# A Car is Faster than an Airplane?

## The Activities

2. Topic: Addition: We used Base Ten Blocks to do some simple additions, such as 3 + 4.
3. Topic: Numbers: We practiced making moderately large numbers, such as 17 and 51, using blue blocks rods and unit cubes.
4. Topic: Counting: We played the “secret number game”, with one secret number — when it’s your turn, you say one number to yourself, and the next out loud (i.e., counting by twos).
5. Topic: Probability: Each kid individually did one or two probability races (download chart here), where you roll the dice repeatedly, fill in a box for the appropriate outcome, and see which number gets to 5 first (or 6, if you color in the box containing the number, like some kids did).

Obviously, they colored over the numbers in this one.

6. Topic: Architecture: 13 Buildings Children Should Know by A. Roeder.
7. Topic: Building: Each kid got 6 Keva blocks and had to build as tall a structure as possible.
8. Topic: Sorting: We had various sets of clip-arts, which you can download here.  For each set, I handed one out to each kid, and they had to sort themselves by their pictures.  The sets we did this week were:
1. Worm, cardinal, goat, ant, spider, from fewest to most legs
2. Bicycle, Boat, Car, Train, Airplane, from fastest to slowest

## How Did It Go?

All five kids attended.  It was a good circle, there was some “when will it be over” at various points, but they stayed engaged and on-task for all but part of the last activity.

The kids aren’t that familiar with addition — it was hard for them to explain what it was before we read the book.  But I think they were able to understand the book.

We did about 4 problems.  For example, say we were doing 3 + 4.  I would say “make one pile of 3 and one pile of 4”.  Once they did that, “Now squish them together and count them”.  They were reasonably good at this, although they sometimes made errors both in making the initial piles, and in counting the combined result.

#### Base Ten Blocks Numbers

This is still pretty hard for most of them.  Understanding that a 10-rod counts as 10 and not 1 is tricky.  I had to give most of them help to be able to make 17.  Some of them started by grabbing about 5 10-rods; some started by getting one 10 rod and 10 unit cubes.  Only one kid could do it without help.  The second number we did was 52, which we all did together.

#### Counting by Twos

The kids were noticeably better at this than last time.  They’re getting better at counting to 30, and they’re better at the skipping part as well.

#### Probability Race

The kids sat scattered around on the floor of the kitchen, each with their own set of dice.  There was a wide variance in the speed that kids went; two kids finished one and a half charts; two kids finished one chart, and one kid finished half a chart.  This was partly due to how fast they counted the dice, and partly about how distracted they were by the other kids.  As I mentioned above, some of the kids colored the boxes with the numbers (so they needed to have 6 to win), others didn’t.

#### 13 Buildings Kids Should Know

The kids only recognized the Eiffel Tower out of the 13 buildings, one of them had been there.  They were interested in some of the architectural features I pointed out, including the “circles” (columns) and “points” (spires).  One kid said the minarets on the Taj Mahal were like Rapunzel’s tower.  I said the Guggenheim looked like a layer cake and they thought that was very funny.

#### Tallest Tower

Lots of interesting designs.  One of the first buildings to be completed was only about an inch high (almost all of the blocks were flat on the table).  The first round, the tallest building was slightly taller than one block-length tall.  I asked if they could do better; and the second time, the tallest was two block-lengths tall.  They definitely don’t have a great grasp of which blocks are useful and which aren’t.

#### Sorting

This activity got a little out of control, with not all the kids concentrating on the activity.  Still, they did well on the number of legs, got the right answer without many difficulties.  However, the slowest to fastest was more problematic, partly because not all the kids were paying attention, partly because they didn’t work together well, and partly because they have very little clue about relative speeds.  They all were pretty convinced a car was faster than an airplane.  The airplane ended up slower than a boat, but mainly because they weren’t paying enough attention to make sure their final order made sense.

# Princess of an Alien World

I led the older circle this week. 4 kids attended.

## The Activities

1. Topic: Multiplication: Book: Amanda Bean’s Amazing Dream, by Neuschwander. 2. Topic: Proofs: Prove that after every odd number comes an even number. Then prove that even + even = even.

A visual proof that Even + Even = Even

3. Topic: Permutations: You are astronauts who have landed on an alien planet. The aliens do not like you.  However, if you learn their language their feeling will improve. Each alien word is made of 4 letters: ABEK. Every permutation is a legal word. How many alien words can you find?  You start as an enemy. If you learn 9 words, you become a visitor. If you learn 16 words, you’re a friend. 20 = Lord/Lady. 24 = Prince/Princess.

The clip art chart showing how the aliens feel as you learn more alien words.

A close-up of some of the alien words we discovered.

4. Topic: Tesselations: Make a tesselation pattern out of a square.  A tesselation is a shape that can cover a plain with no gaps.  You can make a tesselation pattern piece out of a square by cutting a shape into one side, and taping it to the opposite side. This also works with diamonds and hexagons.

My daughter’s pattern piece, made from a square.

My daughter’s completed tesselation.

## How did it go?

This was another review circle where we repeated activities we have done in the last 2 years.  The kids were very excited to see the alien clip art on the wall, and they all remembered (and enjoyed) making tesselations.

#### The Amanda Bean Multiplication Book

This book is about Amanda who loves to count. She learns in the book that it is important to learn multiplication facts so she can count faster.  All  6 kids in circle go to different schools. One girl’s school is working on multiplication memorization, so at various times during the book she tried to remember a multiplication fact like 8×4. It turns out the other kids are actually faster at counting 8 four times, so I’m not sure if the message of the book was fully convincing 🙂

#### Even Number Proofs

I started by asking the kids if 2 is even or odd.  They all shouted “Even!”. 3? “Odd!” 4? “Even!”  I got up to about 12 before one girl said, “It’s a pattern! Even then odd!”. I then asked if they could prove that after every even number comes an odd number. I put 2 equal piles of cubes in front of me and said it was an even number. One of the kids then said, “If you add one more cube it will be odd because you can put it in either pile, because they won’t be equal”.  Another kid joined in, “If you add one more, then it will be even again!”. Next I asked them if you add two even numbers is the result even or odd?  They all said even.  I asked them to prove it.  Again I started with an even number of cubes, divided into two equal piles.  Then I gave them another even number of cubes in two equal piles.  One kid then said the result must be even because you can put the same number of cubes in each of the original number’s piles, and you know you can do it because the number is even. I then asked what about odd + odd? A couple kids immediately said odd, but then everyone thought a bit and said even.  We did a couple examples and saw that looked right. However, the kids got a bit restless before they could explain how to prove it.

#### Alien Permutations

I showed them the clip art chart of alien words. They were very excited by the different pictures and all wanted to get a high alien title, though one kid pointed out that she doesn’t care about princesses 🙂 They all started randomly looking for permutations. As they found new ones, they got to add it to the chart on the wall.  We got to about 21 numbers before it became pretty hard to find new entries.  At that pointed I asked how many they had found that started with A.  We counted and found 4.  There were 6 that started with K, 6 for E and 5 for B.  One kid then said we must be missing 2 that start with A and 1 that starts with B. They started searching for these missing words, and soon found them all!  This was a significant improvement in search strategies from when we last did this activity. At that time they got stuck around 18 or 19.  The kids were all happy to have reached the highest alien title of Prince/Princess.

#### Tesselations

The kids all remembered tesselations, and were all very impressed by my sample tesselations.  They jumped right in cutting pieces from two sides of their squares. They had to wait a couple minutes while I taped the pieces to the other sides, one by one, but as soon as their patterns were ready they started tracing.  Several kids stayed late so they could start coloring in their tesselations.  This really is a fun activity.

# Math Circle Turns Two!

## The Activities

This was the second anniversary of our first math circle, so we decided to revisit some memorable activities from past circles.

1. Book: Brian Wildsmith’s Puzzles by B. Wildsmith.
2. Topic: Conservation of Quantity:  This was one of Piaget’s conservation problems.  I laid out two parallel lines of different colored stones, each with the same number.  I asked which had more, then spread out one of the lines and asked again.  Then I removed a few to make them the same length, spread that one out again, etc.
3. Topic: Conservation of Quantity:  I had some number of blocks, and arranged them in different shapes, including stacking, asking whether there were more or less than before.  I also cheated and (attempted to) remove or add blocks without them noticing.
4. Topic: Logic:  We have some small Wizard of Oz dolls (I believe they originally were prizes in McDonald’s Happy Meals).  The set has Dorothy, Scarecrow, Tin Man, Cowardly Lion, Glinda, Wizard of Oz, Wicked Witch of the West, and Wicked Witch of the East.  In each problem, the characters had a race, and there are clues about what order they finished in.  Note that Glinda is a witch, and I abbreviate WWotW or just West for Wicked Witch of the West.
 Using all but tin man and lion: Wickeds were first and last. The East was next to Glinda. The hats of same color were together (Scarecrow and WWotW). Dorothy held the Scarecrow’s hand. The wizard finished just before Glinda. Using all but tin man and lion: The Wickeds were next to Dorothy on each side. Scarecrow next to Glinda. East finished after Dorothy. The boys were together. Glinda was first. Using all but tin man and lion: Pointy hats finished together (West, Scarecrow, East).  A boy won the race. Dorothy cried because she was last. East pulled Glinda’s hair and finished before her. The two with lightest hair finished together. Someone with black shoes was second. Using all 8: Every other person had a pointy hat. Red shoes won and lost. Boys were together. Witches were together. Scarecrow held hands with the wizard and the lion. Green skin and green pants were together. No hat won the race (Lion or Dorothy). Using all 8: The wizard and the lion are (directly) surrounded by mean witches. Dorothy didn’t win. 2nd place had red shoes. Someone with a dress won. The witches are apart. The WWotE finished right ahead of the wizard. The tin man cried and rusted because he was 2nd to last.

Solution for puzzle #4

5. Topics: Sorting, Numbers:  I gave them a shuffled deck of cards from 1-104 (our deck is from the game Category 5, also available as 6 Nimmt!).  They had to sort the deck as fast as possible, laying it out in a line.

## How Did It Go?

We had four kids this week.

#### Brian Wildsmith’s Puzzles

A nice book with fairly simple puzzles, but with a nice amount of variety.  The kids had no problems at all solving the puzzles.

#### Conservation of Quantity

As expected, both the conservation of quantity activities were trivial for the kids.  They weren’t confused for even a second about which had more, and they pointed out that the spread out line had more spaces between the stones.  They were even less fooled by the blocks activity, rearranging the blocks didn’t trick them, and they actually saw when I tried to cheat.  Even when I managed to get one away without them seeing, they had no doubt that I had stolen it and tried to find it under the table.

#### Wizard of Oz Logic

Last time we did this, some of the kids were pretty good, but not all of them.  This time, all the kids were pretty good at it.  Still, it wasn’t trivial.  Last time, I often directed their attention to particular clues they had already read in order to speed things up (it’s not even clear they would have finished without the hints), this time I only did that a couple times.  Another big difference is that this time, they could read the clues themselves.  They were much better at incorporating clues even when they didn’t uniquely determine anything; for example, if they knew that the wicked witches were on each side, they would go ahead and pick one of the two possibilities, and switch it later if they needed to.

#### Sorting 1-104

This also went better than last time we did it, but not by a lot.  They were able to do the first 60 cards in 10 minutes, and the whole thing in 13:24.  They still had some issues of coordination.  Most of the kids spread out their cards on the ground, but one held her cards in her hand and cycled through them.  At first, they didn’t skip, which slowed them down a lot, but after a while they started skipping, but still usually only one at a time, which slowed them down a lot.  After they got to about 20, someone suggested looking for all the 30’s.  Meanwhile, two of the other kids had started at 50 working upward one at a time.  One funny thing was that the kid who had the pile of 30’s got distracted trying to help those two find the 53.  If they did it again, I think they would be quite a bit faster, since they were quite a bit more robust to missing single cards by the end.

# Is the Gingerbread Boy a person?

I hosted the younger circle this week.  All 5 kids attended, despite the holiday weekend.

## The Activities

1. Topic: Counting by 5s: Book: Leaping Lizards, by Murphy
2. Topic: Counting by 5s: The 5s and 10s game (suggested by Murphy at the end of the book): Combine two decks of cards, removing all face cards and aces. The players take turns flipping over cards. If the card is a 5 or a 10, you keep the card. At the end you add each player’s 5s and 10s and the player with the highest sum wins.
3. Topic: Venn diagrams: Venn diagrams with fairytale cards.
1. Put People in one circle, and Animals in the other
2. Put Monsters in one circle and Girls in the other.
3. Put Animals in one circle and Flying things in the other.

Monsters in the left circle, Girls in the right circle. The Wicked Witch is a girl monster.

4. Topic: Logic: Some people are crossing a river.  The boat can hold two things.
1. How can 4 people get across the river?
2. What about 2 parents and 2 kids? The kids can’t drive the boat.
3. What about 2 people, a cow, and a carrot?  The cow will eat the carrot if there is no person to stop her.
4. The blue person is taking the carrot across the paper towel river in the boat. The green person and the cow are waiting on the other side.

5. Topic: Sorting: Book: Sorting by Size by Marks
6. Topic: Sorting: Give each kid cards with the numbers 1,2,3,4,5,6,7,8,9,10.  Put them in sorted order.  Then give them 10, 20, 30, 40, 50, 60, 70, 80, 90,100.  Then 5, 10, 15, 20, 25, 30, 35, 40, 45, 50.

The sorted number cards.

## How did it go?

#### The 5s and 10s Game

The kids enjoyed flipping over the cards, and seeing if they got a 5 or a 10.  Everyone was rooting for the two kids who hadn’t found any 5s or 10s yet, which was nice. No one got upset about the number of cards they got.  At the end, we added up everyone’s numbers. The kids were pretty good at adding 5 + 5 = 10, and 10 + 10 = 20, but bigger than that was tough. One kid was able to add all the totals.

#### Venn Diagrams with Fairy Tale Cards

Everyone was excited to see the fairy tale cards again.  They jumped right in when I asked the first problem: people in one circle, animals in the other.  There was an interesting debate about whether the Gingerbread Boy is a person or not.  The kids voted 3 to 2 that he is not a person, he is a food.  One kid noticed that the Cinderella carriage had a person (the coachman) and an animal (the horse). Another kid immediately suggested overlapping the circles and putting the carriage there. After the cards were all sorted, I asked questions like: How many people did we find? How many animals?

Next we did monsters and girls. The only girl monster was the Wicked Witch.

Finally we did Animals and Flying things. This is the first one where they put some things in the wrong section. We found several flying animals in the ‘Animals’ section, and some flying animals in the ‘Flying’ section.

#### Crossing the River

We had a boat with 2 seatbelts that can carry two people at a time. How do we get 3 people across the river?  Immediately one of the kids started moving the boat. She took two people across first, and then started the send the boat back across, empty.  I said the boat would float down the river if there was no one to drive it.  She quickly suggested sending one of the people back with the boat, to pick up the third person. Another kid was able to replicate this solution.

Next I changed the problem so we had to get 2 parents and 2 kids across the river. The same girl started working on this problem. She understood right away that babies can’t drive the boat. She and another kid were able to both solve this problem.

Finally, we had 2 parents, a cow and a carrot. The cow will eat the carrot if they are left alone.  Again, the kids easily solved this, first taking a cow across, then a person, then the carrot.  When the second kid tried to replicate this answer, I suggested taking the cow, then the carrot, but they saw that the cow would eat the carrot, and fixed the answer.

#### Number Sorting

First we read a book about sorting, then I spread the kids out, and gave each kid a set of cards with the numbers 1 – 10 on them.  I asked the kids to sort them. Several kids finished immediately, but a couple kids were initially confused about what it means to sort from ‘1 to 10’. They just found the 1 and the 10 card and thought they were done.  They quickly caught on.

Next I handed out cards with 10,20,30,40,50,60,70,80,90,100.  This was much harder for the kids since they are not as familiar with counting by tens. 3 of the kids were able to make progress on their own. The other 2 waited until friends had finished, who came over to give help.

At this point, one kid was begging for an even harder set, and another kid also wanted to try, so I handed out cards with 5,10,15,20,25,30,35,40,45,50. My son was able to sort these but the other kids were sure of the order when counting by 5s.

That was the end of circle, so the kids excited ran around the house screaming, especially when the big kids also had finished their circle. Maybe we can figure out some way to keep the after-circle wildness under control next time.

# How Many Kittens?

## The Activities

1. Topic: Even and Odd: Book: Even or Odd? by J. Mattern.
2. Topic: Even and Odd: Using one color of glass beads, I took out various numbers of beads between 1 and 10, asked them how many there were, and then asked one of the kids to say whether it was odd or even, and prove it (by grouping it into pairs).
3. Topic: Venn Diagrams: I selected some random attribute blocks and cut out two large posterboard rings.  I started using only one ring, and said “Put all the blue shapes inside the ring.”  Then I cleared everything, and said “Put all the thin shapes inside.”  After a few of those, I asked more complicated ones, like “Put all the red triangles inside.”  I also asked what was left over outside the ring for each one.  Then, I added a second ring (NOT overlapping the first), and had instructions for both rings.  I started with disjoint things (one ring is thick, one is thin), but the final few they overlapped (e.g., one ring is red, the other is triangles).  I didn’t point out they might need to overlap the rings, they had to figure that out themselves.
4. Topic: Combinations: I gave each kid a sheet with 12 uncolored kittens; each kitten has a large bow (download here).  I gave them each 6 crayons, 3 for the body (black, brown, yellow) and 3 for the bow (red, green, blue).  I asked them to color as many different kittens as they could.  After they had been going a while, I cut out unique cats from each kid and taped them to the wall until we had all of them.
5. Topics: Place Values, Counting, Numbers: I first introduced the kids to Base Ten Blocks. We went through ones, tens, and a hundreds, I showed them that 10 ones = 1 ten bar, 10 ten bars = 1 hundred square, 10 hundred squares = 1 thousand cube.  We also practiced counting by tens and hundreds.  After that, I had each kid choose up to 9 each of ones, tens, and hundreds, and then as a group we looked at each one and figured out how much it was.
6. Topic: Ordering: Book: Henry the Fourth by S. Murphy and S. Nash.
7. Topic: Sorting: First, I gave each kid a tile with a number from 1-5 on it and had them sort themselves in a line against the wall by their numbers.  Next, I took the 2 and 4 and replaced them with 8 and 11.  After that, I had them sort themselves by height, number of letters in their name, age, hair color (lightest to darkest), and hand size.

## How Did It Go?

All five kids attended.  It was a good circle, all the kids concentrated most of the time.

#### Even or Odd?

A very simple book, it explains clearly what even and odd mean and then has examples.  None of the kids knew what even and odd meant, but they picked it up quickly.

#### Even or Odd, pt. 2

I gave each kid one chance to say whether some number of beads was odd or even.  Almost everything was easy for them, except 1.  The definition in the book was that even means you can group into pairs, and odd was that if you grouped into pairs, you’d have one left over.  Since 1 involves having zero pairs, it’s rather tricky.

#### Venn Diagrams

All the kids helped put things in the circles.  One tricky question was “What’s left?” in the case where you had multiple attributes, e.g. “All blue thin shapes in the ring.”  When we got to overlapping attributes, we had the nice interaction where kids put things back and forth between the two circles.  After a little while one of the kids suggested putting it in the middle so it overlapped both circles; I then moved the circles so there was an overlap.  The kids got the concept of the overlapping circles pretty quickly.

#### Cat Combinations

Two of the kids got 4 unique kittens fairly quickly.  One kid colored the same combination three times in a row, then got a new one.  Another kid was distracted watching the other kids and only finished one.  The last kid didn’t finish any.  Between the kids, they got all 9 pretty quickly.

#### Base Ten Blocks

Most of the kids didn’t know how to count by tens or hundreds, but once I started doing it they were able to recognize the pattern and join in.  Figuring out what 3-digit number corresponded to some number of 1’s, 10’s, and 100’s was pretty hard, only one of them was able to do it consistently.

#### Henry the Fourth

Very simple book about ordinal numbers, but I had the kids to the actions the dogs were doing, which was fun.

#### Kid Sorting

They were better than I expected at the first two activities, partly because one of the kids took charge and helped people find the right places to stand.  They did pretty well on the heights as well.  For ages, they sorted by year without problems, but two of them didn’t know their birthdays, and had to go ask their parents.  Some of the kids understood that an earlier birthday meant you were older, others didn’t.  Hair color (lightest to darkest) was pretty funny, they had no idea.  One of them said right away “My hair is dark” and stood on the dark side; but he probably had the lightest hair. And the one with the darkest hair was in the place that should have been lightest.  Hand size was also hard but they did ok.

# Binary Fox Hunting

This week I led the big kids circle. All six kids attended.

## The Activities

1. Topic: Money. Book: Tightwad Tod by Skinner.

2. Topic: Money, Counting. Count a huge box of change.

3. Topic: Binary Search. In pairs the kids played the “Fox Hunting Game” which I made up.  There’s a fox out on a road and the farmer is trying to catch it. The fox is running all around the road (which has squares numbered from 1 – 40). Each turn, the farmer gets to build one fence that will block the fox.  The fox can choose which side of the fence to run to.  Once the fence is built, the fox can never get past it. The game is over when the fox is trapped on one square, and we count how many fences it took to catch the fox.

The fox is trapped between the fences at 26 and 37. Now the farmer is building a new fence at 29 and the fox has to decide whether to run to 27-28, or 30-36.

4. Topic: Geometry. Show the kids the compasses again, and let them draw circles.

## How did it go?

#### Counting Money

The kids were all impressed by my box of 10 years worth of change.  I asked them to guess how much money was in the box, and half the kids guessed \$1000, and the others just said “a lot”.  We reviewed the names of the coins, and how much each was worth.  Then I dumped out the box, and said we’d count it all. Every time a kid collected a dollar of change, I’d check it and then add it back to the box.

First the kids all just grabbed 4 quarters at a time.  We counted about \$30 that way.  Then some kids decided to try other ways to get a dollar.  Some did 10 dimes.  One kid made a dollar out of mixed denominations.  Several kids successfully counted 20 nickels.  However, 100 pennies was quite challenging because there were not that many pennies, and it was very hard for the kids to keep track of how many they had collected. They tried to work together but that also didn’t go so well, so eventually I stepped in to help.

Everyone was excited by my ever-increasing tally of dollars, and kids wanted to keep counted how many they had so far.  In the end, we had counted \$146.69.

#### Binary Fox Hunting

I assigned pairs for this activity, and each kid took several turns being the farmer and the fox.  At first there was a lot of confusion because the kids didn’t understand that the fox was roaming over a whole range of the row, not stuck on one square.  Many farmers thought you had to put the fence whereever the fox was, and some foxes thought you could jump past old fences that were supposed to block you.

I eventually sat with each pair and walked through the game and the choices the farmer and fox can make.  At the end, everyone seemed to understand the rules. It also looked like the foxes were pretty good at choosing the side of the fence that had more open squares. However, the farmers were not very consistent about where they would build the fences.  The best strategy is binary search – split the range in half each time to catch the fox quickly.

I picked up all the kids’ materials, and then we all sat down to discuss strategy.  I asked them “When you were foxes, which side of the fence did you pick?” My daughter and a couple other kids suggested they would choose the side with more numbers.

Then I asked, when they were farmers, how did you pick where to put the fence? Most kids said they picked randomly.  Some said they put the fence whereever the fox was.

Next I played the farmer and demonstrated a very bad fence-placing strategy, first putting a fence on #1 (the fox moved to 2-40), then putting a fence on #2 (the fox moved to 3-40).  I asked the kids how many fences I would have to build to catch the fox, and they said “a lot”.  One kid looked at the number line and said it would take 39 fences.

Even after this demonstration, the kids still couldn’t explain a good strategy for the farmer.  We should play it again.

#### Compasses

We had about 5 minutes left, so I handed out the compasses and showed them to the two kids who hadn’t seen them before.  Again everyone loved trying to draw circles, but 2 of the compasses broke during this circle, so we need sturdier compasses.