Reciprocal : A number that has a relationship with another number such that their product is 1. This means that, when you take a number such as 5 and then multiply it by its reciprocal, you will end up with an answer of 1. We could also write the number 5 as a fraction. To find the answer, we have to go back to multiplying fractions.
Remember that, when we multiply fractions, we just multiply the numerators together and then multiply the denominators together. From this, we can conclude that:. In the end, to find the reciprocal of a fraction, we simply take the numerator and make it the denominator and take the denominator and make it the numerator.
Essentially, we are just flipping the fraction around. Here are some more examples of reciprocals. Okay, so now that we have the reciprocal issue out of the way, the question then becomes, why do we need reciprocals in the first place? Well, the answer lies in the rule for dividing fractions. The rule for dividing fractions is you take the first fraction and multiply it by the reciprocal of the second fraction.
Yes, you heard that right: to divide, you end up multiplying, but only after first flipping the second fraction around. Flipping the second fraction around finding its reciprocal changes the value of the equation.
In order to keep the equation mathematically the same, we have to change the division question into a multiplication question. When multiplying fractions, the numerators top numbers are multiplied together and the denominators bottom numbers are multiplied together. To divide fractions, rewrite the problem as multiplying by the reciprocal multiplicative inverse of the divisor.
To add fractions that have the same, or a common, denominator, simply add the numerators, and use the common denominator.
However, fractions cannot be added until they are written with a common denominator. The figure below shows why adding fractions with different denominators is incorrect. As I read the post, I thought about how students could also arrive at invert and multiply by a fraction by exploring the patterns of dividing whole numbers by fractions.
Coupled with a context, students will obtain a great grasp of this concept. The idea of creating that shortcut is also based on the understanding of fractions as division as Julie mentioned above and is why in 5th grade we limit students to division of whole numbers by unit fractions and unit fractions by whole numbers.
Great post, Graham. One important thing to note, I think, is that the models you created for the understanding of your readers were created based on you trying to understand this better. For students to do this, they need a context — and you get this more than just about anyone I know. Teaching these procedures without building the understanding behind it can be as damaging as K-C-F. Spot on Mike! The context makes this idea much more accessible for students.
Glad you chimed in here bud, cheers! Thanks for this thoughtful progression — as well as the others that have been beneficial to teachers. In your picture you are using a proportional rectangle to your originals, but that concept comes later. Thanks for your work on this. Such a great point Kristin. You can lead a horse to water….. I personally like this, but I feel dubious about how it would work with my sixth grade students. In fact, they seem shocked that fractions and division are even related even though, as you point out, 5.
However, they do generally have a good grasp of visual models of fractions and the meaning of fractions as a share of one or more wholes. I agree with the first two comments — whole number divided by unit fraction is an excellent place to start. Then I have them go through whole number divided by non-unit fraction e.
Many of the ingredients of this answer are already present in some of the other answers to this question, but are rearranged here in a fashion that I hope is significantly novel. Multiplication and division of fractions is, in the United States at least, typically first encountered in either third or fourth grade; that is, at age 8 or 9. I want to take a minute to talk about representations of fractions.
Given the age of the students in question, I do not think there is much value in formal explanations, i. They may learn that the algorithm is the correct one, but I don't think they will be convinced by it, in the sense of believing that the algorithm has to be that way and makes sense. And to do that you have to try to see things from the perspective of the learner.
The fractions students most often perhaps exclusively? The order in which I list the representations above is not accidental; it corresponds roughly to the order in which these representations actually occur in the classroom.
That is, the most commonly used representation is the "circle cut into wedges", and the least commonly used representation is the "small scoops that make up one big scoop". Since the question is about conviction , I think it is important to realize that not all representations are equally valuable for that purpose. In particular, I think the third representation above, combined with the measurement interpretation of division, is actually the most useful for the purpose of explaining how division of fractions works.
With all of that established as preamble, here is my strategy for convincing a novice that division of fractions works the way it ought to:. Since most students reflexively think of division in terms of equal-sharing, it is a good idea to start by explicitly activating the measurement interpretation, which is less common. Suppose you want to measure six cups of flour, and all you have is a two-cup scoop. How many scoops do you need?
Most 8- to 9-year-old students will be able to answer "Three" immediately. To do so, they do not need to explicitly translate the problem into "Six divided by two equals what? So you have to ask them to make that translation explicit, with prompts like:.
Now you are ready to move to the next step:. Students who do not have such automaticity may need to count by threes. Either way they will get the answer very quickly. Ask them how they know. Guide the conversation to the following, very important summary:. To the extent possible, try to get the student to be the one who says this, or something like it.
Don't say it for them, but do revoice their statement of it to make it more succinct and coherent, if necessary. Once students agree with this basic idea and once said it usually seems completely obvious to them, so much so that they wonder why you were making such a big deal about it , you can move on to the final variation:. Some students will immediately try to do some kind of paper-and-pencil computation, whether they know an algorithm or not.
Discourage this. Ask them instead to just think about the previous problem. If they are still stuck, prompt: How do the new scoops compare to the old scoops? Usually students will eventually come up with an answer like this one:. Because the scoops are twice as large, we only need half as many of them, so we need just 9 scoops instead of Once they have said this, or something like it -- but not before!
Notice that there are two basic principles interacting here:. At this point, the "rule" that "To divide by a fraction, you invert the divisor and multiply" should not seem like a "rule", but merely a summary of something that should have been obvious all along. I've gone into a lot of detail about this, and I think it's worth pointing out that it probably takes more time to read the above description than to implement it. I've tutored dozens of kids in precisely this manner; some of them are elementary school kids learning fractions for the first time, others are high schoolers who "learned" how to divide fractions back when they were 8 or 9 but no longer remember what to do, or remember it imperfectly.
I've had very good success with this method. If working with a student one-on-one, it usually takes no more than five minutes, start to finish. I think there are two reasons why it is effective:. First, it begins by announcing the problem, and then immediately putting the problem on hold and instead considering simpler problems: first division by whole numbers, and then division by unit fractions. This models an important problem-solving heuristic: When you encounter a hard problem, consider a simpler one and see if you can get any insight from it.
Second, it concludes by looking back at a single example and trying to understand the general principles that make it work. This models a second important problem-solving heuristic: When you have solved a hard problem, take a moment to look back at it and see if the perspective of hindsight reveals any general arguments to you.
And while these heuristics are common to most problem-solving contexts, and may even be naturally-occurring for many students, it is worth recognizing that conventional school instruction which I would caricature as "Teach the rule, then do examples, then have students do many exercises, then provide an explanation of the rule" does not provide a lot of space for this kind of slow, reflective consideration.
That change was not accidental. Nobody makes or uses measuring scoops like that; the artificiality of the problem stands out and is distracting. I imagine that this whole instructional sequence would have to be reconsidered from the ground up and may be completely unworkable if one were teaching in a context in which the metric system is consistently used.
Addendum, added September 18, This afternoon I received an email from a middle school student who had found this answer online and was hoping for some further explanation.
Specifically, she wanted an explanation of how to think about division when both the dividend and the divisor are fractions in the example above, the dividend was a whole number. First let me say that I think the existence of the email itself is proof that what seems obvious to the experienced person may not at all be obvious to the novice.
My first instinct in reading the email was to respond "Well, it's exactly the same! Here is the essential idea that you need in order to generalize the work that was already done:. Let's see how that is useful here. Okay, now let's look back and think about what we did. In our example, we have:. In particular, I will try to answer this using a measurement interpretation, and then again with an equal sharing interpretation.
I prefer the former, but include the latter for completeness. Some of the key terms in unpacking this are measurement , equal sharing , and missing factors , which are up to some name-changing the standard three interpretations of division of whole numbers, and which can be extended to discuss the division of rational numbers, as well.
Two additional important terms are partitioning and chunking. What is the effect of this chunking? More generally: Making sense of the invert-and-multiply algorithm is very difficult, and may require a fair bit of scaffolding. Certainly it cannot be done justice in this one brief response. Instead, I hope you will consider the above as a sketch of how the topic might be broached.
For students really to encapsulate this idea will likely take a fair amount of time, and probably experience with standard division and its interpretations , whole numbers divided by unit fractions, whole numbers divided by non-unit fractions, and so forth.
I have no experience teaching fractions, but I think moving away from using the divide symbol makes things easier. It doesn't get used at university level but exponents start being used, so there are still two notations.
When manipulating fractions, students quickly get comfortable with the idea that to combine two fractions they have to manipulate to get the denominators the same. So we multiply numerator and denominator by the reciprocal of that denominator. Some students will be stuck on this point and not really mature to the next step, unfortunately. The ones who quickly get comfortable with this will see the step of showing the denominator multiplied by the reciprocal can be skipped in favor of just using that number the reciprocal to multiply the numerator.
I suspect the main trick will be going from division by unit fractions to division by non-unit fractions. This can be simplified further by considering each group on its own, and then it becomes obvious why division by unit fractions results in multiplication. If you have some apples and chop them into seven pieces, how many pieces do you have? Well, seven times as many as you had apples, of course!
What if you pair those pieces two-by-two? Well, okay, you now have half as many pairs. Did it matter that we used seven?
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