Key to Fractions workbook

[ChrisB]

If you have worked through the Key to Series and/or just the Fractions workbook, can you tell me if it covers in detail why, when dividing a fraction by a fraction, the quotient is larger?  Not just the process of dividing fractions, but the WHY behind it.  Thanks!

[AK_Mom4]

Do you mean something like .... I have 8 slices of pizza.  If I divide each piece in half, I end up with 16  slices?  The pieces are smaller (1/2 slices), but there are more of them

 

No, I don't think that Key to Fractions explains why that works.  DD12 went thru book 1-4 and it was mostly just explanations on HOW to do the work.

 

Wheat really set fraction multiplication and division into the brain for DD was cookie math.  We broke cookies in half, broke the 1/2-cookies into thirds, etc.  She could see (and taste!) that the more we divided by a fraction, the more pieces of cookie we got.  I suppose you could do it with paper cutouts too.

 

[ChrisB]

No.  I mean actually dividing a fraction (not a whole number) by a fraction.  3/4 divided by 1/2 for instance.  I'm trying to find something that has a good "handle" on the WHY, not just the mechanics of multiplying by the reciprocal.  Hope that makes sense.  

 

That's good to know about Key to Fractions.  Thanks!

[AK_Mom4]

I'm not sure if you have seen this yet - it has a pretty good explanation of why division works for fractions.

 

http://www.mathsisfun.com/fractions_division.html

 

[ChrisB]

Nice!  That resource gives a nice overview.  Thanks!

[Momling]

I teach my kids to think of division like 12 divided by 3 as asking 'how many 3s are in 12?' It's easy then to say 3/4 divided by 1/2 is 'how many 1/2s in 3/4?'

Keys to fractions is very procedural and not very conceptual. It's good practice, but Math mammoth does a better job if you are looking to supplement.

[Farrar]

I have them and it's in the second book toward the end, but there's not much on the why end. I like the way they're laid out and I feel like, at this level, I can explain all the whys just fine.

[kiana]

Here are some visuals I like for fractions:

 

Cutting fabric. If you have 9 feet of ribbon and you want it cut into 6-inch (half a foot) pieces, how many pieces do you get? This is especially helpful for when you're working with mixed numbers.

Pizzas: If you cut a pizza into 6 pieces, each piece is 1/6 of the pizza - if you cut each of those pieces, each piece is 1/12 of the pizza. Pizzas are especially helpful for my developmental math students.

[nansk]

Pizzas: If you cut a pizza into 6 pieces, each piece is 1/6 of the pizza - if you cut each of those pieces, each piece is 1/12 of the pizza. Pizzas are especially helpful for my developmental math students.

Sorry, but doesn't this show one-half of one-sixth? (i.e. 1/2 X 1/6)

 

I prefer the other example in this thread: how many one-sixths will fit in a one-half?

[kiana]

Sorry, but doesn't this show one-half of one-sixth? (i.e. 1/2 X 1/6)

 

I prefer the other example in this thread: how many one-sixths will fit in a one-half?

 

Yes it does. They were different examples of things I like to use when teaching fractions.

 

Pizzas are great for when you're dividing fractions by whole numbers, because people can visualize the pieces being cut into smaller pieces. The pizza I was giving was supposed to be an example for 1/6 divided by 2, which, of course, is the same thing as 1/6 times 1/2.

 

Ribbons are great for when you're dividing fractions by fractions. For many people it is much easier to think about a physical strip of fabric when you are cutting pieces 1/6 of a foot long and how many pieces you can get from a 1/2 foot ribbon than it is to just think about sixths into halves.