# Could somebody who understands multiplying fractions explain something to my son?

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Consider 1 3/4 x 2 1/2.

We know the rule is that you convert both to improper fractions...

7/4 x 5/2

Multiply numerators x numerators and denominators x denominators...

35/8

Then you simplify...

4 3/8

(I hope I got al that right.)

My son wants to know why he can't instead say do it this way...

1 3/4 x 2 1/2 =

(1x2) + (3/4 x 1/2)=

2 + 3/8 =

2 3/8

Why does this yield the wrong answer?

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Probably someone here will have a better explanation, but the way I explained it to my son is this.

You can only rearrange parts like that if all of the parts are multiplication OR all of the parts are addition, but not if some of the parts are multiplication and other parts are addition.

1 3/4 is the same as 1 + 3/4, not the same as 1 x 3/4. We leave out the plus sign, but when we say "one and three quarters" the "and" means "plus". So your problem would really be (1 + 3/4) x (2 + 1/2). You have to do the addition before you do the multiplication. We don't need to write the parentheses because the order of operations tells us we always do addition before multiplication. Does that make sense?

ETA: No, this doesn't make sense...lol. Ignore this. I need sleep in a big way.

Edited by MamaSheep
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Consider 1 3/4 x 2 1/2.

We know the rule is that you convert both to improper fractions...

7/4 x 5/2

Multiply numerators x numerators and denominators x denominators...

35/8

Then you simplify...

4 3/8

(I hope I got al that right.)

My son wants to know why he can't instead say do it this way...

1 3/4 x 2 1/2 =

(1x2) + (3/4 x 1/2)=

2 + 3/8 =

2 3/8

Why does this yield the wrong answer?

Think of the fractions like this:1 3/4 = 1 +3/4 and 2 1/2= 2+ 1/2, right?

Use the distributive property:

(1+3/4)(2+1/2)= 1x2 +1x1/2 + 3/4x2 + 3/4x1/2 =

= 2 + 1/2 + 6/4 + 3/8

= 4 3/8

The way he is doing it, he is only multiplying part of the first term times part of the other term.

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okay, you actually CAN do it that way. He just isn't doing it properly.

Look at it this way, if you were doing a problem with ones and tens.

12

X12

You have to first multiply the ones times the ones and the tens, which in reality gives you 20+4 (or 24)

Then, you multiply the tens times ones and the tens, which in reality gives you 20+100 (or 120). Last, you add all those together.

With fractions if you set it up that way it would look like this

1 3/4

X2 1/2

You would have to multiply the 1/2 times the fraction (3/4) AND the ones (1) which would yield 3/8+1/2. Then, you multiply the ones (2) times the fraction(3/4) and the ones (1) which would give you 6/4+2. Last, you would add it all together (first changing all fractions to a common denominator of 8ths) which would be 3/8+4/8+12/8+2=4 3/8.

(If you think of the fractions as decimals you can easily see why this is necessary.)

It is just much, much easier to change to an improper fraction!

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Because when you are multiplying 1-3/4 X 2-1/2, you need to think of it like this:

(1 + 3/4) X (2 + 1/2)

So he has to multipy 1 times both the 2 and the 1/2, then 3/4 times both the 2 and the 1/2.

You need to do that using the distributive property (which he may not have learned yet) which is

(1X2) + (1X1/2) + (3/4 X 2) + (3/4 X 1/2)

Doing this math it will be 4-3/8.

Jeri

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Think of the fractions like this:1 3/4 = 1 +3/4 and 2 1/2= 2+ 1/2, right?

Use the distributive property:

(1+3/4)(2+1/2)= 1x2 +1x1/2 + 3/4x2 + 3/4x1/2 =

= 2 + 1/2 + 6/4 + 3/8

= 4 3/8

The way he is doing it, he is only multiplying part of the first term times part of the other term.

Or better yet,

(1 +3/4)x2 + (1 +3/4)x1/2

=2 + 6/4 + 1/2 + 3/8

= 4 3/8

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We don't need to write the parentheses because the order of operations tells us we always do addition before multiplication.

Just an FYI- I'm pretty sure the order of operations tells us to always do the multiplication and division before the addition and subtraction. You might want to look that up.

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Just an FYI- I'm pretty sure the order of operations tells us to always do the multiplication and division before the addition and subtraction. You might want to look that up.

You are absolutely right. I am very sleep deprived today and probably shouldn't have even tried...lol. And fwiw I don't think I used that part back when ds and I were discussing it, I think I just stuck to the 1 3/4 = 1 + 3/4 bit. It's been a little while. He did his own math today and the answers were right when I checked. My math lesson with dd today was all about how 8 wants to be 10 and has two vacuum nozzles to slurp units from the other number you're adding it to. Probably it's a good thing it wasn't more invovled than that...lol...

Edited by MamaSheep
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Probably someone here will have a better explanation, but the way I explained it to my son is this.

You can only rearrange parts like that if all of the parts are multiplication OR all of the parts are addition, but not if some of the parts are multiplication and other parts are addition.

1 3/4 is the same as 1 + 3/4, not the same as 1 x 3/4. We leave out the plus sign, but when we say "one and three quarters" the "and" means "plus". So your problem would really be (1 + 3/4) x (2 + 1/2).

BTW. This is algebra. -- Or rather, this has some really interesting and useful functions in algebra and factoring polynomials.

Later on, your son will learn

(a+b)*(c+d) <----- FOIL principle First, Outside, Inside, Last

=ac + ad + bc +bd

(if it is a-b, then treat it as a + (-b) and you don't need to learn a second rule)

And that leads to interesting things like

(ax-b)(ax+b)

=a^2x^2-b^2

so you are supposed to be able to see 4x^2-9 and think "Oh, that factors as (2x-3) and (2x+3)"

Edited by vonfirmath
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This is what I would say to explain: (I would probably have some graph paper cut up into strips of 8 squares (each strip being a whole, each square being 1/8th). )

Because 1 3/4 times 2 1/2 means one and three quarter two and a halfs. Your idea is almost right but not quite.

To make up one and three quarters of a chunk of something that it two and a half big, you could take one two and a half and then add another three quarters of a two and a half. In written math, that would look like this:

(1 x 2 1/2) + (3/4 x 2 1/2)

To figure that out what 3/4 of 2 1/2 is (remember that multiplication means "of"), you would first find out what a quarter of 2 1/2 is, and then take three of them:

(1/4 x 2 1/2) x 3

To figure out what 1/4 of 2 1/2 is, you would first think of the two and a half as five halves:

5/2

Next, you would figure out one quarter of the five halves. To do this, you would figure out what 1/4 of 1/2 is and multiply that by five (take five of them).

(1/4 x 1/2) x 5

In order to split off 1/4 from 1/2, you would first have to divide the half into four pieces. Each of those pieces is going to be an eighth of the whole thing, right? You can probably picture that in your head. (This is the "multiply the denominators and the numerators part".)

1/4 x 1/2 = 1/8

Ok - so putting all those pieces back together again:

1/4 x 1/2 = 1/8

5 x 1/8 = 5/8 (Five of one eighth is five eighths.)

You need three of those. Five eighths plus another five eighths plus another five eighths is fifteen eighths. (This is another example of the multiply the numerators and denominators rule.)

3 x 5/8 = 15/8

And finally, we need to add those fifteen eighths to two and a half. You can't add unlike things, so you need to figure out what two and a half is in eighths. How many eighths are there in two and a half? Each one has eight eighths in it, and a half has four eighths in it, so there are 8+8+4 eighths in two and a half. That means twenty eighths is the same as 2 1/2.

20/8 = 2 1/2

We want to add our twenty eighths together with fifteen eighths to get our answer. That would make thirty five eighths (since twenty of something and fifteen of something added together make thirty five of that something).

20/8 + 15/8 = 35/8

And that is your answer. If you want to know how many wholes are in that 35 eighths, you start grouping the eighths into bundles of eight. You can get four bundles and have three left over. That means you can say the answer a different way: four and three eighths.

As you can see, the whole thing is rather complicated when you break it down into each step. It is no wonder that people find fractions confusing and resort to just memorizing how to do it and ignore why.

Nan

Edited by Nan in Mass

my brain hurts!

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Because doing it the right way gets you the right answer and doing it the wrong way gets the wrong answer. Or "because I said so."

Many have explained it very nicely. :)

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Thanks everybody. I think your explanations helped.

It's fascinating how early curriculums get difficult. I have two masters degrees but can't explain everything in elementary school.

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I have nothing useful to add. I just wanted to say that I am seriously going to need some math review before we get to that point. I'm not afraid to admit it. Disappointed, yes, but not afraid.

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I have nothing useful to add. I just wanted to say that I am seriously going to need some math review before we get to that point. I'm not afraid to admit it. Disappointed, yes, but not afraid.

Dawn, I cannot tell you how much I have learned because I homeschool my kids. I was a straight A student. Always right there at the top of my class. But, I really didn't have a clue what was going on. :lol: As I am learning along with my kids, I have found that I understand the why behind all that work I was just churning out while I was the one in school. I'm afraid that education is wasted on the young. (Like the old saying goes!) We older people seem to be able to enjoy and learn it better.:D

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Ahhhhhhh....I now remember why I always hated math.

:D

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Dawn, I cannot tell you how much I have learned because I homeschool my kids. I was a straight A student. Always right there at the top of my class. But, I really didn't have a clue what was going on. :lol: As I am learning along with my kids, I have found that I understand the why behind all that work I was just churning out while I was the one in school. I'm afraid that education is wasted on the young. (Like the old saying goes!) We older people seem to be able to enjoy and learn it better.:D

Lolly that is so true! I'm finding the same thing - everything clicks now.

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I have nothing useful to add. I just wanted to say that I am seriously going to need some math review before we get to that point. I'm not afraid to admit it. Disappointed, yes, but not afraid.

You'll do fine. See what good help you can get right here on this board? I've been homeschooling 6 years, and I've never come to a point where I couldn't get someone to give DS the explanation he needed. And that includes Latin and Spanish -- neither of which do I know! Sometimes I have to get creative to come up with the right resources but I've always been able to find help when I need it.

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