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?


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.



30 Different Ways to Say “I Love You” (Age 7)

The Activities

  1. Topic: Measurement: Book: Taro Gomi’s Playful Puzzles for Little Hands.  We only did a few puzzles towards the end of the book, most of them involved measurement and were pretty hard!
  2. Topics: Geometry, Graphs: I made a set of Valentine’s Day themed arrow direction drawings, downloadable here.  The rules are, using graph paper (ideally with fairly small squares), you start at a vertex and have one of 8 directions and a distance.  I introduced something new this time, which is some of the instructions were in red, which means you moved your pencil but didn’t draw a line.img_2465
  3. Topics: Counting, Graphs: I gave each kid a box of the kind of candy hearts that have messages like “Be Mine” or “Sweet On You” printed on them.  Each kid sorted their box by heart color, and then we made a combined graph with how many there were of each color.  Then we found as many distinct hearts (message + color) as we could.

How Did It Go?

We had all five kids this week.  It was a high energy circle, partly because of candy and partly because four of the five kids had just been to Cirque du Soleil.  We spent five minutes at the beginning of circle so each kid who had been to the circus could say their favorite part, and then we got through the rest of circle without any mention of the circus!

Taro Gomi

One of the problems asked which of a bunch of hats was the shortest and tallest — we tried to find some kind of trick (e.g., number of stripes), but in the end all we could figure out was the measure.  Similarly, the next page had two different colored poles cut into pieces and asked which pole (when put together) was longest, which seemed really hard as well.

Arrow Drawings

The kids did pretty well on these, but there was a pretty big spread in ability.  Most of the kids made a small mistake from time to time, usually either going the wrong distance or not doing a diagonal at 45 degrees.  One kid was noticeably better, going faster and without mistakes.  I was worried the red instructions (pick up your pencil) would be confusing, but they understood it easily.

Candy Hearts

I was originally planning to have them sort by message and make a graph that way, but when we opened the boxes, it turned out that the printing quality on the hearts is really bad — probably at least 1/3rd of them have missing or unreadable messages.  Also, it turned out there are a TON of different messages (“Be Happy”, “Nuts 4 U”, …) — we counted 30 different ones — which would have made it hard to make a graph.  So we did color instead.  And then there was another surprise — there were FAR more oranges than anything else — 3 times as many as most of the other colors!  And it was consistent across boxes as well.  Seems like a pretty solid result that I’d expect to hold up across many boxes.  The kids were pretty excited to find all the different messages and laughed every time we found a new one.  The kids were also REALLY excited to eat some of the hearts, but as far as I know they listened to me and didn’t eat any until the end (they got three each).

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).


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.

Tricky Towers (Age 6)

The Activities

  1. Topic: Time: Book:  At The Same Moment, Around The World by C. Perrin.  After we read the book, I asked the kids for different places they had visited and we figured out what time it was there.  Also I asked why sleeping was more difficult after a long trip.
  2. Topic: Probability:  We did probability charts with two six-sided dice.  Each kid had a chart, and repeatedly rolled the dice and filled in a box (from bottom to top) for that number.  Once one of the numbers gets to the top (5 rolls) that number “wins”.  Most of the kids did 2-3 charts, and then we checked to see what numbers had won most often.img_2316
  3. Topic: Puzzles: Each kid got a Tower of Hanoi set, and they tried to solve as many discs as they could.

How Did It Go?

We had four kids this week.  The kids were all very engaged the whole time.

At The Same Moment

The kids got the idea of the book and liked naming the places they had been.  All of them have been on very long trips so the idea of time zones and figuring out the time in another place was pretty natural for them.  They were also quite familiar with jet lag, particularly the ones who had gone halfway around the world.

Probability Charts

This is always a popular activity and this week was no exception.  One of the kids immediately asked why there was a 13 and 14.  The kids went at wildly different speeds — in the 20 minutes we did this activity, one kid finished 4 and another finished only 1.  The kids are pretty good at adding up the dice now, but some still need to think a bit for the bigger numbers.  At one point, one kid noted that someone else’s chart looked like a pyramid (which is exactly what it looks like “in expectation”).  As expected, we had lots of charts with 6, 7, and 8 winning; I asked them why and they didn’t have a good answer.  One kid noticed that a different kid had two winning charts with “8”, so I asked whether it mattered who rolled the dice.  The kid said “I don’t know” and then “I don’t think so?”

Tower of Hanoi

The kids really made a lot of progress during circle.  Two of them started with 5, solved 6 without too much trouble; one of them finished 7 by the end of circle while the other almost did.  Another kid started with 4, and after getting the hang of it solved up to 6.  The final kid had a lot of trouble with 4, so I helped them with 3, then 4, and they were able to solve 5 by the end of circle.  They all worked hard solving the problems and were clearly getting the idea of moving piles in order to clear up the discs they needed to move.

Trick or Treat Math (Age 6)

The Activities

  1. Topic: Counting. Book: How Many Donkeys? An Arabic Counting Tale, by MacDonald. In this simple book a man can’t figure out if he has 9 or 10 donkeys because he keeps forgetting to count the donkey he is riding. The kids caught on quickly and laughed whenever he got it wrong.
  2. Topic: Maps, Spatial Reasoning, Logic: Fill in a map of a treat-or-treating neighborhood based on the following clues. Here is the clipart we used: halloweencharacters.
    1. Directly to the West of your house is the Witch’s house.
    2. The Zombie house is 2 houses West of the Witch’s house.
    3. Olaf’s house is across the street from the Zombie’s house.
    4. Elsa’s house is directly South of the Witch’s house.
    5. The pumpkin house is directly East of your house.
    6. The Spider house is on the very West end of the South side of the street.
    7. The Butterfly is scared of the Spider. The Butterfly’s house is on the same side of the street as the Spider’s, but as far away as possible.
    8. The Goblin is between the Zombie and the Witch.
    9. The skeleton is directly across the street from the Spider.
    10. Next to Elsa’s house is a Graveyard that takes up two houses.
    11. The Ladybug’s house is right next to the Butterfly’s.
    12. The Fairy can fly right across the street to the Ladybug’s house.
    13. The Wizard’s house is East of the Fairy’s.
    14. Anna’s house is next to Elsa’s house.

      The completed puzzle

      3. Topic: Estimation, Subtraction. Guess how much candy is in a container. Then put the same candy in a shallower container and guess again. Then count the candy and figure out whose guesses were the closest.


      The Candy

      4. Topic: Logic. Tape a Halloween character to each kid’s head. Then the kids ask each other yes/no questions to figure out who they are. The hardest part of the game is not telling your friends what is written on their heads.

      How did it go?

      I wore my witch costume during circle, and I organized it so the kids would get to ‘trick or treat’ after completing each activity from my bucket of small prizes and candy.

      Halloween Logic

      Each clue was pretty easy for kids, especially after they understood what phrases like “directly West” means. The hardest clue was: “The Spider house is on the very West end of the South side of the street.” Two of the kids figured it out on their own. The other two needed some help from their friends to understand the “south side of the street”.

Candy Estimation

The kids were very excited to see so much candy, especially when I told them that the person who guesses closest would get to trick or treat twice after the activity. Interestingly, the guesses did not get closer after I spread out the candy. Most second guesses were at least as wrong as the first guess. I guessed after the candy was spread out (and I got within 2 of the correct number).

After everyone wrote down their guesses I asked the kids to count the candy. They immediately began discussing counting strategies. They eventually decided to sort the candy by type and then count each type. However, they soon realized that some types had too many pieces to be easily counted, and they didn’t know how to add the results anyway. So they switched to counting each piece of candy as it was thrown back into the tub. Two kids both wanted to throw in candy and everyone ended up missing a bunch of pieces when the two throwers could not coordinate. They came up with 67 pieces, but I counted it again and found 72 pieces.

Halloween Twenty Questions

The kids loved seeing costumes taped to their friends’ heads, especially when one boy got ‘Princess Leia’. I told them at the start that it is very important not to tell your friends what is written on their heads, and the kids did pretty well at this. However, some kids asked questions like “Am I a zombie?” because they saw “Zombie” on their friend’s head. The hardest to get turned out to be superman. The kid knew he was a strong hero who wears red and blue, and has an S, and has a cape, but couldn’t think of superman.  Everyone else figured theirs out eventually (with some hints from me about what questions to ask). Everyone really enjoyed this activity. At the end, we had five minutes left so one of the kids moms played and had to figure out she was a pumpkin. The kids loved hearing her questions and shouting out answers. “Can you eat me?” “Yes, but it’s yucky and too chewy!”

Impossible Flips (Age 6)

The Activities

  1. Topic: Puzzles:  Book: Taro Gomi’s Playful Puzzles for Little Hands.  Still haven’t quite finished, this time it was mostly mazes.
  2. Topics: Number Line, Number Recognition.  We revisited higher/lower number guessing again, mostly from 1-100.  As usual, the theme of the game was a bear who wants to steal our picnic food.  But the bear print-out was missing so kids took turns standing next to the wall and using their finger as the bear.  I did a few numbers, and then each kid took turns thinking of their own number.  At the end, we had a discussion about what makes a good or bad guess, and then I did one from 1-1000.
  3. Topics: Combinatorics, Geometry:  Using wooden pattern blocks, find as many ways as possible to make a 2×2 diamond.patternblockdiamonds
  4.  Topic: Logic:  We did the Seven Flipped activity from  Starting with 7 shapes face-down (we used Scrabble tiles), you could flip 3 tiles at a time.  The goal is to flip all the tiles face up.  After they solved that, I switched to 7 tiles, flip 4 at a time (which is impossible) and then 5 tiles flip 2 (also impossible), and we discussed why it might be impossible.

How Did It Go?

We had all five kids this week.

Number Guessing

There is a very wide range of abilities in this game.  By the end, three of the kids completely understood how the game worked, and during the discussion two of them worked together to figure out that they should guess half-way in between each time.  One of the other kids usually made proper guesses, but the final kid frequently made guesses outside of the current range (even when they were just reminded of what the current range was).  I also made a couple “illegal” guesses when I was playing, but was called out on it.  1-1000 is still pretty challenging even for the kids that get it.

Diamond Variants

The kids weren’t as in to this activity as I expected.  A couple of them went off task pretty quickly, building whatever they felt like.  One kid tried hard to use the skinny white diamonds, which doesn’t work.  Another kid was trying but kept building diamonds that were 3 units on the side.  One kid tried for a while to use squares, without success, but then eventually figured out a key insight for building different diamonds, which is that you can swap two adjacent triangles for a diamond, or vice-versa.  So that kid generated more than half of the variants we found.

Seven Flipped

The kids each had their own set of tiles.  There was lots of cheating, but it didn’t matter because I would just ask them to show me again.  At first the kids decided it wasn’t possible, but after a few minutes one of the kids figured it out.  Another kid watched them demonstrate, and then the two of them taught the other three.  Then I switched to 7/4.  There were lots and lots of claims of having done it, but it’s impossible :).  After a while, I asked them to try 5/2 instead.  A couple of the kids started to get the idea that it was impossible.  I myself made a bunch of moves on this problem with the kids watching, and we kept track of how many were face up.  With a hint the kids noticed is was only 0, 2, and 4.  I made a set of maybe 11 tiles with 6 flipped up, and then showed them all the possible moves (2 down -> 2 up, 2 up -> 2 down, and 1 up, 1 down -> 1 down, 1 up), and they saw that it could only be +2, -2, or 0.  One or two of the kids might have understood this proof that 5/2 is impossible.

Taking a Finger Walk (Age 6)

The Activities

  1. Topic: Puzzles: Book: Taro Gomi’s Playful Puzzles for Little Hands.  We’re most of the way through now, probably one circle left of puzzles.
  2. Topic: Logic: I printed sheets with 6 uncolored flowers on one side, and 9 on the other.  There were two puzzles: For 6 flowers, “There are more red flowers than purple, and more yellow flowers than red.  For 9 flowers, “There are more red flowers than purple, more blue than red, and the same number of blue and yellow.”IMG_2108
  3. Topic: Spatial Reasoning:  Corey and I built a number of models out of Legos.  The kids each picked a model and had to copy it exactly.  They could pick it up and look at it from any angle.  Each kid copied several models.IMG_2109
  4. Topic: Attributes, Games:  We played a couple rounds of Set with just the solid cards.

How Did It Go?

We had four kids this week.  One kid had been gone for a couple months, but now everyone is back from summer trips.  This circle went well, the kids were all interested in all the activities.

Playful Puzzles

We spent quite a bit of time on a puzzle with two kids at either end of a very windy path, with the question “Where will they meet?”  We measured by placing coins from either end.  They also enjoyed a page where you were supposed to “take a walk” with your finger by tracing a path and following various instructions along the way (e.g., “Take a rest here” or “Go around this corner really fast”).  Every kid did it once.

Flower Logic

The kids figured out the answers pretty quickly.  Interestingly, different kids figured out the second one from the first.

Lego Models

Different kids definitely had different skill levels on this one.  One kid breezed through a whole bunch, while others took quite a bit longer.  The trickiest ones were the dinosaur, because it was irregular and a bit complicated internally, and an offset colored square because it was tricky to get the right pattern of blocks on the bottom row (the kid working on it initially had the colors going the inverse rotation).


We’ve played before with this circle, which the kids remembered, but not all kids remembered the rules.  Our son has played a lot, so after he got a few I said he had to let other kids get sets.  I was happy because all the other kids got at least one set on their own.