# Micheltello, the Fifth Ninja Turtle

## The Activities

1. Topic: Story Problems: Book: How High Can a Dinosaur Count? …and Other Math Mysteries by Valorie Fisher.
2. Topics: Codes, Combinations, Story Problems: We created a Teenage Mutant Ninja Turtles “adventure” with three stages.  All materials for this activity can be found here.  The story is that Shredder has discovered some ancient machines that, when activated, will cover the world in slime (I printed out some pictures of the characters for those less familiar with ninja turtles).  The turtles obviously want to stop him, and they find Beebop and Rocksteady, defeat them, and take from them a coded message from Shredder.  This was a number-letter substitution code, but unlike in the past, I did not give them the key.

Deciphered message

Then I said that Splinter remembered a legend of an ancient fifth ninja turtle, but he couldn’t quite remember his name.  However, he did remember that the name was a combination of two of the first four turtles’ names (e.g., Raphangelo).  If they can list out all the possible names, he will remember.

Now that they know the fifth turtle’s name (Micheltello), April O’Niel goes on TV and says that Micheltello should come help them.  He shows up, and says that that he knows where the slime machines are.  The turtles go to the slime machines, but each one has a story problem that the kids must solve in order to turn it off (see link to PowerPoint above for the story problems).  Once they solve the problems, the world is saved and the mayor gives them each a prize!

3. Topic: Origami: We made the pinwheel from Easy Origami by John Montroll.

## How Did It Go?

All six kids attended.  This circle went well, all the kids were engaged the whole time — having a theme definitely helps.

#### How High Can a Dinosaur Count?

This book has a bunch of story problems, mostly about addition but a few others, including a couple about money.  Most weren’t too hard for them, but they usually couldn’t solve them instantly.  I had the kids raise hands to answer, all the kids gave at least one answer.  The hardest one was to count up how much money two dimes, two nickels, and three pennies was.

When I first announced the topic, one of the girls said “I’m not interested in boy things.”  Fortunately, once we started working on the problems, she jumped right in :).

The first breakthrough for the code was that one kid said that one of the three letter words could be “the”.  I pointed out there were several three letter words.  They noticed that two of the four three letter words were the same, and tried filling in “the” for that one (which was correct).  They made a bit more progress after this but then got stuck.  One kid kept asking “What are the two little dashes at the bottom?”  I just said “Part of the message.”  Eventually, that kid said “Maybe this says Shredder!”  They didn’t quite get the concept of making sure the pattern of the word worked (they certainly didn’t notice that certain words had double letters until after they filled them in).  Once they got Shredder and filled in the matching letters elsewhere, they guessed “turtle” and “ninja”, and got most of the rest.  They ended up having the whole message decoded except for “_i_th”.  One kid suggested “ninth”, but n was already used.  I suggested they try similar things, and they got “fifth” after a bit.  I had the kids take turns being the writers, since they all wanted to fill in the letters (4 kids writing at the same time is pretty tricky).

They were slower than I expected at generating names.  In similar activities in the past, there’s usually been a flood of suggestions, but I guess the concept of combining a prefix and suffix was trickier.  However, almost immediately, one of the kids suggested drawing a line between the prefix and suffix whenever they thought of a new name.  This was obviously extremely helpful for searching, although they still weren’t that good at that.  After a while, a different kid noticed there were a different number of lines coming from various suffixes, and found things to even it out.  However, they still didn’t realize there needed to be three coming from each one; after a while one of the kids suggested there should be four lines, but I pointed out that you wouldn’t have one for the actual real names, so only three.  All the kids came up with at least one name and wrote it on the chart.  This was the first time in a combinatorics activity that I think they had some concept of having found the all possibilities.

The story problems were somewhat difficult, in particular, understanding what the problem was asking was sometimes tricky.  Their reading is pretty good, but long complicated questions are still difficult.  I had to help almost all the kids understand their questions.  About half the kids finished their own questions fairly quickly, and then I had those kids help the others with their problems.  There some interesting incorrect answers.  For the problem “There are 17 pizzas with 10 slices each, Michelangelo eats one slice from each pizza, how many are left?” the initial answer was 7 (17 – 10).  When another kid came to help, they realized it should be 9 * 17.  I suggested that they might take the initial pieces (170) and subtract the eaten pieces (17), which they could do.  Another interesting problem was “Leo has 2 swords, Raph has 2 sais, Dona has 1 bo, and Michel has 2 nunchuks.  Also, each turtle has 3 ninja stars.  How many total weapons?”  The initial answer was 10, adding all the numbers in the problem.  So, there’s clearly room to improve in translating story problems into math expressions.

#### Origami

The pinwheel wasn’t too bad for them, we can probably do something harder.  But the final step where you pull out the points was tricky for some of them.  One of the kids was eager to tell me that she could make an origami dragon, but we didn’t have time for a demonstration.

# 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.

# 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.

# 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.

# Stepping on a Dragon’s Butt

## The Activities

1. Topic: Large Numbers: Book: Millions, Billions, & Trillions by D. Adler.
3. Topic: Measurement: I bought two standard latex balloons filled with helium at the grocery store.  The problem was to figure out how many balloons it would take to lift one of the children.  The available materials were paper clips and a kitchen scale.

## How Did It Go?

All the kids attended.  Unfortunately, this circle was pretty wild; I almost stopped it early because many of the kids weren’t paying attention during the final activity.  The programming went very well and the kids liked the book, though, so it was still a moderately successful circle.

#### Millions, Billions, & Trillions

This was a pretty interesting book that talked about how big a million, billion, and trillion were.  The kids were particularly interested in how long it would take to count to each of these numbers.  One of the kids didn’t believe that a trillion dollars in \$100 bills would make as tall a stack as it would (700 miles).

#### Programming

The kids were all quite comfortable with putting things in boxes and then using them, definitely better than last time we did programming.  Nothing was too tricky, but I did vary the order that I used the boxes in, I had one where you set from one box into another, and one with a loop.  No one had any problems.  There was some variation in speed, which depended entirely on how fast each kid wrote.  The kids loved “Ask A Friend”, but it also got pretty wild.  Some of the supplied answers weren’t entirely appropriate, which is how we ended up with “[Anonymized kid’s name] stepped on a dragon’s [butt].” Needlessly to say, they thought this was hilarious.

#### How Many Balloons?

This is a pretty nice activity that just didn’t go as well this time; I think the lesson for next time is to do the group activities earlier and save individual stuff for last, since they have less problems staying on task with individual work.

# A Octopus Kissed A Ninja

## The Activities

1. Topic: Addition: Book: Math Man by T. Daniels.
2. Topic: Programming: We did two types of programming this week.  First, we reviewed how loops work, and practiced assigning a variable to itself inside a loop (X = X + 1).  Second, we introduced choosing a random noun/verb/number by drawing a word from a bag — a random version of Ask_a_Friend.  Download the programs here.
3. Topic: Algebra: I attempted to show the kids how to solve algebra problems involving addition and simple multiplication using Base Ten Blocks.  For example, in “X + 17 = 68”, you can first make 68, divide it into a pile of 17 and a pile of 41, and then conclude that X is 41.
4. Topic: Reflections: A while ago I drew a picture that had every uppercase letter hidden in it, duplicated and reflected.  E.g., reflecting “B” around the vertical line gives you a butterfly.  You can download the picture here.

## How Did It Go?

All six kids attended.

#### Math Man

The kids liked the part where Math Man adds a whole bunch of prices in his head.

#### Programming

Last week, many of the kids had trouble with loops that had more than one statement inside; and also with self assignment like “X = X + 1”.  We walked through a program with both of these features: “Box_X = 1; Do 5 times { Print Box_X; Box_X = Box_X + 2 }”.  I think by the end of it most of the kids understood what was going on; we’ll try another in a couple weeks and see how it goes.  Our daughter already had a firm grasp of these two concepts, so I gave her some harder problems to work on (second page of PowerPoint above).  She got the first two but got stuck on the nested loops.

As expected, the kids really liked the programs with random nouns/verbs/numbers.  For the most part, they had no problems, but one thing that turned out to be very hard/confusing was “Do Box_X times {Print “really”}”.  Almost all the kids wanted to print the number; and one of the kids ended up print “2 2 really really”.  So clearly they need more practice using variables in control flow rather than just printing them.

#### Algebra

One issue with this activity is that some of the kids can solve problems like “X + 17 = 63” in their head, but I had everyone work through it methodically.  My approach was a bit wrong, I think; I gave everyone a paper square with “X” on it, and had them make 17 and 63.  Then, I said they should split 63 into two parts, 17 and what’s left, and from that, they can conclude that X is 46.  I think it would have been better to simply have them make 63, and then break it into 17 and 46 and conclude that X = 46.  The problem is, I don’t think they grasped the idea that the two sides were the same, so the extra 17 didn’t really help.  Another problem is that it’s a bit hard to see how to extend this to X – 17 = 63; you need to make a 17 and a 63, put them next to each other, see that if you subtract 17 from this new number, you get 17, so the sum must be the answer.

There’s a different approach as exemplified in the very nice iPad game Dragonbox Algebra which I may try next time I do this activity.  The focus there is much more on the fact that both sides have to be the same.  I also like how they use pictures of monsters instead of numbers, at least at first, which helps emphasize the nature of “subtracting from both sides”.  I think Dragonbox Algebra still gets something wrong, which is that instead of opening the box and revealing what’s inside, the box instead eats the other side, which doesn’t make much sense.  So my ideal version is to have some objects hidden inside the box, emphasize that both sides have the same objects, have them isolate the box, and then reveal it to show that it has the same thing as what’s on the other side.

#### Reflection Picture

The kids really enjoyed doing this again, working together they were able to find all the letters.

# Walking Through Paper

## The Activities

1. Topic: Shapes, Geometry: Book: Zachary Zormer, Shape Transformer, by Reisberg.
2. Topic: Shapes, Geometry: Make a Mobius Strip. Cut a piece of paper so you can walk through it.

My daughter walks through a piece of construction paper.

3. Topic: Proofs, Geometry: Using 1×2 rectangles to try to cover various checkerboard shapes. If a shape is impossible to cover, prove why. Here are the checkerboard patterns they worked on.
1. 4×4 checkerboard.
2. 5×5 checkerboard (impossible)
3. 4×4 checkerboard with two corners missing (impossible)
4. two connected crosses (impossible)
4. Topic: Combinatorics, Geometry: Using wooden pattern blocks, find as many ways as possible to make a 2×2 diamond.

## How did it go?

I led the big kids circle this week, 5 kids attended.

#### Zachary Zormer

The kids all enjoyed this book. Afterward they were excited to make the mobius strips and walk through paper the way Zachary did in the book.  Everyone paid careful attention on both parts of this activity, and successfully completed both.

#### Checkerboard Proofs

All the kids happily covered the 4×4 checkerboard paper with the 1×2 rectangles. I handed out the 5×5 checkerboard next, and everyone thought it would be easy.  Soon they found that it was no so easy afterall.  Some kids tried moving pieces around, but there was always one square left over.

After a couple minutes of trying, one kid said they thought it was impossible. Another kid said the number of squares was odd, and the rectangles could only cover two squares.  We all counted the checkerboard squares and found there were 25.

Next I handed out the 4×4 checkerboard with two diagonal corners cut out. I pointed out that this had an even number of squares.  The kids set to work, trying various patterns. Eventually a couple kids thought is was impossible, but couldn’t explain why. I pointed out that both remaining squares were always white. I asked if one piece could ever cover 2 white squares. The kids checked and decided it was impossible. Someone asked if the leftover squares would always be white, so we counted the whites (8) and blacks (6), and finished our proof.

Finally we worked on the connected crosses. The kids quickly decided it was impossible, because as soon as you use the middle of the cross, the other squares are orphaned.

During this activity, 2 of the kids were really engaged in the proofs. The other 3 kids happily worked on covering the squares, but were more interested in silly solutions (like putting pieces diagonally), than thinking about the proof.

#### Diamond Pattern Blocks

The kids were really excited to play with the pattern blocks. One kid told my daughter that she was so lucky because she could play with them after circle 🙂

After a minute of free play, I showed them a 2×2 diamond, and asked them to find as many different ways to make that shape as possible. Two kids started in right away, and came up with many different options. My daughter didn’t actually find any…I think she was trying to use novel shapes like the square in her diamonds, which doesn’t work.  The other two kids sometimes worked on the activity, and sometime just made their own shapes.

We ended up finding 14 different ways to make the diamond. My daughter found a 15th after circle, which made her happy, since she hadn’t found any during circle.