# True Lies, Archimedes, and Pokemon (Age 8)

## The Activities

1. Topic: Logic. Book: True Lies by Shannon.  This book is from a series of logic / out-of-the-box-thinking books that I love. It has 10 or 15 short stories that each contain something that sounds like a lie. However, if you think carefully, you can find that the characters actually told the truth. For example, a kid baked cookies and was only allowed to each one cookie before dinner, but her appetite was ruined. How? Well…maybe the kid baked a huge cookie.
2. Topic: Archimedes, Volume. I re-read part of the Archimedes chapter in Mathematicians Are People Too, the section about the discovery of volume and buoyancy. Archimedes figured out that a crown contained silver instead of gold because it displaced too much water (silver is less dense than gold). I told the kids that I was now the king, and they were Archimedes. I showed them two ‘amulets’ made of aluminum foil wrapped around quarters. I said I was afraid the craftsman had stolen some of the quarters I gave him, and used extra foil instead. I then asked how we could find out?

The two ‘amulets,’, and the water we used to test them.

3. Topic: Decision Trees. I made a huge decision tree that classified 40 different Pokemon cards into a different pile for each kid. This decision tree spanned three pieces of paper! Here’s part of it:

## How did it go?

We had only three kids this week. The other two were out sick.

#### True Lies

The kids all loved this book, and wanted to keep going.  We did the first 5 lies in about 15 minutes.  I was able to solve all the lies pretty quickly. The kids solved several of them, after some discussion. For example, they came up with the idea of a big cookie, in the example above.

The hardest one was a king said someone could have land that could be surrounded by an ox hide. I helped the kids solve it by asking how they could make a piece of paper surround a large area? They still didn’t get it, so I said, what if you had scissors? Then they thought of cutting up the paper into strips.

#### Archimedes

Last week we read a chapter about the discoveries of Archimedes. A couple kids remembered it, but one of the kids this week was absent last week. So I re-read the 3 or 4 paragraphs about Archimedes, the king, and the crown that could have silver inside instead of gold.

Then I told the kids I was the king, and I had a craftsman make me an aluminum foil amulet out that should contain a certain number of quarters. I was afraid the craftsman had kept some of the quarters for himself, and used more foil so I wouldn’t know. I had an amulet that I had made, that definitely had the right number of quarters. How could we tell if the craftsman had lied?

One kid immediately suggested putting the amulets in water to see how much they raised the level. I asked if there was a simpler test we should do first? She suggested we should weigh the two amulets to make sure they were the same. I got out a kitchen scale, and we found they each weighed 3 oz.  They both looked about the same size too, though they were different shapes.

Next I filled up a two-cup glass measuring cup. We tested the good amulet first, and saw the level raised a very small amount. Next we tested the unknown amulet, and we could see the water went up significantly more. The craftsman must have stolen some of the quarters! We then opened the two amulets and saw that the good amulet had 9 quarters inside. The bad amulet had lots more aluminum foil, and only 6 quarters. The craftsman had stolen 3 quarters.

I finished up by showing the kids that aluminum foil is much less dense than quarters. In fact, foil floats. They were pretty interested in this, and several kids tested out how much foil one quarter could sink.

Several kids said “That was fun!” as we were finishing this activity. After circle, my daughter continued to play with aluminum foil and water, wrapping various things in foil, and seeing if it would float or displace a lot of water.

#### Pokemon Decision Trees

The kids were really interested in the big decision tree, and the Pokemon cards. They were quite happy to see their names in the tree, and all quickly went to work. They wanted to know which Pokemon would be on their team.

I had made the tree so that every kid would end up with 8 Pokemon. However, after the first round, two kids had 10 Pokemon each. We then reclassified the two piles with too many, and ended up with a third pile that now had too many. I kept handing the cards out to different kids to check, but we ended up with one pile that kept having 9 cards in it. I personally checked each card, and finally found the one that was mis-classified.

The kids were all happy to look at which Pokemon they had randomly been assigned. My daughter really wanted to play some kind of math game with the Pokemon, but I didn’t have one prepared, so I just had them pick out their favorite and least favorite Pokemon (based on whatever characteristics they liked).

# How Much For The Whaleshark? (Age 6)

## The Activities

1. Topic: Money: Book: Follow the Money! by L. Leedy.
2. Topics: Money, Addition, Subtraction: I ran a “store” where the kids could (pretend) buy various small toys.  Each round, I gave each kid 4 or 5 “dollars” (play money), and then they chose something to buy and payed me for it.  Some things cost more than \$5, so if they wanted to buy them they had to buy something cheap so they’d have enough next round.  We did three rounds, then reset.  The second run through, I paid them using \$5 bills so they had to make change.  The third run through, I had them keep track on a “ledger” in addition to paying me with physical play money.
3. Topics: Counting, Sorting: We repeated the activity where the kids sorted dominoes according to the sum of the spots.  I gave them 6 dominoes to start, and gave them 3 more at a time when they finished.  I went up to dominoes with 9 spots on each side.  One of the kids has learned his times tables, so he sorted by product instead.  After we used up all the dominoes, I had the kids sort the entire set (by sum).

## How Did It Go?

We had all five kids this week.

This book didn’t have a ton of math, but it did show making change a number of times.  It also introduced the various dominations and was a nice introduction to the next activity.

#### Play Store

The kids got the hang of buying things quickly.  However, they weren’t very good at making change — I think several of the kids didn’t ever quite understand what it meant.  Also, some of them can’t do subtraction, so they also couldn’t do it directly.  That is, if you have a 5 dollar bill and need to pay \$3, either you can make change into ones and pay 3 of them, or you can subtract 3 and get back 2.  Some of the kids didn’t understand either; some understood the change method, and some understood both.  The ledger was also a challenge for many of them, both because they didn’t know subtraction and they didn’t understand how it related to paying for things.  A few got it, but at best the others understood the mechanics but not the meaning.  So we can definitely explore more both the ideas of making change and keeping track of quantities using a running tally.

After we finished the money activity, I had each kid make up a story about a simple math problem.  For example, I would write “3 + 2” and they would say “There were 3 birds sitting on a branch, and then 2 more came, so there were 5 birds”.  Some of the kids closely followed earlier stories, but we got several different types of stories by the end.

#### Sorting Dominoes

The kids were pretty good at this.  All of them were able to finish on their own, with only a few mistakes here and there.  They even handled the 9 spot dominoes pretty well.  For the group sorting, it took them a while to get organized, but once they had at least one domino in each position they made faster progress.  Not everyone participated the whole time but they all contributed.

# Sorting Extravaganza! (Age 6)

## The Activities

1. Topic: Numbers. Book: Missing Math: A Number Mystery, by Leedy. This is a fun, but simple book about all the numbers disappearing. It illustrates various places we depend on numbers, like calendars and phone numbers. In the book, one character can’t remember how old she is because the numbers have disappeared. One kid from circle said she would remember how old she is, even if there are no numbers.
2. Topic: Counting, Adding, Sorting. I brought out a set of dominoes that went up to 6 on each end. We played two different games, suggested by the book, Dominoes, by Planet Dexter
1. Lost Domino: Pick a domino and count the spots. Then put it face down with the other dominos and mix them around. The kids take turns flipping over one domino and counting the spots to see if it is the lost domino.
2. Domino Sort: Mix the dominoes face down. Each kid takes 6 dominoes and then sorts them by the total number of dots on the domino.
3. Topic: Sorting, Teamwork. I made cards numbered from 1 – 99 out of poster board. I shuffled the cards, then gave a stack to each kid. The kids had to work together with no help from me to sort all the cards. First we did it with cards 1 – 50. After that went well, I added in cards up to 99.

## How did it go?

This was the first circle in a few weeks for some of the kids. All five kids were here. Three of the kids had just had a week-long winter break from their schools. There was a lot of energy, but they were also amazingly focused, especially during the team sorting activity. Overall, it was a great circle.

#### Domino Sorting

Lost Domino was very easy for everyone, so we played just one round.  Domino sorting with dominos with up to 6 dots per side went pretty easily for all the kids. They all wanted to add in dominoes that go up to 9 dots per side, which made the activity much harder for most of the kids. Two could still sort them independently, but the other three needed more help.  Everyone was pretty interested in this.

Each kid ended up individually sorting 1/5 of the dominoes.  I suggested that we could sort all the dominoes together in the middle, and I organized this by asking “who has a 0 domino?” “A one?” Most the of kids kept their dominoes sorted, but one kid mixed her stack up, which made it much harder for her.  Several kids kept asking me and other how many dots various dominoes had.  In the end, we had a very nice layout of the sorted dominoes.

#### Sorting Together

The kids’ number recognition has really improved in the last few months, so they were ready to try sorting a bunch of cards together. I try not to help during this activity, because I want the kids to come up with their own strategies, and own evaluations of what worked and didn’t work.

First we did cards 1 – 50. It took the kids about five minutes to sort. Most of the kids laid their cards out on the floor so they could all search for the next number. This works very well, if you are using brute search.

The kids then all really wanted to try the cards up to 99. I warned them that it is *much* harder, and that the bigger kids couldn’t do it the first time they tried. Everyone still really wanted to do it. I shuffled the cards and gave each kid a stack, the said “Go!”.  The kids worked really well together, and stayed focused the whole time, even when the big kids finished their circle and came down and ran around a bit.

Again, things went best when the kids put their cards out on the ground. Some kids did not want to do this, which definitely slowed everyone down.  I even suggested to my son that he should put down his cards, but he felt like he owned those cards, and didn’t want to share them. Eventually he did put them down, but got a bit upset when someone else picked up some of his cards. “55 is mine, I remember!”.

Two kids really took the lead in calling out what number was needed next. 4 of the 5 kids helped search for the numbers. The fifth held onto his cards mostly, but was still very focused.  One kid would find streaks of numbers (51, 52, 53, 54, 55) and hand them off to others to add to the chain. Unfortunately, this meant double the sorting work, since the numbers would get mixed up in the transfer.

After 19 minutes, all the cards were sorted, stretching all the way out of the kitchen, under the dining room table, to the wall. Everyone cheered!

We’ll have to do this again in a couple weeks, and see if they can beat their time.

# Shappy Valentine Day (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 3 (Archimedes).
2. Topics: Factors, Logic:  We played the Big Bad Wolf game (idea from youcubed).  It’s really a puzzle, not a game, since the moves of the Wolf are deterministic.  You start with the numbers from 1 to N (we started with 6 and then moved on to 10 and 15).  Each turn, the player picks a number, and then the Wolf gets all the factors of that number.  The Wolf always must get at least one number.  At the end, the Wolf gets all the numbers that are left.  Whoever has the highest total wins.
3. Topic: Codes:  We finished decoding the little kids’ Valentine’s message from last week.

## How Did It Go?

We had four kids this week.

#### Archimedes

The kids seem fairly interested in this book, but the chapters are a bit too long — they take at least 20 minutes to read.  From now on, I’m going to figure out sections to skip so the book takes no more than 10-15 minutes — probably a bit more than half of each chapter.

The kids are good enough at factors of smallish numbers that they can play this game.  Some of the kids realized you should pick numbers that only give the Big Bad Wolf one number — but they didn’t realize you also want the things you pick to be as large as possible (as far as I know the greedy strategy of picking the largest number that only gives one thing to the Big Bad Wolf isn’t optimal, but it does pretty well).  A couple of the kids got tired of the game after a bit and stopped making progress, but one of the kids who was playing it for the first time really liked it, I heard later that they mentioned it at home.

#### Decoding Valentines

The kids had trouble getting started decoding the two harder messages.  I helped direct the work, and we were able to decode one of them without too much trouble.  The main challenge with the first of the two was that you couldn’t tell the boundaries between numbers (i.e., 26 vs. 2 and 6).  The other was considerably harder because it was written in a color changing pencil, was hard to read, and the rows of numbers overlapped.  As a result, we thought the first letter of the second line was part of the first line, giving us “Shappy Valentine Day”.  The kids thought this was hilarious.

# Math Circle Valentine’s Party (Age 6 and 8)

## The Activities

1. Topic: Addition, Subtraction, Charts: Party Plans: Math Concept Reader, by Harcourt School Publishers. Kids use math to plan a 100th day of school party. It’s a pretty dry book, but the kids were still interested in solving the simple math problems.
2. Topic: Codes. Secret Valentine’s Buddies. The two circles worked together this week. Each kid randomly picked a kid from the other circle, and made them a Valentine that contained a coded message, using a simple letter to number code. Then the kids exchange cards and if they can decode the message they get a small present chosen by their buddy.

## How did it go?

My daughter was very excited to have a Valentines party so she decorated our house. 🙂

We only had 6 kids total this week, three from each circle. They were excited to read the book together. The older kids were especially excited to show off they addition and subtraction skills.

Then we split up to work on the secret Valentines. I worked with the older kids who were quick and efficient, and decorated their cards nicely.  David worked with the younger kids, some of whom were quick, and some slow.  The older kids finished encoding their messages early, so I started teaching them an interesting math game involving factors, that we’ll follow up on in another circle.

A Valentine that contained a coded message.

Next we switched valentines. The younger kids got the key to the code, but the older kids had to decode their messages without the key. This was extra challenging because the coded messages were somewhat illegible.  It was hard to tell words apart, and also where the number breaks were: 2, 24  or 22, 4 or 2, 2, 4.  We focused on the most legible letter, and were able to decode most of it: I Love You _o _u__  Be My Valentine. We figured out the last two words as we were telling the little kids.  We saved the other two messages for next week. It should be easier now that we know some of the number – letter pairs.

A message from an older kid, decoded by a younger one.

We intentionally used two different number to letter codes this week so the older kids would have to work harder. However, it turned out that I was 12 in both codes, the older kids kept expecting to find more overlapping letters, even though I kept saying the codes were different.

We ended circle with chocolate covered strawberries that my daughter and I made.

# A Jellyfish Doesn’t Weigh More Than Me! (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 2 (Pythagoras).
2. Topic: Algebra:  Using a bunch of colored glass beads, we did some simple algebra problems.  I had a chart with boxes numbered 1-8, and then by placing a stone in one of the boxes I could indicate how much that color was worth.  I started with some problems where all the values were known, and asked which pile was worth more.  For example, if Green = 3, Yellow = 2, and Blue = 5, which is bigger, 3G + 2Y or 3B?  Then I made it a bit harder, e.g., 13G + 2Y vs. 13G + B — the goal is to introduce them to canceling equal quantities from both sides.  Next, I changed the problems so that one (or sometimes two) of the values are unknown, and I give them two piles with equal values, and they have to figure out one of the unknowns.  For example, if G = 2 and Y = 3, and G + B = 3Y, how much is B?  Harder, if G = 1 and Y = 4, and G + B = Y + 2B, how much is Blue?  Even harder, if Y = 3, and G + 4Y = G + 2B, how much is Blue?
3. Topic: Decision Trees:  We continued the activity from last week.  First, we did a couple more tracing exercises with playing cards and more complicated trees (Tree 4 and Tree 5 from last week’s post).  Next, we did more tree building using random selections of cards, but this time, instead of using playing cards, we used Pokemon TCG cards.  Besides being an interesting theme, these cards are nice because they have a lot of different attributes (color, HP, damage, weight, size, and more).

## How Did It Go?

We had all five kids this week

#### Pythagoras

Just like last time, the kids were pretty interested in this story.  It took about 15 minutes to read — I tried stopping partway through, but they insisted I finish the chapter.  There wasn’t a great deal of math content — but there were good lead-ins to either the Pythagorean theorem or Platonic solids.  Pythagoras definitely had an interesting life, essentially starting a cult…

I started with one shared problem for the whole table, but that didn’t work well at all — one or two kids were engaged, but the rest started drawing, writing their names in different styles, etc.  I switched to having two groups each doing the same problem, which helped, but I think I should have had a copy for each kid.  Also, I set up the problems each time — I probably should have had the kids make the problems themselves.

One of the kids (Kid A) already knew how to solve algebra problems such as 5X + 3 = 28, subtracting and then dividing.  I didn’t ever write down the problems in this way, but a different kid (Kid B) wrote one of the earlier problems down like this, and then Kid A immediately solved it.  However, Kid A wasn’t able to generalize to some of the other patterns.

For the initial problems with only given quantities, the kids quickly realized you could ignore an equal number of the same color stones from both sides — although they quickly fell back to calculation if it got more complicated.  Later, when there were unknowns, they weren’t able to apply this idea without help.  Perhaps the hardest problem was B=7, 4B + Y = 4Y + B — one of the kids solved it with the intuition that it would only work if B and Y were equal.  I pointed out the idea of grouping one B and one Y on each side, which some of them understood, but I’m not sure they could apply it.

#### Pokemon Trees

Tracing went well, one of the kids had been gone last week but picked it up quickly.  The kids sorted quite a few cards in a short period of time.  They also noticed when certain letters happened less frequently, and made some inferences about the output.  The most interesting part of this section was that the kids thought the cards smelled bad, calling them stinky cheese cards (mostly they smelled like plastic).

The Pokemon theme definitely interested them.  They had varying degrees of success writing down a tree.  They all understood how to make a tree, but some of them had trouble making a tree that matched the cards I dealt them.  The kid who missed last week didn’t understand that they weren’t supposed to rearrange the cards — their tree is the second picture above, which mentions A, B, and C even though I dealt only A and B piles.  Some kids used the different attributes more effectively, finding patterns in the HP, for example.  One of the kids used a range 175 lbs – 251 lbs in order to pick the middle two out of four blue Pokemon.  One kid thought it was funny that Tentacool, a jellyfish pokemon, weighed 100 lbs, thus the blog post title.  One of the kids finished two trees quickly (picture 1 above), each time sorting a bunch of other Pokemon after building the tree.

# Map of the Green Slime (Age 6)

## The Activities

We spent the whole circle drawing a decision tree map of the Choose Your Own Adventure book, The Green Slime, by Saunders.  I did this activity with the older circle 2 years ago. Here’s the blog post.

## How did it go?

We had 4 kids this week, and also an observer who has started his own math circle for his daughter. He was interested in the mechanics of running a circle, and how to keep all the kids involved.  This is one of the most popular activities with the kids, so it was quite easy to keep them all involved.

In the story, you are a babysitter for your 4.5 year old cousin, Stevie. Stevie finds your chemistry set, and makes a batch of growing green slime. What should you do?

Here’s the map we built while we read the book.

The kids all loved the story.  It’s a Choose Your Own Adventure: Young Readers book, so it was just scary enough to be interesting, but nothing really bad happens. We first read through to one ending, without making a map. As soon as we reached an ending, a couple kids started asking if we could read the whole book.  I said we would, and that we should make a map so we make sure we don’t miss anything.

I started by mapping out our whole initial path. We named the ending (“Goo outside”), and then I asked how we could find new parts of the book.  We decided to make a different initial decision (don’t check on Stevie), and got to a new ending.

Eventually we reached the choice about whether to get a ladder or not.  If we didn’t get a ladder we should go to page 6.  One of the kids immediately remembered that we had already turned to page 6.  6 does not appear on our tree, so we traced back our initial path, and found that we got to page 6 if we decided to pour the goo down the sink. So we drew a line from “ladder?” to the next choice after page 6.

There were many interesting discussions during the book…one of the best was about what you should do if your little brother or sister tries to drink a bowl of green goo.  The suggestions ranged from “Kick the bowl out of his hands”, to “Grab the bowl, put your brother on a very high shelf, and tell him he will be dead if he drinks it.”

The kids were all interested in naming the different endings, and showing me where there were new choices. One kid was concerned that we might miss some pages, so at the very end we went page-by-page through the book to verify that we had seen them all. The kids remembered every page, so they were convinced.

To wrap up, I asked how many different endings were there in the book? Most of the kids immediately started counting the boxes, and figured out there were 7 choices.  We also found there were 14 branches on our tree.