I was really upset while I was trying to explain for my daughter that $\frac 23$ is greater than $\frac 35$ and she always claimed that $(3$ is greater than $2$ and $5$ is greater than $3)$ then $\frac 35$ must be greater than $\frac 23$.
At this stage she can't calculate the decimal so that she can't realize that $(\frac 23 = 0.66$ and $\frac 35 = 0.6).$
She is $8$ years old.
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I don't have any experience with kids, so I have no idea if this would just make things more confusing. But you could try taking advantage of common denominators:
Assemble two piles that each contain $15$ identical somethings (paper squares?). Now we can talk about coloring $2/3$ the squares black in the first pile, and coloring $3/5$ of the squares black in the second pile. To do this in a intuitive way, explain that this means, in the first pile, "two out of every three squares are shaded". So to demonstrate visually, count out three squares at a time from the first pile, and for each three counted out, color two of them. Do this until the entire pile has been accounted for.
Then move on to the second pile. Again, "three of every five squares are shaded", so count out five squares at a time, and for every five counted out, color $3$ of them. As before, do this until the entire pile has been accounted for.
Now reiterate that $2/3$ of the squares in the first pile were shaded black, and likewise for $3/5$ of the squares in the second pile. Lastly, actually count the total number of black squares in each, and of course the $2/3$ pile will have the most.
Somewhat unfortunately, the only reason this works so well is precisely because $15$ is a number that is divisible by both $3$ and $5$. If she tries to investigate, say, $6/11$ and $5/9$ using a similar method (but an "incorrect" number of pieces), it won't work out as nicely. So you'll want to look at other answers or meditate further on how to convey the ultimate idea that $x/y$ is answering, in some sense, "how many parts of a whole?", and why that makes "less than" and "greater than" comparisons trickier than with the integers.
One way of going about this would be to explain that the bottom number of a fraction isn't actually counting anything at all. Instead, it's indicating how many equal pieces the "whole" has been broken up into. Only the top number is counting something (how many pieces of that size). It's tougher comparing $x/y$ and $w/z$ given that the "whole" has been broken into pieces of different sizes depending on the denominator. Perhaps this can be demonstrated with a traditional "cut the apple" / "cut the pie" approach. Show, for example, that $1/2, \ 2/4, \text{ and } 3/6$ are the same thing despite the fact that $1<2<3$ and $2<4<6$.
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Buy $2$ cakes of the same size (preferably not too big), let's say cake $A$ and cake $B$.
Cut cake $A$ into $5$ equal pieces.
Cut cake $B$ into $3$ equal pieces.
Give your daughter a choice: she can either have $3$ pieces of cake $A$ or $2$ pieces of cake $B.$ To make the choice easier, put those pieces next to each other, so that she can see that choosing $2$ pieces of cake $B$ is more beneficial.
If she still chooses pieces from cake $A$, at least you get more cake than she does.
EDIT $1$: As @R.M. mentioned in his comment, this "exercise" is supposed to help your daughter understand that her reasoning is not fully correct. After she understands that, it will be much easier to show her a more proper "proof" that generalizes to all fractions.
EDIT $2$: As some people mentioned in the comments, you do not need to use cakes. You could use $2$ chocolate bars or anything of your (or her) liking that can be easily divided into equal pieces.
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A $3\times 5$ cake into fifteen $1\times1$ squares in the normal way. Two thirds is taking 10 pieces, three fifths is $9$ pieces.
This generalizes because if you have $p/q$ and $m/n$ (at least with $0<p<q$ and $0<m<n$ you cut a $m\times q$ cake into $mq$ squares. Then either take $p$ of the $q$ rows (for $pn$ pieces,) or $m$ of the $n$ rows, for $mq$ pieces. So $p/q$ is bigger than $m/n$ if and only if $pn$ is bigger than $mq$.
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Draw out a 15cm line with notches every cm and show $1/5$ of this is 3cm and $1/3$ is 5cm. Then mark the position of $3$ lots of the $1/5$ at 9cm and mark the position of 2 lots of the $1/3$ at 10cm.
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If you want her to understand the general concept, so that she can work out every case by herself, then I suggest both ideas in the other answers:
Step 1: Together with her, draw accurate pictures of 2/3 and 3/5. For example, as parts of a circle (cake or pizza), or as parts of a rectangle (chocolate bar? [but the ones with long tablets(?), not the ones with little squares; that is, drawing a fraction is just splitting the rectangle in two]). I think there is no doubt, visually, that 2/3 is the longer piece/slice.
Step 2a: "How do you work it out with numbers?" Well, 1/3 and 2/3 are easy, because 2/3 means getting 2 of the pieces, and 1/3 is just only one.
Step 2b: [Here comes the crucial intermediate step.] What about 2/3 and 5/6? [Let her think about it!] Well, if you split each "third" in half (there are two "thirds"), you get 4 "halves of one-third". But having half of one third is the same has having one sixth. So 2/3 is the same amount of cake as 4/6. So 5/6 is greater.
Step 2c: Another example: 3/5 and 4/10 [note that here 3 < 4 and 5 < 10, but 3/5 > 4/10; I hope she will be able to understand this].
Step 3. With 2/3 and 3/5, you can neither split thirds to get fifths nor split fifths to get thirds. BUT you can split both to something common to both! [Let her do the math, if possible. The rest is basically the same as other answers.]
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I think, the more convenient way to compare "little fractions" is to apply them to hours because $60$ has so many divisors that it fits many examples.
$\frac 23$ of one hour is $2\times 60/3=40 \min$
$\frac 35$ of one hour is $3\times 60/5=36 \min$
Now to make her understand the ratio, you can remark that a third of an hour is $20 \min$ while a fifth is only $12 \min$ which is almost half of it. Yet $2$ compared to $3$ is bigger than the half ratio, so $\frac 23$ feels bigger than $\frac 35$. This is a very rough argument, but it aims at showing that a bigger denominator matters.
If you want to explain deeper, you can go on with the cross product, showing that $\frac 23>\frac 35$ because $2\times 5>3\times 3$.
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Sometimes a picture can say more as hundred words.
Of course, do it much more precisely, as on the picture - the difference you have to show is only 6%.
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The whole point of fractions is to allow for formal computations like:
$(2/3)\div (3/5) = (2/3)\times (5/3) = 10/9 > 1$
Now, the steps here may need further explanation to an 8 year old learning about fractions. However, that must then be done, because this is the whole point of using fractions. It's all about learning to use formal manipulations so that you don't need to engage in elaborate arguments involving objects cut in 3 pieces and in 5 pieces to get to the answer.
Now, 8 year olds are perfectly capable of mastering the rules for arithmetic with fractions. We tend to forget that using a smartphone or playing computer-games requires using a lot more rules, but the typical 8 year olds are quite proficient with these activities (sometimes more than adults). So, the real problem here is that we postpone teaching math properly to children.
The fact that we are able to see the answer without using the formalism of calculus of fractions, does not mean it's a good idea to confront an 8 year old with that to let it learn about fractions. It's better to let the 8 year old understand the rules for manipulating fractions using very simple examples. When the child gets proficient with using these rules then one can let the child compute the result of this more complicated example.
The child will then have learned not just the formal rules of manipulating fractions, but also that this is a powerful method allowing him/her to get to results that are too complicated to get to without using these rules. In case of this example, it may well be the case that at age 10 the child will find it a lot easier to see that $2/3 > 3/5$ without using the formal rules.
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It's great your daughter has thought this out and come up with a precise reason for her answer. Now you can just show her the reasoning is wrong and then take it on from there.
According to your daughter, $a / b < c / d$ because $a < c$ and $b < d$. Now ask her if she'd rather have $1/2$ or $2 / \text{(biggest number she can think of, let's say a trillion)}$. Show her a pie or something and ask, "if I chop this up into a trillion pieces and you get only two of those incredibly tiny pieces, you'd rather have that than half the pie?"
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If you can go to a pizza parlour after the lunch crowd has passed, and each of you order a small pizza, unsliced, and ask if you could borrow a knife to slice it yourself. (It may as well be fun, right?) Ask your daughter if she wants to eat two pieces of a pie cut into three, or three pieces of a pie cut into five. At 8 she should be able to visualize, but may wish to draw it out too, so bring a pen or pencil and paper..as we all learn differently... she will quickly realize that cutting her pizza into three yields larger pieces than cutting it into five, and therefore eating 2 of 3 pieces is more than 3 of 5 smaller pieces. Then eat as much as you like! Have a fun and educational lunch together!
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Take a square and divide it into $15 \cdot 15$ equal pieces with horizontal and vertical lines parallel to the sides of the square. Now $\dfrac {2}{3}$ means that from the three rows we should paint two. Since there are 15 rows if we paint $\dfrac {2}{3}$ of them we will paint them 10. Exactly the same logic leads us to the conclusion that if we paint $\dfrac {3}{5}$ of the rows we will paint them 9 so $\dfrac {2}{3} > \dfrac {3}{5}$.
This can be generalized on two arbitrary positive fractions, if we have $\dfrac {a}{b}$ and $\dfrac {c}{d}$ then we can make a square divided into $bd \cdot bd$ pieces in the above described way and paint the rows and after they are painted just count how many of them are painted with first and then with second fraction to decide which fraction is bigger (of course, they can also be equal).
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I'm with the answers from Pawel and Sheilagh Peters but my suggestion involves less cooking. :-)
From (preferably different coloured/patterned) sheets of paper, cut out 10 circles the same size. Cut each circle into a different number of equal sized slices from 2, to 10, leaving one whole circle.
From this, you can easily demonstrate the way different numerator => denominator combinations are bigger or smaller than others, by placing the two compared combinations on top of each other. If you can see the bottom combination it is bigger, otherwise the combination on top is bigger. In terms of the question, the two slices from the circle cut into three will be larger than the three slices from the circle cut into five.
This additionally lets you show the fractions that equal each other (for eg, 3/5 and 6/10)
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First ask her about the fractions in terms of their initial representations but with some concept she is interested in. For example: "Okay, so for case ONE 3 out of every 5 times you cast your fishing line, you catch a fish". And "Or for case TWO, for 2 out of every 3 times you cast your fishing line you catch a fish".
Next, go onto the next fraction representation which is just double both the top and bottom of initial fraction: "Okay, so for case ONE, 6 out of every 10 times you cast your fishing line you catch a fix. Do you agree this is the same as 3 out of 5?". Make sure she understands that it is. Explain to her until she understands this.
"Or for case TWO, 4 out of 6 times that you cast your fishing line you catch a fish.
Alright, same thing again. "Okay, so again, for case ONE, you catch fish 9 out of 15 times that you cast your rod".
"Or, for case TWO you catch fish 6 out of 9 times you cast your rod. You also catch fish 8 out of 12 times you cast your rod. And finally, you catch fish 10/15 times that you cast your rod".
"Is is better to catch nine or ten fish for every fifteen times you cast your rod?"
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I'd say that the thing to emphasize is that it is about comparing. It's relative. Both numbers must be accounted for in one view.
It's not about how big either number is, it's about how they relate.
You can't look at just the top number or the bottom number alone. Are 3 pennies worth more than 2 nickels? You can't say 3$>$2 and be done. And you also can't look at pennies$<$nickels and be satisfied (since of course 6 pennies $>$ 1 nickel). You have to look at both parts together.
Emphasize this phrase/concept... it's about how full it is. Show her a big auditorium or stadium... if 20 of the seats have people in them, would you say it is very full? More so than 12 people packed into a tiny room or a closet?). It can be something you keep coming back to in your everyday life until she's got it. Do you feel like this bus is pretty full? Is this parking lot more completely full than that one? Is this bowl of soup more full than that one? Is your backpack more full than that cubby hole? Is this notebook quite full? That helps appreciate the idea of comparison (and build up better insight into the relative sizes of fractions/percents... one of the most useful foundations to interacting with our everyday world throughout our lives)
It quickly comes to the fact that while adding more items on top (increasing the numerator) DOES make any situation to be more full... adding more slots to the bottom (increasing the denominator) makes it LESS full. So if you want to go that direction, you can help her understand making the bottom number bigger actually winds up always making a fraction worse. But emphasizing what's key overall is not really how many pieces there are, how many "slots" there are, or how big each piece is, but about the completed picture, how "full" it really is in the whole combination, to really aid understanding.
The best help for quite a few kids may indeed be the classic shapes. The pie diagram (or a similar block diagram). The better, winning, one is the more "complete", full drawing. Is 3/5 really more complete than 2/3?
If you'd like larger images, which could be printed and then laid on top of each other, here are 3/5 and 2/3.
If you want to hammer home the point that it's not about numerator size ask her if like 25/100 would be more full and look at that picture. It has got a lot of slices (or seats in the room) filled in. But it's got an awful lot of empty ones too.
Now, if you're asking specifically about comparing 2/3 with 3/5 as the specific skill, indeed because they are so very close in relative size... I'd suggest that there may not be any very useful way for the an 8 year old to directly compare them without being given the picture beforehand... and that's not entirely a bad thing. It's good to show people there's questions they can't answer, to make them look forward to expanding their horizons, and learning new tools. Recognition of those fraction sizes comes with practice of seeing them, just like tying one's shoes.
That's why percent/decimals are to come very soon on behind this lesson in the curriculum. It may well be coming up right after this workbook. At most, I'd expect it's about a year away.
Until then? I'd suggest the only hopes are memorizing or visualizing it. But that it's really not a question she needs to be pressed hard to be perfect at yet. If she doesn't remember, she could try making careful drawings of equal sized slices/blocks, and having her see if she can be precise enough to estimate which one is more full. But indeed these particular fractions are just so similar it's pretty difficult to draw - I for one certainly found thirds in particular to be very hard to draw well as a kid.
You could try looking at both pictures in 15ths... but I think that the math to make sense of that can lose many children, especially if not explained with great care and precision. And it still wouldn't be a very useful tool for answering such future questions for a typical 8 year old. It would introduce conversions/common denominators and a lot of useful math... but I'm a firm believer that you REALLY don't want to get kids too lost trying to perform complex arithmetic for a problem too early, before they understand the basis imagery fully enough, or they can end up getting lost in the blind "because I'm supposed to" instead of really having an understanding of what they're doing. She really has to understand that fractions are relative comparisons, all about fullness, before she can best wield the arithmetic on them sharply and to large benefit.
Indeed, I'd argue that this question isn't given to students at this age seeking so much for them to answer well. But that instead the point very much is: "this is pretty tough to get precise enough for, and that's why we need another way". Making children yearn for that way, and therein look forward to and welcome decimals/percentages (as well as fraction math) is a most beneficial thing. Indeed, these coming topics are probably the one math subject that the greatest percentage(!) of students struggle with in all of basic schooling. Especially before algebra. When I was a student, we spent three or four years going back and hammering at them... and many still didn't get them. There's a lot of different concepts and situations to learn to deal with in fraction/decimal/percentage math, and that can overwhelm many. And so having a great foundation on what the basic ideas mean really aids in learning when to use what. Plus, then being good at these will offer a solid foundation for better success in algebra and beyond.
So no need to cause undue stress. But instead an opportunity to reroute it towards encouraging a better understanding of how "full" things are, a tool which cannot be underestimated. Soothe her that it's OK if she doesn't easily/correctly get the answer to some of these more difficult questions for a little while. And her frustration now will prove useful in the long run. As long as you keep encouraging her that she has the tools to answer most such questions (such as by using rough sketches), she'll benefit both now and later to be seeing a question like this!
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Some brands of chocolate bar are actually divided into five rows of three "squares". Offer to share the bar with your daughter: she can choose either two thirds or three fifths, whichever is the most. But first she must convince you that it is the most.
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There are some great ideas here, but I feel most are fundamentally wrong. To explain something new and complex to someone you have to take the core concept of the problem and apply it to something she already understands. It sounds like the problem here is her understanding “Relative Size” and not “Fractions”.
A toy horse in a shoe-box takes up more of the box, than a real horse would a warehouse. Relative to the box, the toy horse is bigger and takes up more space, regardless of the fact that both the real horse and the warehouse are bigger than the toy horse and the shoe-box. Use a few more examples and ask her to come up with her own, to see if she understands. Then, once you think she's got it, try re-introducing the fractions, keeping in mind 2 is a bigger part of 3, than 3 is of 5. Just remember, you're trying to tell an 8 year old that bigger doesn't mean bigger, which is actually extremely confusing.
It's my opinion on learning, but always extract the concept into a series of relationships and then find something that they already understand and explain the relationship in terms of those. Once that’s sunk in, then re-introduce the semantics. The LAST thing you should do is be overwhelming someone with both a complex relationship and semantics that they’re unfamiliar with.
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Does she understand that fractions can be simplified? If so, then just multiply both sides so that they have the same denominator:
$\dfrac{10}{15}:\dfrac{9}{15}$
Then she can clearly see which numerator is bigger. As long as she buys the equivalence between the reduced and non-reduced forms of the fractions, she should be able to transfer the comparison of magnitudes.
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Simply multiply and divide $\frac23$ by $5$ and $\frac35$ by $3$. $$\frac{2\cdot5}{3\cdot5}=\frac{10}{15}$$
$$\frac{3\cdot3}{5\cdot3}=\frac9{15}$$
Thus $\frac23$ is greater than $\frac35$ (as $10 > 9$)
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Take $15$ items (counters, candy, whatever). To compute $2/3$, divide them into three groups of $5$ and take two such groups; you have $10$ items. To compute $3/5$, divide them into five groups of $3$ and take three such groups; you only have $9$ items.
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It's quite impressive that an 8-year old can state a clear reason (even if wrong) for a mathematical conclusion. I suggest you should be pleased rather than upset, and should begin with the respectful approach of addressing her reasoning.
So you might try asking her which is the greater of $1/3$ and $2/5$. If she chooses $2/5$ you might then explore with her the implication that:
$$3/5 + 2/5 > 2/3 + 1/3$$
when in fact:
$$3/5 + 2/5 = 1 = 2/3 + 1/3$$
If that proves unpersuasive I would suggest representing fractions as sections of a straight line (rather than of rectangles, circles, cakes etc which introduce further complications). So draw (or perhaps ask her to draw) a line of (for convenience) $15$ cm and mark against it the thirds (every $5$ cm) and fifths (every $3$ cm). You can then ask whether the line section representing $3/5$ is longer than that representing $2/3$.
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Take a crosslined paper you know those with grids. Now take $3\times 5=15$ squares on your paper. Write lines at the thirds and 5ths. Shade with a different colored pen or crayon or whatever.
First thirds divided by horizontal line: $$\left[\begin{array}{ccccc|ccccc|ccccc} 0&0&0&0&0&0&0&0&0&0&1&1&1&1&1\\ \end{array}\right]$$ And with the fifths divided by horizontal line: $$\left[\begin{array}{ccc|ccc|ccc|ccc|ccc} 0&0&0&0&0&0&0&0&0&1&1&1&1&1&1 \end{array}\right]$$
There is one more colored ( zero ) in the first one than in the second one.
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This is not an answer to the main question, as good ones have already be given (such as starting from $15$ pieces), this is an argument to refute the $3>2,5>3$ reasoning.
Consider $$4>2,$$ and $$6>3$$
but
$$\frac46=\frac23,$$ (split every third in two).
Also
$$3<4$$ and $$6>5$$ but
$$\frac36\ne\frac45$$ and the inequality must go in "some" direction.
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In a first step you can observe with the child that $$ \frac{2}{3} = 1-\frac{1}{3} \quad \text{ and } \quad \frac{3}{5} = 1 - \frac{2}{5}. $$ In words, you are looking at what is remaining of the pie when you have taken either one third or two fifths.
Then, you can show that cutting a third into two pieces gives two sixths, hence $\frac{1}{3}= \frac{2}{6}$ and then $$ \frac{2}{3} = 1-\frac{2}{6} \quad \text{ and } \quad \frac{3}{5} = 1 - \frac{2}{5}. $$
You can then conclude by the fact that sharing fairly between more people leads to smaller shares, thus $\frac{2}{6} < \frac{2}{5}$. Hence, the comlplement to two shares is larger: $$ \frac{2}{3} = 1-\frac{2}{6} > \frac{3}{5} = 1 - \frac{2}{5}. $$
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This reminds me on Israel Gelfand's thoughts on how to explain mathematics: "You can explain fractions even to heavy drinkers," he told an interviewer in 2003. "If you ask them, 'Which is larger, 2/3 or 3/5?' it is likely they will not know. But if you ask, 'Which is better, two bottles of vodka for three people, or three bottles of vodka for five people?' they will answer you immediately. They will say two for three, of course."
Maybe you can replace vodka for cakes when explaining to your daughter?
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Make two drawings, typical of a cake or something. Divide one into three parts and the other into $5$ parts, color the corresponding parts and that way it looks great. In this case as a small difference between $0.66$ and $0.6$ you would have to be detailed for it to be appreciated.
The drawing you take should be predivided in $15$ parts to make it much easier.
The concept of "bigger denominator makes smaller fraction" is easily understood and explained to a toddler. Just show her 1/5th of a pie versus 1/3rd. Or explain that when you divide among more people, there is less per person. So you can explain why her explanation must be wrong without any reference to decimal expansion.
Because 3 is greater than 2, it follows that 3/3 is greater than 2/3. Because 3 is less than 5, it follows that 3/3 is greater than 3/5. Combining the two, we can compare a fraction with larger numerator and smaller denominator with one with smaller numerators and larger denominator. But when one has larger both numerator and denominator, we cannot make this comparison on this basis alone and need more information.
To your specific example, to see why 2/3 is greater than 3/5 without decimal expansions, a more detailed picture (slices of a pie) can work. Or arithmetic like cross multiply the inequality, though that may be above toddler level.