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Much Ado about Watermelons

Day 2 of #summermathcamp

To feed our math brains at 8 am on a Tuesday morning in the summer, we showed this photo of some watermelons in the hopes of generating some conversation.

How many watermelons are there? How do you know?

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http://ntimages.weebly.com/

 

This was just going to be 10 to 15 minutes of a notice and wonder conversation. Yea. We were wrong. Fifty minutes later we were still chatting about watermelons. Who knew a pile of cut up watermelons could keep 45 educators engrossed. Really, what is there to talk about—it’s just a bunch of watermelons, people.

Car full of watermelons

Image from: weknowmemes.com

 

So, here’s what we chatted about.

We predicted that the most common answer would be to cut up two of the one-half sized pieces to make the missing one-fourth pieces, slide those around to make 4 whole watermelons. Then, add the other two halves to make another whole watermelon. So, the sum of 4 whole watermelons and the other whole make 5 watermelons. We thought that some version of that idea would be a good start to the day.

And that’s exactly what happened. One of the campers shared her version and just about every person in the room said, “Yup, I thought about it that way, too.”

answer #1

 

And then Shannon said, “I saw it another way. I saw 4 groups of 3/4 of a watermelon and then I added the 4 one-half sized pieces. So, 3 whole watermelons and two more means that there are 5 watermelons in the picture.” The conversation then moved to connecting Shannon’s use of the algorithm to the photo and then adding in the notation.

answer #2

Since many of the campers were happy to think about the quantity of watermelons using the algorithm in Shannon’s explanation (4 x 3/4), we thought we would pause here and push on the idea of equivalent representations. [We much preferred thinking of the watermelons as 3 groups of 1—written as 3 x 4/4.]

This provoked lots of conversation on how come we can do this.

  • How does the image show (4 x 3/4) = (3 x 4/4)?
  • How does the picture show Shannon’s equation: (4 x 3/4) + (4 x 1/2) = 5?

 

The third way that surfaced was to simply add all the pieces of the watermelons visible in the photo. We recorded it like this: ¾ + ½ + ¾ + ½ +¾ + ½ + ¾ + ½

answer #3

 

Here are some of our takeaways.

If it’s important to the kids it MUST be important to us

  • We need to listen to what our kids are saying and not be on the lookout for the answer listed in the TE or for the student response that matches our preferred strategy.

Questions and their power

  • Your questions need to offer ALL kids a place in the conversation. So, in this situation we could have asked a couple of questions.
    • How many watermelons are there?
    • How many whole watermelons are there?

           Which question lets the kid who says there are 8 watermelons be in the conversation?

Dots on ten frames and Photos of watermelons, almonds, and tangram puzzles

  • Math is an active subject—it’s interesting, irritating, perplexing, confusing and invigorating. It makes your head hurt when you are in the midst of the struggle and then you get to embrace the high fives when that last piece falls into place and the connection appears as a result of your hard work.

 

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Reflections on Summer Fun

The last few days spent in classes have been lovely. I knew they were going to be. Spending time in classes with kiddos and their teachers trying out some ideas that were new to my colleagues, and then chatting about what we saw and heard, and what we learned was just wonderful.

1.  Kids (and their teachers) are super interested in containers filled with stuff. It is nearly impossible to resist joining in the conversation about “how much stuff” is in the container, especially when the stuff is the lovely orange cheese balls that you can buy in the mega-container at Target.

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It offers kiddos who really, really, really struggle with math anxiety a way to join in the conversation. Everyone made guesses and built number lines and laughed and shared ideas.IMG_2161

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These are the sights and sounds of what learning should be for our students.

2.  Games provide an awesome arena for kids to explore numbers and their structure, especially the game we shared on Monday morning, What Fits Between?  We taught them how to play the game. They had to figure out how the three numbers created are related (largest number, smallest number, and the number that fits between) and organize the results.

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Then, we put the students into groups, distributed the record sheets and the sets of UNO cards, and they were off and running. We listened in on their conversations and watched to see what decisions they made about their numbers. Occasionally, we answered questions about situations that arose in their games that hadn’t in the few rounds we modeled. There were lots of conversations: kiddos asking each other for help, doing the if I had made 48 with my cards instead of 84, my number would have been in the middle and I would have won, and deciding what to do if two of the three players in a group made the same number. (They gave each player one point for the round because no one had created a number that fit between the other two.)

  • We learned that if we leave them alone to figure things out, they do. The adage, be less helpful, comes to mind.
  • Kids are able to explore ideas more deeply when they engage in conversation with classmates than when they engage, individually, with worksheets.
  • It’s great fun to have a group of teachers in a classroom when you want to try out new things especially when you are able do some planning together before hand.
  • It’s really nice to have a couple of days to work on new ideas and have time for people to talk together on each of the days.
  • Kids are amazing. If we choose the right tasks, they aren’t even nervous and anxious about the math part of their day.

3.  Using games as assessment is an informative practice. Notes and photos and conversation bring the next instructional steps to light.

  • Of the 12 kiddos playing, nine were ready to move onto playing the next level of the game where each player must build a three-digit number and determine who had created the number this fits between the other two. Three students needed additional opportunities to play the original version of the game.
  • The “top” math student got to play and interact with the ideas just like everyone else, as did the kiddos who are often not interested in math. All 12 students were invested in the game. The students who usually need to get a drink, sharpen a pencil, or use any number of distractions to help them get through math did not need to employ any of those strategies.
  • During the reflection time at the end of the class period the kids decided that they wanted to change the criteria that determined who won the game. Now, in their class, the winner is determined after each player counts up the number of rounds s/he won. The player who has the number of wins that is between the other two, wins.

How do we use the learning we saw today to support tomorrow’s instruction?

The collaborative work started when we noticed that kids were having a difficult time estimating height in an estimation warm-up. We decided to address the manner in which they worked with the concept of estimation. We selected a graphic organizer for them to use, determined that the quantity of objects for them to estimate would be less than 100, provided some boundaries to the quantity, and put the cheese balls in clear, plastic containers so the students could see them. With these adjustments the students were more comfortable in making estimates and discussing their ideas. After the team talked about the work the students did that day, it was decided our next move would be to use a series of photographs from Estimation180.com. We selected the group that asks kids to estimate the number of cheese balls on plates and trays. The students will play What Fits Between? and do some other math work.

I am looking forward to hearing how the day went, and what the team is going to tackle next.