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.
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  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.

Trick or Treat Math (Age 6)

The Activities

  1. Topic: Counting. Book: How Many Donkeys? An Arabic Counting Tale, by MacDonald. In this simple book a man can’t figure out if he has 9 or 10 donkeys because he keeps forgetting to count the donkey he is riding. The kids caught on quickly and laughed whenever he got it wrong.
  2. Topic: Maps, Spatial Reasoning, Logic: Fill in a map of a treat-or-treating neighborhood based on the following clues. Here is the clipart we used: halloweencharacters.
    1. Directly to the West of your house is the Witch’s house.
    2. The Zombie house is 2 houses West of the Witch’s house.
    3. Olaf’s house is across the street from the Zombie’s house.
    4. Elsa’s house is directly South of the Witch’s house.
    5. The pumpkin house is directly East of your house.
    6. The Spider house is on the very West end of the South side of the street.
    7. The Butterfly is scared of the Spider. The Butterfly’s house is on the same side of the street as the Spider’s, but as far away as possible.
    8. The Goblin is between the Zombie and the Witch.
    9. The skeleton is directly across the street from the Spider.
    10. Next to Elsa’s house is a Graveyard that takes up two houses.
    11. The Ladybug’s house is right next to the Butterfly’s.
    12. The Fairy can fly right across the street to the Ladybug’s house.
    13. The Wizard’s house is East of the Fairy’s.
    14. Anna’s house is next to Elsa’s house.
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      The completed puzzle

      3. Topic: Estimation, Subtraction. Guess how much candy is in a container. Then put the same candy in a shallower container and guess again. Then count the candy and figure out whose guesses were the closest.

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      The Candy

      4. Topic: Logic. Tape a Halloween character to each kid’s head. Then the kids ask each other yes/no questions to figure out who they are. The hardest part of the game is not telling your friends what is written on their heads.

      How did it go?

      I wore my witch costume during circle, and I organized it so the kids would get to ‘trick or treat’ after completing each activity from my bucket of small prizes and candy.

      Halloween Logic

      Each clue was pretty easy for kids, especially after they understood what phrases like “directly West” means. The hardest clue was: “The Spider house is on the very West end of the South side of the street.” Two of the kids figured it out on their own. The other two needed some help from their friends to understand the “south side of the street”.

Candy Estimation

The kids were very excited to see so much candy, especially when I told them that the person who guesses closest would get to trick or treat twice after the activity. Interestingly, the guesses did not get closer after I spread out the candy. Most second guesses were at least as wrong as the first guess. I guessed after the candy was spread out (and I got within 2 of the correct number).

After everyone wrote down their guesses I asked the kids to count the candy. They immediately began discussing counting strategies. They eventually decided to sort the candy by type and then count each type. However, they soon realized that some types had too many pieces to be easily counted, and they didn’t know how to add the results anyway. So they switched to counting each piece of candy as it was thrown back into the tub. Two kids both wanted to throw in candy and everyone ended up missing a bunch of pieces when the two throwers could not coordinate. They came up with 67 pieces, but I counted it again and found 72 pieces.

Halloween Twenty Questions

The kids loved seeing costumes taped to their friends’ heads, especially when one boy got ‘Princess Leia’. I told them at the start that it is very important not to tell your friends what is written on their heads, and the kids did pretty well at this. However, some kids asked questions like “Am I a zombie?” because they saw “Zombie” on their friend’s head. The hardest to get turned out to be superman. The kid knew he was a strong hero who wears red and blue, and has an S, and has a cape, but couldn’t think of superman.  Everyone else figured theirs out eventually (with some hints from me about what questions to ask). Everyone really enjoyed this activity. At the end, we had five minutes left so one of the kids moms played and had to figure out she was a pumpkin. The kids loved hearing her questions and shouting out answers. “Can you eat me?” “Yes, but it’s yucky and too chewy!”

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.

Leo the Rabbit (Age 8)

The Activities

  1. Topic: Logic. Book: Still More Stories to Solve by Shannon, Stories 11 – 14. The kids absolutely love this book of brain teaser stories, like what can you say to your two enemies to make them fight each other and leave you alone? Or how can a man get two wishes fulfilled when the genie only grants one wish? We spent about 25 minutes discussing the four stories we read. Most of them we could not solve on our own, but I would read the answer and give hints. Everyone understood the answers at the end.
  2. Topic: Logic, Combinations. We got this problem from the awesome site YouCubed.org. Leo the Rabbit is at the top of a staircase of ten steps. Leo can h0p down either one or two steps at a time. How many different ways can Leo hop down the stairs?
  3. Topic: Counting, Geometry. How many rhombuses are there in a heart made out of the YouCubed logo?

 

How did it go?

This was our first circle in a month, due to traveling. All five kids attended. This was a very high-energy circle, especially for my daughter who was having trouble staying on task. For each activity there were a couple kids complaining they were bored, but also at least a couple who stayed interested and learned something. I had a lot fun actually, because I intentionally didn’t solve the Leo the Rabbit problem ahead of time, and it was exciting to figure it out during circle.

Leo the Rabbit

First we started by drawing the rabbit at the top of a set of 10 stairs. We assigned a letter to each stair, and then each kid wrote a bunch of letter sequences representing the hops the rabbit makes. Kids came up with about 10 paths each before they started to want to find a faster way. My daughter suggested that you could first find all the paths that start with AB, and then all the paths that start AC.  I used this as a starting point. I asked the kids to consider just the last three steps in the staircase. If Leo is on step H, how many ways can he get to the bottom? Several kids were able to enumerate the 3 possibilities: HIJ, HI, or HJ.

Then I added step G. Now how many ways?  I pointed out that if the rabbit hops to step H, then his choices are now the same as the three ways we found for step H, namely: GHIJ, HI, GHJ. But Leo could skip step H, so we have to add in GIJ and GI as possibilities. Some kids understood this, but most did not. So I started even simpler.

What if there is only one step, step J? Then there is only one choice: J.

I ended up drawing a picture similiar to this:

img_20161016_173856

At least one kid really seemed to understand that to get the ways for Step N, you add together the ways down from N-1 and N-2 (since Leo could hop down to either of those). All the kids soon saw that to fill in the next step, you should add the numbers from the two steps below, but many of them probably did not fully understand why. We were all impressed to get 89 ways, and were glad we didn’t try to enumerate them all.

Everyone started out quite engaged during this activity, but people started dropping off and getting distracted. In the end, 3 of the kids were still paying attention and 2 were quite ready for the activity to end.

YouCubed Heart

I intentionally made this activity much easier. YouCubed has a number of interesting questions about the picture, but I just asked how many rhombuses there were, and then let them color the picture for the last five minutes of circle.

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Finishing the Fives (Age 8)

The Activities

  1. Topic: Logic. Book: Sideways Arithmetic From Wayside School, by Sachar. Chapter 2.
  2. Topic: Arithmetic, Patterns. Make the numbers 1 – 60 using only fives. For example, 26 = (5×5) + (5/5).

How did it go?

We had three kids for circle. Everyone was very focused, and followed my rule that we cannot draw Pokemon during Math Circle.

Sideways Math

The kids love the story of this book, and the problems were great for sideways thinking. First we reviewed the chapter from last week, since David led that circle. The kids were able to quickly explain why elf + elf = fool.  The rules are that each letter stands for one number 0..9, and each letter is a different number. The key to this one is that fool has an extra digit than elf, so we know e + e results in a carry. This means f must be a 1, which can be used to solve all the other letters too.

After this review, we read chapter two. In this chapter, Sue gets very upset because she thinks it’s weird to add words. Instead you should add numbers! Like 1 + 1 = 2!

The teacher writes that problem on the board as:

one + one = two

Sue says no! You should put the numbers there, not words! The teacher says, what numbers? Sue says “1 and 2!”. The teacher laughs: but there are no 1s or 2s in the answer!

Then we worked together to solve this problem, knowing that none of the letters stands for a 1 or 2.

We figured out that o must be < 5 because it shouldn’t cause a carry. o can’t be 1 or 2. It can’t be 0 because that would force ‘e’ to also be 0.  o can’t be 3 because there’s no number such that e + e = 3. So o must be 4.

At that point we got stumped because e + e = 4, but e is not allowed to be 2. We couldn’t figure out our mistake, so I checked the solution at the back, and realized, of course, that e + e must be 14. Then we quickly solved the rest of the problem.

Sue, in the book, then starts shouting out a bunch of math facts like: one + two = three, four + seven = eleven, and all the kids laugh at her. The teacher laughs to and says it’s impossible. So our next task was to prove that those problems are impossible.

one + two = three.  We quickly realized that there are no three digit numbers that add to a five digit number.

four + seven = eleven. This one was much trickier. Our intuition was that too many numbers have to be zero for this to work, e.g. u + e = e, o + v = v, f + e = e. But we had trouble proving it was impossible because what if there were carries involved? In fact, I thought I had proved it, until a kid explained that maybe u + e = e because r + n caused a carry. So really 1 + u + e = e + 10, which is possible if u is 9 and e is 7, for example. So we didn’t quite get a satisfying proof.

five + two = seven. I did most of this one myself. First I figured out that i + t must carry to the f, so that f + 1 = se. That means f = 9, s = 1, e = 0.  But we also know e + o = n, but that’s impossible if e is zero, because e + o must then equal o.

At this point the kids started to get a bit antsy. Some kids wanted to read the next chapter because the story is so funny, but no one really wanted to work on any more problems, so I ended the activity here.

Fives Chart

Two weeks ago, the kids filled out about half of chart where you compute the number 1 – 60 using only fives. For example, twenty is (5 x 5) – 5. We promised them a small prize if they could get 40 of the numbers completed, and another prize if they could fill them all in. This week, one of the kids realized that if the chart contains the answer for a number like 40, you can easily compute 39 and 41 by adding or subtracting 5/5.  This allowed them to quickly finish the whole chart. They still were pretty interested in using smaller numbers of 5s when possible, recognizing that it is not very elegant to write 5/5 fifty eight times to get 58.

Here’s a part of their chart:IMG_20160905_173605.jpg

 

 

 

 

Taking a Finger Walk (Age 6)

The Activities

  1. Topic: Puzzles: Book: Taro Gomi’s Playful Puzzles for Little Hands.  We’re most of the way through now, probably one circle left of puzzles.
  2. Topic: Logic: I printed sheets with 6 uncolored flowers on one side, and 9 on the other.  There were two puzzles: For 6 flowers, “There are more red flowers than purple, and more yellow flowers than red.  For 9 flowers, “There are more red flowers than purple, more blue than red, and the same number of blue and yellow.”IMG_2108
  3. Topic: Spatial Reasoning:  Corey and I built a number of models out of Legos.  The kids each picked a model and had to copy it exactly.  They could pick it up and look at it from any angle.  Each kid copied several models.IMG_2109
  4. Topic: Attributes, Games:  We played a couple rounds of Set with just the solid cards.

How Did It Go?

We had four kids this week.  One kid had been gone for a couple months, but now everyone is back from summer trips.  This circle went well, the kids were all interested in all the activities.

Playful Puzzles

We spent quite a bit of time on a puzzle with two kids at either end of a very windy path, with the question “Where will they meet?”  We measured by placing coins from either end.  They also enjoyed a page where you were supposed to “take a walk” with your finger by tracing a path and following various instructions along the way (e.g., “Take a rest here” or “Go around this corner really fast”).  Every kid did it once.

Flower Logic

The kids figured out the answers pretty quickly.  Interestingly, different kids figured out the second one from the first.

Lego Models

Different kids definitely had different skill levels on this one.  One kid breezed through a whole bunch, while others took quite a bit longer.  The trickiest ones were the dinosaur, because it was irregular and a bit complicated internally, and an offset colored square because it was tricky to get the right pattern of blocks on the bottom row (the kid working on it initially had the colors going the inverse rotation).

Set

We’ve played before with this circle, which the kids remembered, but not all kids remembered the rules.  Our son has played a lot, so after he got a few I said he had to let other kids get sets.  I was happy because all the other kids got at least one set on their own.