Although, this is the performance standard (a very simple standard at first glance) that students need to be able to do, my mathematical goal was different. Further down (like about 3 pages later) in the CA math framework, I find this beauty of a mathematical goal: (Mathematical goal meaning - What is it about math you want students to understand?)
"Students need experiences with different-sized circles and rectangles to recognize that when they cut something into two equal pieces, each piece will equal one half of its original whole (1.G.3). Children should recognize that the halves of two different wholes are not necessarily the same size. They should also reason that decomposing equal shares into more equal shares results in smaller equal shares." Van de Walle also chimes in about early fraction work and said, "Given children's experiences with fairly sharing items among family and friends, sharing tasks are a good place to begin the development of fractions. The sharing tasks allow children to develop concepts of fractions from an activity that makes sense to them, rather than having the structure imposed on them." (Teaching Student-Centered Mathematics K-2)
|Yes, I have a cookie crumb on my lip!|
Kids kept asking me why I was wearing an apron. I told them I had been up all night baking cookies, which was true. So, I called two students up to the front of the room to be my helpers. I told the kids I had two halves of cookies. I asked them if it would be fair if I gave them each half of a cookie. The cookies are hidden in a paper bag. The class all shouts out, "Yes!" So I give half of a small cookie to Amore and half of a huge cookie to Shawn.
Don't worry, this is only the launch of the lesson. I know from listening to upper grade teachers and reading Van de Walle that students have a difficult time understanding fractions as numbers and not only parts of a whole. So I stole an idea that Andrew Stadel shared and used it in a different way. (BTW - this is an example of how elementary, even primary, can steal ideas from middle and high school teachers!) I put zero and one on a string number line and asked the kids where half would go. They partner talked their disagreements, but in the end it came out that half should be equidistant between zero and one.
Time for a little estimation...
I held up one small cookie and one extra large cookie and asked the kids to choose a just right brave estimate for the question, "How many small cookies equal a large cookie?" Students wrote their estimations on their desk with a marker.
Then we laid out small cookies on the large cookie to see how many it would take. It turned out to be 10 so we added 2 - 10 to the existing number line.
Then I strung up another number line directly above it and hung zero up above the other zero. This number line represents the large cookies.
"Boys and Girls, if the yellow number line represents the small cookies and the pink represents the large cookies, where will the one go on the large cookie number line?"
Kids whispered to each other. Brave ones called out, "Above the 10!"
Me, "How do you know?"
Kids, "Because it takes ten cookies to equal one big cookie."
Hmm "Well, if that's true then where would one half of a large cookie go on the number line?"
Kids, "Above the 5! Half way between the zero and the ten!"
Me, "How do you know?"
Kids, "Because 5 is half of ten. 5+5 is 10."
Which lead to this idea:
Kids disagreed on this one. Some thought between half and one, but after some kids convincing each other, and a demonstration of breaking half into half they came to consensus that half of half is half way between zero and one half. Kids created some examples and non examples of half and divided half into half again.
The next goal will be: "They should also reason that decomposing equal shares into more equal shares results in smaller equal shares."
Thinking - What is the pattern and/or relationship between the number of parts and the size of the parts. We will continue to work on equal sharing tasks as Van de Walle suggested.
I'm wondering how shallow my student's understanding would be if I didn't read the whole framework, and just glanced at the standard.
This lesson planted seeds for fractional reasoning, proportional reasoning and algebraic understanding. I'm sure I'll continue to chip away at this lesson, but I think we are off to a great start.
P.S. Of course, the kids had to have cookies!
Thanks Matt Vaudry and John Stevens for the fun inspiration!