Help me with this very basic maths problem

[bookbard]

So, my daughter had a maths question which she got right by estimating. I wanted to try to do it longhand to work it out. In short, it was turning fractions into percentages. Easy, right? You multiple the top by 100 and then divide it by the bottom.

So how come I get a different answer if I do it via calculator? 

7/8 via the calculator is 87.5 

But if I do it using long division, I get 87.4 - am I doing it wrong? Is the last 4 a remainder and not a point and am I supposed to do something different with it?

Thank you. 

[Shoeless]

I got 87.5 both ways?

[bookbard]

OK, so I needed to bring another 0 down to make it 40 and then divide it by 8 to get 5. Phew, I feel a lot better now that it makes sense.

[HomeAgain]

3 hours ago, bookbard said:

OK, so I needed to bring another 0 down to make it 40 and then divide it by 8 to get 5. Phew, I feel a lot better now that it makes sense.

To be fair, I think your only mistake was changing the remainder to a part of the decimal.  Your way would have gotten 87 and 4 parts of 8 left, or 87 4/8, or 87 and 1/2 when reduced.

[bookbard]

58 minutes ago, HomeAgain said:

To be fair, I think your only mistake was changing the remainder to a part of the decimal.  Your way would have gotten 87 and 4 parts of 8 left, or 87 4/8, or 87 and 1/2 when reduced.

Oh, thanks - that also makes sense. 

[Not_a_Number]

5 hours ago, bookbard said:

So, my daughter had a maths question which she got right by estimating. I wanted to try to do it longhand to work it out. In short, it was turning fractions into percentages. Easy, right? You multiple the top by 100 and then divide it by the bottom.

So how come I get a different answer if I do it via calculator? 

7/8 via the calculator is 87.5 

But if I do it using long division, I get 87.4 - am I doing it wrong? Is the last 4 a remainder and not a point and am I supposed to do something different with it?

Thank you. 

I think this actually comes down with the double way we do division.

What do we mean by, say, 700 divided by 8? Are we splitting the whole 700, or is a little bit going to be left over? Depending on our definition, we get different answers, but we almost never talk about that.

[bookbard]

15 hours ago, Not_a_Number said:

I think this actually comes down with the double way we do division.

What do we mean by, say, 700 divided by 8? Are we splitting the whole 700, or is a little bit going to be left over? Depending on our definition, we get different answers, but we almost never talk about that.

Thanks, I spent a bit of time today doing a few other sums. 8/9 is an interesting one. It's an interesting activity because it involves understanding fractions, division, and percentages. Not sure when they do this at school - I personally have no memory of it - and I don't know how Australia tends to tackle these things, because they introduce the calculator so early here. The actual question my daughter was working on was something like John ate 8/10 of his ice cream, Jane ate 8/9 of her ice cream, who ate more (it was slightly more complex but that's the gist). And so my daughter could estimate easily that Jane ate more. This was on a year 7 test but I would assume it would be about year 5 level USA? 

[Not_a_Number]

9 hours ago, bookbard said:

Thanks, I spent a bit of time today doing a few other sums. 8/9 is an interesting one. It's an interesting activity because it involves understanding fractions, division, and percentages. Not sure when they do this at school - I personally have no memory of it - and I don't know how Australia tends to tackle these things, because they introduce the calculator so early here. The actual question my daughter was working on was something like John ate 8/10 of his ice cream, Jane ate 8/9 of her ice cream, who ate more (it was slightly more complex but that's the gist). And so my daughter could estimate easily that Jane ate more. This was on a year 7 test but I would assume it would be about year 5 level USA? 

I think that last one doesn’t require any calculation, right? The one with the ice cream, I mean.

[HomeAgain]

9 hours ago, bookbard said:

Thanks, I spent a bit of time today doing a few other sums. 8/9 is an interesting one. It's an interesting activity because it involves understanding fractions, division, and percentages. Not sure when they do this at school - I personally have no memory of it - and I don't know how Australia tends to tackle these things, because they introduce the calculator so early here. The actual question my daughter was working on was something like John ate 8/10 of his ice cream, Jane ate 8/9 of her ice cream, who ate more (it was slightly more complex but that's the gist). And so my daughter could estimate easily that Jane ate more. This was on a year 7 test but I would assume it would be about year 5 level USA? 

This is about where my extra littles are working (around year 4/early year 5).  They are still using visual cues and saying it out loud: "Jane ate 8 of the 9 parts of her ice cream, John ate 8 of his 10 parts of ice cream.  The same item cut into less parts has MORE per part, so Jane ate more in her 8 parts than John did in his 8 parts."

They cannot do percentages from that yet.

[Not_a_Number]

3 minutes ago, HomeAgain said:

This is about where my extra littles are working (around year 4/early year 5).  They are still using visual cues and saying it out loud: "Jane ate 8 of the 9 parts of her ice cream, John ate 8 of his 10 parts of ice cream.  The same item cut into less parts has MORE per part, so Jane ate more in her 8 parts than John did in his 8 parts."

They cannot do percentages from that yet.

That’s definitely “mental model” practicing 🙂 . I like doing a LOT of it.

[HomeAgain]

48 minutes ago, Not_a_Number said:

That’s definitely “mental model” practicing 🙂 . I like doing a LOT of it.

Same.  We use a lot of physical models first before they get to that point, and we're still working on how something like 2 parts of 4 is the equivalent of 2 things divided into 4 parts.  It requires a slight shift into a more abstract division model, I think, but that's our first step before percentages from fractions.  If you have any ideas to make it click, I'd be open to them.  It may just be a maturity thing and need to explore concretely more.

[Not_a_Number]

27 minutes ago, HomeAgain said:

Same.  We use a lot of physical models first before they get to that point, and we're still working on how something like 2 parts of 4 is the equivalent of 2 things divided into 4 parts.  It requires a slight shift into a more abstract division model, I think, but that's our first step before percentages from fractions.  If you have any ideas to make it click, I'd be open to them.  It may just be a maturity thing and need to explore concretely more.

Honestly, this is the reason I start fractions as division. My personal sequence is to work really hard on division of integers (no remainders -- just splitting in genuinely equal parts), observing lots of patterns, and then shift to fractions as a natural extension. In the same way, I do negative numbers as a natural extension of subtraction questions we can't answer with the non-negative integers... 

I've only tried this with DD8, who's very mathy, so I'll report back about my extras in a year or two. It worked super well for DD8, though, and she never needed to adjust to fractions at all -- they felt very natural to her. 

I'd probably do a lot of actual modeling of splitting things into parts and observing. I'd also work on integer division concurrently, making lots of observations -- I do think our lizard brains are simply better with integers, so it's best if a kid fully internalizes that division is the opposite of multiplication with integers before really messing with fractions. 

But then I'm weird and think that division with remainders dilutes the model for lots of kids. I only did it with DD8 after fractions and decimals were safely out of the way, and I never actually use the division symbol for that operation.