- Topic: Measurement: Book: Measuring Penny by Leedy.
- Topic: Combinatorics: Given a chain of 5 circles, how many different ways can you color in 0 circles? 1 circle? 2 circles? How can you know if you’ve found them all? Do you notice any patterns? Here is the Powerpoint file for the circle coloring chart.
- Topic: Engineering, Building: Build the tallest tower you can out of 20 Keva blocks.
How did it go?
We had all 6 kids at circle again this week. Overall, it was a really successful circle…Most of the kids were engaged most of the time. All the kids especially enjoyed the building competition.
This is a book about a girl measuring different aspects of her dog Penny and other dogs. She uses lots of different units (inches, centimeters, # of dog biscuits), and dimensions (height, nose length, how high the dog can jump, weight, etc.). The kids really loved this book. It combined story-telling and math in a really nice way.
They especially loved when the book discussed how much money Penny was worth. $1000 as a burglar alarm, $20/day in entertainment, and 1 million dollars in love. Several kids commented afterward about how much they had enjoyed the book.
We did this activity a 1.5 years ago, but I wanted to see if they could go a bit deeper this time.
First I asked how many different ways there are to color in zero circles. All the kids saw there was only one way…not coloring anything. So on the wall chart I wrote 0 at the top, and 1 at the bottom for the number of ways.
Then we moved on to coloring in 1 circle out of 5. Kid #1 and Kid #2 immediately said there were 5 ways. I let Kid #1 color the first row on the wall chart…She colored in the 1st circle. Then I asked Kid #3 if she wanted to do the next, but she wasn’t sure what to do yet, so I asked Kids #4 and #5, but they also didn’t want to answer. At that point, Kid #2 volunteered so I had her go. She colored in the 2nd circle.
I checked Kid #6 paper, and she had a new way, so she colored in the 3rd circle. Then Kid #3 and #5 caught on and added a row with just the 4th circle colored, and another with 5th circle colored. Then the kids said that was all. I asked how they were sure, and they explained that they had colored in each circle, so there was no other way. I said, ok, and wrote 1 (for the number of circles) at the top, and 5 (for the number of ways we found) at the bottom, on the wall chart.
Next was much harder: color in 2 out of 5 circles. The kids started doing this on their own paper, and if they found a new way, they would stand in the line to add it to the wallchart (after I checked it). At this point most of the kids seemed to understand the activity, and were excited to contribute. For the quieter kids, I tried to explicitly check if their paper had a new coloring, since they were less likely to yell for me to check.
After awhile we had 10 ways (which is all of them). I asked the kids if they thought we were done. They weren’t sure. I asked if there was any way to check…did they see any patterns here, the way they did with one circle colored in? Someone said that you could check if the 1st and 3rd were colored, and the 1st and 5th. I said it was a good idea and structured it a bit differently so we checked if we had 1st and 2nd, 1st and 3rd, 1st and 4th, 1st and 5th. Three of the kids definitely understood this.
Next I said we had checked all the ways that had the 1st box colored, so we should check the ways that have the 2nd box colored. The kids then suggested checking 2nd and 3rd, 2nd and 4th, 2nd and 5th. We continued on with this pattern until we verified that we had all possible colorings. So I wrote 2 at the top of the chart, and 10 at the bottom.
Next we colored in 3 circles at a time. By this point, the kids really wanted to add to the wallchart, so they would show me as soon as they had a new coloring. After we got 10 different 3s, I suggested we should check if we had all ways. We tried to do this, but it was tricky to describe having the 1st 2nd 3rd colored in. After a bit of trying, I wrote 3 at the top, and 10 at the bottom.
For coloring in 4 circles, the kids did not immediately see that there were 5 ways to do it. However, they quickly came up with all 5 ways. I asked how we could tell if we were done, and someone said that in each row only one circle was empty, and we had all 5 choices for the empty circle. I wrote 4 at the top, and 5 at the bottom.
For 5 circles, everyone knew there was only one way.
Now I asked if anyone saw any patterns. Kid #1 was really excited to see 1 5 10 10 5 1 for the number of ways to color. I asked the kids why there was 1 way to color 0 and one way to color 5. One kid explained that either you must color all or none. I said that was a good point, it was like 0 and 5 were opposites of each other.
Next I asked about 1 and 4. Are they opposites of each other? Kid #1 and #2 were very excited about this, and pointed out that with 1 circle, there was always one that was colored, while for 4 circles there was always one that was not colored. Then I spent some time asking different kids if they could find the corresponding rows between 1 colored circle and 4 colored circles. All the kids were eventually able to find the “opposite” row in each chart. I saw several nice “A-ha!” moments where a kid really understood how it was working.
Next I said we should check if 2 & 3 circles also had opposites. We found that they are opposites. Every kid wanted a chance for me to point to a row in the 3 circle chart, and them to find the corresponding opposite in the 2 circle chart.
I started by telling the kids that everyone would get a treasure at the end of this activity. There was tons of screaming and yelling and excitement. Then I said they would pick the treasure in the order they finished in the activity. Each kid gets 20 Keva blocks. Whoever builds the tallest tower gets to pick their treasure first.
I counted out the blocks and put each kid in a separate spot on the floor. The kids then built their first tower. 3 of the kids built a tower that had 2 blocks on each side, turned up on their edges. They all ended up 9.5” towers. The other 3 kids built a straight stack of blocks on their flat side. This was a 7” tower.
As soon as I measured, I encouraged the kids to try to build a new, taller tower. After about 15 minutes of building, the winning tower ended up being 29” tall, 2nd place was 24.5”, last place was 7”. It was quite fun making the rounds measuring each kid’s tower. Some kids came up with lots of novel ways to build, and other kids mostly learned strategies by watching.
Then all the kids put back their blocks. I congratulated them on their hard work, and their good sportsmanship (much improved in the last 2 years!), and told them what place each kid got. They really were surprisingly good sports about this activity!
Everyone was ecstatic to get to pick a treasure. They definitely took their time evaluating all the treasure options. (Monetary value ~25 cents 🙂 ).