comparing fractions

Part of our curriculum is to teach our students how to compare fractions.  We focus a great deal on equipping students with strategies for doing this mentally.  They take part in an investigation where they 'discover' these mental strategies.  Though many students dislike the initially struggle, they realize that developing these mental strategies saves them time in the end.  If they just stop and THINK they can often discover which fraction is larger without ever even thinking about 'common denominators.' 

One reality is that sometimes the use of equivalent fractions is necessary and we ask the kids to use common denominators as a last resort. 

This foldable was created with the students AFTER the investigation and BEFORE the weekly quiz.  It was intended to be a summary reference piece.  I created it with the intention of it being used as a FLOWCHART.  If you look closely, students are asked to try one of the mental strategies at the top and then it is suggested that they move to the bottom of the foldable flowchart if one of the mental strategies did not apply.

{all folded up}                     {as you open}

 

{close up of mental strategies-outside}

  {close up of mental strategies-inside}

 

 

{what to do if one of the mental strategies doesn't work}

pi, good ole pi!

Remember when I said earlier that there can be a lot of pressure associated with being the 1st to teach something to the kids. I found the concept of PI to be the perfect example. Pi is a difficult concept for most adults to wrap their heads around, much less students, and 10-12 year old students at that. My PLC partner and I must have grappled with the concept of pi for an hour one day, going back and forth about is pi a rational number since it's the ratio of the circumference of a circle and that circle's diameter, yet when you look at the numbers they appear to be irrational. Then we got into how pi is really the ratio of two measurements, which forces it to always be an approximation because measurements are always approximations. whew. All of this as we were trying to figure out how to present the concept of PI to students who have limited ratio, proportion, geometry and measurement knowledge.

We tried very hard to choreograph the perfect dance of accurate knowledge without completely blowing their minds!

here is the foldable I came up with in an effort to do just that:

{front}

Just like any other math teacher, I am very hesitate to attach a number with pi, I much prefer to refer to it as the ratio of circumference to diameter (there is a hilarious segment of a stand up on pi that my PLC partner sent me that acknowledges the face that math teachers refuse to refer to pi as anything other than the ratio :-) but that's a post for another time). Okay, back to the point...sorry...I wanted to emphasize the ratio, so I typed up the ratio in fraction notation and made the fraction bar really thick so it wouldn't get overlooked. The kids were responsible for folding the paper in half (hot dog style) and then cutting on the thick fraction bar. I fit 4 of these to a page, so it was a real paper saver, for once. Notice, I also put the pi symbol as a watermark on the front. I created this foldable using ActivInspire software as well as Microsoft Word.

{inside}

The ratios on the right of the foldable (535/170 and 526/168) are both data from a Twizzler activity my PLC partner put together which allowed the students to discover the ratio of circumference to diameter of the same circle. I was trying to point out that neither of these are 'perfectly pi' because they are measurements, but they are close to pi. We also discussed as a class that the more accurate your measurements, the closer your number should be to pi.