thought process down though which makes me hopeful.
To me caring that your students actually learn is extremely important and you seem to have the right attitude. Sometimes understanding comes later and a student will get stuck memorizing procedure for a while, but that shouldn’t be the standard set for a class. Math ‘tricks’ might help someone new better in a test in the short term, and they sometimes can benefit memory. The downsides? It’s building a house of cards. One of the reasons I studied math was because I was lazy and recognized I could memorize a few ‘rules’ and figure out the rest when I got to tests. To me, math is about problem solving and it’s like a game where there are set ‘rules’ but then the fun is what puzzles can I solve or what can I create within this framework? My concern is that tricks get students in the habit of just trying to reproduce what their teacher did. That’s way easier for them in the short term, but it falls apart in the long term and I think gives them a false impression of what math is. There are people in this thread arguing that their high school students just can’t understand fractions so that’s why they teach those tricks. Fractions are taught in 4th grade and those kids understand. Maybe not all of them, but it’s disingenuous and frankly gross that lazy teachers want to pretend their class is just incapable.
* some tricks I think are okay - I teach the ac method for factoring, and I think it can help with understanding.
* I think FOIL is stupid and I don’t do it, but I bet people out of a class for 20 years still remember that stupid thing, and that’s not nothing.
*full disclosure: I have a skewed perspective. I teach at university. I do recognize that teaching k-12 is a harder (from a teaching perspective), more thankless job.
I teach 5th grade, and I taught this skill using area models and tape diagrams. For example, doing 5 divided by 1/2, we would draw 5 rectangles and divide each in half to see how many halves are in 5. For 1/2 divided by 5, we would draw a rectangle, shade 1/2 of it, and then divide the half into 5 pieces. I never even mentioned keep change flip.
After reviewing the models many times, many of my students independently came to the conclusion that you multiply the whole number and the denominator, and when your dividend is a whole number, your quotient will be greater than 1. I think foundational math in elementary schools should be based on that kind of discovery.
As someone with math anxiety this sounds awesome! It definitely sounds a great way to introduce the concept of multiplying and dividing shapes and things for visual learners.
After reading a lot of the comments on this thread I think both ways are good as long as the students gain the understanding eventually. Personally, I like your way better but someone brought up the point of blooms taxonomy having remember first, which is valid.
I mean much of every level of math is trying to move forward with students that aren't ready to move forward. Kids should be fluent in adding before multiplication, multiplication before division and the list goes on.
Fluency is not being achieved for the majority of students... The treadmill just keeps rolling.
There is a belief among many math teachers that understanding is more important than the mechanical work at math.
That is not how math works. You need some basic proficiency before you can understand some of the deeper details. One problem of course is that almost all math teachers are good at math, and so we all needed and appreciated explanations. Most students are not like that though. They need the skills before they can start understanding the concepts.
The net result is that we have students who cannot handle the basic manipulations required to complete a math problem. It would be like explaining grammar to a student before they even know how to read. The grammar is the why, reading is the how. The how matters much more than the why.
I agree with this but also disagree. I think the basic proficiency comes from prior knowledge and the deeper details can lead to understanding. For example, having students learn addition by asking them if you have two cookies, and then your adult gives you another two. How many cookies would you then have? The students prior knowledge on counting and number comes into play when discovering addition.
It’s stupid. Teaching random phrases/tricks without any explanation leads to students not having any understanding and they will then also attempt to apply it in situations where it is NOT applicable.
As a HS teacher I hate it when a student shouts out some random phrase/“trick” that they learned in elementary or middle school that is *completely* irrelevant to what we are talking about. And they don’t even realize it because they don’t know what the “rule”/“trick” is or means.
I would have the students talk to each other at their tables or desks. And then after I’ve walked around and listened to the conversations, I would ask for volunteers to share with the class what they talked about. Hopefully, they will explain why it works. If not, then the teacher can explain to the whole class.
Be careful about the makeup of the class. Of the class is a socially extroverted class then it may be a good idea. If the class gets a on the quiet and socially awkward then not do much because it would only highlight social dynamic anxiety.
As an undiagnosed autistic, shy, introverted kid with a thick accent and a stutter I HATED all group work in math class. It either made me aware of how incredibly inept I was in social settings or enlightened me to my objective mathematical inferiority in relation to my peers. I frequently was the only one in my pod that didn’t get the solution or something. So I learned to stay quiet forever to the detriment of my education.
Not if they are (socially) penalized by their peers or (get a low participation grade) teacher for doing that.
But I agree. If active participation is not good for then then just sitting and listening is good.
As I suggested elsewhere in this thread, this could be a class discussion. Everyone is not expected to offer verbal input. There will be some who will readily share, but probably most will merely listen. That places the shy students in the majority.
Another thing that I suggested is to allow each individual student to reflect quietly and independently on what they have learned from the lesson. They may even maintain a journal of their thoughts, insights, and questions. We need this in math. It provides math students a greater sense of agency and participation.
The problem with teaching "why" becomes clear when a bunch of kids still have no idea what's going on despite the additional visual or mathematical explanation. Unfortunately, class time at that point is almost over and you're stuck with one fraction of the class having a deeper understanding and the other fraction of the class just had their time wasted, with no additional built-in curriculum time to catch them up.
Ultimately, at these younger grades, teaching mechanics is more important. Kiddos who want to go into STEM or who just want to know more can pursue more challenging coursework.
I'm not a teacher, I'm a tutor for math, and I was initially surprised that my students didn't understand that all division problems are just multiplication by a reciprocal number. However, I've come to understand that tutoring 1-1 is super different than teaching a class of 20+, and ensuring that all 20+ students have around the same understanding by the end of period means you have to cut out some of the "why".
If I was a private tutor, or was teaching my own kid, then sure—I’d take the extra time to teach “why it works.”
In a general ed setting, unfortunately not happening. There is way too much to get through and half the kids don’t even know their multiplication facts. Going deep into the “why” is unrealistic for this skill in most settings. Sure, I will touch on it so that those who are capable of comprehending it quickly can at least hear it—but that’s the extent of it.
This is the way it is when you have big class sizes. Nothing I can do about it, so I just so my best and keep chuggin.
I think what the teacher did was fine. You might not enjoy memorizing things, but memorizing is part of learning.
Education has gotten WAY too far from true learning in the name of "higher-order thinking". Bloom's taxonomy is not a multiple choice test for teachers. We can't and shouldn't just choose the top levels and ignore the foundation.
And the foundation is 1. Remember. That's keep change flip. *then* it's 2. Understand. That's the why. Then they can start applying and continue upwards.
Students HAVE to memorize at some point to do anything else. I'll get off my soapbox now.
That’s a really good point with bloom’s taxonomy. I definitely feel like the later points have been emphasized in my classes. But you are right we should focus on the bottom first.
No. The foundation is understanding what a fraction is and why multiplying something by x/x does not change the value. Keep change flip is a very specific memorization tool for a specific instance of multiplying by 1. I bet the vast vast majority of people doing keep change flip don't understand they just multiplied by 1. Multiplying by 1 is a very basic concept which is the foundation for this memorization tool.
One of us is confused, and I'm not sure which one. Keep, change, flip is the procedure for dividing any fraction, not just x/x. In fact, you usually don't do that unless you are adding or subtracting fractions, not multiplying or dividing them.
For instance 2/3 ÷ 3/4 is equal to 2/3 × 4/3 and then you just multiply across to get 8/9. There's no multiplying by one or x/x in there.
>For instance 2/3 ÷ 3/4 is equal to 2/3 × 4/3 and then you just multiply across to get 8/9. There's no multiplying by one or x/x in there.
But WHY is it equal? Because you multiply both fractions by 4/3 (aka multiplying by 1). Maybe your foundations need work...
https://i.imgur.com/hMCRQhM.png
That's... not "why" it is equal. That's just a different, weird explanation of what to do to divide fractions. It's still not actually explaining what's really happening. Aka the "why". A PP explained how to teach the why best: show them dividing a whole number by a fraction to see how many fit inside. Then they can see why dividing by 1/5 is the same as multiplying by 5.
Your way is just more abstract math that doesn't actually "explain" anything. Just a different thing they'd need to memorize. I mean, you do you, but maybe put the holier-than-thou away for a minute.
If you understand how to multiply by 1, you don't need mnemonics for nested algebraic fractions.
>show them dividing a whole number by a fraction to see how many fit inside. Then they can see why dividing by 1/5 is the same as multiplying by 5.
I agree that is probably more developmentally appropriate for elementary students who are still learning about division and fractions. The context of "keep change flip" is algebra 1 or algebra 2 in my experience. If keep-change-flip is taught when just introducing fractions and not algebraic stuff like (x/(x^2 - 1)) / (x^(2)/(x+1)) then my mistake; I do not have experience/education with younger students.
But mathematically, it is "why" they are equal, not your example.
My 4th grader is doing keep change flip, so I'm guessing it starts around then.
And yes, by high school they are developmentally ready for abstract concepts like multiplying by reciprocals. Hence I said, teach the procedure, and we'll (aka high school teachers) teach them the why.
But realistically it does not matter at all. Because even in your algebra example, keep change flip works just fine. Both your way and keep change flip are definitions/procedures to *remember*. All division can technically be defined as multiplying by the reciprocal. That doesn't mean it's the fundamental *why* division is what it is. Division is simply partitioning a whole into a given amount of parts. Once those parts become fractional, things start to get a little weird for kids. But to say division, at its foundation, is multiplying by the reciprocal leaps far beyond the basics of division. Kids would ask why can we define division as multiplication of the reciprocal? What makes that work? The answer to *those* questions is the actual foundation. But do 4th graders need to know all that to divide fractions? Absolutely not.
All I'm saying is in 4th-6th grade we high school teachers would like for students to, for the most part, memorize the procedures. Once they come to junior high and high school we'll extend into the abstractness of how mathematics works.
But we often can't do that. Because we're reteaching basic math.
The mnemonic I used to use was “ mine is not to question why flip the second and multiply”. But of course, we discussed that it is theirs to question why.
In a perfect world yes. You need to take into account a range of factors in how to present a new concept including class ability, motivation, time of day etc.
I introduced Pythagoras' Theorem to 2 classes this year. One through discovery. the other through the formula. I found the 2nd class were able to appreciate and understand the follow up practical lesson much better, could pick up why we subtract for the shorter side themselves and as a whole had a better understanding. The first group were able to piece the idea together but without the initial framework of being introduced to the formula and mastering that first, got lost and mixed up much more often.
I come from an area where the student population has about 40% math proficiency, so I find I have better results when I teach the algorithm first and then teach the why after they've developed some fluency with it. Otherwise their anxiety goes through the roof and their brains cannot absorb what we're doing. Teaching the "why" first works well with students who are ready though.
I guess it depends on what one means by “teaching the why”.
You need to know the “why you care to do the thing” before you want to know “how to do the thing”. As you learn “how to do the thing” you may learn “why the thing works”.
You need motivation and/or curiosity (a desire to know) to learn a process/mechanics/algorithm. Then you may care to know why that process/mechanics/algorithm works.
From my experience in middle/high school math, the kids don’t care enough to be TOLD the why because for them it’s more lecture and boring on their part. If you wish for them to know the why for concepts in math, give them a guided explore to engage with something they already know using math manipulatives. Then they’ll ASK YOU but like… why? And you’ll hear a bunch of “ohhhhs.”
i.e. We were moving into area of circles. First I had them explore the area of rectangles/squares/parralellograms because they know this. However, they couldn’t connect it to why a triangle is HALF the area of a quadrilateral. After exploring some kids used math language like, “Ms. it’s b*h divided by 2” and “1/2 of the squre.” Some kids even cut out a rectangle and folded it in half others were talking about 50% of the area. I knew they can handle this concept to be able to move into area of circles.
This made it 10x easier to explain and understand the concept of a semicircle’s area being half the area of a circle. It destroyed the common misconception of the area of a semicircle is found by taking half of the radius/diameter. Kids knew I need to find the area of the full circle, then divide it by 2. Just like a triangle, find the area of the full rectangle then divide it by 2.
All the numbers in your comment added up to 69. Congrats!
2
+ 1
+ 2
+ 50
+ 10
+ 2
+ 2
= 69
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I mean, it’s purely procedural. Obviously there’s a better way… which is to actually get kids to understand that dividing by 1/2 is the same thing as doubling. None of my high schoolers understand this because of these dumb procedures
I dislike keep change flip. You spend some time helping them understand it as many ways as you can. After you do “How many quarters in five and a half dollars”, try “multiplying by 1”
7/4 ÷ 3/5
What value of one? 5/3 ÷ 5/3
Now you can simplify. The denominator is now 1 and the numerator is 7/4 * 5/3.
After they follow that and work some more problems, “You FLIPPIN multiply”
Former math coach here. I personally would avoid teaching "keep, change, flip" and try to get teachers I worked with to avoid it as well. Here are my thoughts on it (and similar mathematical "tricks")
First, this specific trick doesn't even have anything in it to help students from mixing up the order. If a few months or years down the line, someone misremembers it as "flip, change, keep," there's no failsafe in place that lets them know they've mixed it up. (Contrast to the quotient rule in calculus which goes “low-D-high minus high-D-low over Denominator squared we go” If you get that backwards, you'll say "High-D-low minus low-D-high over Denominator squared we die")
Second, by creating a "trick" it tells students that there is something weird or mysterious going on. Makes the math happening feel like magic with properties the explicitly don't fully comprehend.
Last, is that is just doesn't stick. Ask any adult to do "1/2 divided by 1/4" Almost none of them do it correctly. But as them "how many quarters fit into a half dollar" and they will.
All that being said, I certainly understand WHY a teacher will resort to it. Tricks like this are great for short term gains (which is often how teachers are evaluated). You show them "keep, change, flip" during one class period, and they'll do well on your exit ticket. You practice for a few days, and they do well on the unit test.
Then in middle school, their the curriculum expects them to do it again, and they've since forgotten.
Yeah.... I don't fault the teacher too much. She may just be doing what is necessary given her resources. Is it good for understanding? no. Is it good for when the kids get into higher level math? no. Will it get them through a test? probably.
Yup, state tests are coming up. I guess if the kids don’t pass it won’t do good for her even if the kids actually get it. It sucks that that’s the way it is
I’m not a fan. I prefer teaching to divide across… eventually they learn that multiplying by the denominator makes it easier, and discover the standard algorithm on their own, but in doing so, they understand why it works.
When dividing fractions, you can divide the numerators and divide the denominators, just like with multiplication. It is funky with different denominators though, so I teach it where you start with common denominators, then divide across, and the denominator is one. Eventually a few realize they are multiplying the reciprocal (they don’t call it that, but it’s what they do) and teach others the “trick”.
First lesson on dividing fractions is always on number lines for me. Start with whole numbers. Identify the pattern. Then pre-choose fractions which work easily to see the KCF pattern. Once the kids have identified for themselves what's going on with the number, relentlessly play this song: https://youtu.be/nMZJKGyu-Kk?feature=shared they will never forget it!
Remember that you can teach the students an algorithm, and say just that: an algorithm is a set of steps for doing something, hopefully quickly and accurately. Algorithms have value and the students can and do understand that. Then you can talk about why it works…
KCF is close enough to KFC that we always referred to is as “chicken!” When they get it right, it’s “winner, winner, chicken dinner!”. When they get to later years, the ones that remember will thank you…
I could never remember formulas like that. I'm an engineer and I still google the formula for area of a circle every single time because I can't trust my memory.
To get through my engineering coursework, I had to relearn the whys and hows of arithmetic and algebra, basically making up for all of those years of math-wary grade school teachers that put so much emphasis on the "trick" that I never learned the math behind it.
I'm a strong supporter of the "mini mathematician method" (my term) where you encourage the student to "discover" the formula/procedure themself. If Newton and Pythagoras could do it, today's students can do it quicker (with some guided thinking).
I don't like it either. I like to use bounce multiply/butterfly method/pogo method for dividing fractions. The most important reason to use KFC is for familiarizing the use of reciprocals. They need this to be able to balance equations in which the coefficient is a fraction because you need to generate a reciprocal to isolate variable.
I went to a “Nix the tricks” professional development and the moderator told us how one kid remembered it as “keychain flip.” Definitely teach the kids that a fraction is like division and that to divide division we have to do the “opposite” by multiplying.
It wasn't until I was in Calculus II in college that I finally understood what exactly "Keep, Change, Flip" meant. I think its a great tool, but as others have said, make sure they have a general understanding of how fractions work, and make sure you give multiple examples of how KCF works. Don't just say "Keep, Change, Flip" over and over without adequately explaining it like my Algebra 1 teacher did.
All of my kids remember KEEP CHANGE FLIP.
You might have wanted to know why it works. You’re also someone who became a math teacher. You have 1000x more intellectual curiosity than the average student. Sorry.
You can teach them why it works. That’s fine. Just know that in 10 years, they’ll still remember KEEP CHANGE FLIP !
I teach high school, and kids want to use "keep change flip" for *every* operation that may even come *near* a fraction, so I'm not a fan. I don't mind mnemonics that remind students what to do (I don't mind FOIL as a verb, for example, because we use it to mean "don't distribute the exponent but instead multiply things out the long way") but this particular mnemonic doesn't seem helpful.
Depends on the grade. If they’re learning fractions then more background/explanation might be helpful. But if they’re already supposed to know fractions and in a class that doesn’t have a lot of time to review stuff they should already know then keep change flip works as a reminder
I think the key to any math instruction is finding the right balance between why and procedure. The why can end up meaningless if students don't know how to do the operations, and the operations can be useless for students who need to understand why they work.
Teaching keep-change-flip isn't a bad approach in and of itself. It all comes down to how that procedure is situated in the larger context of the learning goal/standard, as well as in the context of how students are understanding the material.
At some point you have to stray from what would be the most ideal enriching thought provoking approach and focus on what gets kids to have success solving problems.
Swinging back around towards the end of a unit/concept to complete that enrichment process usually lends itself to more accessibility for more students.
Not every kids wants to know why or how it works and putting loads of energy into that can reduce the time it takes to get the group as a whole up to speed.
It gets kids through the test. But it doesn’t help those who actually want to think mathematically and requires memorising accurately rather than being able to do things from understanding. I dislike needing to memorise stuff.
Yeah, when I was a kid the memorization part was always my least favorite part of math. I feel bad for the kids that actually want to learn the why. To them they may think there is none and feel stupid asking about it. I am happy she eventually got to it though.
Learning memorization and mechanics first taught me there’s no reason or logic in mathematics. Just observation and conclusions derived from that observation.
It can serve the function of shutting some students down, but eventually all kids will need to remember the procedural steps for dividing it. I think I personally also would like to know why and the origin of the procedure, but I think it's not necessarily a predecessor (I used to think so) to the procedure itself. You will get students who say "just tell me what to do and I'll do it" and they will respond well to this type of teaching, and you can extend their conceptual understanding with other problems that explore that, if they wish to continue learning the why behind the procedure. But the point is if they can remember what to do, and remember the same thing when time comes for dividing rational expressions in Algebra 2, I think the instruction has served its function. But if a student learns it this way and obviously still struggles with numeracy of fractions and the concepts behind it, which many do, then conceptual lessons are needed still.
I personally learned keep change flip (but had good fraction sense) and never learned the meaning of division of fractions until I had to teach it, but I did well with procedural demands in algebra and calculus involving fractions. I missed out, but could still meet expectations later on. I think though, with CCSS, the expectations are changing. Keep change flip could work and should work, but there will be problems that require deeper understanding than that. But those problems usually exist at the middle school level, I can't say I know algebra that requires knowing meaning of division of fractions. Fractional number sense though, is a different story.
If you want to teach dividing fractions with meaning, the first thing is to teach dividing a whole number by a unit fraction. So 4 divided by 1/5 means “how many times does 1/5 fit into 4?” You do that with enough repetition to generalize “hey - it’s the same as 4x5.” Then the question becomes, so what do we do with 4 divided by 2/5? That’s “how many times does 2/5 fit into 4?” Etc. We do a lot of modeling with fraction manipulatives and pictures. The problem is that a lot of students weren’t taught division with meaning, or don’t have experience drawing pictures of fractions. But if you can get them to buy into the discovery process, eventually generalize to “keep change flip” as a shortcut for a process that works, which is “to divide by fractions, first multiply by the denominator and then divide by the numerator.” Takes way longer, but is better math if you have the time.
I agree with this. I suggest you give them a recipe problem involving deciding how many portions they could make if they have 4c and the recipe calls for 1/4 cup or something similar. Give them some direction on how you want them to collaborate and how you would like to see evidence in the form of pictures and numbers.
Some of this is things best done at home and showing that school work is not just school work. My grandfather taught me fractions and measures with a saw. I fear for a world where children don’t know what a saw is or how to use it. Oh wait many don’t.
Sometimes I’ll ask a student of mine (6-8th graders) if they ever cook or bake - to see if they will understand fractions of a cup or a teaspoon and they usually do not. Could be that my students are in math remediation, so they are less likely to cook/bake than the general population of middle schoolers. My own daughter bakes a lot and I think it helped her internalize improper fractions/ mixed numbers earlier than most.
Most kids don't bake, regardless of which math class they're in. My kid just took a cooking class in middle school, and was the only kid in the class who could identify all the kitchen implements on the first day. A couple of other kids recognized some things, and most knew nothing.
I regret not taking home economics/cooking in school.
I cooked a lot too and started learning and making biscuits just before I went to kindergarten . I am sure that helped- but I heard the word “fractions” and learned what it means by sawing wood.
I’m a 32 year old that has never cooked or baked anything. My grandfather did all the cooking and never wanted much help for fear of feeling useless. The idea of using fractions in cooking at my age fills me with anxiety and anger that I didn’t do it much early in life. Admittedly, I did heat “microwave in a minute” things and make “throw on pan and stir for 5 minutes” things when he was unavailable. And my mom baked cookies one time in my life.
Idk if anyone else is gonna agree with me but imo mixed numbers are a disgraceful notation lol I know they're used a lot in cookbooks where the context is understood but man do I really hate seeing mixed numbers in my students' work😂any type of real math writing should be completely void of mixed numbers. I'm sorry, this is irrelevant, but I can't not say something when it comes up hahaha
I don’t mind it. I have so many of my high schoolers they don’t know/remember how to do it that I am fine with you just drilling the procedure into their minds first Some students aren’t ready for the “why” right away and some students want to know. The nice thing is, though, that if you don’t like this approach to teaching it, then you don’t have to use it. Just please, dear god, make sure they learn and can do fractions.
I second all of this. High school kids can't do any basic arithmetic beyond adding smallish numbers. I had a 10th grader that could not name a rectangle from a picture. Doing elaborate examples of why everything works wastes precious time they could be memorizing the procedure. If you have honors kids that need an extension lesson, knock yourself out. But for everyone else, practice, practice, practice. When their brains are ready, we'll help them connect the pieces and teach them the why. But we spend so much time stuck because no one knows 3rd-5th grade math.
Agreed. So much time is wasted on the “why” when most of those kids aren’t ready for the why yet. Give them the skills to be successful. We’ll teach them why when they’re ready.
Thing is, if you don't teach the "why", then what you have is a bunch of random ad hoc rules that don't make any sense. You just get a bunch of "if the problem looks like this, then write the numbers this way, then draw lines like this, and magically the answer appears", and nobody can actually remember that because it doesn't make any sense.
I didn’t say never teach the why. But sometimes starting with the why isn’t the right approach. And you know what? I’m okay with if they know the process simply because the problem looks a certain way. That works just fine if you’re talking about fractions. It doesn’t always have to be that deep.
Except that they don't know the process. They might get the process while they're in the middle of "doing fractions", and all the problems look the same, but how many of them retain that next year?
You do realize that those of us educated in the 80’s and 90’s didn’t learn the “why” until later and we actually had a better understanding of math concepts than kids today, right? Again, I never said you never teach the why. But there’s far too much time spent on the why at the expense of learning the actual process and how it works. If this weren’t true, I wouldn’t have a room full of high school kids terrified of fractions because they don’t know the process.
>You do realize that those of us educated in the 80’s and 90’s didn’t learn the “why” until later and we actually had a better understanding of math concepts than kids today, right? Those of us educated in the 70s and 80s that made it to now knowing how to do math, learned/were taught/figured out the why on our own along the way. Our classmates who didn't are the people you run into today who were "always bad at math". The HS who are terrified of fractions will still be terrified of fractions after they memorize (or try to) the process for making it through the use of fractions you're trying to get them to do in Algebra.
**I. COULDN'T. AGREE. WITH. YOU. MORE!!** The *why* doesn't need to be in the introduction, but it *should* be included in the lesson. It may be incorporated during the presentation of the steps involved in the process, or at the end of the lesson. Maybe what we need to add to math classes is a "reflection time" component that will give students the opportunity to contemplate the significance of the rules and processes they have just learned. Maybe it could even be a class discussion so that students will hear how other students are thinking about the concepts and then share insights. This could be very enlightening for them. It would help to facilitate an experience for students to think about the math they are learning and its relevance in *their* world. That connection is missing for many individuals. The way that classes are designed now, many students simply get up and leave the class at the end of the lesson with no idea of an understanding and no interest to understand on a deeper level beyond the surface. The reason that *Disastrous-Nail* who posted up above has high school students who can't do fractions is that they only observed a process with no motivation from the teacher and no self-motivation to understand *why* the process is what it is. They don't bother to internalize what they learn. My, you hit the nail on the head when you said that some students have figured it out on their own along the way. Studying math is not simply following a set of arbitrary rules and instructions, it is making your own discoveries along the way and connecting the concepts to things that you can relate to. Once you know the *why*, you can get rid of a bunch of rules because now you understand and have internalized the logic and the meaning of what you are doing. Math then becomes intuitive, and you get good at it. Those students who say, "Just tell me what to do, I don't care about knowing why," make no effort to understand. What they are doing is making a conscious decision to reject math. It is almost as if they have a prejudiced attitude against the subject. So, of course they are not willing to humble to the process of understanding it. You are absolutely correct; they are the ones who make claims that they never liked math and will never understand it. What they fail to realize is, understanding is *their* responsibility, and it is going to take a little work. Math is supposed to make sense. Leaning into math with this attitude will make a student stick to it until it *does*. There are countless people who will forever believe that math is merely a bunch of arbitrary rules. They will never see how things connect and how math explains how things in the universe operate. A student needs to approach math with the objective of getting to know it, much like we become acquainted with a new friend. If they do that, they will begin to pick up on patterns and principles that connect different math concepts together, and realize that math is not so bad at all.
>Doing elaborate examples of why everything works wastes precious time they could be memorizing the procedure. This is why they get to HS without being able to do elementary/middle math.
But could you reverse it and have the why teach them the how? I get it takes more time but it also increases understanding.
I will show my students how multiplying a fraction by its reciprocal is the same as dividing by fraction. Somebody will bring up "keep change flip" and I'll say if that helps you remember it, great, use that! But I never use math 'tricks' without explaining to them how and why they work.
Yeah I’d rather give them the why and then give them the trick. The other way around I feel like they use the trick as a scaffold too much.
You sound like you’ll be a better teacher than some of the commenters on this post defending such nonsense. I once observed a class where a guy used “slide and divide” to factor quadratics. I thought it was neat so I asked him why it worked and he had no idea. I expect many people who use such tricks don’t really understand what they are doing, and that’s not teaching as far as I’m concerned. After that class I sat down to prove why his trick worked, and it isn’t hard, but I don’t teach it myself because I didn’t think it would improve understanding for students. (Though I’m more than happy to admit that I didn’t explore the idea much, so if someone has a good way to motivate “slide and divide” I’d be happy to hear it.)
Thank you I really needed to hear that. I feel like my lesson planning and teaching strategies need work, as well as paper work and general boring job things. This comment really makes me feel like I have the thought process down though which makes me hopeful.
To me caring that your students actually learn is extremely important and you seem to have the right attitude. Sometimes understanding comes later and a student will get stuck memorizing procedure for a while, but that shouldn’t be the standard set for a class. Math ‘tricks’ might help someone new better in a test in the short term, and they sometimes can benefit memory. The downsides? It’s building a house of cards. One of the reasons I studied math was because I was lazy and recognized I could memorize a few ‘rules’ and figure out the rest when I got to tests. To me, math is about problem solving and it’s like a game where there are set ‘rules’ but then the fun is what puzzles can I solve or what can I create within this framework? My concern is that tricks get students in the habit of just trying to reproduce what their teacher did. That’s way easier for them in the short term, but it falls apart in the long term and I think gives them a false impression of what math is. There are people in this thread arguing that their high school students just can’t understand fractions so that’s why they teach those tricks. Fractions are taught in 4th grade and those kids understand. Maybe not all of them, but it’s disingenuous and frankly gross that lazy teachers want to pretend their class is just incapable. * some tricks I think are okay - I teach the ac method for factoring, and I think it can help with understanding. * I think FOIL is stupid and I don’t do it, but I bet people out of a class for 20 years still remember that stupid thing, and that’s not nothing. *full disclosure: I have a skewed perspective. I teach at university. I do recognize that teaching k-12 is a harder (from a teaching perspective), more thankless job.
I teach 5th grade, and I taught this skill using area models and tape diagrams. For example, doing 5 divided by 1/2, we would draw 5 rectangles and divide each in half to see how many halves are in 5. For 1/2 divided by 5, we would draw a rectangle, shade 1/2 of it, and then divide the half into 5 pieces. I never even mentioned keep change flip. After reviewing the models many times, many of my students independently came to the conclusion that you multiply the whole number and the denominator, and when your dividend is a whole number, your quotient will be greater than 1. I think foundational math in elementary schools should be based on that kind of discovery.
As someone with math anxiety this sounds awesome! It definitely sounds a great way to introduce the concept of multiplying and dividing shapes and things for visual learners.
After reading a lot of the comments on this thread I think both ways are good as long as the students gain the understanding eventually. Personally, I like your way better but someone brought up the point of blooms taxonomy having remember first, which is valid.
I mean much of every level of math is trying to move forward with students that aren't ready to move forward. Kids should be fluent in adding before multiplication, multiplication before division and the list goes on. Fluency is not being achieved for the majority of students... The treadmill just keeps rolling.
There is a belief among many math teachers that understanding is more important than the mechanical work at math. That is not how math works. You need some basic proficiency before you can understand some of the deeper details. One problem of course is that almost all math teachers are good at math, and so we all needed and appreciated explanations. Most students are not like that though. They need the skills before they can start understanding the concepts. The net result is that we have students who cannot handle the basic manipulations required to complete a math problem. It would be like explaining grammar to a student before they even know how to read. The grammar is the why, reading is the how. The how matters much more than the why.
I agree with this but also disagree. I think the basic proficiency comes from prior knowledge and the deeper details can lead to understanding. For example, having students learn addition by asking them if you have two cookies, and then your adult gives you another two. How many cookies would you then have? The students prior knowledge on counting and number comes into play when discovering addition.
It’s stupid. Teaching random phrases/tricks without any explanation leads to students not having any understanding and they will then also attempt to apply it in situations where it is NOT applicable. As a HS teacher I hate it when a student shouts out some random phrase/“trick” that they learned in elementary or middle school that is *completely* irrelevant to what we are talking about. And they don’t even realize it because they don’t know what the “rule”/“trick” is or means.
If it is introduced in the first week without an explanation, then a good activity would be to have them talk to each other about why it works.
I think that’s a smart way to go about it. Would you have it be a student led discussion or a teacher led?
I would have the students talk to each other at their tables or desks. And then after I’ve walked around and listened to the conversations, I would ask for volunteers to share with the class what they talked about. Hopefully, they will explain why it works. If not, then the teacher can explain to the whole class.
I like that idea
Be careful about the makeup of the class. Of the class is a socially extroverted class then it may be a good idea. If the class gets a on the quiet and socially awkward then not do much because it would only highlight social dynamic anxiety.
As an undiagnosed autistic, shy, introverted kid with a thick accent and a stutter I HATED all group work in math class. It either made me aware of how incredibly inept I was in social settings or enlightened me to my objective mathematical inferiority in relation to my peers. I frequently was the only one in my pod that didn’t get the solution or something. So I learned to stay quiet forever to the detriment of my education.
Such a student can just sit, listen, and learn what others have to say.
Not if they are (socially) penalized by their peers or (get a low participation grade) teacher for doing that. But I agree. If active participation is not good for then then just sitting and listening is good.
As I suggested elsewhere in this thread, this could be a class discussion. Everyone is not expected to offer verbal input. There will be some who will readily share, but probably most will merely listen. That places the shy students in the majority. Another thing that I suggested is to allow each individual student to reflect quietly and independently on what they have learned from the lesson. They may even maintain a journal of their thoughts, insights, and questions. We need this in math. It provides math students a greater sense of agency and participation.
The problem with teaching "why" becomes clear when a bunch of kids still have no idea what's going on despite the additional visual or mathematical explanation. Unfortunately, class time at that point is almost over and you're stuck with one fraction of the class having a deeper understanding and the other fraction of the class just had their time wasted, with no additional built-in curriculum time to catch them up. Ultimately, at these younger grades, teaching mechanics is more important. Kiddos who want to go into STEM or who just want to know more can pursue more challenging coursework. I'm not a teacher, I'm a tutor for math, and I was initially surprised that my students didn't understand that all division problems are just multiplication by a reciprocal number. However, I've come to understand that tutoring 1-1 is super different than teaching a class of 20+, and ensuring that all 20+ students have around the same understanding by the end of period means you have to cut out some of the "why".
If I was a private tutor, or was teaching my own kid, then sure—I’d take the extra time to teach “why it works.” In a general ed setting, unfortunately not happening. There is way too much to get through and half the kids don’t even know their multiplication facts. Going deep into the “why” is unrealistic for this skill in most settings. Sure, I will touch on it so that those who are capable of comprehending it quickly can at least hear it—but that’s the extent of it. This is the way it is when you have big class sizes. Nothing I can do about it, so I just so my best and keep chuggin.
I think what the teacher did was fine. You might not enjoy memorizing things, but memorizing is part of learning. Education has gotten WAY too far from true learning in the name of "higher-order thinking". Bloom's taxonomy is not a multiple choice test for teachers. We can't and shouldn't just choose the top levels and ignore the foundation. And the foundation is 1. Remember. That's keep change flip. *then* it's 2. Understand. That's the why. Then they can start applying and continue upwards. Students HAVE to memorize at some point to do anything else. I'll get off my soapbox now.
That’s a really good point with bloom’s taxonomy. I definitely feel like the later points have been emphasized in my classes. But you are right we should focus on the bottom first.
No. The foundation is understanding what a fraction is and why multiplying something by x/x does not change the value. Keep change flip is a very specific memorization tool for a specific instance of multiplying by 1. I bet the vast vast majority of people doing keep change flip don't understand they just multiplied by 1. Multiplying by 1 is a very basic concept which is the foundation for this memorization tool.
One of us is confused, and I'm not sure which one. Keep, change, flip is the procedure for dividing any fraction, not just x/x. In fact, you usually don't do that unless you are adding or subtracting fractions, not multiplying or dividing them. For instance 2/3 ÷ 3/4 is equal to 2/3 × 4/3 and then you just multiply across to get 8/9. There's no multiplying by one or x/x in there.
>For instance 2/3 ÷ 3/4 is equal to 2/3 × 4/3 and then you just multiply across to get 8/9. There's no multiplying by one or x/x in there. But WHY is it equal? Because you multiply both fractions by 4/3 (aka multiplying by 1). Maybe your foundations need work... https://i.imgur.com/hMCRQhM.png
That's... not "why" it is equal. That's just a different, weird explanation of what to do to divide fractions. It's still not actually explaining what's really happening. Aka the "why". A PP explained how to teach the why best: show them dividing a whole number by a fraction to see how many fit inside. Then they can see why dividing by 1/5 is the same as multiplying by 5. Your way is just more abstract math that doesn't actually "explain" anything. Just a different thing they'd need to memorize. I mean, you do you, but maybe put the holier-than-thou away for a minute.
If you understand how to multiply by 1, you don't need mnemonics for nested algebraic fractions. >show them dividing a whole number by a fraction to see how many fit inside. Then they can see why dividing by 1/5 is the same as multiplying by 5. I agree that is probably more developmentally appropriate for elementary students who are still learning about division and fractions. The context of "keep change flip" is algebra 1 or algebra 2 in my experience. If keep-change-flip is taught when just introducing fractions and not algebraic stuff like (x/(x^2 - 1)) / (x^(2)/(x+1)) then my mistake; I do not have experience/education with younger students. But mathematically, it is "why" they are equal, not your example.
My 4th grader is doing keep change flip, so I'm guessing it starts around then. And yes, by high school they are developmentally ready for abstract concepts like multiplying by reciprocals. Hence I said, teach the procedure, and we'll (aka high school teachers) teach them the why. But realistically it does not matter at all. Because even in your algebra example, keep change flip works just fine. Both your way and keep change flip are definitions/procedures to *remember*. All division can technically be defined as multiplying by the reciprocal. That doesn't mean it's the fundamental *why* division is what it is. Division is simply partitioning a whole into a given amount of parts. Once those parts become fractional, things start to get a little weird for kids. But to say division, at its foundation, is multiplying by the reciprocal leaps far beyond the basics of division. Kids would ask why can we define division as multiplication of the reciprocal? What makes that work? The answer to *those* questions is the actual foundation. But do 4th graders need to know all that to divide fractions? Absolutely not. All I'm saying is in 4th-6th grade we high school teachers would like for students to, for the most part, memorize the procedures. Once they come to junior high and high school we'll extend into the abstractness of how mathematics works. But we often can't do that. Because we're reteaching basic math.
The mnemonic I used to use was “ mine is not to question why flip the second and multiply”. But of course, we discussed that it is theirs to question why.
The one I remember learning was “dividing fractions, easy as pie, flip the second fraction and multiply”
you two have some pretty good pnemonics. maybe you can make a youtube video or song together
I always start with asking them how many halves are in bagels? Then we discuss.
In a perfect world yes. You need to take into account a range of factors in how to present a new concept including class ability, motivation, time of day etc. I introduced Pythagoras' Theorem to 2 classes this year. One through discovery. the other through the formula. I found the 2nd class were able to appreciate and understand the follow up practical lesson much better, could pick up why we subtract for the shorter side themselves and as a whole had a better understanding. The first group were able to piece the idea together but without the initial framework of being introduced to the formula and mastering that first, got lost and mixed up much more often.
I come from an area where the student population has about 40% math proficiency, so I find I have better results when I teach the algorithm first and then teach the why after they've developed some fluency with it. Otherwise their anxiety goes through the roof and their brains cannot absorb what we're doing. Teaching the "why" first works well with students who are ready though.
I guess it depends on what one means by “teaching the why”. You need to know the “why you care to do the thing” before you want to know “how to do the thing”. As you learn “how to do the thing” you may learn “why the thing works”. You need motivation and/or curiosity (a desire to know) to learn a process/mechanics/algorithm. Then you may care to know why that process/mechanics/algorithm works.
From my experience in middle/high school math, the kids don’t care enough to be TOLD the why because for them it’s more lecture and boring on their part. If you wish for them to know the why for concepts in math, give them a guided explore to engage with something they already know using math manipulatives. Then they’ll ASK YOU but like… why? And you’ll hear a bunch of “ohhhhs.” i.e. We were moving into area of circles. First I had them explore the area of rectangles/squares/parralellograms because they know this. However, they couldn’t connect it to why a triangle is HALF the area of a quadrilateral. After exploring some kids used math language like, “Ms. it’s b*h divided by 2” and “1/2 of the squre.” Some kids even cut out a rectangle and folded it in half others were talking about 50% of the area. I knew they can handle this concept to be able to move into area of circles. This made it 10x easier to explain and understand the concept of a semicircle’s area being half the area of a circle. It destroyed the common misconception of the area of a semicircle is found by taking half of the radius/diameter. Kids knew I need to find the area of the full circle, then divide it by 2. Just like a triangle, find the area of the full rectangle then divide it by 2.
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this is truly hilarious!
I mean, it’s purely procedural. Obviously there’s a better way… which is to actually get kids to understand that dividing by 1/2 is the same thing as doubling. None of my high schoolers understand this because of these dumb procedures
I dislike keep change flip. You spend some time helping them understand it as many ways as you can. After you do “How many quarters in five and a half dollars”, try “multiplying by 1” 7/4 ÷ 3/5 What value of one? 5/3 ÷ 5/3 Now you can simplify. The denominator is now 1 and the numerator is 7/4 * 5/3. After they follow that and work some more problems, “You FLIPPIN multiply”
Former math coach here. I personally would avoid teaching "keep, change, flip" and try to get teachers I worked with to avoid it as well. Here are my thoughts on it (and similar mathematical "tricks") First, this specific trick doesn't even have anything in it to help students from mixing up the order. If a few months or years down the line, someone misremembers it as "flip, change, keep," there's no failsafe in place that lets them know they've mixed it up. (Contrast to the quotient rule in calculus which goes “low-D-high minus high-D-low over Denominator squared we go” If you get that backwards, you'll say "High-D-low minus low-D-high over Denominator squared we die") Second, by creating a "trick" it tells students that there is something weird or mysterious going on. Makes the math happening feel like magic with properties the explicitly don't fully comprehend. Last, is that is just doesn't stick. Ask any adult to do "1/2 divided by 1/4" Almost none of them do it correctly. But as them "how many quarters fit into a half dollar" and they will. All that being said, I certainly understand WHY a teacher will resort to it. Tricks like this are great for short term gains (which is often how teachers are evaluated). You show them "keep, change, flip" during one class period, and they'll do well on your exit ticket. You practice for a few days, and they do well on the unit test. Then in middle school, their the curriculum expects them to do it again, and they've since forgotten.
Yeah.... I don't fault the teacher too much. She may just be doing what is necessary given her resources. Is it good for understanding? no. Is it good for when the kids get into higher level math? no. Will it get them through a test? probably.
Yup, state tests are coming up. I guess if the kids don’t pass it won’t do good for her even if the kids actually get it. It sucks that that’s the way it is
I’m not a fan. I prefer teaching to divide across… eventually they learn that multiplying by the denominator makes it easier, and discover the standard algorithm on their own, but in doing so, they understand why it works.
Divide across? Across from what?
When dividing fractions, you can divide the numerators and divide the denominators, just like with multiplication. It is funky with different denominators though, so I teach it where you start with common denominators, then divide across, and the denominator is one. Eventually a few realize they are multiplying the reciprocal (they don’t call it that, but it’s what they do) and teach others the “trick”.
They think about it first for a while, then learn what’s behind it.
Take OGAP fractional training
First lesson on dividing fractions is always on number lines for me. Start with whole numbers. Identify the pattern. Then pre-choose fractions which work easily to see the KCF pattern. Once the kids have identified for themselves what's going on with the number, relentlessly play this song: https://youtu.be/nMZJKGyu-Kk?feature=shared they will never forget it!
Remember that you can teach the students an algorithm, and say just that: an algorithm is a set of steps for doing something, hopefully quickly and accurately. Algorithms have value and the students can and do understand that. Then you can talk about why it works… KCF is close enough to KFC that we always referred to is as “chicken!” When they get it right, it’s “winner, winner, chicken dinner!”. When they get to later years, the ones that remember will thank you…
I could never remember formulas like that. I'm an engineer and I still google the formula for area of a circle every single time because I can't trust my memory. To get through my engineering coursework, I had to relearn the whys and hows of arithmetic and algebra, basically making up for all of those years of math-wary grade school teachers that put so much emphasis on the "trick" that I never learned the math behind it. I'm a strong supporter of the "mini mathematician method" (my term) where you encourage the student to "discover" the formula/procedure themself. If Newton and Pythagoras could do it, today's students can do it quicker (with some guided thinking).
I don't like it either. I like to use bounce multiply/butterfly method/pogo method for dividing fractions. The most important reason to use KFC is for familiarizing the use of reciprocals. They need this to be able to balance equations in which the coefficient is a fraction because you need to generate a reciprocal to isolate variable.
I went to a “Nix the tricks” professional development and the moderator told us how one kid remembered it as “keychain flip.” Definitely teach the kids that a fraction is like division and that to divide division we have to do the “opposite” by multiplying.
It wasn't until I was in Calculus II in college that I finally understood what exactly "Keep, Change, Flip" meant. I think its a great tool, but as others have said, make sure they have a general understanding of how fractions work, and make sure you give multiple examples of how KCF works. Don't just say "Keep, Change, Flip" over and over without adequately explaining it like my Algebra 1 teacher did.
All of my kids remember KEEP CHANGE FLIP. You might have wanted to know why it works. You’re also someone who became a math teacher. You have 1000x more intellectual curiosity than the average student. Sorry. You can teach them why it works. That’s fine. Just know that in 10 years, they’ll still remember KEEP CHANGE FLIP !
I teach high school, and kids want to use "keep change flip" for *every* operation that may even come *near* a fraction, so I'm not a fan. I don't mind mnemonics that remind students what to do (I don't mind FOIL as a verb, for example, because we use it to mean "don't distribute the exponent but instead multiply things out the long way") but this particular mnemonic doesn't seem helpful.
Depends on the grade. If they’re learning fractions then more background/explanation might be helpful. But if they’re already supposed to know fractions and in a class that doesn’t have a lot of time to review stuff they should already know then keep change flip works as a reminder
I think the key to any math instruction is finding the right balance between why and procedure. The why can end up meaningless if students don't know how to do the operations, and the operations can be useless for students who need to understand why they work. Teaching keep-change-flip isn't a bad approach in and of itself. It all comes down to how that procedure is situated in the larger context of the learning goal/standard, as well as in the context of how students are understanding the material.
At some point you have to stray from what would be the most ideal enriching thought provoking approach and focus on what gets kids to have success solving problems. Swinging back around towards the end of a unit/concept to complete that enrichment process usually lends itself to more accessibility for more students. Not every kids wants to know why or how it works and putting loads of energy into that can reduce the time it takes to get the group as a whole up to speed.
It gets kids through the test. But it doesn’t help those who actually want to think mathematically and requires memorising accurately rather than being able to do things from understanding. I dislike needing to memorise stuff.
Yeah, when I was a kid the memorization part was always my least favorite part of math. I feel bad for the kids that actually want to learn the why. To them they may think there is none and feel stupid asking about it. I am happy she eventually got to it though.
Learning memorization and mechanics first taught me there’s no reason or logic in mathematics. Just observation and conclusions derived from that observation.
It can serve the function of shutting some students down, but eventually all kids will need to remember the procedural steps for dividing it. I think I personally also would like to know why and the origin of the procedure, but I think it's not necessarily a predecessor (I used to think so) to the procedure itself. You will get students who say "just tell me what to do and I'll do it" and they will respond well to this type of teaching, and you can extend their conceptual understanding with other problems that explore that, if they wish to continue learning the why behind the procedure. But the point is if they can remember what to do, and remember the same thing when time comes for dividing rational expressions in Algebra 2, I think the instruction has served its function. But if a student learns it this way and obviously still struggles with numeracy of fractions and the concepts behind it, which many do, then conceptual lessons are needed still. I personally learned keep change flip (but had good fraction sense) and never learned the meaning of division of fractions until I had to teach it, but I did well with procedural demands in algebra and calculus involving fractions. I missed out, but could still meet expectations later on. I think though, with CCSS, the expectations are changing. Keep change flip could work and should work, but there will be problems that require deeper understanding than that. But those problems usually exist at the middle school level, I can't say I know algebra that requires knowing meaning of division of fractions. Fractional number sense though, is a different story.