The Bird and the Bikes (Age 9)

The Activities

  1. Topic: Money. Book: The Story of Money by Maestro.  We continued where we left off last time, and got as far as early money in the Americas.  My favorite part was the discussion of why paper money caught on better in China than in Europe (the government was more stable in China).
  2. Topics: Algebra, Arithmetic:  I wrote down the equation (5789 + 1286) x 549 = 3,884,175.  I used my phone to compute the right hand side.  Then I asked them a series of questions: What is the answer if you change 5789 to 5790 (the answer increases by 549)?  What if you change 549 to 550?  What if you change 549 to 1098?  Each time, using my phone, I checked that you got the same answer by evaluating directly vs. evaluating incrementally.IMG_2535
  3. Topic: Probability: I attempted to teach the kids how to flip a coin properly, and then each kid (and me) spent 5-10 minutes flipping coins and writing down the sequence.  Then, I asked several questions: “Do you expect more heads or tails?  Is heads more likely after you’ve just gotten three tails in a row?  Is heads-tails more likely than heads-heads?”  For each one, we counted in our sequences to see whether the results matched the kids’ intuitions.
  4. Topic: Logic: I drew a picture of two bicycles riding toward each other at 5 mph, starting 10 miles apart, and asked them how long before the bicycles met.  Then, I added a bird flying at 20 mph back and forth between the bicycles, turning around and going back whenever is met a bicycle, and asked how far the bird flew before the bikes met.IMG_2533

How Did It Go?

We had four kids this week.  It was a pretty good circle, some distractions as always but a lot of good thinking as well.

The Story of Money

This book has been going well, our daughter was able to explain the China vs. Europe paper money difference later that day when we were talking about circle.

Incremental Algebra

The kids did quite well on this activity, many of them were comfortable with parentheses and they didn’t have much problem getting the right answers.  By the end they were getting confident enough they thought the checking was a waste of time.

Coin Flip Sequences

When I started asking questions, I got some answers like “more heads after 3 tails in a row” — but I was surprised that after hearing each others’ answers they quickly converged to 50/50 no matter what.  So they seem to have a decent grasp on the idea of independence of coin flips.  Flipping was kind of hard for them, they really wanted to move their whole hands instead of just their thumbs.  For this reason the results were slightly suspect.  And of course this is a probability exercise so the results never come out perfect.  But by the end I felt pretty confident that some of the kids understood the idea of evaluating probabilities by counting occurrences from a sequence of trials (i.e., statistics).  The trickiest part was that if you have a sequence of, say, 5 heads in a row, and you’re counting outcomes after 3 heads in a row, you use this sequence 3 times (first 3, second 3, and final 3).

The Bird and the Bikes

One of the kids got the clever answer to the full question almost immediately.  Partly this was because I made it easier by asking the bikes only question first.  But still, I was impressed.  We also started computing the “brute force” way where we figured out how far the bird flew before meeting the first bike (8 miles).  The kids did okay at this too even though it’s a bit tricky.


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.

Building Dice (Age 6)

The Activities

  1. Topic: Logic, Puzzles. Book: Playful Puzzles for Little Hands, by Taro Gomi. This is a really cute book with lovely illustrations. The puzzles cover mazes, dexterity, counting, logic, subtraction, and find-the-differences.  We did about 10 pages in 15 minutes.
  2. Topic: Cubes, Spatial Reasoning. I gave each kid a cube pattern and a die. I drew six dots on one of the sides of the cube, and asked the kids to copy the die onto the cube so it would be exactly the same. Then we cut out the cube and glued it together to make a die.IMG_20160522_195041
  3. Topic: Probability. 
    1. I gave the kids some colored stones and a bag. They put 10 stones in the bag that were a mix of red and green (e.g. 9 red and 1 green).  Then I repeatedly drew out a stone, and replaced it in the bag, trying to figure out how many of each color the bag contained. I made silly guesses along the way, to entertain the kids.
    2. I divided the kids into two teams which each had a bag. The bags each contained 10 stones, but one bag had more red stones than the others.  One team member repeatedly drew out one stone and then put it back in the bag. The other team member kept track of how many reds vs. greens were drawn out. In the end the two teams presented their chart, and then tried to guess which bag had more red stones.


      The bag with 10 stones.

How did it go?

All five kids came to circle this week. There was lots of extra energy because we had a picnic afterward, so the kids were excited, and there were lots of younger siblings playing in a nearby room. Even so, everyone generally paid attention, except for my son who drew pictures instead of participating the in the team probability activity.

Building a Die

The kids were surprisingly good at this. Once I drew the six side on their cube printout, they could generally figure out where the 1 should go (with a few mistakes), and then most of them even figured out where to put the remaining sides, with only a bit of help from me. Everyone was quite good at cutting out the shape, except my son who got frustrated and cut his in half…I gave him the one I had been cutting out.

Another parent helped with gluing the cubes together since the kids all needed help at the same time.

Overall, I was impressed by their spatial abilities.

Probability Trials

First I had one kid put 10 stones in the bag without telling me the colors. The kids LOVED this.  Then I started drawing and replacing one stone at a time. First I drew a green stone, so I said: “Oh, I think they’re all green!”.  The kids giggled. Then I drew a red stone, and said “Oh no, I think they’re all red!”. The kids corrected me, pointing out I just drawn a green stone the time before. Then I started drawing more, keeping track of how many more reds than green I drew out.  After 20 or so trial I made a final guess. This is probably not enough trials, because I never actually got it correct.

We let 3 different kids put stones in the bag for me, but then the kids started to lose interest, so I told the last two kids that they got to be captains of the next activity. This really pleased them.  Being captain meant they got to pick their teammate, and decide whether they wanted to be the note taker or the one who drew out the stones. One captain decided to be the stone person, and the other wanted notetaker, so both positions were apparently desirable.

Four kids were actually pretty efficient at drawing out stones and making tally marks to count them.  My son was on a team of three, and decided to draw pictures instead of participate. I let him because he wasn’t distracting anyone else, and we were running out of time.

After about 30 draws, I called the kids back to the table.  One team had drawn 33 times and gotten 16 greens and 17 reds.  The other team had drawn 25 times and gotten 12 greens and 13 reds.  The kids decided that the team with 33 draws must have more reds — because they had drawn red 17 times and 17 > 13.  After circle, David pointed out that I should have then asked which bag had more greens? and complain if they again said the team that had drawn out 33 times.  But I just let it go because I didn’t think of it at the time.

In reality one bag had 3 reds and the other had 5 reds, but 25 – 35 draws were not enough to distinguish the two cases.  So we’ll have to repeat again, and give more time for repeated draws.

Misguided Record-Keepers (Age 6)

The Activities

  1. Topic: Puzzles: Book: Taro Gomi’s Playful Puzzles for Little Hands by T. Gomi.  We did the first 10-15 puzzles.
  2. Topic: Probability:  I had a cloth bag and two colors of glass beads, red and green.  Secretly, I chose four total beads and put them in the bag (3 red and 1 green).  The kids took turns drawing out one bead at a time, checking its color, and putting it back.  After the kids had done this a couple times each, I had them guess how many green and red there were (I told them there were four total).  We did that again, with 4 green this time.  Finally, I took another bag, filled bag A with 8 green and 2 red, bag B with 2 green and 8 red, and had them try to figure out (using the same draw one and put back) which bag had more green beads.
  3. Topic: Geometry: I downloaded some cube net diagrams from the internet (a cube net is an “unfolded” cube).  I had one full sheet for each kid, each kid getting one of two diagrams.  I showed them how to cut out and fold the cube, and then they each made their own (I helped them assemble it once they had put glue on the tabs).  After that, I took mine (which wasn’t glued) and wrote a letter on each face.  Then I asked “What letter is on the opposite side from the A?”  After they answered, I folded it up and we checked.  After we did a few different questions, I switched to the miniature cube diagrams from page 3 of the PowerPoint.  I wrote letters again and asked the same kind of questions, and then checked by folding up the cube.IMG_1904

How Did It Go?

We had four kids this week.

Playful Puzzles

These activities went really well, the kids were all into it.  Different kids were good at different puzzles — one kid knew right and left, another is good at counting, another was the fastest at finding things in a picture.

Drawing With Replacement

The kids were better than expected at not looking in the bag — when we did this with the older circle, people tried to cheat.  The first time they drew 9 reds and 1 green.  Two of the kids thought there would be 2 greens and 2 reds; the other two thought 1 green and 3 reds.  I played the fool and said there could be 4 reds, but they were onto me.  In retrospect, I should have gone farther and had them make a puzzle for me and then I could make a bad but not impossible guess (e.g., “I think there are more reds than greens because the first thing I drew was red” or guessing more red than green after drawing more green than red).   The second time there were no reds, and everyone guessed correctly (they drew about 10 times).

Then I introduced two bags, asking which had more greens.  They took a bit to get going, but then one kid took charge of one bag and another the second.  I had given them paper to keep track, but didn’t tell them what to keep track of.  They ended up all making a chart to track the TOTAL number of reds and greens across both bags, which obviously doesn’t help much.  The bags were skewed enough that the kids in charge of each bag could tell that their own bag was mostly one color, so in the end they all guessed the right answer.  But the record keepers had no idea without listening to the bag holders :).

Cube Nets

The kids were pretty good at cutting out their diagrams.  They were decent at folding as well.  I helped assemble because that part is pretty tricky.

The first time I asked them a “what’s on the opposite side” question, they had no clue.  But they quickly noticed how the line of four squares formed a ring around the cube with the two other flaps on opposite sides, so by the 2nd or 3rd question they were all correct.  And when I switched to a new diagram with the same backbone of four consecutive squares plus a slightly different arrangement of the opposite flaps, they still got it right.   However, once I switched to one where the opposing flaps were at opposite ends (a sort of 2 shape), they got it wrong.  And I didn’t have a chance to get to a diagram that was a zigzag, with no backbone.

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 🙂

I Want To Go Last! (Age 7)

The Activities

  1. Topics: Division, Primes: Book: The Number Devil by H. Enzensberger, first half of third chapter.
  2. Topics: Primes, Multiplication: Following the chapter from the Number Devil, each kid did a sieve of Eratosthenes up to 70.
  3. Topics: Games, Probability: Using percentile dice (two 10-sided dice which together roll a number from 0 to 99), we played this game: going around the circle in turn, each kid picks a number.  I roll the dice, and whoever is closest gets a point (if there’s a tie, each kid gets half a point).  After doing this a few times, we did the same thing except that instead of rolling the dice, we computed how many numbers would make each person win, and they got that many points.  E.g., if the numbers were A: 10, B: 45, and C: 85, then A wins from 0-27 for 28 points, B wins 28-65 (tie on 65) for 37.5, and C wins 65-99 (tie on 65) for 34.5 points.

How Did It Go?

We had 4 kids this week.

The Number Devil

This chapter talks about the connection between multiplication and division, and about prime numbers.  It introduces the sieve of Eratosthenes.  One interesting thing that came up is one of the kids, who knows division already, first said that they hadn’t done division this way before, but then later said that they probably knew this way of doing it because they knew how to do division.

Sieve of Eratosthenes

We’ve tried this before, and this time the kids were definitely better.  But some of the kids still made multiple mistakes, particularly when counting by threes.  I tried to explain why it makes sense to cross out every third number, but I’m not sure they fully understand that counting by 3’s gives you multiples of 3.

Dice Guessing

The number picks were pretty random for a while; one of the kids guessed lucky numbers, and most of them liked to pick larger numbers.  They did all realize they should pick between 0-99.  After a bit, one kid realized that guessing right next to another guess might be a good idea — but it then backfired on them when the next person did the same thing.  They soon decided that they all wanted to go last — with good guessing, it’s not an advantage to go last, but with the way they were guessing, it definitely was.  I had initially planned to use the dice the whole time, but quickly realized that the variance was too high — one of the kids was winning by a sizable margin despite not having made the best picks.  So I switched to giving points based on number of ways to win (I did all the calculations, it would have been hard for them).  Some of the kids understood this pretty well, but some of them were pretty confused and didn’t know what I was doing.  For one thing, they hadn’t seen notation like 45-58 before, and the idea of writing down all the numbers that would win for them wasn’t obvious.