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.

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).

A Bag Full of Dice (Age 9)

The Activities

  1. Topics: Geometry, Three Dimensional Shapes: Book:  Sir Cumference and the Sword in the Cone by C. Neuschwander.
  2. Topics: Geometry, Three Dimensional Shapes:  A while ago we bought 5 full sets of “D&D dice” (4, 6, 8, 10, 12, and 20 sided).  We counted the edges, faces, and vertices for each of these and made a chart like in Sir Cumference, showing that “Faces + Vertices – Edges = 2”.  I also pointed out the dual relationship between 6 & 8 and 12 & 20 sided polyhedra (i.e., 6-sided has 6 faces, 8 vertices, and 12 edges; 8-sided has 8 faces, 6 vertices, and 12 edges; you can switch between the two by putting a vertex in the middle of each face and connecting adjacent vertices).img_2431
  3. Topic: Numbers: We did What’s the Secret Code? from  There are some clues about what the secret number is like “The digit in the hundreds place is ¾ the digit in the thousands place.”  There is more than one answer which is cool.
  4. Topics: Origami, Geometry: We did Paper Folding from  There are a number of folding challenges like “Construct a square with exactly ¼ the area of the original square. Convince yourself that it is a square and has ¼ of the area.”img_2432

How Did It Go?

We had four kids this week.  As usual some kids followed along better than others, but most people were engaged for both the dice activity and the paper folding.

Sir Cumference and the Sword in the Cone

The kids liked the book, they laughed at quite a few of the math puns.

Euler’s Polyhedron Formula

The kids definitely enjoyed making the chart.  They did a pretty good job staying on task (it was easy to get distracted and start rolling the dice).  Counting the edges on some of the dice was fairly tricky but was much easier with good grouping strategies.

What’s the Secret Code?

The kids did well on this except that they had trouble with the decimals.  They did find one of the decimal answers, because they knew that .5 = 1/2, but I believe there were other possible decimal answers as well.

Paper Folding

The kids solved all the tasks except the last one, which was making a non-diagonal 1/2 area square.  I figured out a pretty complicated way to do it (by transferring the side length of the diagonal answer onto a horizontal edge), they copied what I did but it was pretty tricky (see picture above).

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.

Robots, Planes, and Pie (Age 8)

The Activities

  1. Topics: Puzzles, Arithmetic: Book: Edgar Allan Poe’s Pie: Math Puzzlers in Classic Poems by J. Patrick Lewis.  We read 5 or 6 of the poems and they solved the math puzzle.  For some of the poems, I found the original version and read it to them first.
  2. Topics: Logic, Hard Problems: You have available an unlimited number of airplanes.  Each airplane can hold 12 units of fuel, and the airplanes can refuel each other in midair.  Each unit of fuel lets an airplane go 1000 miles.  Airplanes can only land at the starting line — if they run out of fuel anywhere else they crash.  I asked the kids to try to get as far from the starting line as possible without having any planes crash.  I created a powerpoint with planes and distance track as a visual aid — the planes show the fuel units and the kids could fill in the units in pencil as they simulated their solution.
  3. Topics: Counting, Factors: We did the Robot Stepper activity from  I made a square grid of the numbers from 1-100 for the kids to fill in, and gave each kid a different starting number and number of steps.  After each kid had done several different charts, we looked at them as a group to see what kind of patterns we could find.

How Did It Go?

We had all five kids this week.

Edgar Allan Poe’s Pie

The kids liked the puzzles and did a pretty good job listening and trying to solve them.  However, they weren’t very interested in hearing the original poems, some of them said they were boring or “Why are we doing this?”  I was surprised because I thought they might like the change of pace.

Long-Range Airplanes

As is often the case on this kind of problem, a couple of the kids tried hard and the rest were distracted most of the time.  They all liked the planes — one kid was even grabbing other kids’ planes :(.  One of the kids made quite a bit of progress.  I gave the kids a way to get to 7 using 2 planes (they both move 4 spaces, one plane gives 2 fuel and returns home, other plane has enough to get to 7 and then back home); the one kid quickly figured out you can get to 8 using 2 planes, and kept improving until they got a plane to distance 12 and back (using 5 or 6 planes, can’t remember).  Framing the problem as “How far can you get?” rather than “Can you get to X?” was good, I think, because it took the pressure off.

Robot Stepper

Everyone was into making the charts.  One kid made a couple mistakes, decided to X out the mistakes, and then decided to go ahead and X out every skipped square.  All the kids noticed patterns as they were coloring, and often stopped actually counting and just used the pattern instead.  The best insight on this problem was one kid was able to explain why stepping by 9 created a backwards diagonal (going down adds 10, going to the left subtracts 1).  Unfortunately the kids weren’t super interested at the end when we laid out all the diagrams and analyzed them, but maybe it’s just because circle was almost over at that point.

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.

A Trick-or-Treat Circle (Age 8)

The Activities

  1. Topics: Proofs, Time, Logic:  I asked the kids to determine whether or not every year has at least one Friday the 13th.
  2. Topics: Geometry, Time:  I made a map of a neighborhood for trick-or-treating.  The red house is your house.  Each inch of road is one block, and trick-or-treating along one block takes ten minutes.  First, I asked them how long it would take to trick-or-treat on every block (this requires repeating a few blocks).  Next, I asked what the most number of different blocks you could visit if you had 3 hours.  Finally, I asked how many different blocks you could visit if you had to return to your house to drop off candy every 5 blocks (i.e., after 40 houses).  A bonus question I didn’t get to was, if you wanted to minimize the time to visit every block, and you didn’t have to start at your house, where should you start?
  3. Topics: Combinations, Combinatorics, Logic:  I had a list of 10 possible trick-or-treaters:
    1. Evil Queen — Baddie, Girl
    2. Bride of Frankenstein — Baddie, Girl
    3. Vampire — Baddie, Boy
    4. Mummy — Baddie, Boy
    5. Princess — Goodie, Girl
    6. Fairy — Goodie, Girl
    7. Wizard — Goodie, Boy
    8. King — Goodie, Boy
    9. Alien — Neither, Neither
    10. Slime — Neither, Neither

    First, I asked how many ways there were to pick three trick-or-treaters.  Then I asked how many ways to pick three trick-or-treaters, with the requirement that there’s at least one Baddie, one Goodie, one Boy, and one Girl.  Note: Picking groups is much harder than picking ordered line-ups (where Evil Queen, Princess is different from Princess, Evil Queen).  If I were doing this again I would stick with ordered line-ups, it’s hard enough already.

How Did It Go?

We had all five kids this week.  This was a pretty hard circle; 2 of the kids were engaged through-out, with one saying how they liked the hard problems; the other 3 were distracted a lot of the time.

Friday the 13th

This is a pretty tricky problem, it’s not immediately obvious how to do it even for adults.  The kids made some good progress and had some interesting ideas.  First, one kid figured out that for there to be a Friday the 13th, the 1st had to be a Sunday.  Another kid wrote down the years starting with 2000 (she wanted to check “all the years”).  I used my phone to look up the calendars for each year, and we checked which months had a Friday the 13th each year.  One kid was really excited to try to find a year with no Friday the 13th, because then they’d be done.  But there is indeed a Friday the 13th each year, so we didn’t find one :).  At this point, I gave them a hint, which is to draw a pie chart like in the picture above.  The idea is to go through an entire year starting with January, assume that the 13th in January is, say, a Sunday, and then figure out what day of the week the 13th is in each month.  If you do this, you’ll find that every single piece of the pie is filled, which is what you need to prove that there’s always a Friday the 13th.  Unfortunately, the kids were not good at doing the calendar arithmetic to figure out what day of the week Feb 13 is given the day of the week for Jan 13.  So, we didn’t get that far, and since we had already spent 25 minutes I moved on to the next activity.  Most of the time, two of the kids were working on the problem while the others were drawing, etc.

Trick-or-Treat Optimization

The kids liked the theme of optimizing trick-or-treating.  Unfortunately, I made an error in how I set up the problem.  My intention was that they should concentrate on how many blocks you’d have to walk, but I drew the houses big enough that they focused on visiting houses instead of walking along blocks.  The map I included above I redid afterwards to make it clearer that it’s about blocks, not houses.  The problem with houses is that if you have houses on the corners of streets, it makes the counting a lot messier.  And counting houses is a bit more intuitive, so that’s what they defaulted to.  The result of this was that about half the kids thought I meant that it took 10 minutes to visit three houses, when I actually meant it took 10 minutes to walk one block.  All the kids paid attention during this activity.

The kids figured out that you’d have to backtrack or at least revisit some blocks.  They were all pretty comfortable with figuring out how long it would take to visit all the blocks, but the idea of the best route wasn’t as compelling.  They did understand the idea of visiting as many as possible in 3 hours.  The final problem, about returning home each time, isn’t actually that interesting with the map I had, but they still had to think about it some to figure out how to do it.

Picking Trick-or-Treaters

This problem turned out to be harder than I expected.  I just forgot that they weren’t that comfortable with combinations yet.  Even if I had done the ordered line version, they still didn’t immediately remember how to do the multiplication to figure out the answer to the unconstrained version.  They did figure out this part, and we moved on to the constrained version.

I actually gave them a four person version that required 2 baddies instead of 1 — it turns out to be a lot harder than the three person version.  Also, the non-ordered version is a lot harder to think about.  With the three person version, it’s not so bad to reason along the lines of “Let’s pick the baddy first, and the goody second.  For each of the possible combinations (there’s only 4 distinct ones), we can figure out what the third person can be.”  The four-person version gets a lot more complicated, so I switched to the three-person version — we made some progress but didn’t solve it.

Again, two of the kids worked hard, while the other three were distracted.