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Part-Part Fractions!

We usually think of fractions as part to whole.

However, fractions can be part to part fractions. Here are a few ideas.

Stating part-to-part relationships as ratios is more common. Part-to-part relationships can be stated (written) as fractions, as well.

From the Common Core State Standards:


3 cups flour to 4 cups of sugar      3 to 4     3:4    (written as ratios)

Written as a fraction it would be 3/4.

Another way to think of it is 3/4 cup of flour per cup of sugar  (only changing one word from 6.RP.2).

My twitter friend posted the following https://twitter.com/mathhombre/status/743624326344220680

It was complaining about the Ministry of Education’s 28-page booklet explaining how to teach fractions.  I do agree that they may have made some poor choices and didn’t explain things as well as perhaps they could have, in the part-part concept of fractions section, but I don’t feel what they write is mathematically incorrect.

The 7/2 is 7/2 non-oranges per orange. It is a force (& a poor choice), but it works. “3.5 non-oranges per orange.”

Part-to-part fractions behave like rates (and it’s no accident that the other example in 6.RP.2 is a rate!). Ironically, the units *are* the same (cups/cups).  I’ve found that when dividing when the units are the same you can let them cancel *or* keep them.  Keeping them may help (because you have some labels).

Finally, I’ll end with one more part-to-part fraction.  The oil-gas mixture (for a chainsaw, for example) is 1 part oil to 50 parts gasoline.  It might be 1 oz to 50 oz. or 1 gal to 50 gal.  (1:50)  As a fraction we get 1/50 ounce of oil per ounce of gas.

The 6-7, Ratios & Proportional Relationships progressions document has many good examples of part-to-part ratios. https://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

“For each,” “for every,” “per”