# Leo the Rabbit (Age 8)

## The Activities

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

## How did it go?

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

#### Leo the Rabbit

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

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

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

I ended up drawing a picture similiar to this:

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

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

#### YouCubed Heart

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

# 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).

# Number Magic (Age 8)

## The Activities

This whole circle is built from activities described in the book Games for Math by Peggy Kaye.

1. Topic: Reducing Fractions.  Book: Fractions in Disguise by Einhorn. A millionaire collects fractions for fun, but then a villain steals a rare fraction and tries to disguise it. Only reducing the fractions to their true values can find the lost fraction.
2. Topic: Addition, Subtraction, Number Properties. I performed a math magic trick. Each kid picked a three digit number where no two digits could match, e.g. 581. Then I turned all their numbers into 1089 by:
1. Reverse the kid’s number.
2. Subtract the smaller number from the larger. (e.g. 581 – 185 = 396).
3. Reverse the result and add it to the result (e.g. 396 + 693 = 1089).
3. Topic: Logic, Strategy.  I taught the kids “Tapatan” a tic-tac-toe like game. Each person takes turns placing one of their three stones on the board. After all six stones are placed, you take turns sliding the pieces from point to point along the board lines. You cannot jump over another piece or land on top it. The first person to get their three stones in a line (vertical, horizontal, or diagonal) wins.
4. Topic: Logic, Addition. I gave the kids a series of ‘number bubble’ puzzles. Place the given digits in the bubbles to make each row add up to the required sum.

A solved puzzle, placing 1,2,3,4,5,6 so that each side adds to 12.

## How did it go?

There were only two kids this week, so it was a good, focused circle. My daughter had a few angry moments, but settled down after some warnings.

#### Fractions in Disguise

Both girls really enjoyed this book. The mystery of the stolen fraction was quite compelling, but they were each a bit reluctant to spend energy trying to reduce the fractions in the book.

#### Number Magic

The kids were quite impressed by this trick. Right away they started trying to figure out how it worked. One girl noticed that the middle digit is always 9 after the initial subtraction. Both kids wanted to try again several times.

I told them there is one class of numbers that the trick does not work for. Eventually, my daughter stumbled upon it. If the first and last digits are consecutive, then the final answer will be 198 instead of 1089. For example: 231 – 132 = 99, 99 + 99 = 198. We noticed how the subtraction always results in 99 in this case.

#### Tapatan

This game proved to be pretty fun. The girls quickly started thinking a move or two ahead to make sure they didn’t let their opponent win. My daughter was quite a poor sport whenever she lost, crumpling up the board, or throwing the pieces. The other girl was very calm during these tantrums. There is a lot more to this game than to tic tac toe. We added one extra rule: you cannot undo a move on your next turn, i.e. you can’t move a stone back to the same place it had been the previous turn. This helps prevent stalemates.

#### Bubble Logic

Both girls quickly got the idea of these problems, and had some good insights. On the first class of problems, with the 4 bubbles in a cross shape, my daughter quickly noticed that you should always put the middle two numbers together, and the largest and smallest together.

Later, when we switched from the L-shaped 5-bubble puzzle to the cross-shaped 5-bubble puzzle, both girls independently realized that they could reuse their answer from the L-shape. This was a great insight.

We stayed late at circle a few minutes, because both girls wanted to finish all the bubble puzzles.

# Love And Hate (Age 8)

## The Activities

1. Topic: Logic: Book: Still More Stories to Solve by G. Shannon, stories 9-10.
2. Topics: Simulation, Triangular Numbers: I did a variant of last week’s Star Wars battles activity.  This time though, I used Pokemon theme and had 1 big Pokemon (e.g., Charizard) fighting a bunch of small Pokemon (we used small blue cubes, so they were Merills).  Each small Pokemon had 1 hit point and 1 attack, and the Charizard had lots of health, e.g., 25, and did 5 damage (but could only attack one Merrill each turn).  Each round, the Merrills each do 1 damage to the Charizard and the Charizard kills one Merill.  The question was, how many Merills does it take to knock out the Charizard?  Once they figured this out, I asked 50 HP and 100 HP.  Finally, I changed it so the Charizard also had a fire breath that did 1 damage to all enemies that could be used every 3 turns, but now the Merills didn’t have to all attack at once (some could stay back and be protected from the fire breath).  The goal was to figure out how much damage 6 Merills could do.
3. Topics: Drawing, Scale: I picked two line drawings from the Internet, scaled them to the size of a sheet of paper, and overload a 12 x 16 grid.  Then, I gave each kid a piece of graph paper (with much smaller squares), and asked them to shrink the picture, treating each square on the big picture as one square on their graph paper.

## How Did It Go?

We had four kids this week.  This circle didn’t go that well — most kids weren’t trying to solve the Pokemon problem, and only a couple kids tried to use the graph paper to shrink the drawing.

#### Still More Stories To Solve

These two were probably too hard to actually solve.  However, it was still interesting to see if they could understand why the solution worked.  The second one involved someone making a clever lie to trick another of the characters, and the two characters having different information states.  One kid got the idea, but the others didn’t understand the solution — the first kid explained, and I think eventually one of the others understood but I’m not sure the other two ever did.

#### Pokemon Battles

Only one kid really tried on this activity.  They solved the 25 HP problem by trying different numbers of Merills, and I pointed out that 7 + 6 + 5 + 4 + 3 + 2 + 1 is the same as 1 + 2 + 3 + 4 + 5 + 6 + 7.  A couple of the kids remembered how to compute this, and I made a chart up to the 7th number.  I reminded them that you didn’t need to compute from scratch to get the next triangular number on the chart, and then that kid was able to solve 50 HP and 100 HP.  For the advanced version, the one kid tried out a few different strategies for the Merrills and found the best one for 6 Merrills.

#### Shrunken Drawings

Most of the kids didn’t try, one specifically said they didn’t want to use the graph paper and just wanted to copy it.  I worked on it as well, as an example of how to do it, hoping they would see how I was doing it.  One kid made a good effort to shrink the drawing using the graph paper, two made fairly nice looking, somewhat shrunken copies (but probably only 80% size, rather than 50%), and one drew a different-looking robot.  Our daughter said she “loved AND hated” the activity — afterwards she said it was because she liked the robot but the girl was too hard — but I think actually she liked that it involved drawing but thought the shrinking was too hard.  I asked her how she could love and hate something, and she said “It’s just like I love and hate Mommy.”  (She’s been experimenting with emotions recently).

# Spoons Full of Beads (Age 8)

## The Activities

1. Topic: Estimation. Book: Betcha by Murphy. In this book two friends estimate the number of various objects, e.g. cars on a block, people on a bus. It ends with guessing the number of jelly beans in a large jar.
2. Topic: Estimation. Guess how many beads will fit on each of three different spoon sizes. First the kids guessed by just looking at the spoon. Then we piled on the beads, and I let them make a second guess based on the beads on the spoon. Then we counted to see how many there really were.
3. Topic: Building. Next I gave each kid the small size of spoon and challenged them to fit as many beads on the spoon as they could.
4. Topic: Charts. Each kid tracked a different attribute of a Pokemon: color, height, hit points, number of abilities. We flipped over 10 random Pokemon cards and each kid updated their chart. Then we looked at our charts and predicted how the chart would change when we add in the next 10 Pokemon cards.

## How did it go?

We had four kids this week, and it was a rowdy but good-natured circle. Several kids got off topic at various points but they generally came back on track after some warnings from me. This morning was my daughter’s 8th birthday party, and she had come home completely exhausted and grumpy. I was quite worried she would want to sit out circle again, but she actually did ok, though she was gigglier than usual.

#### Estimation

The kids all enjoyed the book, and played along making their own guesses. The spoon activity was also fun. It was a bit tough to get kids to actually make their guesses, but once they were written down, everyone enjoyed helping to put the beads on the spoon and counting the beads to see who was closest.

We started with the medium sized spoon, and they realized the smaller one would hold fewer, and the bigger one more. Their second guesses (after seeing the beads on the spoon) were generally more accurate than the guess based just on the spoon.

The kids were not satisfied that I had really gotten the maximum number of beads on each spoon, so I gave them a chance to do better. I gave each kid a small-sized spoon and we sat on the floor piling beads on the spoon.  Initially we had gotten 30 beads on the spoon. I managed to get 39 on during this activity, which is one more than the 38 that had fit on the medium sized spoon. Two other kids got 34 and 36 beads. At first my daughter was messing around and giggling but eventually she got quite serious and managed to get 41 beads on her spoon, beating me by two.  The last kid never really tried and mostly threw beads around the room or put them in pockets.

#### Pokemon Charts

4 of the 5 kids are obsessed with Pokemon Go, so last week David promised them a Pokemon activity. We thought making charts of various Pokemon attributes would fit into this lesson because we could predict the attribute distribution after some Pokemon to the chart. However, this turned mainly into a looking at Pokemon and making tally marks activity. Not sure how much we really learned here, but the kids enjoyed it.  Two kids did notice that their attributes were closely related: nearly every pokemon with 40 or fewer hit points is also shorter than two feet.  We also noticed that nearly every Pokemon card has two attacks. A few have one attack, and none seem to have three or more attacks.

# Should Have Done Something About Pokemon

## The Activities

1. Topics: Logic, Puzzles:  Book: Still More Stories to Solve by G. Shannon, stories 6-8.
2. Topics: Optical Illusions, Geometry:  We did several activities from The Usborne Optical Illusions Activity Book, by S. Taplin.  The first activity was about coloring a diamond grid — the well-known illusion about two different ways to see a pattern of cubes.  The second activity involved a pattern with several rows of arrows, odd rows point left and even rows pointing right — once colored, it can either look like, say, red arrows pointing right on a blue background, or blue arrows pointing left on a red background.  The third activity was a circle of dots which when connected in the specified way generated a circular hole in the middle in the shape of a circle.  I extended this activity by showing the kids how to draw a line drawing of a star: Draw a cross on a sheet of graph paper, and then draw a line from (0, X) to (12 – X, 0) for all X.

## How Did It Go?

We had all five kids this week.  I realized 15 minutes before circle we should have done an activity involving Pokemon, and indeed there was a lot of talk about Pokemon Go during circle while the kids were coloring.  We’ll definitely do something about Pokemon soon.

#### Still More Stories to Solve

The first puzzle was a variant of “This sentence is a lie.” — awfully hard for an 8-year-old to guess.  The second was about a king saying “Don’t do X until you see my face” (meaning, “until we meet again”) and then someone sees the king’s face on a coin so they do it earlier.  With some clues the kids realized that people’s faces were on money, and then they figured out the answer exactly.  The last one also went pretty well, they needed a lot of hints but they figured it out.

#### Optical Illusions

The first two only went ok, the kids were fine coloring the pictures, but then they weren’t impressed by the illusion at all.  For the cube one, I’m not sure if they were actually seeing it both ways, or if they were just uninterested; it’s very hard to tell the difference.  Most kids said something similar for the arrows: they said “They go both ways.”  One kid was quite sure that they were blue arrows going left, because the top and bottom row were red, so red looked like the background.  I added a row of blue arrows to the top, and then they said that the arrows didn’t go either way.

The third activity (circle of lines) was fine, not too hard and a nice-looking result — but still not that much excitement.  However, the star-shaped pattern was much more interesting to them.  It was tricky to do correctly — many of the kids repeatedly forgot to move one of the endpoints of the line.  In the end, all of the kids asked for me to make another cross on grid paper so they could take it home and try it again.  At first, I was doing the wrong thing — connecting (0, 12) to (1, 0) when I should have connected (0, 11) to (1, 0).  One very interesting thing about this activity is that there are several closely connected curves.  Besides the one we did, there’s also one where the length of the line you draw is constant (draw all possible lines of length 12 connecting the X axis to the Y axis), and there’s also connecting (0, X) to (1/X, 0), which makes a hyperbola.