Lots of Bunnies (Age 7)

The Activities

  1. Topics: Fibonacci, Sequences: Book: The Number Devil by H. Enzensberger, night 6, which covers Fibonacci numbers.  We explored a number of the properties of Fibonacci numbers (such as that fact that the squares of two consecutive Fibonacci numbers is also a Fibonacci number), and drew “Bunny diagrams” to see how Fibonacci could arise in nature.IMG_1726
  2. Topics: Graphs, Coordinates: The kids each did another worksheet from worksheetworks.com.  This time the coordinates included negative values.
  3. Topics: Games, Addition: We played Clumsy Thief, which involves finding pairs of numbers that add to $100.

How Did It Go?

We had only 3 kids this week.


The kids were able to write out the Fibonacci sequence until about 1000.  The kids varied in how interested they were in the various properties; one kid was “bored”.  One of the kids really understood the bunnies diagram and did quite a few rows.

Coordinate Pictures

As usual, the kids enjoyed doing the pictures.  One of them had brought back their homework from a previous week (another coordinate picture) and got a prize.  The negative coordinates confused them for a while, but still the actual difficulty is knowing whether the first coordinate is horizontal or vertical.  By the end they were doing pretty well, and they all finished one picture.

Clumsy Thief

Two of the kids liked the game, the other wasn’t so sure; I think this corresponded roughly with how quickly they were able to find the pairs that added up to $100.

Cargobot for Pre-Readers (Age 5)

The Activities

  1. Topic: Tallying, Counting. Book: Tally O’Malley by Murphy. In this book, a family plays a counting game where each person picks a color of car, and then makes a tally mark when they see a car of that color. The book shows how to ‘bundle’ the marks together with the fifth mark.
  2. Topic: Tallying, Number Recognition. I put tiles with the numbers 1 – 100 in a bag. Each kid picked a digit from 1-9. Then we took turns drawing numbers out of the bag. If the number contained your digit, you got a tally mark.

3. Topic: Programming. We played Cargobot with our hands today. Cargobot is an on-line programming game where you control a robot arm, moving it left or right and picking up or dropping boxes. We played this using our own arms, and colored stones as the commands.


Your arm starts above the blue dot. The program of stones moves both boxes to the square closest to the blue dot.

4. Topic: Attributes, Set. We played a couple rounds of Set with just the solid cards.

5. Topic: Attributes, Venn Diagrams. Using the fairytale bingo cards, we categorized cards in two ways: Things that Fly vs Thing that go in Water. Girls vs Scary Things.

Girls vs. Scary Things. The witch and the three bears fit in both categories.

Girls vs. Scary Things. The witch and the three bears fit in both categories.

How did it go?

There were only 3 kids this week.


The kids all enjoyed the book. They were interested to see who would win the family’s games. Drawing numbers out and making tallies was good practice for them. I had the kids read out the name of each number they drew. There was lots of excitement when someone drew 22, since it gave the kid who had chosen ‘2’ two more tallies.


First I demonstrated how to use the stones to make a program.  Then I gave each kid one box, and asked them to move it to the square closest to the blue dot.  It took each kid a couple tries, but soon they caught on.  They were all pretty good about fixing bugs and not giving up…though my son was a bit more fragile than the others.

As each kid finished, I gave them a new task. Move two boxes to the square closest to the blue dot.  One girl quickly wrote the program, but it turned out she had expected to be able to pick up two boxes at once.  My son teased her saying of course you can’t pick up two!  I assured her that her program worked and made sense, but I asked her to update it so it would work if you could only hold 1 box at a time.

Next I checked my son’s program. It turns out that he expected that the hand could hold 2 boxes, but that it would take two ‘red’ bead to pick up two boxes. He was very upset when I tried to explain that the hand could only hold one bead at a time.

Meanwhile, the girl had a new idea. She suggested we should take the program that moves one box, and do it twice instead.  I said this was a good idea, and helped her add a second line to her program, that was identical to the first one. We tested out the program, and found that the second time the hand ended up going too far left.  She fixed it by removing one green bead from the second line. I asked why it hadn’t worked, and we figured out it was because the hand originally started above the blue dot, but after dropping the first box, the hand was above the square where the box was dropped.

Next my son and the other kid both independently had the same idea that we should repeat the first program. We all worked together to try it, and then fix it.

This activity went very well, except that the kids all wanted me to check their programs with them at the same time.  I’ll have to figure out some way to remove that bottleneck. Perhaps the kids can work in pairs to check each other’s programs, now that they get the basic idea.


I let the kids vote for the next activity. Two kids voted for Set, and one voted for Venn Diagrams.  In Set the kids were fairly even, though there were many incorrect Sets picked up still.

Venn Diagrams

One kid said Venn Diagrams was boring, and I said I thought they would be fun. Once we started, everyone seemed pretty into it.  There was some disagreement on the Girls vs. Scary Things category, because one kid wanted to put the Castle and the Crown in the ‘Girls’ circle, but the other kids didn’t.

After we categorized all the squares I asked questions like: How many scary things were there? How many scary girls? How many things that were not scary and were not girls? These were all pretty easy. The only hard one was “How many things were either girls or were scary but were not both?” Some kids wanted to count the scary girls, some said they didn’t understand the question.

Venn Fairy Tales (Age 5)

The Activities

  1. Topic: Word Problems: Book: The Case of the Missing Zebra Stripes, “Henry’s Tale”.  We stopped to solve all the riddles in this story.
  2. Topic: Numbers: Using a deck of cards from 1-104 (e.g. Slide 5), we did several activities.  First, I gave each kid 10 random cards and asked them to find the largest one.  Then I asked what the largest one was overall.  Next I gave them each a stack of cards and asked them to find all the 50’s.  After that, they had to find all the numbers that ended in 2.  Finally, they needed to find pairs of cards that were flipped (e.g., 68 and 86; also 55 by itself).
  3. Topics: Odd One Out, Venn Diagrams, Attributes: Using a set of fairy tale character cards, I dealt sets of 4 cards, and then asked the kids to come up with ways to group them (e.g., these are animals, the others aren’t).  There were usually 3-4 different ways the kids came up with per set.  After doing that for a while, using big paper circles to make a 2-circle Venn diagram, I asked them to place all the cards where one of the circles was “Magic things” and the other was “Animals”.

How Did It Go?

We had 4 kids this week.

Henry’s Tale

The riddles were a good level for the kids; they solved them all but it took a while.

Number Cards

There’s still a wide variety of abilities for recognizing numbers; most but not all of the kids now recognize two-digit numbers. The kids found the numbers 100+ to be very amusing. Finding mirror images was pretty hard. One of the kids convinced everyone else to help find the number they were looking for. At one point, two different kids were each looking for the other’s card. One kid decided they really wanted to find the sum of all the cards, which actually was one of the activities in the big kids circle!

Venn Fairy Tales

The kids did a good job coming up with different ways to group the cards. The most interesting one was when we had gone up to 6 random cards, and the kid decided to split into “real” vs. “not real” — but 5 of the 6 cards were (probably) in the wrong group! (Gingerbread Man => real, Golden Goose -> real, Flute Player => not real). The Venn diagram went well, they didn’t use the overlapping region though until I asked them specifically where the dragon should go. We had to have a vote about whether the queen was magical or not (she looks like the wicked queen from Snow White).

Losing at My Own Game (Age 7)

The Activities

  1. Topic: Triangular Numbers.  Book: The Number Devil by Enzenberger, Chapter: “The Fifth Night”, pages 89 – 101.
  2. Topic: Triangular Numbers. Compute the 100th triangular number.
  3. Topic: Big Number Guessing. Think of a number between 1 and 1,000,000, how many guesses does it take to get it.

How Did it Go?

Number Devil

My daughter whined when she heard we were reading this book, but a couple other kids cheered, so it’s a mixed review.  This chapter was about triangular numbers.  Here is wikipedia’s picture of the first six triangular numbers: 

In the book, Robert learned that he could compute the 12 triangular number (1 + 2 + 3 + 4 + …+ 12) as:

1 2 3 4 5 6
12 11 10 9 8 7

Each column in this table adds up to 13, so the 12 triangular number is 13 * 6 = 78.

The kids all enjoyed seeing the pattern of how to compute the next triangular number from the previous (i.e. the 12 triangular number is the 11th + 12 more).

The 100th Triangular Number

After the book, I asked the kids what they thought the 100th triangular number was. The kids said the 10th triangular number was 55, so the 100th should be 55 * 10 = 550.  I said we should compute it similarly to the the way they figured out the 12th in the book.

The kids then suggested I should group the numbers into sets that add up to 13.  Instead I said we should pair up the smallest and the largest numbers.

1 with 100, 2 with 99, etc.  I asked how many pairs of numbers we would have? The kids at first said 100, or said they couldn’t figure it out.  Then I showed how there had been 6 pairs when there were 12 numbers, and one girl said “Oh it should be 50!” and explained that she divided 100 by 2.

Next we checked how much each pair added up to: 101.  So the 100th triangular number should be 101 * 50.  One girl tried to compute this long-hand, but got immediately frustrated because she had only done 2 digits times one digit in school. She felt that this would be impossible until her school taught it.

I suggested first calculating 100 * 50. Eventually someone said that was 5000.  But we were supposed to compute 101 * 50, so how many more 50s do we need to add? My daughter said one more, so the 100th is 5050.

Big Number Guessing

First we warmed by up having the kids guess my number between and 1 and 20. This was easy. Then I had one kid write down a number between 1 and 1,000,000. He chose 456, 276.  The other kids thought it would take 800 guesses to find his number.  I gave them 25.  They got down to 456,267…456,299 before they ran out of guesses.  This was definitely NOT easy for them to think of a number between 461000 and 447,000.

Next I claimed that I could guess their number between one and one million in just 25 guesses. They wrote down a secret number, and I started guessing, doing binary search.

It went well at first, but I wasted a few guesses by getting mixed up with whether they had said higher or lower. Then when I had 10 guesses left, I got mixed up again and wasted about 6 guesses going the wrong direction.  I hit 25 guesses and did NOT get their number.  My daughter kindly gave me two extra guesses, and I got it, but I totally lost.  The kids enjoyed that 🙂

This was a good activity because it works on their sense of big numbers. But it was hard enough that some kids (including my daughter), didn’t really want to participate, so I had 2 – 3 very interested kids and a couple more who were busy drawing.

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.

Pompeii Money (Age 5)

The Activities

  1. Topic: Lines, Shapes. The Case of the Missing Zebra Stripes by Time-Life Books. (Page 40 – 47). This section was about lines: parallel, intersecting, shapes.
  2. Topic: Money, Addition. I showed the kids glass beads that I said were money from a different country.  Green beads were worth $3, red beads were $2, and yellow beads were $1. First I handed out small handfuls of money, and helped the kids add it up.  Next we figured out how much more money each kid needed to have $20. Finally, I opened up a Math Circle store where the kids could spend their money on tiny toys.

    My kids playing 'store' after circle.

    My kids playing ‘store’ after circle.

  3. Topic: Attributes. I taught the kids Set.  We only used the solid colored cards. First I explained what a Set is: a set of three cards where all three cards either match or are different for each attribute.  Next we looked at two cards and figured out what 3rd card would make a set.  Finally we played Set, laying out 9 cards at a time. Each kid would raise their hand if they thought they saw a Set.

    Each row is a Set.

    Each row is a Set.

How did it go?

Pompeii Money

The kids all said they had seen money before.  I told them that we would play with pretend money from a different country.  I asked what our country should be called? One boy immediately proposed “Bumpitup” which he said was a country that had been destroyed by a volcano.  I asked if he meant “Pompeii”, and he said yes.

I showed the kids the values of the different colors of money, then handed a few beads to each kid.  The kids really varied in their ability to add up the money.  Two of the kids just wanted to count the beads, not add up the dollar amount.  They seemed to understand what was going on, but needed one-on-one help to add 3+3, etc.  Another kid could pretty much do it on their own, with just a little help.  My son could quickly add it all up himself.

We did two rounds of adding small groups of money.  Everyone soon understood that greens were the best, since they were worth three.  Next I worked with the kids to add money so that each had $20.

I asked “What’s the fun thing about money?” One kid said the fun was that you can use money to buy more money.  I then opened the store, which had small stickers and toys available for $2, $5, or $10.  The kids took turns picking an item and paying me the correct amount (often needing help).


My son had played Set before, but the other kids had not.  We practiced finding Sets, and then eventually played a couple rounds. If one kid fell behind, I let that kid have extra time (and clues) to find the next Set.  All 4 kids caught on by the end of circle, and were excited to play.

JPM (Jumps Per Minute) — Age 5

The Activities

  1. Topic: Time: Book: The Case of the Missing Zebra Stripes, “Monkey with a Minute”.
  2. Topic: Time: We figured out how many times we could do various things (draw triangles, jump) in one minute.  We counted to 60 seconds using “1-Mississippi”, and watched the second hand on an analog watch for one full rotation.  We also tried to stand on one leg for a minute.  Finally, we timed 4 different sand hourglasses (1 minute, 2 minutes, 3 minutes, 5 minutes).IMG_1709
  3. Topics: Attribute Blocks, Sequences: Each kid made a chain of 10 attribute blocks, varying only one attribute per step.  Then they had to make it into a loop.  Finally, the whole group worked together to make a loop out of all 60 blocks.
  4. Topics: Geometry, Optimization: We revisited the activity with pastures and animals.  The rules are only one type of animal per pasture.  Each 1×1 square can hold two animals, and each equilateral triangle can hold one animal.  The goal is to hold a certain number of animals using as few fences as possible.  Initially, each kid had their own set of animals, but at the end the kids worked together to make one farm for a bunch of animals.

How Did It Go?

All five kids attended this week.

Monkey with a Minute

This was only a couple of pages, but it gave the idea for the next activity.

How Much in a Minute?

I used an iPad to do all the timing, they all enjoyed watching the digital display.  As usual they loved jumping — but one minute is a pretty long time to jump with such high energy!  Only one of the kids managed to stand on one leg without touching their other foot for a full minute.  The hourglasses turned out to vary in accuracy: 1:05, 2:01, 2:45, and 4:55.

Attribute Block Sequences

Most of the kids have a pretty good handle on this now, but many still make mistakes.  The kids are more inclined to vary shapes than any other attribute, followed by color — so there were lots of long sequences of thick shapes, for example.  After they had gotten a full loop using most of the shapes (which they could do mostly by themselves, with a few mistakes), they weren’t that great at finding places to add the few remaining shapes.

Fencing Pastures

All the kids understood the rule about only one kind of animal per pasture, and most understood the rules for how many animals there could be per pasture (although they’re not that great at actually counting up larger pastures).  They were less consistent about trying to use as few pastures as possible; and none of them really understood that you could save fences by putting all animals of the same type close together.  In the group activity, we ended up with one region built by a couple of the kids that was pretty efficient, and then a few outlying pastures — initially there had been a bunch of large unused pastures connecting the outlying pastures, but the kids realized they could take away all those fences.