# The Bird and the Bikes (Age 9)

## The Activities

1. Topic: Money. Book: The Story of Money by Maestro.  We continued where we left off last time, and got as far as early money in the Americas.  My favorite part was the discussion of why paper money caught on better in China than in Europe (the government was more stable in China).
2. Topics: Algebra, Arithmetic:  I wrote down the equation (5789 + 1286) x 549 = 3,884,175.  I used my phone to compute the right hand side.  Then I asked them a series of questions: What is the answer if you change 5789 to 5790 (the answer increases by 549)?  What if you change 549 to 550?  What if you change 549 to 1098?  Each time, using my phone, I checked that you got the same answer by evaluating directly vs. evaluating incrementally.
3. Topic: Probability: I attempted to teach the kids how to flip a coin properly, and then each kid (and me) spent 5-10 minutes flipping coins and writing down the sequence.  Then, I asked several questions: “Do you expect more heads or tails?  Is heads more likely after you’ve just gotten three tails in a row?  Is heads-tails more likely than heads-heads?”  For each one, we counted in our sequences to see whether the results matched the kids’ intuitions.
4. Topic: Logic: I drew a picture of two bicycles riding toward each other at 5 mph, starting 10 miles apart, and asked them how long before the bicycles met.  Then, I added a bird flying at 20 mph back and forth between the bicycles, turning around and going back whenever is met a bicycle, and asked how far the bird flew before the bikes met.

## How Did It Go?

We had four kids this week.  It was a pretty good circle, some distractions as always but a lot of good thinking as well.

#### The Story of Money

This book has been going well, our daughter was able to explain the China vs. Europe paper money difference later that day when we were talking about circle.

#### Incremental Algebra

The kids did quite well on this activity, many of them were comfortable with parentheses and they didn’t have much problem getting the right answers.  By the end they were getting confident enough they thought the checking was a waste of time.

#### Coin Flip Sequences

When I started asking questions, I got some answers like “more heads after 3 tails in a row” — but I was surprised that after hearing each others’ answers they quickly converged to 50/50 no matter what.  So they seem to have a decent grasp on the idea of independence of coin flips.  Flipping was kind of hard for them, they really wanted to move their whole hands instead of just their thumbs.  For this reason the results were slightly suspect.  And of course this is a probability exercise so the results never come out perfect.  But by the end I felt pretty confident that some of the kids understood the idea of evaluating probabilities by counting occurrences from a sequence of trials (i.e., statistics).  The trickiest part was that if you have a sequence of, say, 5 heads in a row, and you’re counting outcomes after 3 heads in a row, you use this sequence 3 times (first 3, second 3, and final 3).

#### The Bird and the Bikes

One of the kids got the clever answer to the full question almost immediately.  Partly this was because I made it easier by asking the bikes only question first.  But still, I was impressed.  We also started computing the “brute force” way where we figured out how far the bird flew before meeting the first bike (8 miles).  The kids did okay at this too even though it’s a bit tricky.

# Robots, Planes, and Pie (Age 8)

## The Activities

1. Topics: Puzzles, Arithmetic: Book: Edgar Allan Poe’s Pie: Math Puzzlers in Classic Poems by J. Patrick Lewis.  We read 5 or 6 of the poems and they solved the math puzzle.  For some of the poems, I found the original version and read it to them first.
2. Topics: Logic, Hard Problems: You have available an unlimited number of airplanes.  Each airplane can hold 12 units of fuel, and the airplanes can refuel each other in midair.  Each unit of fuel lets an airplane go 1000 miles.  Airplanes can only land at the starting line — if they run out of fuel anywhere else they crash.  I asked the kids to try to get as far from the starting line as possible without having any planes crash.  I created a powerpoint with planes and distance track as a visual aid — the planes show the fuel units and the kids could fill in the units in pencil as they simulated their solution.
3. Topics: Counting, Factors: We did the Robot Stepper activity from youcubed.org.  I made a square grid of the numbers from 1-100 for the kids to fill in, and gave each kid a different starting number and number of steps.  After each kid had done several different charts, we looked at them as a group to see what kind of patterns we could find.

## How Did It Go?

We had all five kids this week.

#### Edgar Allan Poe’s Pie

The kids liked the puzzles and did a pretty good job listening and trying to solve them.  However, they weren’t very interested in hearing the original poems, some of them said they were boring or “Why are we doing this?”  I was surprised because I thought they might like the change of pace.

#### Long-Range Airplanes

As is often the case on this kind of problem, a couple of the kids tried hard and the rest were distracted most of the time.  They all liked the planes — one kid was even grabbing other kids’ planes :(.  One of the kids made quite a bit of progress.  I gave the kids a way to get to 7 using 2 planes (they both move 4 spaces, one plane gives 2 fuel and returns home, other plane has enough to get to 7 and then back home); the one kid quickly figured out you can get to 8 using 2 planes, and kept improving until they got a plane to distance 12 and back (using 5 or 6 planes, can’t remember).  Framing the problem as “How far can you get?” rather than “Can you get to X?” was good, I think, because it took the pressure off.

#### Robot Stepper

Everyone was into making the charts.  One kid made a couple mistakes, decided to X out the mistakes, and then decided to go ahead and X out every skipped square.  All the kids noticed patterns as they were coloring, and often stopped actually counting and just used the pattern instead.  The best insight on this problem was one kid was able to explain why stepping by 9 created a backwards diagonal (going down adds 10, going to the left subtracts 1).  Unfortunately the kids weren’t super interested at the end when we laid out all the diagrams and analyzed them, but maybe it’s just because circle was almost over at that point.

# 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:

# Elf + Elf = Fool (Age 8)

## The Activities

1. Topics: Codes, Arithmetic, Logic:  We did the first activity from Sideways Arithmetic from Wayside School by L. Sachar.  In this activity, you have to solve letter-number substitution problems like “elf + elf = fool”, “egg + egg = page”, “top + tot = opt” and “ears + ears = swear”.
2. Topic: Arithmetic:  We revisited the activity where you have to make as many different numbers from 1-60 using only 5’s and the four basic arithmetic operations (plus parentheses).  This time I gave them a prize based on how many they could come up with working together.

## How Did It Go?

I started circle with a talk about the goals of math circle, with three points: 1) The activities are supposed to be hard and strengthen your brain (mentioning that just like soccer, you practice to get better and stronger), 2) The activities often will be things you haven’t done in school, and 3) Even if you think an activity is boring, if you say that it might affect the other kids, plus if you try hard you might find it’s actually interesting.

Whether due to this talk or not, circle went much better this time.  The activities were fairly tricky and all five kids contributed to both activities and paid attention most of the time.

#### Elf + Elf = Fool

The first one I had to give them some fairly strong clues before they realized that ‘f’ had to be 1.  After that they mostly figured out the rest.  “egg + egg = page” is quite a bit harder, the kids came up with all the key ideas but I steered them some.  “top + tot = opt” went even better, and by the 4th or 5th one some of the kids were getting pretty comfortable.

#### Formula 5

We’ve done this a couple times before with less progress than I had hoped, but this time went much better.  Besides the pep talk, I did two things differently: I had a chart on the wall where they could add their answers, and I gave them 1 prize at 25 answers, 2 at 40, and 3 if they got all 60.  In about 25 minutes they got 32 unique answers.  They also figured out the idea of adding or subtracting (5 / 5) repeatedly, but they didn’t use it to grind out all 60 numbers.  Initially they had (5 / 5) + (5 / 5) + (5 / 5) for 3, I challenged them to find a better way and they eventually got (5 + 5 + 5) / 5.  We’ll probably give them a chance to get the rest next time.  Also, a good variant we should do in the future is giving them a challenge like “Make 5, 13, 19, 27, and 41 using as few total 5’s as possible” with prizes based on how few — there’s lots of really interesting math in figuring out the factors and figuring out what nearby numbers can be made cheaply (similar to dynamic programming if you’re familiar with that concept from computer science).

# Self-Portrait in Blocks (Age 7)

## The Activities

1. Topic: Arithmetic. Book: Alice in Pastaland, by Wright. This book is a re-telling of Alice in Wonderland with a light math theme. The kids loved the theme of the book, and were happy to work out the various math problems. It’s a longer book, so we only read the first half.
2. Topic: Programming. We brought back our old pen-and-paper programming language.   Here are the programs.
3. Topic: Geometry. We gave each kid a strip of paper with a word on it, like cat or crab or rainbow. The kid then tried to make that object using wooden pattern blocks. When done, the other kids tried to guess what it was.

## How did it go?

We had 4 kids this week. It was a very focused and fun circle. Everyone stayed on task the whole time.

#### Programming

Three of the four kids immediately remembered how the language worked. They zoomed through the programs quickly, making progress on their own.  There were still a few mistakes, but very few.  The trickiest program we had today had nested loops:

Do 4 times {

Do 2 times {

Print “B”

}

}

One kid figured this out completely independently. Two others got it with help.

The fourth kid was less comfortable with programming. They understood assignment and printing, but still didn’t fully understand the concept of a loop.  For example, which lines should you repeat? What’s inside the loop and outside?  With some one-on-one help they completed five or six programs, but they still need more practice.

#### Pattern Block Pictures

Everyone loved this activity. The kids quickly made surprisingly good pictures of cats, crabs, tree, flower, star, airplane.  The hardest ones were: pizza, dragon, and car.

We had a few minutes left after we finished all the words I had prepared. Three kids decided to write their own cards for me and the other kid to make.  One of the cards I got said “Corey” (my name).  So I made a lovely self-portrait out of blocks:

# Furry Monsters Hate Raisins (Age 5)

## The Activities

1. Topics: Arithmetic, Counting: Book: Math for All Seasons by G. Tang.  We read about half the pages, solving each problem.
2. Topics: Combinations, Venn Diagrams, Attributes: The kids made pumpkins using 2 different kinds of eyes, 1 kind of nose, and 2 kinds of mouths, each with 2 orientations (right-side up or upside-down) — download templates here.  The goal was to make as many different pumpkins as possible.  After that, we made Venn diagrams with attributes like “Half-circle Eyes” or “Upside-down Mouth”.
3. Topic: Logic: We solved several Halloween-related logic story problems.  We’ve done this activity before with this circle, but this time we did all seven puzzles instead of just three.
4. Topics: Estimation, Counting: We have a small plastic skeleton bucket (about the size of a baseball).  One at a time, I filled the bucket with several different types of item: glass beads, paper clips, etc.  For each item type, he kids all guessed how many there were in the bucket, and then we counted as a group.

## How Did It Go?

We had four kids this week.

#### Math for All Seasons

The kids liked counting the items.  I don’t think they ever did the clever counting methods until after I suggested it.

#### Pumpkin Combinations

The kids did a pretty good job coming up with different pumpkins.  I taped each one to the wall after checking whether it was different.  There were a few duplicates which we fixed by turning one of the parts upside down (since we were using glue sticks, this was easy to do).  They ended up making 18 of the 32 possible pumpkins; they didn’t get blocked, they just got tired of making pumpkins.  Around this time, one of the kids started to get distracted and remained that way for the rest of this activity and the next as well.

The Venn diagrams went pretty well — they weren’t all that great with the overlaps, but they understood the idea.  We did a three-circle Venn diagram, and they did alright with that as well.

#### Halloween Logic

I still helped them some, keeping things moving, but they were much faster than last time we did this (in March).  They were more likely to guess and fix things later, and they understood all the clues right away.  So we were able to get through all 7 of the puzzles.  Two of the kids (including the distracted one) thought the clue “Furry monsters hate raisins” was hilarious.

#### Skeleton Estimation

They guessed low every time, except for a guess of 90 for small plastic acorns when the answer was 36.  In general I’ve found that they almost always guess too low.  They were particularly far off for paper clips (largest guess was 60, answer was 179).  They enjoyed counting the items as a group.

# Fractal Hearts (Age 7)

## The Activities

1. Topics: Geometry, Fractals: Book: Mysterious Patterns: Finding Fractals in Nature by S. Campbell.
2. Topics: Geometry, Fractals: I showed them how to make some standard fractals (tree, snowflake, Sierpinski triangle), and we spent some time drawing each of them.  Some of the kids also designed their own fractals.
3. Topics: Arithmetic, Order of Operations: We revisited the fives activity, where we tried to make as many numbers as possible using only fives and standard arithmetic operations.  Here’s the progress so far:

## How Did It Go?

#### Mysterious Patterns

The kids were pretty interested in this book, and it led well into…

#### Drawing Fractals

We spent quite a bit of time on this; the kids would have happily done this the whole circle. Some of the invented fractals were pretty nice.

#### Fives

Some progress was made, but the kids aren’t as inventive as I might have expected. They’re especially hesitant to use subtraction, interestingly.