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Friday, March 20, 2015

Fraction Progress: Referring to the Same Whole

I've been doing some personal professional development about the fraction progression in the Common Core Standards.  

I was thinking about this standard:
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

When I came upon this task at Illustrative Mathematics.
If you were to choose the two pictures that best compares 2/3 and 2/5, which two illustrations would you choose?

Your first reaction would be that students could choose either illustration 3 or 4 to represent 2/3.  That is true, but why must you select illustrations 4 and 5 together?

This question highlights the fact that in order to compare two fractions, they must refer to the same whole. 
Do we spend enough time on the fact that both fractions must refer to the same wholes?
     When we use pre-made worksheets that compare fractions, they usually have a pre-drawn "Whole."  

Using this task from Illustrative Mathematics, in a small group format, could lead to an amazing discussion of the importance of the "same whole."

Here at home, I think I will give my soon-to-be Third Grader 
a cookie like this and ask her to share 1/2 of the cookie
 with me. 

I will promise to share 1/2 of my (unseen ) cookie with her.

My cookie will look like this:
What do you think her reaction will be?  Will our discussion lead to the importance of the "same whole?"
P.S.  If I had a multi-grade 2/3 classroom, would this still be an appropriate lesson, using cookies instead of a worksheet?  What questions would I ask of a 2nd Grader versus a 3rd Grader? 

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