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Teaching fractions - need some help!


ondreeuh
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My son is in 6th grade and is working on a chapter on ratios, proportions, and percents. This has mucked up his understanding of fractions. Previously, we had always talked about fractions being part of a whole. 4/7 meant that one whole unit was divided into seven pieces, and we are using four of those pieces. He understood how to multiply and divide fractions, and how to do cross multiplication to solve for a missing value. He was on the right track.

 

Now with ratios & proportions, we are talking about the fraction representing a relationship between two numbers. So these fractions he can't simplify to a single number - he understands this, but forgets because it has become automatic to simplify 3/1 to 3.  Even though we are no longer talking about parts of a whole, we can do cross multiplication to solve for a missing value. This is getting kind of abstract to him (a fraction that acts like his previous understanding of a fraction in one way but not another), but he is getting good at setting up proportions in similar figures and solving for the missing value.

 

Now percents are completely confusing him. He totally understands that percent means per hundred, and can convert between decimals and percents. He can convert easy fractions like 1/2 = 50/100 = 50%. As long as he can make it an equivalent fraction of 100, he is fine. But no matter how many times we go over it, he cannot look at a fraction like 4/7 (that can't be set up as a simple conversion to x/100) and remember to divide it in order to get to a percent. He sees 4/7 as "four out of seven," or MAYBE "the ratio of four to seven," but DEFINITELY not "four divided by seven will get me a decimal which I can convert to a percent."

 

I think the sticking point is that when we are dividing 4/7, we're not talking about parts-to-whole in the same way we were before. Previously, we were talking about 1 whole pizza divided into 7 slices, and then Bob ate 4 of those slices. Now we're talking about 4 pizzas divided between seven people, and Bob eats 4/7 of a pizza. It's a conceptual jump that he isn't making.

 

We have done a lot of balancing simple equations, and we write division as a fraction ( so 2x divided by x is written 2x/x) but his brain isn't connecting the two things.

 

Any suggestions on where to go from here?

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I'm not sure if this helps at all, but I like to think about the conversion of fractions to percentages this way:

 

To convert the fraction 4/7 to a percent, think about your fraction as representing 4/7 of "the whole thing" - or 100 percent.

 

So, written mathematically, this would be 4/7 x 100% = 400/7 = 57.14%

 

This allows you to avoid going through the step of making a decimal and then multiplying by 100 to get back to a percent; he'll eventually want to get comfortable with that process, but this strategy should help him see that with percents, you think of your fraction as some portion of 100 (as opposed to some portion of 1).

 

 

(Or, put another way, 4/7 of 1 would be 0.5714 while 4/7 of 100 would be 57.14....)

 
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I'm not sure if this helps at all, but I like to think about the conversion of fractions to percentages this way:

 

To convert the fraction 4/7 to a percent, think about your fraction as representing 4/7 of "the whole thing" - or 100 percent.

 

So, written mathematically, this would be 4/7 x 100% = 400/7 = 57.14%

 

This allows you to avoid going through the step of making a decimal and then multiplying by 100 to get back to a percent; he'll eventually want to get comfortable with that process, but this strategy should help him see that with percents, you think of your fraction as some portion of 100 (as opposed to some portion of 1).

 

 

(Or, put another way, 4/7 of 1 would be 0.5714 while 4/7 of 100 would be 57.14....)

IIRC, that's how Saxon teaches it.

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OK, I tried explaining it to him. He still doesn't "get" why you divide the numerator by the denominator, though.

 

It got to the point that I was asking "Is this a problem you keep having? Is there a phrase I keep saying? Maybe the phrase I keep saying is relating to the problem you keep having?"

 

"'A fraction is a division problem.' So I divide?"

 

Yes.

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OK, I tried explaining it to him. He still doesn't "get" why you divide the numerator by the denominator, though.

 

It got to the point that I was asking "Is this a problem you keep having? Is there a phrase I keep saying? Maybe the phrase I keep saying is relating to the problem you keep having?"

 

"'A fraction is a division problem.' So I divide?"

 

Yes.

 

How about breaking it down even further and making it pictorial so he can really see why he's dividing his fractions:

 

Draw a rectangle - the whole thing is 100%.

 

Divide it into seven equal pieces - explain that each of these pieces represents 1/7 of the 100% or 100 "divided by" 7 - have him divide 100 by 7 to get the value of each piece (14.3).

 

Now, shade in 4 of those pieces - he can see that 4 of the pieces will add up to 57.2% (14.3 x 4).

 

Finally, show that you can directly calculate 4/7 of 100 in one step (100/7 x 4) -- conceptually, you are taking 100, dividing it into 7 pieces, and then multiplying it by 4. (For these types of problems, when he sees "/" he should say "divided by").

 

Have him continue to go stepwise through pictorial representations of fractions/percents until it makes sense intuitively.

 

(Also, if this idea of using division to calculate fractions is the real hang-up, then you don't need to start with percents; he can start by calculating fractions of any number, then move on to percents once he's mastered that).

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