Multiplying fractions with mixed numbers

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apple2357

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When multiplying fractions with mixed numbers, the standard approach is to convert both to improper fractions and then multiply in the usual way. However, how do you explain/convince a 10-year-old the following method ( which some tend to opt for) is wrong, rather than just tell them it doesn't work:

For example:

[math]2 \frac3 5 *4\frac56=8\frac{15}{30}[/math]
 
apple2357 said:
When multiplying fractions with mixed numbers, the standard approach is to convert both to improper fractions and then multiply in the usual way. However, how do you explain/convince a 10-year-old the following method ( which some tend to opt for) is wrong, rather than just tell them it doesn't work:

For example:

[math]2 \frac3 5 *4\frac56=8\frac{15}{30}[/math]
Click to expand...
Well, you can show them that the result is wrong, on a calculator (using decimals)!

Or, you can explain that they are claiming that [math]\left(2+\frac{3}{5}\right) \times \left(4+\frac{5}{6}\right)=2\times4+\frac{3}{5}\times\frac{5}{6}.[/math] If they happen to have seen the distributive property, they would know that's wrong; if not, you can show that other examples don't work, such as [math](1+2)\times(3+4)\ne1\times3+2\times4.[/math]
This shows why what they did doesn't work. And then you might carry out the distribution for each example to show what is missing. (This also shows why using improper fractions is more efficient, though not the only correct method.)

ADDENDUM: If they don't believe in calculators, or want something related to what they've learned about multiplication, you could use an area model, making a rectangle 2 3/5 by 4 5/6 and seeing that the area is bigger than 9. Or use a simpler example, like 3 1/2 by 4 1/5, and actually count up the total area, which illustrates the distributive property. For a demonstration of these ideas, see here (in the last half of the page).
 
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Dr.Peterson said:
Well, you can show them that the result is wrong, on a calculator (using decimals)!

Or, you can explain that they are claiming that [math]\left(2+\frac{3}{5}\right) \times \left(4+\frac{5}{6}\right)=2\times4+\frac{3}{5}\times\frac{5}{6}.[/math] If they happen to have seen the distributive property, they would know that's wrong; if not, you can show that other examples don't work, such as [math](1+2)\times(3+4)\ne1\times3+2\times4.[/math]
This shows why what they did doesn't work. And then you might carry out the distribution for each example to show what is missing. (This also shows why using improper fractions is more efficient, though not the only correct method.)

ADDENDUM: If they don't believe in calculators, or want something related to what they've learned about multiplication, you could use an area model, making a rectangle 2 3/5 by 4 5/6 and seeing that the area is bigger than 9. Or use a simpler example, like 3 1/2 by 4 1/5, and actually count up the total area, which illustrates the distributive property. For a demonstration of these ideas, see here (in the last half of the page).
Click to expand...
Thanks Dr P.

My initial thought was like you to use

[math]\left(2+\frac{3}{5}\right) \times \left(4+\frac{5}{6}\right)=2\times4+\frac{3}{5}\times\frac{5}{6}.[/math]
But i don't think many 10 year olds have been taught how to multiply out terms like this and it often comes about when teaching quadratics much later. Many 10 year olds would have no issue with the above! So i steered clear of that.

I then thought about offering simple counter examples where it fails but to me it doesn't tell them why their case doesn't work. But i like the area idea because that at least gives it some meaning.

I think one of the biggest challenges we have in teaching young children is to stop them from overgeneralising and i think this is an example of that. Maybe the source of the issue is the following...

2.5 which is 2 and 1/2 which is 2+1/2 and not 2*1/2 even though it is written like this?
 
apple2357 said:
But i don't think many 10 year olds have been taught how to multiply out terms like this and it often comes about when teaching quadratics much later. Many 10 year olds would have no issue with the above! So i steered clear of that.
Click to expand...
Yes, that's why I said, "If they happen to have seen the distributive property". They may not have seen the term, but it's entirely possible that they have seen the idea in learning about multiplication. (This is mentioned near the end of the page I referred to, and also, for example, in these U.S. grade 3 standards. It isn't just used in algebra any more!)

So it's worth trying, at least with the whole number example.

apple2357 said:
I think one of the biggest challenges we have in teaching young children is to stop them from overgeneralising and i think this is an example of that. Maybe the source of the issue is the following...

2.5 which is 2 and 1/2 which is 2+1/2 and not 2*1/2 even though it is written like this?
Click to expand...
Yes, I agree that is a big part of this.
 
Yes, that's a good point. When children learn multiplication i have seen some do the 'grid method' which involves partitioning numbers into tens and digits and using a table which is essentially the equivalent of multiplying out using the distributive property.
That can then be used to multiply two mixed numbers. It less efficient but at least its consistent to a previous method
 
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