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.


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 youcubed.org.  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.

Mother’s Day Origami (Age 8)

The Activities

  1. Topics: Combinatorics, Combinations:  Anno’s Three Little Pigs by M. Anno.
  2. Topic: Origami:  To celebrate Mother’s Day, we made two different models: a double heart from Essential Origami and a rose from Origami Made Easy.IMG_1898

How Did It Go?

We had four kids this week.

Three Little Pigs

We spent some time looking at and understanding the pictures.  We’ve read this book before, and while I still don’t think they fully understood it, they understood a lot more than last time.


The double heart model was fairly tricky and they needed some help.  But it’s a pretty cool model!

Eggs and Boxes (Age 6)

Age 6

The Activities

  1. Topics: Numbers, Codes, Algebra:  The Cat in Numberland, Chapter 3, by I. Ekeland.  In this chapter the letters come to visit the numbers, and we learn about letter/number ciphers and letters standing in for numbers.
  2. Topic: Algebra:  I made problems of the form “X + 3 = 5” using unit cubes from Base Ten Blocks and a small cardboard box.  I.e., I would secretly put 2 blocks into the box and close it, put 3 blocks next to it, and then say “There are 5 blocks total, how many are in the box?”
  3. Topic: Primes:  I introduced the idea of primes using Base Ten Blocks: a number N is a prime if the only rectangle you can make using N blocks is 1 X N.  I gave different numbers to each kid and had them figure out whether it was prime or not.
  4. Topics: Combinations, Combinatorics:  I printed a bunch of “Easter eggs” with a top and bottom section.  Using five different colors of crayons, I asked the kids to make as many different eggs as they could, coloring each section in solid colors (not stripes/dots/etc.).  I taped each one to the wall (stacking repeats).IMG_1886

How Did It Go?

We had four kids this week.

Cat In Numberland

The algebra in this chapter is tricky because it includes addition, subtraction, multiplication, and division; most of the kids don’t know multiplication or division yet.

Box Algebra

This worked pretty well.  The kids understood what was going on right away, and they were always excited when I opened the box and dumped out the blocks inside to see if their guess was correct.  At the end they made a problem for me, which was something like “X + 3 = 39” (of course, they used as many blocks as they could).

Rectangle Primes

We did up to about 14.  I kept track of each result.  The only odd composite number <= 14 is 9, so for the most part they just needed to check a 2 row rectangle.  Proving something is prime is tricky, of course, and whenever a kid said that something was prime, I always asked them “did you check 3-wide”?  Whoever had 9 didn’t initially check 3×3.

Easter Eggs

The kids were really into this activity and worked very hard to get all the combinations.  They got all 10 two-color combinations pretty quickly and without help (first two rows in picture above) — but there was no pattern to which color was on top vs. bottom.  Then one of the kids realized that you could flip the colors.  They quickly got 6 more, but the next 3 took them a lot longer to find, and I had to help them find the last one.  This got them to 20, but they didn’t think of having the same color on top and bottom.  I suggested it to them and they quickly made the last 5.  Then I rearranged them so that there were same color tops along the rows and same color bottoms along the columns.  I realized afterwards that I should have made this chart before I gave them the hint about same color top/bottom, because then there would have been gaps and I could ask them what went in the gaps.


Jump, Jump, Floopsy!

The Activities

  1. Topic: Counting: Book: Prairie Dogs Perching: Counting By 3s by A. Tourville.
  2. Topics: Numbers, Counting: We recently bought a 100 tile board, so we did a few activities with it.  First, taking turns, I pointed to a number and had the kid say the number.  Next, still taking turns, I gave a random number tile to the kid and had them place it on the board.  Finally, we used the colored tiles to count by 3’s.
  3. Topics: Combinatorics, Combinations: We had print-outs of Easter Eggs with two halves, top and bottom, to color.  There were four colors of crayons, you could use the same for top and bottom, and the rule was you couldn’t personally make the same combination twice (it was okay if you repeated what someone else did).  Each time someone finished an egg, I taped it to the wall.
  4. Topics: Reflections, Symmetry: I had printed out individual squares with letters in large font, and then folded them along a line of symmetry.  Taking turns, the kids had to guess what letter it was.  We used a hand-mirror to check before I unfolded it (by holding the folded symmetry line flush against the mirror at a 90 degree angle).  Then, I drew a couple half shapes (e.g., semi-circle) on a piece of paper and asked each kid to draw a completed reflection.  We discussed the several different ways to reflect each shape.  Finally, each kid drew their own shape which I reflected in the mirror.
  5. Topic: Counting: Book: How Many Sharks in the Bath? by B. Gillham.  Each page had 4 counting problems, and a place on the right with numbers 0-10 so you could point to your answer, so I did “counting twister” where one by one 4 different kids answered the questions and then held their finger on the answer.
  6. Topic: Programming: We did “Dance Programming” with 4 instructions: Jump, Up, Down, and Spin.  The kids stood in a line facing me, and I explained each of the instructions.  Then, I gave them sequences of up to 4 instructions, which they did (sometimes I did it as well, but not always so they couldn’t just copy me).  Next, I introduced the idea of functions and gave them two, “Moopsy” which was “Jump, Jump, Jump”; and “Floopsy”, “Spin, Down, Up”.  I then gave them some programs including these two functions plus basic instructions.  Finally, each kid gave me a program with up to 5 instructions that I had to do.

How Did It Go?

All 5 kids attended.  There was a bit of “when will circle be done?” after the egg coloring, but they liked the reflections and dance programming enough that it stopped after that.

Prairie Dog Counting

There was a lot of counting by 3’s in this book, I had the kids count along with me on many of the pages.

100 Tile Board

The kids are in pretty different places on recognizing, saying, and finding numbers above 20.  Some don’t know to say “fifty-four” for 54, saying “five four” instead.  Most of them didn’t realize that the rows went by tens, in increasing order; and most don’t have the sense that 84 is large and 18 is small.  But they were all happy to try and find the numbers on the board, so this activity went well, and they were improving as we went on.  After we had done two rounds of placing random tiles on the board, I gave them each 3 to place on the board (and then gave more to the kids who finished first).  The skip counting also was pretty helpful, some of the kids who have trouble counting by 3’s in the air were able to do it on the board.

Easter Egg Coloring

The kids got the idea right away.  After a while, a couple of them started looking for eggs that were new on the wall (as opposed to just making sure they were new for themselves).  In the end we got 13 different eggs out of 16, not bad.  They seemed to understand the difference between top red/bottom blue and top blue/bottom red.  We also got a few “rainbow eggs” that didn’t follow the rules — I made the mistake of taping those to the wall in a different place, which didn’t discourage making them.


At first, for each letter, after the kid guessed I would first show it held up against the mirror, then unfold it.  They were all surprisingly good at this activity, and we got through a whole pile of ~ 20 letters.  They also were pretty good at completing the shapes that I drew.  4 kids completed a semi-circle as a circle, the 5th reflected vertically.  I drew half a diamond, which was harder — most of the kids drew a triangle, while one drew an X.  I also drew a random shape which was harder still, but several of the kids successfully reflected it.  The kids were not as into drawing their own shapes as I expected, and they mostly didn’t get the idea of drawing half shapes (so, for example, one shape was a house, rather than a half-house).

How Many Sharks in the Bath?

Since the kids were doing the counting, each page took a little while; so we only did about half the book.  It was interesting watching the kids arrange themselves so they could all touch their numbers at the same time.

Dance Programming

Everyone liked this activity and followed along with the instructions; some of them were definitely able to follow successfully, but some may have been copying other kids.  Similarly, some of them understood the functions, but probably not all.  I got a little dizzy doing the kids’ programs :).

1 Trillion Easter Eggs

The Activities

  1. Topic: Ratios: Book: Beanstalk: The Measure Of A Giant by A. McCallum.
  2. Topics: Combinations, Combinatorics:  I had printouts of eggs with the top and bottom separated by a line.  First, I gave them 3 colored markers and asked “If you have 3 colors, how many ways can you color the eggs?” (Answer: 9).  Then I asked about 5 colors, without actually giving them 5 colors, to see if they could figure out the pattern without actually doing the coloring.IMG_1328
  3. Topic: Multiplication: I had a bag of plastic Easter eggs with a slip of paper with a number from 1-9 inside each one.  At first, each kid drew out two eggs and had to figure out the product of the two.  After a few rounds of this, they started drawing out three eggs and multiplying them all together.
  4. Topic: Tesselations: Using pattern blocks, we worked together to make this pattern:IMG_1329

How Did It Go?

Four kids attended this week.  Everything went pretty smoothly this week.  At the end of circle, all the big kids went and voted on the birthday party activity in the little kids circle.

Beanstalk: The Measure of a Giant

This book was about ratios and was a good level for the kids.

Egg Coloring

When I asked about 3 colors, several of the kids immediately began coloring, and quickly found all the combinations.  They clearly have gotten better at looking what’s missing rather than just trying random combinations.  I asked how many there would be with 5 colors, the most popular answer was 15 (5 * 3).  Then I arranged the eggs into a grid (as shown above) with same color tops in the rows and same color bottoms in the columns.  I asked them questions about how many were there if you only had 2 colors and 1 color, and arranged the eggs in expanding “rings” to show what gets added each time you add a new color (this suggests another activity, proving that n^2 = sum of first n odd numbers).  I also pointed out how each column and row corresponded to a bottom/top color.  Finally, I asked what shape the chart was for each of 1, 2, 3 colors, and how many eggs on each side; at this point one of the kids saw that the pattern was to make an n x n square.  When I asked about 10 colors, I still got two kids saying 3 * 10, one saying 10 * 10, and one saying “I don’t know.”  After a bit more discussion they all decided it was 10 * 10.  I also asked 1000 colors and 1 million colors just for fun.

Egg Multiplication

A couple of the kids have already memorized their multiplication tables, so the two number multiplication was very easy for them.  However, three numbers was much more complicated.  In particular, they definitely don’t have the idea of multiplying place by place.  One of the largest multiplications was 9 * 9 * 6, quickly reduced to 81 * 6.  I gave them lots of hints for what 80 * 6 was: first I asked what 8 * 6 was (immediately got 48), and then using Base Ten Blocks, I said “If 6 * 8 unit blocks is 48 unit blocks, what is 6 * 8 ten blocks?”  No one ever realized the answer was 48 ten blocks — someone eventually added 80 6 times in their head to get the answer.  They were able to say that 48 ten blocks was 480 once I pointed out that 6 * 8 ten blocks is 48 ten blocks.


I had come up with this pattern when experimenting (playing) with the blocks one day, and thought it was both pretty and somewhat challenging.  At first, they didn’t get the rules and had trouble expanding the pattern.  But once I pointed out that the yellow was surrounded by blues and whites, and the green was surrounded by blues and whites, after some practice they were able to continue expanding the pattern.

Babb Abab Baaa

The Activities

  1. Topic: Subtraction: Book: Subtracting With Sebastian Pig and Friends on a Camping Trip by J. Anderson.
  2. Topic: Addition: I put a bunch of slips of paper with numbers 1-9 inside of plastic Easter eggs.  Each kid drew out 2 eggs, opened them, and added them using Base Ten Blocks.  After a bit, they switched to drawing 3 eggs instead.
  3. Topic: Patterns: We made a number of patterns using Rummikub tiles, which have numbers 1-13 in one of 4 different colors.  Taking turns, the kids completed the patterns in increasing order of difficulty.
    1. 9,10,11, all blue
    2. 2,3,4,5 alternating black and red
    3. 1, 1 tile face down, 2, 2 tiles face down (in a stack), 3, 3 tiles face down
    4. 1,3,5,7, all yellow
    5. 2,4,6, all red
    6. face down stacks with 1 tile, 2 tiles, 3 tiles
    7. 1,4,7, all blue
    8. face down stacks with 1 tile, 2 tiles, 1 tile, 2 tiles
    9. 2,3,4, all yellow, rotating clockwise
    10. 1,1,2,3,5, all yellow


  4. Topic: Numbers: Zero is the Leaves on the Tree by B. Franco.
  5. Topic: Numbers: We discussed some interesting zero facts.  What’s 2+0, 3+0, 1 million + 0, …; what’s 2 – 0, 3 – 0, …; and even multiplying by 0.
  6. Topic: Combinations, Combinatorics: I told the kids that baby language had only two letters, A and B.  The task was to find all the 4 letter words in baby language.  The kids took turns adding new words to a list taped to the wall.IMG_1320

How Did It Go?

All 5 kids attended.

Subtracting with Sebastian Pig and Friends on a Camping Trip

The kids liked finding the mice who had stolen all the pig’s things.  They’re still not that great at problems like 11 – 7.

Easter Egg Addition

This went quite well, all the kids were into it.  Most of the kids needed Base 10 Blocks to do the addition.  A couple of them needed help figuring out how to use them, but got the idea after a while.  Our son has practiced addition a lot, so I gave him multiplication problems instead.

Rummikub Patterns

We took turns with which kid got to touch the tiles to make the solution, but all the kids worked together to figure out the answer.  As we’ve seen in the past, counting by 2’s and 3’s is fairly easy for some kids and difficult for others.  The two trickiest ones were 1, stack of 1, 2, stack of 2, …, because they wanted to just do the stacks, and not include the numbers; and, not surprisingly, Fibonacci, they didn’t figure that one out.

Zero is the Leaves on the Tree

For each picture, I asked the kids why there were 0 of whatever it was (e.g., why were there 0 leaves on the tree during fall?)

Zero Facts

They picked up on the addition and subtraction pretty quickly, although I did need to show it on my fingers at first.  For multiplication, they wanted to say 0 * 5 = 5, not too surprisingly since that was the pattern from addition and subtraction.  It took them several explicit run throughs of “What’s 0 + 0 + 0 + 0 + 0?” before they got the pattern.

Baby Language

They did quite well on this, they got about 12 before I started helping them.  I wanted to make sure everyone got the same number of turns writing on the sheet, so I helped on the last few.  We were running short on time, so I didn’t ask them whether that was all of them and how could they tell.