A Fraction Question

We have a question. One of the simplest possible fraction multiplications is

1/3 x 1/5 = 1/15

The question is, why? What is your simplest explanation of this equation?

To make the task absolutely clear, the explanation is to be suitable for an imagined average primary school student who is just being introduced to the notions of fraction multiplication. Explanations targeted at Peano or a Child Gauss or whomever are way, way off the point.

 

UPDATE (05/01/22)

Thank you to everyone for your comments and suggestions. They were very interesting to read. I’ll now give my own answer to the question, at which, of course, commenters are welcome and encouraged to hurl their objections. Just to be clear, I am, for once,* not pretending to be an expert. As with most commenters here, the teaching of such notions at the (upper?) primary level is well outside of my normal world. The question arose from my general concern with the undeniable awfulness of most primary level education, from specific and very enlightening discussions with Simon the Likeable, and from pseudo-teaching my nine year old daughter.

To be absolutely clear, my question was in no way intended as a trick, or as a surrogate for some other question, even if other questions naturally arose by implication. My question was intended exactly as stated:

What is the simplest explanation of 1/3 x 1/5 = 1/15 for a primary student just being introduced to fraction multiplication?

For this purpose, a deep-end definition of a/b x c/d is obviously inappropriate. At the other pole, the explanation of 1/3 x 1/5 of a pizza does not answer my question, even if it might be a valuable component of such an answer. I’m far from convinced even of that.

I agree with many of the commenters that the natural starting point is to interpret – in effect, define – “1/3 x” as “a third of”. All young children are introduced to fractions as “fractions of”: a third of a cake or whatever. We then use “1/3 x” to capture the notion of “a third of” in a precisely mathematical and symbolically manipulable way. This interpretation/definition is equivalent, as it must be, to defining “1/3 x” as “÷ 3”, but I don’t think that need concern us here; the “third of” expression is more intuitive for the intended audience.

Most easily, of course, children can take a third of suitable integer multiples: three cats or six apples or one Djokovic. That is, quantities that we are happy to divide into three equal groups. This easily extends to fractions of suitable natural numbers: a third of 6, and so on. As such, having students get used to 1/3 x 6 and the like is relatively straight-forward. (1/3 x 7, for example, is an important side story, but it can be left for later.)

But how, then, finally, does one explain 1/3 x 1/5? I’ll take it as agreed that it means “a third of 1/5”, but how do you take a third of 1/5? I think there is a simple and compelling answer.

The problem is that 1/5 is not “three things” (or a multiple of three things), as was the case with the cat clowder above. That leads many to replace 1/5 by a fifth of a pizza, and then out come the knives.

But wait a minute: 1/5 is three things. 1/5 is, of course, exactly three fifteenths. That is, the fraction 1/5 equals the fraction 3/15.** And, a third of 3/15 is obviously 1/15. In symbolic but easily decipherable and easily motivatable form,

1/3 x 1/5 = 1/3 x 3/15 = 1/15

This seems to me, by far, the most natural and simplest introduction to and explanation of non-trivial fraction multiplication. All that is required, which I agree is not a gimme, is a proper understanding of equal fractions and genuine automaticity of multiplication. (On the other hand, if a student does not have this background, the idea of their understanding fraction multiplication, or performing it well, is farcical.) But, except in conversation with Simon the Likeable, I have never seen such an approach clearly expressed.***

That’s it. Have at it.

 

*) Do not expect this to be repeated.

**) People who write that the two fractions are “equivalent” will be summarily shot. 

***) Definitely some commenters were suggesting similar approaches, but these approaches seemed to me clouded with extraneous ideas or activities. My apologies if I short-changed anyone, which is very possible. 

123 Replies to “A Fraction Question”

  1. I would show that breaking the left side of a unit square into thirds and the bottom side into fifths leads to the square being partitioned into 15 congruent rectangles. Each rectangle has area 1/15. Prior to this, show the students that the area of a rectangle is length times width. I would follow that up with a demonstration on the number line. Break the segment between 0 and 1 into 5 equal lengths and then each of these into 3 equal lengths. We now have 15 equal lengths between 0 and 1 and the endpoints of these lengths moving to the right of 0 can be labeled 1/15, 2/15, etc.

    1. OK, so they know what a fraction is. So two things I would try:

      1) By definition: \displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.

      Have at it.

      2) Alternatively: Let \displaystyle \frac{1}{3} \times \frac{1}{5} =?

      Multiply both sides by 3 and multiply both sides by 5: 1 = 15?

      Now a prize for who knows what ? must be.
      (I’m assuming that \displaystyle 5 \time \frac{1}{5} etc. has already been taught)

      But I make no claim to being a decent primary school teacher, which I think is an incredibly difficult speciality (and hats off to all primary school teachers during 2020 and 2021. Very tough times for you).
      Or even a decent lower or middle school mathematics teacher. Actually, now that I think about it … Not even a decent VCE teacher, although I’ve got most of them all fooled (my jokes, anecdotes and hatred of VCAA perform a vital role of misdirection).

      1. Thanks, John. These are obviously more advanced/sophisticated approaches than what other people have suggested, and you state that primary school maths is not your natural domain. Still, I’ll ask.

        Regarding (a), I’m assuming you agree that, to the extent that is possible and the market will bear, definitions should be motivated. So, how might you motivate this definition?

        Regarding (b), it seems more promising, although the algebra is too sophisticated for the intended audience. The approach seems to boil down to using 3 x 1/3 = 1, together with the usual rules/assumptions/axioms of multiplication. I guess that could also be arranged to have “x 1/3” be declared the same as “÷ 3”. Either way, it seems heavy as the *introduction* to fraction multiplication.

        1. Marty, regarding your first question, I’ll get back to you on how to motivate the definition once I have the motivation.

          Regarding your last paragraph, I’m going to disagree. I think many students would figure out that ? = 1/15 pretty quick. Note that I used ? rather than x. But then I might move on to using x (x for the unknown – mysterious) if ? was working OK.

          But I really do prefer an approach that starts from a definition of multiplying fractions. They will meet all sorts of rules for doing all sorts of mathematical things, so why not?

          Then again, we have you first paragraph to quite rightly consider … As a primary school teacher, I’d probably make a good brain surgeon. Abandon hope all ye who enter here.

          Actually, all this is not too far from what’s been occupying my thoughts the last few days … I’m (trying to) preparing to teach Logic this year (thanks, VCAA) and I’ve been thinking a lot about Boolean algebras – in particular how to teach what a Boolean algebra is and what examples to use. No surprise, I’m teaching it as a definition. And I have several examples to show what a Boolean algebra is (and is not). Because it’s more than just 0’s and 1’s and I wonder how many students will come in thinking otherwise. Then switching circuits, propositional logic and digital circuits can be seen as \displaystyle examples of a Boolean algebra rather than THE Boolean algebra …

    2. Thanks, Marc. I see the idea of double-breaking up a thing (square, segment) into fifteenths of that thing is natural. I’m not sure I see how this establishes 1/3 x 1/5 = 1/15. Also, (a) how would you “show” (not sure what you mean by the use of that word) the students that the area of a rectangle is L x W; (b) does your suggestion mean you wouldn’t be addressing multiplications such as 1/3 x 1/5 until after the area of a rectangle is established?

        1. OK, thanks, Marc. That answers (b). How about (a)? Also, (c): given (b), what year level do you consider it appropriate to introduce fraction multiplication?

  2. I’m curious to know what you think of this but I tell my students that we can replace the multiplication symbol with the English word ‘of’. So the question becomes ‘what is 1/3 of 1/5?’ Then a simple pizza/cake drawing where we split the fifth into thirds and can see the result is a 1/15. For whole numbers it needs to be ‘lots of’ …if you have 2 lots of 4 you have 8. If you have 1/2 of a lot of 8 you have 4 etc

    1. Thanks, A. This seems, as tom puts it below, that you would effectively define “1/3 x THING” to be “a third of THING”. (I’m not suggesting you’d be using anything remotely like the formal language of “definition”.)

      Do you think it important that THING be a pizza/whatever? How does that help? Even if it helps, how do you then PizzaLand? How, and when, do you get them to understand the meaning of 1/3 x 1/5?

  3. Similar to Anonymous above I would probably use the word “of” a few times, leading to the question, “what is one-third of one fifth?” Not sure I would use the pizza/cake in place of something like a rectangle (remember those colored rods from primary school…? I do – does that age me?)

    The answer follows.

    And yes, after teaching Year 7s remotely for 2 years (some thrived, some surprisingly so, a lot did not cope so well with my attempts to write on a laptop screen while talking through ideas such as these…) I raise a glass to all primary teachers who are able to equip students with good numeracy skills.

    And to those who continue to use number-based lessons as a punishment (you know who you are)… may you one day find the right career.

    1. Thanks, RF. So, you’d lean towards something like Marc’s second suggestion, the dividing up of (maybe a Cuisenaire incarnation of) a chunk of the number line? Then, I have the same question as for Anonymous: how do you get the kids from a third of a fifth of a thing, to 1/3 x 1/5? (As for JF, I realise this is not your home territory.)

  4. I would also do the same as anonymous and replace the multiplication symbol with “of” when explaining it. I think (?) fractions don’t really get elevated to proper number status until after students are meant to be comfortable with taking fractions of fractions. So relating this to multiplication in general is perhaps not appropriate for young students.

    I’m not sure I would start with \frac{1}{3} \times \frac{1}{5} but rather make sure they are comfortable with taking fractions of whole numbers as well as fractions of sets of objects. They should be cool with “what is one third of this …?” where … is a variety of physical objects or numbers.

    I like taking thin strips of paper as a model. Finding \frac{1}{5} means dividing something into five equal parts and taking one of those parts. I would fold the paper into five equal parts (this is challenging but there are tricks for getting it close). Taking \frac{1}{3} of \frac{1}{5} means taking one of those small parts above and dividing it into three equal parts and taking one of them. Fold the paper into fifths and then fold the fifths into thirds and unfold it. “How many parts have I folded in total? How do you know? Are they all the same size? What would you call that fraction?”

    I like this approach because it really emphasizes that the parts need to be equal in size. Physically folding the paper equally is hopefully an interesting enough challenge to make that memorable. (Who can get the parts the closest to being equal?) Also, you can build on this by getting them to fold a multi-coloured fraction wall. They’ll probably notice the rule by the time they finish it. And, it segues into number lines pretty easily.

    I would expect to teach this concept multiple times and multiple ways for students to get it though anyway. Like everyone else, I’ll add the disclaimer that I’m not a primary school teacher either and wouldn’t expect to make a good one.

    1. Thanks, wst, although you seem to have stolen my question.

      Of course, the very first notion that a kid will see is a fraction of a thing. But, at some stage they are supposed to get (functionally) comfortable with the idea as fractions as numbers, and at some stage they are supposed to get (functionally) comfortable with multiplication of fractions. My question is what to do at the very beginning of that second stage: how do you explain/justify/whatever to a primary school student at the appropriate stage that 1/3 x 1/5 = 1/15?

      And, the same question as to the people above: how do you go from 1/3 of a 1/5 of a piece of paper to 1/3 x 1/5?

      1. I think you just need to say we call taking a fraction of something multiplying the something by that fraction. There are lots of good reasons for it. You could offer some, or ask the student what they think. But on some level, it’s a cultural decision. It works out really well, but I’m not sure it is inherently true. If you stare at the words and symbols long enough, they all start to lose their meaning. What even is multiplication?

        If they are familiar enough with taking fractions of fractions of things, then they will probably understand the rule for multiplying fractions and so it’s just a symbol for that. So I guess I ended up at John Friend’s approach of giving them the definition: \displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}. But they will already know this formula from taking fractions of fractions so it will hopefully seem like a summary rather than something new.

  5. Personally the way I initially learnt it, if I recall correctly, was first by the definition \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}, which I then justified to myself with some thought, exploration and experimentation. So I would suggest that an approach that could work with some students would be first being given the definition and then using the methods that other commenters have given to convince students of the correctness. Care should also be taken that that definition of fraction multiplication is proven to students in general, not just for the specific example given. I feel that John Friend’s (2) approach, with its algebra, might be too abstract and advanced for the average primary student to extend to the general case.

    1. Thanks, Tungsten. You *really* got introduced to fraction multiplication by a general definition? At what year level?

  6. Before a child learns what fraction multiplication is, they first must learn what a fraction is.

    Interestingly I learnt about decimal numbers before fractions, and long division (with recurring numbers) before fractions. From there you can see through experimentation that 1 ÷ 2 = 0.2, that 1 ÷ 5 = 0.5, and that (1 ÷ 2) * (1 ÷ 5) = 0.2 * 0.5 = 0.1 = 1 ÷ 10. Afterwards, when I learnt fractions (as an alternate representation to decimals represented by an integer divided by another integer), the rule that (a/b) * (c/d) = (a*c)/(b*d) was natural as that applied to division as well.

    As an aside I am skeptical of using tricks such as teaching that a fraction can be thought of as a part of a pizza or another object. If you try to force someone to learn through a purely intuition-based approach, they will make false assumptions based off of this model (evidenced by my primary school teachers who insisted that 7/5 is not a fraction because there is no 7/5 of a pizza.) The best approach is to teach a raw fact, and to let the children come up with their own reasoning themselves. This has the added benefit that instead of teaching them two possibly contradictory things, if their own model is false they can easily see that it contradicts the definition.

    1. Thanks, (the same?) Anonymous. I agree totally with your first sentence, and that’s hardly a gimme. Your second paragraph is astonishing: where, and at what age, did you learn about decimals before fractions?

      I have mixed feelings about your third paragraph. Obviously my answers above suggest I am also sceptical of “tricks”, although I’m not sure that’s really the word. But “teach a raw fact” sounds wrong, and possibly a contradiction in terms.

      1. I can’t remember exactly, but I think I learnt decimals somewhere between the age of 4-7. I can’t remember exactly where I learnt decimals, but probably not at school. Interestingly enough I remember being confused by fractions when I first learnt them, as I treated 7/5 as the expression 7÷5, and wondered why you wouldn’t just evaluate 7/5 as 1.4 as you would evaluate 1+1 as 2. I also avoided fractions because it didn’t feel right (instinctually) that you could have two completely equivalent and identical fractions (e.g 1/2=2/4), which didn’t seem at the time proper as an evaluation of an expression, which I thought should have only one answer (a problem you don’t encounter* with decimals). Eventually once I learned of simplified fractions these qualms largely went away, and obviously after learning much later on about the formal definition of rational numbers as equivalence relations it has all become more apparent to me. (Obviously it would be beyond the level of an average primary student to teach that, though it might make for an interesting extension exercise.)

        I probably used the wrong terminology, “analogy” would probably be better than “tricks”, and instead of “teach a raw fact” something like “teaching precise definitions” would be better. Analogies, and other non-precise forms of reasoning perhaps have some place in teaching, though I’ve seen them misused too much throughout my schooling to treat them with anything but cynicism. Perhaps a great teacher could use them effectively, but so far they have only served at best as something confusing and distracting that should be ignored, and at worst as something that is outright false.

        *except for of course expressions like 0.999… = 1, but I don’t remember being too concerned with those.

        1. Thanks again, Anonymous. I don’t think your perspective, or upbringing, is all that common. I’m not sure what, for example, 2.7 means other than 27/10, but I guess you were comfortable with some sense of these numbers. I also have a lot more sympathy with non-precise forms of reasoning in teaching, although there should be some underlying precision and clarity, even if it goes unstated.

        1. OK, sure. You are aware that you could put your name as Fred Flintstone, or whatever, and you’d still be just as anonymous?

  7. Thanks very much for all the contributions so far. I have thoughts and questions (and some criticisms) about the suggested ideas, but I’ll leave responding for tomorrow, when I’ll hopefully be less of a zombie.

  8. Step 1: pull out a paper strip
    Step 2: tear the strip into three identical pieces
    Step 3: for one small piece tear it again into five pieces – this is one fifteenth (of the original one)

    Now hopefully the primary kids would be able to picture that…but I think the ways suggested above by other people are also excellent (possibly way better than mine since the younger kids tend to throw paper tosses to each other. Hope the palate strip strategy will not turn into a disaster. LoL)

    One idea must be reinforced throughout all years – the concept of unity, as well as the concept of the original amount as reference. This must be made clear to students to avoid other confusions.

    I really like the use of proposition “of” mentioned above: “A of B means A times B”. Similarly I also use “A to B” for division and fractions with the later learning stages.

    1. Thanks, Lancelot, although I’m not sure I understand. Are you suggesting that a fraction can only mean a fraction *of* something?

      1. No, I am not. I was just describing a possible starting point.
        In fact, many kids do have this misunderstanding you described. But later they should learn – fractions can be greater than one if it is expressed as a mixed or improper fraction, which is another story then.

  9. Having read what’s been posted so far, I’ll offer two thoughts:

    1) Teaching how to multiply fractions supposes that the idea of a fraction has already been taught. So there is/(should be) a pre-existing knowledge that is/(should be) being built upon.

    2) The use of analogies can be a very powerful teaching tool. However, care must be taken:

    a) An analogy is not *exact* (if it was, it wouldn’t be an analogy). It can therefore cause problems if taken too far (and how does a student know when it’s been taken too far?) and create misconceptions (and misconceptions, once constructed, are extremely difficult to extinguish).
    As noted by Anonymous: “I am skeptical of using tricks such as teaching that a fraction can be thought of as a part of a pizza or another object. If you try to force someone to learn through a purely intuition-based approach, they will make false assumptions based off of this model.”

    b) There is always the danger that the analogy is not understood and so becomes the focus of the teaching. Which creates confusion.
    Using Lancelot’s suggestion as an illustration (and no criticism intended), I can imagine there will be students that might wonder why all three small pieces of paper don’t get torn into five pieces. The paper tearing operations must then be explained and justified and this becomes the focus and a complication.

    Junior science is full of analogies (eg. the teaching of electricity and electrical circuits) and I’ve witnessed many lessons get bogged down explaining the analogy …

    1. I’m not sure I have a compelling sense of the concept of fractions without objects to take fractions of. I guess you do eventually, but that’s not where people start, is it? A fifth is a fifth of something. I don’t consider this a trick. It’s just what it means? And you can have 7/5 of a pizza, if you start with more than one.

      1. Thanks, wst. (And everyone – getting there.) Do you consider that 3 has to mean “three apples” or similar?

      2. Hi wst and happy new year. I must disagree with you. You \displaystyle cannot “… have 7/5 of a pizza, if you start with more than one.”

        1) Already the pizza analogy is becoming stretched like spaghetti. The focus has now become trying to teach the analogy. Trying to make it work beyond the limit of its ‘validity’.

        2) 7/5 of two pizzas is a (poor) analogy for \displaystyle \frac{7}{5} \times 2 NOT \displaystyle \frac{7}{5} \times 1.

        1. But surely you can have 7 quarters of an orange? I didn’t even realise it was an analogy. This is just what fractions mean to me. And numbers. If someone says they have “3”, I think “3 what?”

          1. An orange has four quarters. It does not have seven quarters.

            If you want seven orange quarters you need to cut up two oranges. Now you can choose seven of those eight quarters and ‘prove’, using your analogy/model, that 7/8 = 7/4 …

            Unless you want to add further confusion by saying that we only wanted one orange therefore we only want half of the 7/8 …

            Where’s Abbott and Costello when we need them?

            The analogy/model/whatever is not exact, it has limitations. What we’re now trying to understand (and presumably so are some of these students) is the model, NOT the mathematics. Whatever use the model might have had is gone. Whatever mathematical idea you’re trying to teach is lost. You now have a class of confused students.

        2. I often wonder what the purpose of school mathematics is and should be. I still don’t know, but for me, analogies are a big part of mathematics (and physics). I don’t think they are a distraction or a waste of time, because they’re at the core of these subjects. Physics still seems to me like a just bunch of analogies with equations to match, verified by experiment.

          The point is that you can use the analogy to think about things in a context where things make sense and you can think clearly. They aren’t an excuse to give up reasoning and start making up random things. I think you can have precise rules for what you are allowed to do and so forth while still being flexible about which objects exactly you are working with. In this way, an ‘analogy’ could actually be an exact realisation of certain mathematical rules, at least within the context of a thought experiment.

          \frac{7}{5} is kind of ambiguous in the context of pizzas. You can think of it as 7 \div 5. That is: take 7 pizzas and divide them as a whole into 5 equal parts and choose one of them. Alternatively, you can think of it as 7 \times \frac{1}{5}. That is: divide one pizza into 5 equal parts, then take 7 copies of that part. The fact that you end up with the same amount of pizza could be considered kind of interesting really. You can think about it in terms of pizzas and then bring that back in to thinking about numbers and lengths.

          I don’t see how thinking of 7/5 of two pizzas is in any way relevant.

          1. “And you can have 7/5 of a pizza, if you start with more than one.”

            Your words.

            Now, by saying “I don’t see how thinking of 7/5 of two pizzas is in any way relevant” do you mean that what you said above is no longer relevant?

            You might think I’m being churlish. That’s not my intent. I think the discussion about the pizzas or oranges or whatever is exactly what could (and maybe does) happen in some classrooms. And perish the thought we get a pizza with oranges on top …

            1. You seemed to suggest I’d mess up and confuse \frac{7}{5} \times 2 with \frac{7}{5} \times 1 if I moved to the context of taking fractions of actual objects. And I honestly just didn’t know how that’d happen. But you kind of explained how it could happen from the student’s point of view in the context of oranges above. Thanks.

              I still think the use of numbers for counting actual things is pretty inherent to their meaning and not just an ‘analogy’. People surely worked out rules for operations on fractions before they wrote down the axioms of a field. I believe developing the mental flexibility to decide on a unit of measurement and work with fractions of that unit (like \frac{7}{4} of an orange) is important and not something to avoid because it can be confusing.

              We do it with money and time and lengths (and fungible commodities like oranges – at least I think so).

              1. Hi wst.

                I’ve been wearing the black hat. Yes, this is exactly what I think could happen from the student’s point of view. Now imagine me as the student questioning the teacher (whose maths won’t be anywhere near as strong as yours is) like this and that teacher trying to explain it …

    2. Indeed, JF.
      Using these examples in junior classrooms must come with experience and care in the terminologies.
      I did come across kids confused with different sorts of things when learning fractions (even Year 8!)

  10. Yes. Pretty much. For common everyday usage with small children. Is that weird?

    For other applications and situations, I am happy for people to define 3 however they want.

  11. In answer to the bigger question here (or what I’m guessing it is): for a primary school student, I’m OK with them not realising that a fraction is the result of the division, not the process, IF they are confident working with them.

    Pre-school age students encounter numbers through counting, when 2 apples makes good sense, as they are able to abstract and generalise you can think of times tables without the context of “x groups of y”, but it does help at first.

    Likewise fractions, until students are ready to accept what numbers really are (quite an abstract idea) and that fractions are numbers, I’m OK with them seeing fractions OF things and learning how to work with fractions.

    If the jump has not been made by Year 7, it really needs to be a priority for the Year 7 teacher though…

    1. RF, I very strongly disagree with your first and third paragraphs. The concept of a fraction as a number must be established in primary school. I’m not saying that’s easy or that students will totally get it: who does? But they must be working to seeing numbers, including fractions, as things in their own right, not simply quantities or pieces of something else. At some point, and I think earlier than some people here seem to think, the pizzas have to go.

      1. This gets back to the first of my “two thoughts”:

        “1) Teaching how to multiply fractions supposes that the idea of a fraction has already been taught. So there is/(should be) a pre-existing knowledge that is/(should be) being built upon.”

        I wouldn’t move to arithmetic operations with fractions until I’d taught (and I would need to be convinced that they’d learnt what I thought I’d taught) that a fraction is a number – a new sort of number but in some cases also an old friend (natural numbers as fractions). This is where I’d initially be eating pizza’s, cutting up oranges, ripping up paper etc. Once they were happy with all this I’d call these fractions \displaystyle proper and then introduce the idea of a \displaystyle vulgar fraction. I’d get a lot of mileage from this – the pronunciation alone is worth the money – and why a fraction might be called vulgar.

        Maybe I’d even ask them whether you can have 3/2 of a pizza …

        And once they know that a fraction is a number, why not give a rule for multiplying these numbers? That is, give a definition for doing it.

        But this is not my natural domain (irrationals is closer to my natural domain) so I’m happy to be told that this is dumb and why it’s dumb.

      2. Perhaps I should re-phrase my statement:

        IF the concept of a fraction as a number in its own right has not been established in primary school THEN it must be done in Year 7.

        I am of (no pun intended, happy accident) the opinion that as soon as students are ready for this – give it to them. A skilled teacher should be able to decide when this is the case.

        One of the greatest challenges teaching Year 7 each year is getting students to let go of some of the bad habits they have learned in primary school (BODMAS being a particular big-bear of mine, there are many others) – dealing with the “but my grade 6 teacher said…” is a tiring, annual exercise.

        1. Thanks, RF. We agree on the importance of Year 7 for confirming the solid platform of rational numbers and the arithmetic of these numbers, as the foundation for what is to come. The difference appears to be that you have an exasperated acceptance that, in the main, the platform will *not* be there, and thus you and your colleague Year 7 teachers will have to make that platform, or at least check and tighten a hell of a lot of joins and bolts. I, on the other hand, being a Professional Moon-Howler, refuse to accept this.

          1. In my recent experience, around 40% of students come to Year 7 with said platform. I consider myself very, very fortunate with this percentage based on conversations with teachers in other schools.

            There are some brilliant primary teachers around, but when you are told that a certain primary school “doesn’t believe in times tables” then you know the issue goes beyond teacher quality and into the realm of school leadership.

  12. Interesting that the two fundamental ways of explaining maths both made an appearance. Most have suggested what I call the “bottom up” or “historical” approach of working from examples to arrive at the general rule. The “top down” method delivers the general rule and then justifies it with examples. I favour the first of these, and would get quite chuffed when the students, or at least one of them, discovered the general rule for themselves. But that is just the introduction, most teaching time will be taken up practising the rule or exploring its consequences. As the weeks rolled by my hope was just that the students have remembered that the rule was justified somehow. Perhaps I should have returned to the origin of the rule as in the top down way.

    Recently I found myself teaching fractions (one-on-one) for the first time ever. I remember working several example of multiplication using “of” and the dreaded pizza. Then suggesting “look we can get the answer easily – just multiply the tops and then the bottoms. Which way do you prefer?”. The students happily adopted the rule thereafter, as I expected they would.

    Of course the real difficulty in this topic is justifying the idea that multiplying by a fraction is equivalent to “of”.

      1. Yes as I indicated, that is the real problem. You need to consider both discrete collections and continuous bodies. I had started an explanation of how I was teaching fractions but it was running to a great length, and was not good, so I scrapped it. In the end it comes down to a definition of what it means to multiply by a fraction? We could define 1/3\kern 5pt \times THING to be 1/3 of THING, but why? Because it fits with 3\kern 5pt \times THING equals 3 lots of THING? And before that – what does 1/3 on its own mean?

        My primary school teacher in the 50s seemed to have no trouble with these rather difficult ideas, or is it just my faulty memory. Did the Education Department supply standard lessons in those days?

        1. Thanks, tom. In the end, we have to have definitions, even if we don’t phrase it in that language.

          I’ll check my curriculum guides from the 60s to see what was suggested. I don’t think it was that the ideas were better understood, although I think the explain/get-on-with-it balance was much better then.

        2. I like to TRY to get around that by saying thing = \frac{thing}{1} and therefore \frac{1}{3} \times thing = \frac{thing}{3}

          It never works perfectly, so I’m always looking for other ideas!

          1. Thanks, RF. I don’t think you can get around the need for a definition (even if it is not presented to the students strongly in that form). What you are suggesting here seems to be a special case of John Friends’s a/b x c/d definition. You may be fine with this, and this may be fine.

            Fundamentally, multiplying by 1/3 amounts to dividing by 3, and the question is how to *introduce* that idea in the most understandable manner. The other dimension to this discussion is how much and for how long to rely upon THING being a physical or geometric object, or how much and how quickly to make THING a number.

            I have my own thoughts about all this (because, of course I do). I don’t think the many comments and suggestions have changed my thoughts, but it’s changed my thoughts about my thoughts. The discussion has been very interesting and very helpful.

            1. The best Year 7/8 teachers I have worked with (and I try to emulate with mixed success) all seem to use definitions AT SOME POINT. Exactly when differs and the implicit/explicit nature of the definition also varies a bit.

              The main issue with the \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} definition is essentially the same issue with the \frac{a}{b} = \frac{c}{d} when a \times d = b \times c – students who have never seen an algebraic expression or are not ready for this abstraction will miss the entire point.

              If a student is not able to grasp the abstraction after a reasonable attempt, THEN I am OK with, for the moment, settling for the student being able to work with the fractions numerically and at least be getting correct results.

  13. This is not a simple problem for primary school students. Without the layers of knowledge to answer this the majority of students will struggle. I would not start with the problem 1/3 x 1/5, I would conclude with it after a series of lessons.
    I would have 4 counters and say that I am going to half them and show the two groups of 2. Another example follows, I have 6 counters and I am going to half them and show the two groups of 3. If the class was understanding I would have 8 counters and ask I am going to half them, this time ask what is half of 8? Write/record the fact 1/2 of 8 = 4. Students continue the lesson at their table with counters. Have some problems on the board 1/2 of 10 =, 1/2 of 12 = and maybe 1/3 of 6. They have to concretely show the answers and importantly record the answer. Finish lesson with a discussion that fractions are “parts of.” Equal parts definition will come later.
    Next lesson or concept to learn. Revise the fraction of a group lesson with 1/2 of 8 and 1/3 of 9.
    Introduce the term “of” means times/multiplying. Write on the board 1/2 of 6 is the same as 1/2 x 6.
    Ask what is 1/2 times 6? Write the answer 1/2 x 6 = 3. Repeat with other examples 1/2 x 20 = , 1/3 x 12=
    Students return to their tables, counters available and answer questions 1/2 x 14 =, 1/3 x 15 =, 1,4 x 16 =.
    Finish the lesson with “of” means times/multiplication. The concept of multiplying by fractions is the same as dividing invariably comes up.
    Next lesson or concept, fraction of a whole. Revise fraction means “parts of” and the term of means times/multiplication. Ask questions 1/2 of 18 and deliberately 1/2 x 18, more examples of similar questions.
    Have a square of paper and fold it in two “equal parts” and say this side is 1/2 and the other side is also 1/2.
    Have a fraction chart and color in only 1/2. Repeat by folding another square into 3 equal parts, point to and ask what fraction is this part. Colour in 1/3. Ask how many thirds = 1. Repeat with quarters
    Have a fraction chart and paper for every student. Students work in small groups or independently on fraction of a whole activities. Revise fractions are equal parts
    Next lesson/concept using the fraction wall chart compare equivalent fractions, two quarters = one half.
    Ask for more examples from the class.
    Next lesson/concept. Write on the board 1/2 of 1/2. Using the fraction wall chart show that 1/2 = 2 quarters. Restate “of” means multiplication; 1/2 x 1/2 = 1/4. Other examples 1/2 x 1/4, 1/2 x 1/3
    Using their fraction wall chart students record their answers for 1/2 x 1/5, 1/2 x 1/8, 1/3 x 1/3
    Finish the lesson with a discussion, can you solve 1/3 of 1/4 without using the fraction wall chart?
    Next concept/lesson. 1/2 x 1/4 worked example is solved by multiplying the numerators and then the denominators.
    1/2 x 1/4 = 1/8. Another example 1/3 x 1/4.
    Students solve set problems including 1/3 x 1/5.
    This series of lessons will not be seamless and many misconceptions will need explaining and further practice.
    Do not start with 1/3 x 1/5 unless prior knowledge has been established.

  14. Thanks, John, but it really makes no sense to say “do not start with 1/3 x 1/5”. My question is, once you want the students to understand 1/3 x 1/5 (or a third of a fifth of one or more pizzas), how do you do it?

    Of course one would not want that or attempt that until prior knowledge has been established. And, of course, one can also debate the required prior knowledge and the strength of that knowledge.

    In any case, your answer appears to be: (i) “x 1/3” means “a third of” (for example); (ii) start with easier, more visible or physical multiplications such as 1/2 x 1/2; (iii) move (pretty quickly it seems) on to, e.g., 1/3 x 1/4 as a straight number thing. I’m not sure I understand what you intend in (iii), what “multiplying the numerators and then the denominators” means.

    1. Thanks for the editing Marty.
      From my experience in primary schools only 1/5 of the class 5 or 6 would be able to understand the problem 1/3 x 1/5. The other 4/5 need instruction in graduated steps.
      Multiplying the numerators and denominators is the final step when automation has occurred.
      1/3 x 1/5 = 1 x 1 over 3 x 5 = 1/15.

      1. Thanks, John. I asked this question because it seems to me 1/3 x 1/5 is very not obvious for a novice student to understand. Of course the final automation is easy, but what I want is the clearest way to motivate that automation. It seems to me that you, and others commenting here, make a leap there.

        I have my own answer, and could give that answer, or hints towards that answer. But I’ll keep it for now.

          1. Thanks, John. I’m not a big fan of CLT, although it at least has the atypical benefit of not being false. In any case, I think you may have misinterpreted my (poorly phrased) question, although I think your answer may be implicit in what you wrote earlier.

            The automated thing is to multiply the denominators: 1/3 x 1/5 = 1/(3 x 5). My question is, how do you explain why that procedure works? (“motivate” was a bad choice of words.)

            1. I think this may be the procedure you may be after.
              I would use it later in the steps of fraction development.
              It is certainly important.
              When mutiplying by a fraction the product is less than the muliplicand.
              example 6 x 1/4 = 6/4 = 1 and 1/2
              Also the commutative law applies
              1/4 x 6 = 6/4 = 1 and 1/2

                1. My explanation was trying to figure out what your explanation would be.
                  I am intrigued how you would explain it?

            2. Marty, I don’t like the last paragraph:
              “The automated thing is to multiply the denominators: 1/3 x 1/5 = 1/(3 x 5). My question is, how do you explain why that procedure works? (“motivate” was a bad choice of words.)”

              I don’t like it one little bit. The “automated thing” completely disguises the fact that the numerators must also multiply together. So you end up with students learning a bunch of rules for special cases instead of just one general rule for all cases.

              What comes immediately to my mind – rightly or wrongly – is something I detest and loathe with a passion: Three different formulas often taught for Ohm’s Law for the three different transpositions making V, I and R the subject. I know this isn’t what you mean, but that’s how it makes me feel.

              I will argue that the general rule/definition can be learnt and competently used by most primary school students once the necessary prior knowledge has been taught and appropriate skills practiced.

              The real trouble starts with \displaystyle \frac{1}{3} + \frac{1}{5}

              The more they learn, the more confused they become because they have to start making decisions and choices …

              1. Ugh! Do I need to post the Animals song again? Of course I’m not advocating 1/a x 1/b = 1/ab as a special rule.

        1. Aha … so one of the simplest possible fraction multiplications \displaystyle in \displaystyle form is \displaystyle \frac{1}{3} \times \frac{1}{5}

          So a bit of a trick question, Marty …
          I stand by my initial suggestion (top post): “1) By definition, … so have it.”
          But obviously that definition has been motivated with previous ‘simpler’ examples. Indeed, as Marty already said and which I’ve also remarked:
          “Of course one would not want [to attempt this multiplication] until prior knowledge has been established.”

          So Marty, perhaps your real question is:
          What prior knowledge is needed before asking student to calculate \displaystyle \frac{1}{3} \times \frac{1}{5} and how would you teach this prior knowledge?

          1. Thanks, John. It certainly wasn’t intended as a trick question, although it seems to me some people are tricking themselves. And, no, your suggestion above is not my real question.

            My real question is how do you explain, to a primary student at the stage where it is possible/appropriate/is-useful/is-required, the meaning of and the computation of 1/3 x 1/5? Yes, that question requires making clear what prior knowledge should be understood, and to what level, but my question is quite specifically about a multiplication of the form 1/3 x 1/5: what it means and why, as simply as possible, it equals 1/15.

            Note that I chose the denominators 3 and 5 with some care: small enough that the required arithmetic should be familiar and easy enough, but not reduceable to a short cut “keep halving” thing. Also, as should be obvious, I’m less than thrilled with either pizzas or deep-end definition.

  15. I think you might have unintentionally tricked a lot of people here, because to answer your question requires an explanation of how multiplication and division should be interpreted by a young, beginner mathematical mind in the making. Probably you thought it was an obvious question, but judging by the discussion, (and assuming I am correct) it doesn’t seem that way.

  16. The next question should be how do you explain to a VCE student the meaning of and the computation of 1/3 x 1/5?

    Best wishes to everyone for 2022.

  17. OK… I’ve thought about this a lot today… in between drinks.

    I think JF was really onto something with his very first point but I will make one extension:

    IF a student has properly been taught HOW the simplification of fractions works, then the definition for multiplication falls out very nicely.

    So… my answer is that it really comes down to proper sequencing of the topic.

    1. Thanks, RF. Some questions:

      1) Could you explain JF’s point in your own words?

      2) What do you mean by “how the simplification of fractions works”, and how does this lead to a/the definition of multiplication?

      3) Do you really mean this to be an “extension” of JF’s point, or do you more generally mean “moreover, this …”?

      1. In response to (3): pick one, I’m not convinced the conclusion is any different. Probably the latter though.

        (2) When I show students how to simplify \frac{2}{6} (for example) I have them write \frac{2}{6}=\frac{1\times2}{3\times2}=\frac{1}{3} \times \frac{2}{2} = \frac{1}{3} \times 1 = \frac{1}{3}

        (1) A fraction is made up of two distinct parts (three if you consider the vinculum) and just as we have rules for how to multiply numbers, we have rules about how to multiply fractions. Yes, you can simplify before multiplying if common factors are present, but this comes from how we define the rules for fraction multiplication.

        I could go further, but by this stage, even my better students would have stopped listening.

        By the way, with reference to Terry’s point above, I find by Year 10, for example, students who have never been taught WHY simplification of numeric fractions works have NO CHANCE of simplifying algebraic fractions.

        1. OK, thanks, RF. Regarding (2), you seem to use the definition/property of multiplication to simplify fractions, rather than the other around. The way you have it might make sense for a how-it-all-fits-together lesson in Year 7, but is obviously not how the notion/process would first be introduced/taught. Which raises a question …

  18. Hi,

    Haese in there primary texts introduce fractions by stating “we divide a whole (numbers) into equal portions” ie the pizza(s) approach before expanding into “proper/improper” fractions , rational numbers , numerator defn, denominator define ,simplifying fractions by cancelling common terms,addition ,multiplication,LCDs ,HCFs In the year 7 text for the IB Middle years course

    They then investigate reciprocals , roots of fractions and continued fractions for those who have grasped the above concepts

    Steve R

    1. Thanks, Steve. I haven’t seen the Haese primary texts. I’ll note there’s a big difference between dividing a pizza into equal portions and dividing a whole number into equal portions.

      1. Marti,

        Haese in year 7 doesn’t over complicate… It just defines ” a divided by b” as the fraction a/b

        Or ” we divide a whole into b equal portions and then consider a of them” before define a as the numerator and b as the denominator ( which can not be zero as it is not possible to divide a whole into zero pieces)

        Steve R

      1. Marti,

        Professor Kitaoka is obviously a follower of Escher…

        I find his works fascinating (eg. the concentric circles which appear to spiral) and he shows how easy it is to trick the mind

        Steve R

  19. Well, this one took off. Next post will be “If you had Djokovic in a sound-proof room, and a selection of power tools, what would happen next?”

    Thanks to you all for the comments. I’ll update with my own thoughts tonight, as soon as I’ve drugged the kids.

  20. I like your solution Marty of saying \frac{1}{5}=\frac{3}{15} and therefore \frac{1}{3} of \frac{1}{5} is \frac{1}{3} \times \frac{3}{15} = \frac{1}{15} but I think it becomes a bit of a chicken/egg problem because the student may well then ask WHY \frac{1}{5}=\frac{3}{15} and, as pointed out before on one of my less well-thought-through ideas, the argument gets a bit circular.

    I have racked my brain trying to remember when and how I learned multiplication of fractions at school myself. I really don’t recall. Decimals I remember from Grade 4 because it was taught by a student-teacher (I asked a lot of dumb questions of the poor guy – to the point he pulled me aside for extra assistance during reading time…) but fractions are a bit of a black-hole.

    1. No, it doesn’t get circular. There is no way that the notion of equal fractions should be taught as a consequence of fraction multiplication. Equal fractions must be taught before, and is prior knowledge for, fraction multiplication.

      You might as a secondary teacher, revisit equal fractions in the light of fraction multiplication. But that would be secondary (so to speak).

      1. Fair enough. I’m always looking at these things with the eyes of a secondary teacher, and will undoubtedly miss many, many blind-spots (pun not intended).

        Side question, relating to your (**) above: why do so many teachers AND TEXTBOOKS use the phrase EQUIVALENT for equal fractions when it is (i) wrong and (ii) linguistically stupid to use a big word when there is a simpler (and in this case correct) word to use instead…?

        Genuinely curious. Although it is a battle I know I have lost, despite an annual attempt to rectify.

      2. Marty, your method of “But wait a minute: 1/5 is three things. 1/5 is, of course, exactly three fifteenths. That is, the fraction 1/5 equals the fraction 3/15.” etc.

        is a nice approach. I hadn’t thought of doing/(thinking about) the multiplication in this way. BUT … it requires the student to do/(think of) a number of things:

        1) To know that they’re aiming for a 3 in the numerator. (Lucky the original numerator is 1 and not a number that makes what you’re aiming for less obvious …)

        2) What calculation is required to get the new denominator. I’ll grant that this has probably been taught previously – if you’ve introduce the idea of a fraction you’re probaly going to teach equal fractions before multiplication …? Maybe. But I wonder if equal fractions don’t get taught until prior to addition of fractions …? In fact, I can see equal fractions getting taught \displaystyle after multiplication in order to motivate how to make equal fractions in order to motivate addition of fractions …?

        3) How to do the required calculation to get the required denominator.

        And then I wonder if some students will wonder why you seem to only be taking a third of the numerator.

        Good luck using this for the only-slightly less simple \displaystyle \frac{1}{3} \times \frac{2}{5}. And then \displaystyle \frac{2}{3} \times \frac{2}{5} etc.

        I know the approach you’ve proposed can be tweaked accordingly, but there’s a lot more thinking required that’s by no means automatic (although it might become so for the best students). And it will ultimately all lead to using the definition and I wonder if it will have motivated a ‘proof’ for the definition.

        So Marty, I’m going to disagree with you, being the bad teacher that I am. I much prefer giving and using the definition \displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}, the ‘deep-end’ approach as you call it. The product of any two vulgar fractions can be done using it so you have just the one approach. And with a bit of drill work I think every student will be able to use it successfully.

        But as I said earlier, the real trouble comes with addition of fractions. No way around it, you need to use equal fractions then. And so begins the confusion for many …

        I think it’s important to declare that nothing suggested by any commentator here can be done unless the times tables have been rote learnt.

        1. Thanks, John. A few comments in reply:

          a) The concept of and manipulation of equal fractions is always taught before fraction arithmetic, as it should be. To suggest otherwise is as wrong as can be.

          b) I said explicitly that one needed a solid sense of equal fractions and mental multiplication for this explanation to have any meaning. You might argue that the solid sense is often not there. So what? How does that change the argument for what is required to have a sense of fraction multiplication?

          c) I am not for a minute suggesting my explanation as an alternative to the eventual easy technique for computing a/b x c/d. I am suggesting it as a quick way to give some solid meaning as a foundation for what is to follow.

          d) Given (c), I wouldn’t be looking to push this explanation too far or for too long, but it easy enough to do. Whatever sense you make of 1/3 x 1/5, it is very easy to then make corresponding sense of 2/3 x 1/5 or 1/3 x 4/5 or 2/3 x 4/5. The numerators are easy.

          e) As an introduction to fraction multiplication, the top down approach is wrong.

    2. Well, to be fair, it is often taught to students that given a fraction like \frac15, the way to find equal fractions is to *multiply by 1*, which is to say that (for a given a\ne0)

          \[\frac15 = \frac15 \times 1 = \frac15 \times \frac{a}{a} = \frac{a}{5a}.\]

      So in this sense it is related to fraction multiplication.

      In my view, the way around this is to teach first “multiplication by 1” as something special, and talk through that. Once multiplication by 1 is worked out, you’ve got equal fractions sorted (and some useful fun “strategy” to use, kids love to say “I multiplied by 1” when explaining their solutions) and then questions like the one Marty poses here are a natural next step.

      1. A lot of teachers will say this and not see a problem with it (not even stopping to consider the a=0 issue)

        Marty clearly sees an issue. I look forward to having the annual “discussion” with my colleagues this year – I have a lot more ammunition now. Thanks to all!

  21. I was looking around and found these helpful notes: https://math.berkeley.edu/~wu/CCSS-Fractions_1.pdf, where the author seems to have thoughtfully broken up the topic of fractions into a prescribed order of teaching them, logically, with definitions. They seem to advocate a similar approach to Marty: “interpret the product (a/b) × q as a parts of a partition of q into b equal parts” (p.34). I just thought I’d share in case other people find it interesting too.

    1. An interesting read. Thanks.

      I find the notion of how many compared to how much worthy of some further thought.

      I’ll tell the VIT it is my “professional reading” and took me 200 hours…

    2. Thanks very much, wst. If I’m thinking of the right guy, the author Wu is known, and smart. I’ll take a look.

      1. I think he is. He has a lot on his home page and writes a lot about TSM (Textbook School Mathematics) and its problems. It seems right up your alley.

        1. He uses the phrase equivalent fractions though, so Marty may like him a bit less after reading this paper…

          Ignoring that, it is really well written.

          Thanks for sharing.

          1. I think each section starts with an extended quotation of a curriculum document (the Common Core Standards) and so the expression comes from there.

            1. Wu seems in two minds on that, perhaps intentionally. I note that at least once he has “equal” in parentheses after writing “equivalent”. One problem is that “equivalent fractions” is standard, and Wu is writing for teachers. It is valuable to be understood, and one has to decide whether it’s worth taking a stand. For better or worse, this aspect doesn’t much concern me.

              In general, the Wu document seems very good, but very odd. Would primary teachers really read this? Also, if there’s *really* a clear suggestion of an explanation of 1/3 x 1/5 (or something larger that implies it), I missed it.

              1. Sorry, maybe I overestimated the similarity from being ignorant. I think I found this similar on page 47:

                “It will be best if we begin by explaining a special case:

                    \[\frac{4}{3} \times \frac{5}{2} = \frac{4 \times 5}{3 \times 2}\]


                We have to first find out what one part is when \frac{5}{2} is divided into 3 equal parts… Now we are confronted with computing a third of “5 copies of \frac{1}{2}” but 5 is unfortunately not a multiple of 3. At this point, the fundamental fact about equivalent fractions saves the day. We know that

                    \[\frac{5}{2} = \frac{3 \times 5}{3 \times 2}\]

                and the numerator 3 \times 5 of \frac{3 \times 5}{3 \times 2 } is certainly a multiple of 3.”

                If I was a primary school teacher, I’d read it.

                1. Ah, ok, thanks. Yes, that is in essence the same as what I’m suggesting, although I wouldn’t have chosen that example, and it’s pretty buried.

                  The likelihood that you as a primary teacher would read Wu is evidence of nothing further.

                  1. I think something like this document should be part of the syllabus or its elaborations, to provide clarity about what is to be taught. The elaborations that VCAA provide are all like “students do this” and “students do that”. But they are pretty vague on the actual content and its logical foundations.

  22. I’m late to this party, but the question is something I’ve given a fair amount of thought to in recent months. I do very much like the equivalent fraction approach that marty put forth, but I see just one practical problem with it.

    Students have a hard time grasping the idea of equivalent fractions early on, and I do believe some sense of what it means to multiply fractions should come *before* students have fully grappled with understanding equivalent fractions. I also believe it is helpful to “interpret the product (a/b) × q as a parts of a partition of q into b equal parts” as Wu has, but I don’t always agree with Wu’s approach to the topic of fractions.

    I think it is critical to consider how fractions are introduced in the first place, and I have a few ideas about that. (Disclaimer: My experience is as an enrichment tutor, not a primary educator.)

    1) A key point to establish in the move from natural numbers to fractions is that all the units are going to be treated as uniform. That is a change from natural numbers, which may count varying-sized objects as individuals.

    2) Fractions are a way of indicating and counting pieces that are smaller than one unit. The one rule we must follow is that we can only “count” things when the are the same size, and so we introduce the fraction bar and denominator to say how big the pieces are, where the numerator is just our count. After describing that the denominator calls for splitting the ordinary unit of 1 into its number of equal pieces (the number of parts that are equal to 1), the big lesson is that when all the denominators are the same, we can count, add, and subtract fractions by treating the numerators as natural numbers. They are just counts of pieces. (This makes it intuitively clear that fractions such as 7/5 have meaning, and it avoids the confusion of interpreting 7/5 as a fraction of 2 or some other number that is greater than 1.)

    3) Next we develop multiplying fractions by whole numbers, going back to the definition of multiplication as repeated addition. (Getting tricky, and I wouldn’t say it to the kids this way, left multiplying by a natural number is repeated addition, and right multiplying by whole number is expressed by Wu’s interpretation, since we can’t assume that multiplying a whole number by a fraction is commutative, though we go on to show that it is.)

    4) I also believe it’s very important to use circles to represent units to be split into pie-piece sector fractions, not rectangles. It’s too easy to lose sight of the “whole” in a rectangle model, and to get confused over the sizes of different fractions when the picture of “one” is lost. Sectors make it very easy to see when a unit is complete, and with all circles drawn the same size, they are an excellent scaffolding for later working with mixed numbers and improper fractions. (And please, don’t ever tell me to call them fractions greater than one!)

    I know I haven’t gotten all the way to answering the question, but this is the basic development of fractions that I feel best lends itself to introducing fractional multiplication. It’s the way I was taught, which if I remember correctly was under a balanced approach to New Math in an accelerated school (all kids one year ahead in math) that had excellent math teachers, and we were multiplying fractions in 2nd grade.

    1. Thanks, Mr. Microphone. Ignoring the fact that you’ve been summarily shot for using the term “equivalent fractions”, I’ll think carefully about all you’ve written, and will respond properly later on. But, one comment-question for now.

      You note that

      “Students have a hard time grasping the idea of equivalent fractions [sic] early on …”,

      and I agree entirely. I would word it more simply that students have a hard time grasping the idea of fractions. The fact that the one idea – the idea, for example, of the abstraction of two bits out of six – has infinitely many natural representations, of the exact same thing, is very very difficult. For young and old.

      What puzzles me is your follow-on:

      “… I do believe some sense of what it means to multiply fractions should come *before* students have fully grappled with understanding equivalent fractions.” [sic again]

      I really don’t get this suggestion. The problem is for young (and old) students to understand that 2/6 is the exact same thing as 1/3. We agree that that is very difficult, but how can multiplying either 2/6 or 1/3 by anything possibly assist this foundational understanding?

      Now, to some extent, your suggestion is unarguable, since students have not “fully grappled” with equal fractions until around the second year of university. The overwhelming majority of teachers have *never* seen a proper presentation of fractions, and most who have seen it probably didn’t get it.

      I’m being only slightly facetious. For pretty much all mathematical concepts, and particularly the “elementary” ones, students have to grapple with enough to get a working sense of the meaning, and then accept it, and then get on with *lots* of practice to solidify their sense. The central deceit of modern enquiry teaching is the idolisation of Understanding way, way beyond its worth. We must accept that Mathematics is too hard, and that the Quest for Understanding can only ever bring limited returns, particularly for primary students.

      So, you are correct, that students simply cannot have a great grasp of equal fractions before they start dealing with fraction arithmetic. And, that fraction arithmetic will help solidify the reality of that equality. So, in particular, comparing quantities such as 6 x 2/6 and 6 x 1/3 can definitely help convince students of the equality of 2/6 and 1/3. But, I would claim that such computations are much more usefully, and truthfully, viewed as demonstrating the *consistency* of fraction multiplication with previously (somewhat) understood concepts, rather than somehow being a *proof* that 2/6 = 1/3. The latter framing is way too voodoo-ish. It undermines the (usually best unstated) fact that mathematics is a game with definitions and rules.

      Moreover, once you get up to making sense of 1/3 x 1/5, I think there is absolutely no argument. One cannot make sense of 1/3 x 1/5 without a strong functional understanding that 1/5 and 3/15 are the exact same thing.

      1. All right. I should have said equal fractions, but I was under the influence of Wu and slipped into the general educator parlance. (I could get into a New Math defense of the wording, regarding fractions versus rational numbers, but that would be a waste of time.) And yes, we are agreed that equal fractions are very difficult to comprehend, for people of all ages.

        I would like to suggest that understanding occurs in stages. The first is that n/1 = n (because when 1 piece is a whole unit, it’s the same as counting whole numbers. The next is that n/n = 1, no matter what n is (because n pieces make a whole, so when you have n of them, that’s 1). After that come the fractions that are equal to whole numbers, because (an)/n = a(n/n) = a(1) = a.
        (I do it by factoring because I think it makes more sense to introduce simple fractions before introducing the operation of division. It helps to lay the groundwork for what division means, which will better cement the notion early on that division and rational numbers are the same thing.)

        Now to your point about “fully grappled.” I should have been clearer that in context, what I meant by that term was full awareness that, for instance:

        2/3 = 4/6 = 6/9 = 8/12 = … = (2n)/(3n) = …
        and 4 = 4/1 = 8/2 = … = (4n)/n = …

        I meant exactly that, recognizing that above works for any a, b, and c in place of the starting 2, 3, and 4, and nothing more. (This “full grappling” as I’ve defined it can occur as early as 4th grade for strong students.)

        And sure, you can draw pictures of slicing pizzas into smaller slices until you’re blue in the face, but until students fully understand how to multiply fractions (and more generally, natural numbers), they won’t connect those examples to the general symbolic rule.

        As for your point about *consistency* versus *proof*, I suppose I’m saying that consistency comes first, and proof comes later. But, speaking about proof, even 1st and 2nd graders can learn what I’ll call operational sense (not the same as what “number sense” has come to mean). Before kids are introduced to fractions, they should understand order, know about, and have some basic operational sense with regard to, natural and whole numbers as they are added, subtracted, and multiplied.

        In particular, they should understand that if a known and an unknown, when both are multiplied by the same known factor, produce the same product, then the first known must be equal to the unknown. That is going to be the basis of students learning how to multiply fractions like (1/3)*(1/5).

        I could develop this in a more step-by-step fashion, but I don’t have time to write it up at the moment.

        The crux of the matter will be that I show that 15 times the unknown amount that is (1/3)*(1/5) is equal to 1, and 15 times 1/15 is also equal to one. Also included in my development is the rule that d*(n/d) = n, but I’m leaving out a lot of details. I think you can see where I’m going with this, though.

        What I like about this approach is that it appeals to the essence of what a fraction means, at the level they have currently come to understand it.

          1. In reviewing the entire thread again, I see that my approach is very much like what John Friend suggested (his item 2) in the second response at the very beginning of the comment thread.

            My main departure is that I would hold off on the “multiply both sides by” algebraic procedure, and present it more as “we know this equals that” with a chain of rewriting to accomplish basically the same end. Also, I would have a very specific introduction to fractions already established, as I described earlier.

            I also believe in using the “fraction of” means “fraction x” alternate definition, similar to what Anonymous (1), Red Five, and wst suggested. I have no qualms using “of” with pure numbers when their fractional values support that context (though I do start with THINGS), and I go on to show that the notation with “of” matches the action of multiplication in all such cases where it applies.

        1. Thanks again, Mr. M., and sorry to be slow to get back to you. I’ll probably also reply to your original comment, but a couple quick thoughts on the grappling thing.

          a) You are suggesting, as others have, that understanding the equality of fractions comes from multiplication in the form, for example, 6/10 = (3 x 2)/(5 x 2). Of course that is correct, and is the basis of understanding six pizza slices out of ten is the same as three pizza slices out of five.

          But this multiplication *within* fractions is very different from multiplication *of* fractions. It is the latter that I think must, logically and pedagogically, come after a working understanding of equal fractions. In particular, I fundamentally disagree with any suggestion to somehow “prove” that 6/10 = 3/5 by calculating

          6/10 = (3 x2)/(5 x 2) = 3/5 x 2/2 = 3/5 x 1 = 3/5.

          Sure, such a computation can and should be shown *after* an understanding of both equal fractions and fraction multiplication, to demonstrate the consistency of concepts. But, again, a demonstration of consistency is very distinct from a proof.

          b) Your final paragraphs seem to be suggesting something like

          15 x (1/3 x 1/5) = … = (1/3 x 15) x 1/5 = … = 1.

          This hadn’t occurred to me, and I see it as a possibility, sort of. But anything like this seems to me long and complicated, and to be ducking the fundamental issue, to try to give a sense of the meaning of 1/3 x 1/5.

          1. Thank you, marty.

            So far, I’m with you on (a), regarding the pizza slices, and I agree that it’s a good place to start. I also agree with you that connecting the 6/10 = (3 x 2)/(5 x 2) with the (3 x2)/(5 x 2) = 3/5 x 2/2 = 3/5 x 1 = 3/5 is the exact problem that students have with understanding equal fractions. It’s the reason that they struggle with recognizing them in the beginning, because the equating of the latter from the former is not obvious to them.

            But, yikes! I would never say to “prove” that 6/10 = 3/5 using the multiplication of fractions rule. I would only use that approach to reinforce practice with equal fractions later. My proof is much more fundamental than that, and it’s the same way I show how fractions could be multiplied.

            Basically:
            5 x (6/10) = 30/10 = 3 x (10/10) = 3 = 5 x (3/5), going back to my basic rule that n x (a/n) = a.

            In other words, I motivate equal fractions and the meaning of unit fraction multiplication simultaneously, both stemming from a basic rule of what it means to multiply a fraction by its denominator. I know it’s kind of complicated, as you mentioned in (b), but the approach calls for reinforcing natural number multiplication (and division) practice in the context of fractions, which is the key skill that will allow students to understand the topic well.

            1. Thanks, Mr. M, and sorry for provoking the “yikes”. I didn’t think you were proving 6/10 = 3/5 via multiplication of (as opposed to within) fractions, although I interpreted your very first comment as perhaps suggesting that. But, other commenters have, pretty explicitly, been suggesting something like this. (Unless I misinterpreted them as well.)

              1. Thank you again. Fair enough, I did see many explicit calls in the comments to use the multiplication of fractions rule first, so the advisory is understandable.

                Perhaps I should put it more succinctly in words, to better convey its essence.

                My very first sense of the meaning of 1/3 x 1/5 was taught to me in second grade by a specialist teacher whose name I no longer remember, but whose work I’ve come to appreciate more than ever, especially after learning that such clear instruction is and never was anything near the norm.

                It goes like this: 1/3 x 1/5 is the number which, when multiplied by 15, is equal to 1, and we know that number to be 1/15.

                So the bulk of the thought goes not into “what is 1/3 x 1/5”, but more into “how would you multiply that unknown number by 15, and what would you get.” Then it comes back to an exercise in associativity of multiplication and practice with regrouping. (Of course, it’s essential to cover these skills thoroughly with natural numbers first.)

                Also, I really like the development that you put forward, especially the way it uses the most natural definition of “1/3 x” a number:

                1/3 x 1/5 = 1/3 x 3/15 = 1/15

                But there’s one issue with it. I could see some students who’ve picked up how to multiply fractions somewhere else producing the answer 3/45 and then getting hung up trying to simplify it.

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