# Sorting and Unsorting (Age 7)

## The Activities

1. Topic: Units. Book: Dinosaur Deals by Murphy. In this book, a boy wants to get a T-Rex trading card. He finds a girl who will trade it for 3 Allosaurus cards, but the boy only has 1 Allosaurus. How can he get the T-Rex.

2. Topic: Sorting, Teamwork. We repeated the sorting activity from a few weeks ago, sorting the cards 1 – 104. This time we started by discussing possible strategies based on what went well last week. Then I timed the kids to see how fast their sort was, so we can try to get faster in the future.

3. Topic: Programming. We played the game Robot Turtles, with a few rules changes to make it more cooperative (in past circles some kids have gotten upset if their turtle falls behind):

• All turtles are trying to get to the same jewel.
• Turtles can walk on top of each other.
• Each person gets only one ‘laser’ card, so sometimes you have to work together to rescue a friend trapped behind ice blocks.

## How did it go?

#### Sorting

First we discussed what worked well last time, and what strategies we could use this time. One kid said that sorting goes slowly when people hold cards in their hands (they spend a lot of time rummaging through the cards), so someone proposed laying the cards out on the ground so everyone can see.  Then another kid suggested sorting the cards into groups of 10 at the start (1-9, 10-19, etc), then doing the big sort. I made labels for each group of ten cards, and put them around the table. Then I gave each kid 1/4th of the deck, and started the timer.

Sorting into decades went very smoothly, everyone was working together, and in parallel. There were some mistakes, for example someone misread 72 as 27, but overall progress was quick.

Sorting into piles of 10

Next, two of the kids took the 1-9 pile and started sorting it into the final spot on the ground. The other two kids picked up some random piles. One kid laid out the 100s, 70s, and 90s on the ground, but ended up mixing them all together. The other kid took just the forties, and laid them out in order 40 -49. Then he picked up the 50s and laid them out in order under that 40s.

Sorting the 40s and 50s (soon to be undone by a friend)

Meanwhile the other two kids got up to the 40s. One of them came over and scooped up the row of 40s, completely mixing them up, and then resorted the cards into the final positions. This wasn’t too too slow, but seemed suboptimal 🙂

The sorting really slowed down once we got to the mixed up 80s, 90s, and 100s. Three kids had a bunch of random cards in their hands, and one kid was distractedly counting the already sorted cards.  I pointed out that it seemed slow, and that people were holding cards, so they laid the cards down and eventually finished.

The final time was 16:30, which is not terrible, but definitely can be improved.

Afterward, I asked the kids what went well, and what could have gone better. One kid said it would be better if I didn’t take out some cards. At the start of the activity I randomly pulled out 7 cards from the deck, and told the kids. In practice this speeds up the sort a lot, because they don’t get stuck trying to find one card forever, and just move on, assuming that it must be one of the removed cards.

In this discussion, I demonstrated how one kid had sorted the 40s, and then the work was lost when the friend scooped them up. The kids then suggested picking the cards up in order. We tried this, and found that it was indeed quicker to lay down a sorted pile of ten cards than a shuffled one.

I also pointed out that the beginning was really fast because everyone was able to help at once, but no one had any strong ideas about how to make the full search parallelizable.

#### Robot Turtles

Most of the kids had played this before. Some groaned for some reason, when they saw it, but everyone seemed excited. There was a bit of extra energy left over from sorting, so this was a wild 10 minutes, but we did finish a couple puzzles. The tricky parts were that kids wanted to move their turtles while laying down their programming cards, and also, they would mix up the two turning directions without noticing. But overall they were much better at this than I expected. Their favorite part was using the lasers to rescue their friends.

# Odds & Ends (Age 7)

## The Activities

1. Topic: Probability: Book: A Very Improbable Story by E. Einhorn.
2. Topic: Probability:  First, I secretly put 2 red and 8 blue stones into a small drawstring bag.  Each kid took turns pulling one stone out, looking at it, and then putting it back.  The question was, are there more reds or blues?  I repeated it with 4 red / 6 blue, and also 5 red / 5 blue.  Finally, I made two bags, one with 10 red / 10 blue, and the other with 11 red / 9 blue, divided the kids into two teams, and asked them to figure out which bag had more reds.  I gave the kids paper and pencil and they decided to make charts to keep track of the results.
3. Topics: Numbers, Sorting:  I had about 20 different numbers on squares of paper, 0, 1, 3, 4, 6, 8, 12, 13, 100, 105, 1001, 1052, 1053, 1000000, -5, and -100.  First, I handed each kid one number and asked them to sort themselves.  We did this several times, starting simple and then using some of the trickier numbers.  Then, instead of handing them the numbers, I taped a number to each kids’ back, and without telling each other what the numbers were, they needed to sort themselves.  We did this a few times as well.
4. Topics: Tangrams, Geometry:  I gave each kid six different tangram puzzles.  For the kids who finished earlier, I had them work on the letter “A” from Tangrams: 330 Puzzles.

## How Did It Go?

We had four kids this week.  It was a good circle, a few of the kids got a little antsy when we were discussing the results of the bag counting, but otherwise they were all engaged the whole time.

#### A Very Improbable Story

The kids liked the cat on the head :).

#### Probability Bag

The kids immediately grasped the idea of looking for whichever color came out more often.  Not surprisingly, they were overconfident — once, after only 3 draws one kid concluded red was the winner and dumped out the bag, only to find out that there were 5 of each.

For the team activity, one of the teams delegated one person to pull the stones and the other to record, while the other was taking turns drawing out stones.  The former strategy was about 2x faster, so I suggested the other team use it as well.  It was very interesting to see the two charts (pictured above).  One was a standard tally chart, except with 6 instead of 5 in each group.  For the other, the kid started by writing a bunch of numbers, and then checking them off as stones were pulled out of the bag.  The results came out pretty nicely — exactly 50% for the 10/10 bag, and 55.6% for the 11/9 bag (expected 55%).  However, the kids were a bit confused by the fact that team 1 had counts of 15 red and 15 blue vs. 30 red and 24 blue for team 2 — at one point, one kid concluded that team 2 had more reds AND blues.  In fact, the only way I got them to conclude that team 2 had more reds was to ask them to guess what was in each bag.  Their guess for team 1 was 10/10, while their guess for team 2 was “6 more reds than blues” (not coincidentally, they had drawn red out 6 times more than blue).  I asked them how many reds there would be if there were 6 more reds than blues, and 20 total — this was actually quite hard for them and I had to help them a lot (the initial guess, 16, didn’t work).  Of course, 13/7 doesn’t match their observed results.  So, there’s clearly a lot more them to learn for the fine shades of probability!

#### Number Sorting

This activity was pretty easy for them, even with the numbers taped to their backs.  They had a lot of fun, particularly when I gave them negative numbers or really big numbers.  They did a great job not telling each other — the closest they came was saying one kid’s number was really low (when it was -100).

#### Tangrams

This group has done these puzzles before, but that wasn’t an issue, they didn’t remember the solutions.  They were better than last time, but the puzzles still definitely weren’t trivial.  The bonus puzzle is much harder because it wasn’t to scale, but they made a good effort and made progress.

# The Case of the Missing Blink Cards (Age 8)

## The Activities

1. Topic: Measurement. Book: Measuring Penny by Leedy. Lisa gets a homework assignment to measure something in a variety of ways. She decides to measure her dog Penny.
2. Topic: Sorting, Patterns, Charts. We have a card game called Blink, which has cards of six different colors, with six different shapes, and five different numbers (1 – 5). Here are some sample cards.

We calculated that there should be 6 * 6 * 5 = 180 unique cards. However, the Blink deck only contains 60 cards.  I asked the kids to figure out which cards are missing, and if there’s any pattern.

3. Topic: Protractors, Measurement, Triangles. Each kid got a protractor, and  triangle I had drawn before circle. We measured each angle, and then added them up to see what we got.

## How did it go?

We had 4 kids this week. Overall, it was a fun, focused circle.

#### Measuring Penny

We had read this book over a year ago. Some kids remembered it, but 3 out of 4 kids wanted to hear it again. This time I took several different breaks to discuss the book. For example, when they talked about ‘nonstandard units’, I measured one of the kids’ hair in number of “Corey Hands”. We also measured everyone’s ears in centimeter. The kids had a great time with the book, and stayed focused and interested.

We had told the kids about the missing cards several weeks ago, and they all remembered that there are supposed be 180 cards.  We recalculated it again, just to be sure.  Then I asked if we could figure out which cards are missing?  I started them off by sorting through the cards and showing that there were only two cards that had red lightning bolts on them.

The kids took over from there. At first they just randomly picked a color and shape, and looked for the matching cards. Soon this became unmanageable, so one of the kids suggested moving to the floor, and making a separate row for each color, and use columns for the shapes. This resulted in the following chart:

I then asked the kids several questions about the cards, which were easy to answer with this chart. How many of each color are there? 10.  How many of each shape? 10.

How many of each number? This one was trickier because the chart is not sorted by number. One kid wanted to rearrange the chart, but instead we went row by row looking for each number. We found there are 12 of each. We also found that for each color, there are two of each number.  During this time, two kids counted the attributes, and two kids were keeping notes.

One kid’s notes.

Next I asked how many cards are missing from each row? The kids looked at their chart, and said two are missing from each row. We then calculated there must be 12 missing cards total. But that would make only 12 + 60 = 72 cards, not 180, like we calculated.

I should have asked the kids where the other cards were, but instead I just showed them how to update the chart to sort by number too.  So we had a row for each color, and a column for each number/shape combination. Two kids helped me fix the chart:

The new chart.

Then we counted the missing cards in each row. This time we found there are 20 cards missing in each. 20 * 6 = 120 + 60 = 180!

#### Triangles and Protractors

I handed out several big triangles I had drawn with sharpie before circle. The kids used protractors to measure the three angles, and then add them up.

Using the protractor was still challenging for the kids, but they all made progress when I helped them. The kids added angles up to 180 four or five separate times. We also got 182  and 183 a couple times. Weirdly, when I did it myself, I got 188 for a triangle…Not sure why.  But the kids actually noticed that it was near 180 all the time, so we may be almost ready for the proof that they must always be 180.

My daughter measuring a triangle.

# The Parallelogram and the Pendulum (Age 8)

## The Activities

1. Topics: Logic:  Still More Stories to Solve, by G. Shannon.  We read and discussed the first two stories.
2. Topics: Spatial Reasoning, Tangrams:  We did the same set of tangrams from a few weeks ago (letters, numbers, and things from Cinderella).
3. Topics: Physics, Experiments:  Inspired by the Galileo chapter of Mathematicians are People Too from a few weeks ago, I hung a makeshift pendulum from the ceiling — a roll of tape suspended from an 8′ thread, hanging from a sticky hook attached to the ceiling.  I had pre-marked the 2′, 4′, 6′, and 8′ points away from the center of the roll of tape.  We released the pendulum twice at each length, varying the height that we released it at, and timed how long it took to go 20 swings.
4. Topics: Sorting, Patterns:  We have a card game called Blink — basically a racing version of Uno.  Each card has some number of symbols 1-5, one of six colors, and one of six shapes.  There are 180 possible combinations, but only 60 cards in the deck.  After we figured out there should be 180, I asked the kids to find out which ones are missing.

## How Did It Go?

We sat on the floor this week to make room for the pendulum; this tends to make them a little crazier since they can easily roll around on the floor.

#### Still More Stories to Solve

I wasn’t crazy about the first puzzle, but the second one, about two brothers having a contest to see whose horse would get somewhere LAST, was nice.  The kids figured it out with some hints.

#### Tangrams

Corey and I discovered that I’m better at Tangrams than she is :), so unlike last time, where Corey AND the kids were stuck I was able to help them solve the puzzles.  The main thing I tried to teach them was to figure out where the big triangles go first; it’ll be interesting to see if next time we do Tangrams they remember this.

#### Timing a Pendulum

As you can see from the chart above, we had really reproducible results.  I believe we were actually only counting 19 swings (we started on 1 as we let go and then stopped when we said 20, when we should have let it swing again).  Anyway, I had incorrectly remembered from physics long ago that the time was linearly proportional to the length of the pendulum, so I was initially worried about the timings we were getting — but once they were all in, it become obvious (to me, not the kids) that the time is proportional to the square root of the length.  I asked how long the pendulum should be to get 15 seconds; and also, how long would a 32′ pendulum take.  They were comfortable assuming a linear relationship, but when I pointed out that 8′ was four times 2′ while 60 s was only two times 30 s, they couldn’t really use that information — one kid did guess 1/2 foot for the 15 seconds question, but they didn’t stick to their answer so I think it was just a guess.

One thing that worked out well is that the pieces of tape served as resting spots for the thread so that it would stay at the right length (I didn’t cut the string, we had looped it over the hook like a pulley).  If I hadn’t had the tape sticking out, it would have been hard to maintain a constant length.

I only had time to explain the problem and figure out how many cards there should be before circle ended.  We’ll probably do the main activity next week.

# Merge Sorting 224 Numbers (Age 8)

## The Activities

1. Topic: Logic. Book: True Lies, by Shannon. We did stories 14 – 18 today, finishing the book. Again, everyone loved the stories, and there were lots of lively discussions and theories about how the apparent lies could actually be true.
2. Topic: Gravity, Physics. We discussed the results of dropping things off our balcony last week, and whether Galileo was right or wrong when he said everything falls at the same speed.

Which will hit first? The marker or the dinosaur?

3. Topic: Proofs. We investigated various number properties, and tried to ‘prove’ them using blue blocks as a way to explain graphically:
1. odd + odd = even
2. even + even = even
3. even + odd = odd
4. even * even = even
5. odd * even = even

Visual illustration that even * odd = even.

4. Topic: Sorting, Merge Sort. We tried out the sorting strategy we had used in the Easter egg sorting activity: each kid sorted their own stack of numbers, and the merged all the stacks together at the end.

## How did it go?

We had all five kids this week, and it was a very successful circle. My daughter was very tired from activities earlier in the day, so she caused problems at various points, grabbing away materials or getting off task. Otherwise, everyone was generally engaged.

#### True Lies

The kids really do love this book, and everyone offered lots of interesting theories this time. For one story, I liked our theory better than the one in the book. In “Pockets” a tour guide bets a rich visitor: “I bet I have more money in my pocket than you have”. The visitor pulls out his fat wallet and then says “I accept”, and he loses the bet.

The kids came up with the idea that because the rich man said “I accept” after he had already taken his money out of his pocket, that meant he had zero dollars left in his pocket, so he lost.  The book says the answer is that the guide meant “I have more money in my pocket than you have in my pocket”.

#### Gravity Followup

One kid was very clearly able to recall the story of Galileo dropping objects off the tower of Pisa to prove that everything falls at the same speed.  Everyone also remembered that the Kleenex did not fall at the same speed as the coin, when we dropped them off the balcony last week. One kid was able to explain that the air caught the Kleenex and made it drift. We then tried out a few different dropping tests, in the room (not off the balcony this time). Everyone generally correctly predicted when things would hit at the same time or not. However, one kid initially said the heavy bottle of lotion would fall faster than the coin, but she changed her vote after everyone else said they’d hit at the same time.

#### Even * Even = Even

Next I got out the blue unit cubes. Everyone groaned and said they didn’t like this material.  I said that we were going to the same activity the little circle did last week, and that they had done when they were 5 or 6.  This made them more interested. One girl said “Are we doing primes?”…I was glad she remembered that activity from long ago.

I started by asking whether an even + even number will be even or odd? The kids thought of a couple examples, and then confidently stated that it will be even. One girl was able to explain using the blocks, that each even number could be made of a 2 equal rows of cubes. When you add them together, you know the resulting two rows will also be the same length, so the result is even.

Next another kid demonstrated (with a bit of help), that odd + odd = even. They explained that the two extra cubes from the odd numbers would combine into another pair, yielding an even result.

At this point all the kids had caught on, and many different people want to explain how odd + even = odd.

Next I asked about even * even?  First we tested it by trying out a few concrete examples: 2 * 4, 6 * 4, 8 * 2. They were all even. Is it always true? The kids showed that if you have a bunch of sets of even numbers, then when you combine them, the result must also be even.  This also worked for odd * even.  An odd number of even numbers can still be combined to an even number, with no left over cubes.

#### Merge Sorting 224 Numbers

I asked the kids about the strategy we used in the Easter sorting activity. Everyone remembered how each kid had sorted their own numbers, then made a sorted pile and combined them together into one line. I said we would do the same again, and see if it seemed useful.

I handed each kid a pile of about 20 – 30 numbers, and asked them to sort it. They all begged for more numbers, so I handed out a few more. It was quite fascinating to see the individual strategies they used. A couple kids laid out all the cards around them, so they could see everything. Then they picked them up in order. If they missed a card they had different ways of adding it in: one kid paged through her stack until she found where the number belonged. The other kid dumped all the numbers off the stack in a random pile until they found the number’s place, and then had to sort them all back.

The slower strategies involved lining the numbers up in order, and sliding down the numbers to make space for the next picked up number. The line management took a lot longer than keeping a stack.

One girl actually got a bit stuck in the individual sort. I couldn’t tell what her strategy was, but eventually I helped her by separating out the numbers < 100, 100..200, and 200+. She took by far the longest, so I started handing out extra numbers to the other kids who had already finished. The two kids who kept stacks of numbers were able to quickly add in the new numbers. The ones who used lines took longer as they shifted their lines around.

Eventually everyone finished the individual, and carefully stacked up their cards with the smallest ones on top. Then the merge sort started. The kids were great at working together, especially in the beginning before they got tired. Their stacks ended up having 224 numbers in them, randomly distributed between 1 – 300. This meant there were gaps in the sort order. The gaps really slowed down the sort because the kids weren’t that good and determining that there was a gap, and which number to put down next. Even so, merging 224 numbers took less than 10 minutes, and the individual sort also took about 10 minutes, so that’s reasonably fast for the first try.

# Even + Even = ? (Age 6)

## The Activities

1. Topic: Infinity. Book: The Cat in Numberland, Chapter 2, by Ekelund. In this chapter, all the numbers are living in the infinite hotel,when along comes Zero, who also wants a room. How can they fit him in?
2. Topic: Proofs, Even and Odd Numbers. Define even and odd numbers. Then play with blue blocks adding even + even, and see if the result is even or odd.  Will the result always be even? What happens if you add odd plus odd?
3. Topic: Sorting. Sort the number tiles 1 – 100 on the number board.
4. Topic: Gravity. We took turns dropping objects off the balcony. Which object will hit the ground first?

A balled-up tissue vs. a flat tissue

## How did it go?

We had only three younger kids this week, so it was a pretty easy circle. The big kids were reading a book Galileo, who experimented with gravity by dropping items off the Leaning Tower of Pisa, so David wanted to drop stuff off our balcony. I figured that’d be pretty distracting to the little kids, so we decided to do it as a joint activity.

#### Cat in Numberland

This is one of my favorite books…it really emphasizes the wonder of infinity, and that some questions are not easy to answer.  In this chapter, the number Zero needs a room in the hotel. Ultimately, all the numbers decide to move up one room: One will stay in Room 2, Two in Room 3, etc. This leaves Room 1 empty, so Zero can move in. Finally the numbers each re-label their rooms to match themselves.

The kids were very interested in the book. One girl said she was worried that some very big number would no longer have a room, because they all moved up one room. I said that that big number could stay in the room for the number one bigger.

After the book, I said that the number Negative One knocked on the door and wanted a room. What can we do? At first the kids thought Negative One would have to sleep outside, or build a separate hotel, but then someone proposed moving the numbers up one room again. This seemed to work.

Even and Odds

First I asked the kids what is an even number? Kids started naming numbers: 4, 6, 12. My son started counting by 2s.  I then got out unit blocks and gave each kid a handful. I asked if that was an even or odd number of cubes? Some kids answered by counting the blocks and then knowing that 22 is even.  I showed the kids that instead, you could make two rows of blocks, and if the rows were the same length then number is even.

Even after this explanation, so kids wanted to instead count the cubes, but I kept demonstrating that what I really cared about was not what the number is, but whether it is odd or even. I showed that I could tell this without knowing how many cubes there were.

Next we tried a few addition problems with cubes: four + six, is it even or odd? 12 + 14? Is every even + even always even? Why?

The kids had some intuition about why even + even would be even, but it was hard to put it in words. One girl said something like “same number + same number means the answer will have the same too”.

Next we explored odd + odd, and found that it seemed to always be even. Why? One kid suggested that you could combine the two left over cubes to make even rows.

#### Sorting

We had a few minutes left before it was time to drop things off the balcony, so I got out the 1 – 100 board, and the number tiles. I seeded the board with four or five numbers, and then the three kids worked together to put in the rest of the numbers. They are much better at this than when they started circle. In the beginning it took them forever to sort the tiles on the side of the board that shows the numbers. This time it took about 7 minutes to sort the tiles on the side with no number labels. All three were able to place ’78’ in the right place by finding the ’71’ and counting up.

Dropping Stuff off the Balcony

See David’s blog entry for a full description, but this was fun and wild as you might expect. The older kids expected everything to drop at the same speed because they had read the book about Galileo. The little kids had no expectations, but really enjoyed having stuff thrown off the balcony, and seeing which would hit first.

Everyone enjoyed how slowly the tissue fell.

# Merge Sorting Easter Eggs (Age 6 and 8)

## The Activities

1. Topic: Probability, Impossibility. Book: It’s Probably Penny by Leedy. Leedy is one of our favorite authors. The kids liked this book about what’s possible, probable, impossible and certain.
2. Topic: Sorting. It’s Easter today, so I filled 102 Easter eggs with numbers ranging from 1 – 1011.  I hid all the eggs outside, and then kids did an egg hunt. Then each kid counted their eggs, then opened them and sorted the numbers inside.  Next I had each kid pick up their sorted numbers so the smallest one was on top. We then used merge sort to sort all the numbers together, on the sidewalk.

## How did it go?

We had two younger kids this week, and 5 older kids, so decided to have a fun, combined circle and do an egg hunt since it is Easter today.

The kids loved the egg hunt, racing around at top speed. They especially liked trying to find the 3 golden eggs, which were so well-hidden they required clues. We gave the two younger kids a 30 second head start. In the end, the kids collected between 12 and 17 eggs. My daughter was proud to get the most (though she didn’t find any golden ones).

Next the kids opened the eggs and sorted their numbers. The big kids had no trouble at all. One of the younger kids needed some help recognizing numbers above 100. She seemed to know that 263 was bigger than 207, if I said it out loud, but I’m not totally sure.

Then I showed the kids how to carefully pick up their numbers so the biggest number was on the bottom of their pile and the smallest was on top. We then started sorting all the numbers together. We found 1, 2, 3, and 4 immediately (on top of various kids’ stacks). Then we tried to find 5…my son started sorting through his stack. I asked if it was possible that he 5 somewhere in his stack? He didn’t know, and wanted to look.  One of the bigger kids explained that it was not possible because his smallest number was already on top, and it was 12.

After this, the kids seemed to get the idea of the sort, although it took them awhile to figure out which number to put next, because the numbers were not consecutive. Also various kids dropped their whole stack of numbers several times, requiring re-sorting. Finally we go to the really big numbers (that had been inside the golden eggs), and added 1000, 1010, and 1011 to the end of our line.

I wanted to take a picture of the kids with our nicely sorted numbers, but my son scattered the numbers before I could gather everyone.

After circle today we had a picnic with the other families, which was really fun. At the picnic my daughter organized yet another egg hunt with eggs containing stickers and small treasures from her room.  All the kids played together really well.  At the end of the night the parents took turns playing tag with the kids.  There was sprinting, taunting, crashing and fun 🙂