What were our five stations? I wanted to give students a variety of different materials to use. We had done paper folding on Valentine's Day (I didn't blog about that one--not the most successful lesson--thus this "REDO"!), so I wanted some other visuals. Here they are:
|Fraction Dominos . . . I had two different sets. Students worked to match up the different representations.|
|Smartboard...using "Kidspiration" software|
|Fraction Circles . . . for some students, circles are much harder to visualize!|
|Pattern Blocks. . . more about these later!|
|Fraction Bars...these are "linear" and work really well for lining up equivalent fractions.|
How do you write "four halves"?
Can I use two different pattern blocks to make 1/2?
Do I have to use the yellow pattern block as the "whole"?
How many fractions are equivalent to 1/2?
Is it possible for 2/6 to equal 1 whole? (Wait for this one . . . details down below!)
I also noticed that very few students were considering fractions that were greater than one "whole", so with some gentle nudging and guiding questions, students started thinking outside the box. Here are some of our best examples. Note: If you have an iphone and a projector, math can be AMAZING! I walked around snapping photos of cool discoveries, emailed them to myself, and then projected them. We interacted with them on the Smartboard. . . . kids love it!
Check these out!
|Here is his work on the Smartboard . . . we learned that 1/2 of 24 is 12, that 12/24 is the same as 1/2--and many other equivalent fractions, all with his newly formed "whole" shape!|
|This struggling math student TOTALLY got in the zone on Friday! You could SEE the lightbulb go on! She found all these different ways to make 1 1/2 . . .|
|She was even willing to come up to the Smartboard with me to help explain what she had done!|
But here was my favorite . . . we STILL don't have a clear answer! See if you can follow this student's logic.
This student posed the following question:
"If I cut the extra green pattern block in half, I could fill in the rest of the space on the orange square. I know that one green triangle is 1/6, so using two of them would be 2/6. Is 2/6 the same as one whole square?"
I paused to let myself chew on this one a little before I said something I didn't want to say . . . (I have to do this often, so you know!) I asked her what she thought. She nodded and said that she thought so. I sent her back to her team and they discussed and debated and modeled and drew and then came to me and announced that they were split. So, when all our rotations were done, we posed the question to the class.
The students had some interesting insight--and several of them were able to explain their thinking very clearly. Sadly, not all the students were able to grasp it yet. The turning point came when one of my most QUIET students raised his hand and said . . . I love this . . .
"I'm pretty sure this is like the strawberry and grape problem."
Silence. Whispers. And then . . .
"OH YEAH! The wholes are different shapes and sizes!"
I'm still not convinced that all my students "got it", but I know more are on their way. I'm so glad I didn't brush off the student who came to me with the original question. It would have been SO easy to tell her that you can't compare the "whole" square with a "whole" hexagon. It really needed to come from their own understanding. This "constructivist" approach is really far more meaningful than our attempt to impart our own knowledge! This, coupled with the Common Core's high stress on the use of math language and critical thinking, makes me feel better about how long the unit is taking--I think the next few lessons will go so much faster because of the true understanding my students have developed over the last few weeks.
So--next week? We are going to look at the number patterns involved with equivalent fractions and begin adding and subtracting with like denominators as required by the Common Core. Just a reminder that I WILL be making this unit into a "for purchase" product, but if you want to try some now, consider checking out my fraction sequencing activities.