Birthday Treasure Hunt (Age 6)

The Activities

  1. Topic: Multiplication. Book: Too Many Kangaroo Things To Do, by Murphy. This book is about friends planning a surprise party for Kangaroo, using multiplication along the way. The kids all enjoyed the book, taking turns computing the simple multiplication (1×1 up to 4×4). One kid proudly predicted that the animals must be planning a surprise party.
  2. Topic: Various, Story Problems. I made a grid of hexes that were hidden at first. The goal was to find the hex with a diamond printed on it. Each turn the kids got to move their piece to uncover a new hex and then solve a different type of math problem for each picture type. Here are the hex pictures you need, and the full list of problems is below. We worked as one team, and I asked each kid to try each problem. If someone solved it faster than the others, then they were supposed to whisper the answer in my ear instead of shout it out. As soon as the jewel was uncovered, all 4 kids got to pick a prize from our treasure box.IMG_20160821_174304
    1.  Firefly – square numbers:
      1. First square bigger than 0.
      2. First square bigger than 5.
      3. First square bigger than 10.
      4. First square bigger than 20.
      5. First square bigger than 30.
      6. First square bigger than 40.
      7. First square bigger than 50.
      8. First square bigger than 60.
      9. First square bigger than 70.
    2. Unicorn – fractions:
      1. Divide a circle in half, then split each piece into 3 pieces.  How many pieces do you have?
      2. Divide a circle in half, then split each piece in half, then split each piece in half. How many pieces do you have?
      3. Divide a circle in four pieces. Then split each piece in 3 pieces. How many pieces do you have?
      4. Divide a circle in half. Then split each piece into 3. Then split each piece into 2. How many pieces do you have?
    3.  Dragon – money:
      1. A diamond ring costs $100. How many rings can Hans buy with $125?
      2. Diamond earrings cost $20. How many earrings can Olaf buy with $207?
      3. A diamond necklace costs $11. How many necklaces can Marshmallow buy with $110?
      4. Elsa bought 20 diamond rings that each cost $10. How much money did Elsa spend?
      5. Sven bought 4 bracelets that each cost $32, and 3 rings that each cost $14. How much money did Sven spend?
      6. Anna spent $60 on 5 necklaces. How much did each necklace cost?
      7. Hans spent $39 on 3 bracelets. How much did each bracelet cost?
    4. Troll – story problems:
      1. A troll had 12 muffins. He ate some of them. Now he has 7 muffins. How many did he eat?
      2. There are 20 muffins. Some trolls came. Each troll ate 4 muffins. How many trolls are there?
      3. 4 trolls brought muffins to a party. Each brought the same amount. There are 24 muffins at the party. How many did each troll bring?
    5. Witch square – codes: Figure out what the coded word is by subtracting the given number from each letter. For example, DBU -1 = CAT
      1. -1:  DBU
      2. -2: DTQQO
      3. -1: QPJTPO
      4. -2: JCV
      5. -1: TQFMM
    6. Maze – patterns:
      1.  1 5 9 13 __   __
      2.  1 2 2 3 3 3 4  __  __  __  __
      3. 91 82 73 64 __   __   __
      4. 11 22 33 __  __  __  __
      5. 1 1 2 3 5 8 __  __  __
      6. 1 2 4 8 __  __


    How did it go?


We had four kids today and they were all very motivated by wanting to earn a prize in honor of my son’s upcoming birthday. We played the game with 37 hexes, and the kids got unlucky and didn’t find the jewel until they had uncovered 30 hexes. Toward the end I started letting them move 2, 3, or 4 hexes without solving the problems, just to make sure we found the jewel.

All four kids worked hard on the game questions. My son is quite far ahead of his age in calculation and story problems but he did a really good job not telling the other kids the answers. The other kids stayed involved though, and we made sure to work out each answer as a group, using Base Ten blocks or counting on our fingers if necessary. One kid got bored after 30 minutes but didn’t distract the others. Another kid especially enjoyed problems the required counting by 4, 20, or 11. At first he didn’t think he could count by 11s, but quickly he saw the pattern and took the lead.

The fourth kid is the least comfortable with the number line but he got really excited by square numbers and solved all three square problems before anyone else (smallest square above 0, smallest square above 5,  smallest square above 10). We used Base Ten Blocks to do this. I showed the kids how 9 is a square number because you can make a square out of 9 unit cubes, and he then spent some time making other squares out of unit cubes. He also solved this pattern: 1, 2, 2, 3, 3, 3, 4, _, _, _, _ first.

Everyone enjoyed decoding the witch’s code and trying to sound out the trickier words…pasta? pesto? poh-aye-son? Ooohhhh: poison!

The unicorn fraction problems turned out to be tricky. All the kids could follow the instruction: draw a circle and divide it in half. But “Now divide each piece into three pieces” was tricky. Only my son figured out how to divide each half into three equal pieces. The other kids ended up drawing straight lines and getting three very uneven pieces. Most kids also forgot to divide *each* half, so they would get ‘4’ as the answer instead of 6.

We finally uncovered the jewel, and celebrated. Then everyone picked a prize and ran around outside to get rid of their pent up energy. A very successful circle!



Laziness is a Virtue (Age 8)

The Activities

  1. Topics: History of Math, Mathematicians: Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians (Volume One) by L. Reimer and W. Reimer, Chapter 5 (John Napier).
  2. Topics: Multiplication, History of Math:  I printed out one set of Napier’s Bones per kid.  I showed the kids how to use them to do multiplication of a large number by a single digit number.  A follow-up that we didn’t do this time is to use the bones for multiplying two large numbers.
  3. Topic: Sorting:  The kids attempted to sort cards with all numbers from 1 to 300.

How Did It Go?

We had four kids this week.

John Napier

The kids liked the (non-mathematical) stories about John Napier and the rooster and John Napier and the pigeons.  We’re now thankfully past the part of the book where the mathematicians all die at the end of the story.

Napier’s Bones

The kids understood how to use the bones fairly quickly.  One of the kids is very good at paper-and-pencil multiplication, so while I could slightly win on large number X single digit multiplication, the other kids couldn’t.  I showed them why it worked (see second picture above).  The place where the bones really shine is multiplying two large numbers, since you can use the same arrangement of the bones for each single digit multiplicand, so we should do that race at some point.

Sorting to 300

The kids decided to use labels, this time making sure to have labels bigger than 200.  It took them 10-15 minutes to make the labels, so next time I’m going to count this time as part of the overall time.  In particular, they didn’t think of simply using the cards that were multiples of 10 as “labels”.  They went with the same strategy of running back and forth even though I tried to get them to come up with something better; our daughter said that she WANTED to run back and forth so she could get some energy out.  I let them go for 17 minutes (which meant that circle went over by ten minutes), and then got 200 of the 300 cards sorted.  One of the kids was feeling tired and decided to first find all the numbers between 1-100 and do those first, since that required less running.  If only more of them were lazy…

Counting without Seeing (Age 5)

The Activities

  1. Topic: Zero, Multiplication, Addition. Book: A Place for Zero by Lopresti. This book covers the additive and multiplicative properties of zero in a very flavorful and engaging way.
  2. Topic: Counting. We used the Base 10 Blocks to count to 100 by 10s, and 1000 by 100s and 10s.IMG_20151129_175021
  3. Topic: Search, Pan Balances, Inference. I showed the kids a box that contained some number of plastic leaves.  I asked them to figure out how many leaves were in the box without opening the box.  I gave them a pan balance, and an identical, empty box.IMG_20151129_174846


How did it go?

There were only 3 kids at circle this week due to Thanksgiving travel.  It was a really good, focused circle.  All three kids paid attention and took turns very nicely.

A Place for Zero

The kids really enjoyed this book, about poor Zero who has no place in numberland. One kid predicted that Zero would be paired with a One to make 10 (which turned out to be true).

The book first shows how any number + Zero = the number.  Next Zero headed off to multiplication land to try to multiply himself.  At this point, I paused the book, and used Base 10 Blocks to teach the kids about multiplication.  Two of the kids had not done multiplication before, but I explained that 2 * 3 means you make 2 groups with 3 items in each group, and then count how much you have.

We did a couple small problems like that, and then I asked how much would 0 * 7 be? Two kids thought it should be 7.  My son said it would be 0.  I said, well, if we make Zero groups each with seven cubes, how may cubes do we have?  The kids agreed that would probably be zero, but were not super convinced.  I figured that 7 * 0 was probably easier to explain.

I said, “If I give you 7 bags which each have 0 pieces of candy in them, how much candy did you get?” All three kids immediately saw that they would have no candy.  After circle, I quizzed one girl, in front of her mother: “What’s 1,000,000 times 0?” and she instantly said “0”, and went on to explain that 1 million bags, each with no candy, makes 0 pieces of candy. 🙂

After we got a bit of intuition about multiplication, I finished the books, and the kids were all excited to see that 0 multiplied by any number made 0.  They also loved seeing 0 and 1 pair up to make 10 and lots of other big numbers.

Base 10 Blocks

After the book, I had the kids try out some big addition problems, like 22 + 33, using base 10 blocks.  First you make 22 by getting two 10-bars, and 2 unit cubes.  Then make 33, and combine the two piles and count the result.  All three kids were able to do this.

My son looooves addition and multiplication, so these problems were too easy for him, but he was thankfully very patient during this activity.  He begged the other kids to give him a hard problem, and the hardest one they thought of was 100 – 2.  My son said 98, and we all checked that he was right by counting 2 higher than 98.

Next we used 100 squares to count by 100 to 1000.  Then we used 10 bars to count by 10s to 1000. The kids each made 100 out of 10 bars, and I carefully added it to our stack, so in the end we had a block of 10 bars the same size and shape as the 1000 cube.

Counting Without Seeing

Next I showed the kids the pan balance, and said they should use it to figure out how many plastic leaves were in a treasure box (without opening it). If they could get it right, they would each earn a treasure.

They quickly got the idea of putting a handful of leaves in the empty box and weighing it vs the treasure box. They were very good at interpreting the result, knowing that if their box weighed more than the treasure box, then they should take out some leaves.  They even understood that they should only adjust it by a few leaves if the boxes were close in weight.

At one point, it looked like the boxes weighed the same, but the kids wanted to test to make sure, so they first added one leaf (too heavy), then took away two leaves (too light), then added back a leaf to get a match.

To find our answer, we opened up the kids’ box and made piles with 5 leaves in each. We had 41 leaves.  We then checked the treasure box, and found 41!! Everyone cheered!

Spending a Million Dollars (Age 7)

The Activities

  1. Topics: Multiplication, Division: Perfectly Perilous Math, Challenge 3 — if you spent 50 cents every second, how many days does it take to spend 1 million dollars (to nearest day).IMG_1742
  2. Topic: Combinations: We continued the pumpkin activity from last week.  Last week, they had made a bunch of pumpkins but hadn’t checked for duplicates.  This week, we tried to come up with strategies for finding all the duplicates.  Also, I started with the simplest version of the problem (circular eyes, nose, and mouth), and then gradually added elements (non-symmetric parts, multiple choices), computing how many different combinations there were each time.
  3. Topics: Logic, Measurement: Perfectly Perilous Math, Challenge 4 — if you have one bucket that holds 3 quarts, and one that holds 5, how can you measure exactly 4 quarts using just those two buckets?  We worked as a group to solve this problem using various containers and beads (instead of water).

How Did It Go?

We had all five kids this week. All the activities this week were group activities, which turned out to be a problem. Most of the time, only two or three of the kids were actually working on solving the problem; the others were drawing, making paper airplanes, or otherwise not paying attention. Kid A grabbed and crumpled another Kid B’s paper, and later in circle (unrelatedly) Kid B threw a container of beads at Kid A, which we all had to help pick up. I think another issue was that the problems were all pretty hard; the kids working on them did a good job, but it discouraged the other kids.  One of the kids who drew during the Million Dollars activity said they couldn’t help because they didn’t know how to do multiplication, and then was very attentive for the remaining two problems.

Spend a Million Dollars

One of the kids is a lot better at large multiplication than the others, so that kid did most of the computations.  Another kid helped figure out what the right numbers to multiply were.  The kids figured out how many seconds were in a day, but then they were planning to multiply by 50 (cents) — they didn’t realize that 50 cents was half a dollar.  They also needed help figuring out how to divide 1 million by 43,200 — they know the method of repeated addition/multiplication, but I think their number sense suffers a lot past 1,000.  They got pretty close to getting the right answer, but I had to a help a lot for the final few steps.

Pumpkin Combinations

One of the kids suggested sorting the mouths by orientation, and then by type of mouth.  A different kid decided to implement this, but changed it to have 1 column for each of the combinations of 4 mouths.  Some of the other kids helped sort (there were about 35 pumpkins total), and eventually the pumpkins were all in columns.  There were still 8-9 pumpkins in each column, and they mostly stopped making progress at this point, besides finding a few duplicates ad hoc.  I suggested putting the circle eyes above the triangle eyes, but they didn’t take to this idea.

Then I switched gears a bit and started with a problem with circle eyes, nose, and oval mouth.  This meant there was only one possible pumpkin, which the kids figured out right away.  Then I made the nose triangular.  There were answers of both 2 and 4; the 4 answer was reasonable because they showed how you could put the straight edge in any of the 4 directions (although two of them make pretty weird noses).  I said we should only allow the two directions, and then added triangular eyes.  They got 4 right away.  Then I added a smiley/frowny mouth, and it got much harder.  I got answers of 4, 6, and 8.  6 is an interesting answer, because you could think of that if you considered varying each dimension one at a time.  I had the kids try to write out all the combinations — I really should have had them to all 4 for the previous problem first, because the key insight here is that if you group the faces by smiling vs. frowning, you have the same set of 4 faces from the round mouth case, once for each mouth.  In the end, one kid was able to figure out that there should be 32 different answers for the previous week’s problem, but wasn’t able to figure out which faces were still missing.

Tricky Buckets

The beads worked okay, although one of our containers was more like 2.6 than 3, so the measurements didn’t quite come out right.  Also, we ended up picking up a bunch of beads because of the bead-throwing incident.  One of the kids figured out a solution right away using a third bucket; and then they were able to come up with the 2-bucket solution with only a bit of help.  I asked a follow-up question about a 5 and 7 bucket (trying to make each of 1, 2, 3); the kid who had done the best on the first part was able to make progress and solved 2 and 3 but not 1.

Zombies and Torture Chambers (Age 7)

The Activities

  1. Topic: Multiplication, Story Problems. Book: The Book of Perfectly Perilous Math by Connelly. Chapter 1: The Pit and the Pendulum. In this chapter you are tied to a table as a sharp pendulum swings back and forth, lowering with each swing. Meanwhile a rat is one minute away from gnawing through the ropes that hold you. Will you escape before the pendulum kills you?
  2. Topic: Pendulums. Each kid had a spoon pendulum. We experimented with how many times the pendulum would swing back and forth in a certain amount of time.
    1. All 5 pendulums were the same length, but dropped from different heights.
    2. 5 different length pendulum.
    3. One pendulum that we all counted together, with different length strings.

      Spoon pendulum.

      Spoon pendulum.

  3. Topic: Logic, Story Problems. Chapter 13: The Rope Bridge, from the Perfectly Perilous book. Zombies are twenty minutes away from you. You and four friends have to cross a rope bridge. Two people can go at a time, and  you have to share one flashlight. Each person takes a different amount of time to cross: 1, 2, 3, and 8 minutes. Can you all cross in time?

    Our picture of the problem, complete with a Minecraft zombie.

    Our picture of the problem, complete with a Minecraft zombie.

  4. Topic: Combinations. How many different jack-o-lanterns can you make with a set of eyes, nose, and mouth?  There are 2 different eyes, 1 nose, and 2 mouths. Each feature can be right side up or upside down.

    Our messy pumpkin patch.

    Our messy pumpkin patch.

How did it go?

We had all 5 kids this week. Halloween was yesterday so we had spooky, Halloween themed activities.

The Pit and the Pendulum

The kids instantly loved this book. They wanted to look at all the chapters at once, and were excited (and slightly scared) by the short stories.  The book says the pendulum swings back and forth every 7 seconds, and drops one inch per second. The pendulum is 15 inches above you. How long till it hits you? The kids quickly figured out that the answer would be 15 X 7. One girl computed this using long-hand multiplication. I had the other kids check her work by computing: 10*7 + 5*7, which they could all do.

So the pendulum would drop in 105 seconds, but the rat will untie you in 60 seconds, leaving 45 seconds to escape.

Pendulum Spoons

I gave each kids a pendulum spoon. I wanted each pendulum to be 12 inches long so I had a measuring tape. All the kids wanted to measure their own pendulum, which slowed us down.  Eventually, they were all set. Everyone released their spoon at the same time, and counted their swings for 20 seconds. Three kids got 20 swings, and two got 18 swings, so it was pretty close.

Next each kid chose a different length of pendulum. We had some more trouble because at first everyone wanted a long pendulum, and then as soon as I convinced a couple kids to use a short one, then everyone wanted a short one. Finally I got 2 kids with long strings, and 3 with short ones. We tried to time them for 20 more seconds, but spoons were hitting each other, and everyone was laughing and forgetting to count.

I corralled everyone together again, and we used just one pendulum and counted it together, with a long, medium and short string. We did the long string twice to make sure our result was repeatable. We found that in 10 seconds, the long pendulum swung 6 times, the medium 8 times, and the short 12 times.

I asked why that might be. One girl suggested that different parts of the rope may be pulling on the longer pendulum. My daughter suggested that the longer pendulum travels further on each swing.

The Rope Bridge

Again, the kids loved the themes in this book. They all wanted to draw and talk about zombies, and also to solve the problem to see if the kids lived. Two kids quickly came up with the same idea: the fastest person should go with a friend across the bridge, and then run back with the flashlight to get the next person.

Everyone liked that plan, but one girl was worried it would be too slow. So, we worked it out together. First the 1 minute and the 8 minute person crossed. There were some interesting ideas about how long it would take them to get across. Someone said it would take 9 minutes. Someone else said it should take 1 minute, because the fast person could carry the slow one on their back. We finally decided it should take 8 minutes, becuase the 1 minute would have to slow down to the speed of the slower person.

We worked out it should take: 8 + 1 + 3 + 1 + 2 = 15 minutes to cross, which gave us one extra minute before the helicopter arrived to rescue us from the zombies.

Pumpkin Combinations

My daughter was really excited to do this one, because the older circle doesn’t work with pictures and glue sticks very often any more. This time, I decided not to sort the pumpkins for the kids. I would make them decide if there were any duplicates.

At first everyone just made pumpkins, but it quickly became difficult to tell if the pumpkin was new. A couple kids volunteered to sort the pumpkins. One kid wanted to sort them first by whether the nose was up or down. Another kid wanted to make rows for each kind of eye.

We ended up making rows for each eye, but then the kids couldn’t figure out how to sort within each row. My daughter wanted to make the columns have the same nose and mouth, but many pumpkins were missing, and there were many duplicates which complicated it.

Also, having 3 kids try to sort at once, without a clear strategy was pretty chaotic. The first kid independently tried to come up with a strategy by drawing different types of pumpkins on her paper. The second kid drifted between the sorting and the pumpkin making without really being engaged. The third made pumpkin after pumpkin without checking if there were duplicates. The fourth delivered pumpkins to the floor where we were sorting.  My daughter and I tried somewhat unsuccessfully to sort the pumpkins together. Then we ran out of time.

So…we don’t know how many different pumpkins we made!  Follow up work for next time.

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.

Guessing Games (Age 7)

The Activities

  1. Topics: Geometry, Perimeter: Book: Chickens on the Move by P. Pollack and M. Belviso.
  2. Topic: Binary Search: I printed out some charts with letters A-Z and corresponding numbers between 1 and 99, increasing from A to Z, but with a different pattern of numbers on each chart.  The kids took turns being the “puzzle master” and being the guessers — the goal was to find a particular number (which I specified each time) in as few guesses as possible.
  3. Topic: Probability: Same as an activity the Age 5 circle did last week.  I had two dice, one with twenty sides (1-20) and another with ten (0-9).  Putting the twenty-sided first, if you roll both of them you get a number between 10 and 209.  First game was, write down a number, then I roll the dice, if your number is higher, you win.  Next was, if you’re lower you win.  Finally, the interesting game is you pick one number, I roll 5 times, and you win if at least one roll is lower than your number and at least one is higher.  We also played variants where you needed, say, at least 2 higher and at least 6 lower out of 10 rolls.
  4. Topic: Probability: Each kid had a bag with 10 colored blocks, split between red and green (different number per kid).  They could pull blocks out one at a time, putting them back after each draw, as many times as they wanted.  When they were done, they would guess how many green and how many red were in their bag.
  5. Topics: Multiplication, Commutivity: We took the product 3 * 5 * 7 and evaluated it 3 different ways — (3 * 5) * 7, (3 * 7) * 5, and (5 * 7) * 3).  Then I showed them an informal proof of commutivity, using a 3-dimensional figure made of blocks (changing the order of multiplications corresponds to rotating the figure).
    IMG_1443 IMG_1441 IMG_1444

How Did It Go?

We only had two kids this week.  As usual with small circles, both kids were very attentive.  Several of the activities could be viewed as competitions; and indeed the kids did compete.

Chickens on the Move

A light book that introduces the concept of perimeter.

Binary Search

The kids really liked being the puzzle master.  We used a whiteboard where we started with each letter on the board, and then erased the letter and replaced it with the number after they guessed.  They picked this up really fast.  They didn’t quite do binary search, they guessed where in the interval the next number would be based on its relationship to the endpoints; but it’s not clear that binary search was the best strategy anyway, since the numbers were somewhat uniformly distributed (I did have some fairly significant skew in some of the charts, but since there were only 99 possible values for the 26 boxes, you couldn’t get in too much trouble by not splitting evenly each time).  They got every number in 3-5 guesses.


They also were pretty good at this activity.  One kid chose a really low number on the first round, probably because they were confused about the rules; but after that, they stuck with 209 or 208 for higher (they thought for a while they had to choose a number that was possible; after I said it didn’t, they chose 1 for the next round).  When I said the rules for the final game, they said right away that they wanted to be in the middle.  However, they chose ~ 50 for the middle at first.  Later they realized 100 was the middle.  Since they got this so quickly, I added advanced rounds where they needed, say, 2 higher and 6 lower.  We used colored blocks to keep track of each result (you got a green if your number was higher, red if lower).  Both of the kids got the general idea, but one of the kids chose better numbers than the other.  I also played a couple times; as it happened, even though my numbers were only slightly better than theirs, it made the difference between winning on losing (not that likely that this would have happened).

Bag of Blocks

One of the kids stopped after 15 pulls, 3 red and 12 green.  They guessed 3 red and 7 green, when the answer was actually 1 red and 9 green.  The other kid stopped after 30 pulls (they chose to make it a multiple of ten), got 21 red and 9 green, and guessed 8 red and 2 green (actual answer was 7 red and 3 green — although one of them got dropped on the ground along the way, so there were only 6 red and 3 green).  So there’s still plenty to learn here — they only have a general sense of ratios; and they don’t have a strong sense that more pulls = better results.

Commutivity of Multiplication

They sort of already knew that you can do multiplication in any order, although calculating 3 * 5 * 7 was hard enough that I think they were still a bit surprised it came out right.  The proof by 3D figure seemed to work for at least one of them.