Smaller Than A Number

This post was supposed to happen after the school year was over.  At the end of the year, we spent the last 3 weeks talking about fractions, my favorite topic in grade 3 curriculum. I knew I needed something to keep me going in June.  Then the summer started. My memories are hazy now, but I’ll give it a try. Here are the highlights in (to my best recollection) chronological order.

The Cookie Fiasco

The Cookie Fiasco is a wonderful funny book in which four friends are trying to share three cookies. We read the first half, stopped and I asked my students to help the friends with the cookies. Apparently, they could get 1/2 and 1/4 or 3/4. Some students noticed that these must be the same. I noticed that while everyone was successful helping the friends, some students were calling any equal parts halves.

All About Fractions

After my students agreed that the friends had to use fractions to share their cookies fairly, I asked them to tell me all they know about fractions.

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Some students also added their wonders.

We spent some time working on notation afterwards. And I asked my students to brainstorm what numbers are to see if fractions fit in. I don’t think everyone was convinced though. Yes, you can count them and they fit on the number line. But they are smaller than numbers. I am still unsure how I should have continued the conversation about fraction being numbers, or parts of numbers, or some new weird numberland creatures.

Number Talks Images and WODB

This image from Number Talks Images website appeared to be a great prompt, intriguing and accessible. My favorite solution: put all halves together, cut one corner watermelon into three quarters, and give these quarters to the other watermelons.

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Which One Does Not Belong helped to focus once again on the idea that the same fraction of the same whole can have different shapes.

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From http://www.fractiontalks.com/2016/11/mtbos-connected-fraction-talks-and-wodb.html

Half of a Square

I love the paper folding challenge from youcubed website, and we did it this year again. I simplified it a bit. I asked my students to make a triangle that is half of the original square, a quarter of it, then make a square that is a half and prove it to their classmates and to me. “These 2 squares fit on top of each other, they are the same.” I tried to talk about Plato but the audience didn’t seem excited.

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The Sad Bunny Story

Once upon a time there lived a lazy bunny who went to visit his friend owl. They had a sleepover, and in the morning lazy bunny was very lazy to go home. The owl suggested that the bunny shouldn’t push himself too hard. He should go half the way on the first day, half of the remaining way on the second day, then half of the remaining way again. How many days will it take for the bunny to make it home?

I don’t remember where I got this story but the scenario worked well for eight-year-olds. We modeled the first couple of days on the clothesline and did the initial estimation.

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I offered meter sticks, strips of paper, poster paper, and then in a stroke of insight I taped a couple of number lines on the floor. We ended up getting together and doing a proving part on the large number line. “Three days? Can you show me on this line?”

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bunny extra

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My stories about Zeno were met with much more enthusiasm. Especially the Arrow paradox. My students did a bit of experimenting with throwing non-sharp objects to prove to me that they do indeed hit the target. “The Paradox!”

Clothesline Fractions

Building on our work with the fractions on the number line, I asked my students to put these fractions in order in their table groups, and then we discussed the order and found a correct place for each one. One half and two quarters caused a debate. “Can you have two numbers at the same place on a number line? Is it the same number or different?”

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Quarter the Cross

I was really excited to try this challenge in my classroom. For the first lesson, I followed the lesson outline that David Butler described in his blog. I gave out the templates with the crosses, my students sketched down some ideas, did the gallery walk, shared a few with the class, and then everyone had another try.

 

I’ve decided to end the year with the quick art project. Quarter the cross. Same fraction. Different shape. Two colors.

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Questions and Thoughts

I wonder at which point fractions stop being a wonderful new mathematical concept and start being frustrating and challenging. Is it when students need to start adding, subtracting, multiplying and dividing them? Is there anything I can do to prepare them for this? What intuitions would be helpful?

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