How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$?

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.

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 $2<4<6$.

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.

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.

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!