Wednesday, December 14, 2016

Examples, Counterexamples, and More with Equivalent Fractions

equivalent fractions
As I have been working to build my students' understanding of key fraction concepts, I wanted students to really work to deepen their understanding that fractions can have more than one name--and can be represented in more than one way.

Today's "warm up" was geared toward reminding students that they can represent fractions in many ways...so I gave them 5 minutes to make a "mini poster" (literally 4 inches by 4 inches!) to show as many ways to show 1/2 as possible.  After they worked, we did a little gallery walk.  We came back together and had a discussion about what we saw...different shapes...number lines...fractions of sets...equivalent fractions...
equivalent fractions

equivalent fractions
It was a good warm up to our main lesson which worked to help students derive the "computation" method for finding equivalent fractions. (If you missed my post yesterday about building that understanding, CLICK HERE to read that one!).  After our explorations, it wasn't that big of a stretch for students to recognize that multiplying or dividing the numerator and denominator by the same number generated new fractions that are equivalent.  We proved it with some drawings, some manipulatives, and then moved to bare numbers.

So after working with equivalent fractions for a while, I wanted to put my students to the test to see how WELL they understood the concept!  So often we give students a quick exit slip or something like that--a "fill in the blank" worksheet that follows whatever computation rule we have taught.  If they fill in the blanks correctly, we assume understanding. It just isn't that simple.

To really get students talking, I asked them to do an activity in my big fraction unit....an activity where they need to first generate equivalent fractions (sometimes I do this activity where they can write fractions, draw fractions, etc like the warm up) but today I simply wanted them to use their new algorithm to make a set of 5-7 equivalent fractions for the "unit fraction" I assigned them.  The group below was working to generate a list of fractions equivalent to 1/8.
equivalent fractions
 Here's where the fun (and really deep understanding) kicks in.  The next task is to create ONE more fraction for their poster--that does NOT fit their "team"...in other words, that is NOT equivalent.  I explained that they would be then traveling from team to team to try to find the "mystery" fraction.  I had them write their "counterexample" on the back so students could check their work...and encouraged them to try to be as sneaky as possible when making their "outsider" so other students would really have to work!  Some groups did an AMAZING job...and I heard some great math talk!  This really immersed them in this idea of equivalent fractions and having to think hard about number patterns and the true meaning of equivalence.  We came back together after their gallery walk and discussed their findings--and talked about some of the trickier ones.  It was a ton of fun and a great use of time.  The paper and pencil practice work we did after this was done in a snap--almost all students were really getting it, and those who weren't met with me for a little extra practice.
equivalent fractions

I wrote about this a few years back with a slightly different twist if you want another idea.  Just CLICK HERE for that post!  These ideas are a part of my full fraction unit which you can check out by clicking the image below.  If you are looking for ways to deepen your students' understanding--plus have fun teaching, you might want to check it out!

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