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”.IMG_2067
  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).

Birthday Treasure Hunt (Age 6)

The Activities

  1. Topic: Multiplication. Book: Too Many Kangaroo Things To Do, by Murphy. This book is about friends planning a surprise party for Kangaroo, using multiplication along the way. The kids all enjoyed the book, taking turns computing the simple multiplication (1×1 up to 4×4). One kid proudly predicted that the animals must be planning a surprise party.
  2. Topic: Various, Story Problems. I made a grid of hexes that were hidden at first. The goal was to find the hex with a diamond printed on it. Each turn the kids got to move their piece to uncover a new hex and then solve a different type of math problem for each picture type. Here are the hex pictures you need, and the full list of problems is below. We worked as one team, and I asked each kid to try each problem. If someone solved it faster than the others, then they were supposed to whisper the answer in my ear instead of shout it out. As soon as the jewel was uncovered, all 4 kids got to pick a prize from our treasure box.IMG_20160821_174304
    1.  Firefly – square numbers:
      1. First square bigger than 0.
      2. First square bigger than 5.
      3. First square bigger than 10.
      4. First square bigger than 20.
      5. First square bigger than 30.
      6. First square bigger than 40.
      7. First square bigger than 50.
      8. First square bigger than 60.
      9. First square bigger than 70.
    2. Unicorn – fractions:
      1. Divide a circle in half, then split each piece into 3 pieces.  How many pieces do you have?
      2. Divide a circle in half, then split each piece in half, then split each piece in half. How many pieces do you have?
      3. Divide a circle in four pieces. Then split each piece in 3 pieces. How many pieces do you have?
      4. Divide a circle in half. Then split each piece into 3. Then split each piece into 2. How many pieces do you have?
    3.  Dragon – money:
      1. A diamond ring costs $100. How many rings can Hans buy with $125?
      2. Diamond earrings cost $20. How many earrings can Olaf buy with $207?
      3. A diamond necklace costs $11. How many necklaces can Marshmallow buy with $110?
      4. Elsa bought 20 diamond rings that each cost $10. How much money did Elsa spend?
      5. Sven bought 4 bracelets that each cost $32, and 3 rings that each cost $14. How much money did Sven spend?
      6. Anna spent $60 on 5 necklaces. How much did each necklace cost?
      7. Hans spent $39 on 3 bracelets. How much did each bracelet cost?
    4. Troll – story problems:
      1. A troll had 12 muffins. He ate some of them. Now he has 7 muffins. How many did he eat?
      2. There are 20 muffins. Some trolls came. Each troll ate 4 muffins. How many trolls are there?
      3. 4 trolls brought muffins to a party. Each brought the same amount. There are 24 muffins at the party. How many did each troll bring?
    5. Witch square – codes: Figure out what the coded word is by subtracting the given number from each letter. For example, DBU -1 = CAT
      1. -1:  DBU
      2. -2: DTQQO
      3. -1: QPJTPO
      4. -2: JCV
      5. -1: TQFMM
    6. Maze – patterns:
      1.  1 5 9 13 __   __
      2.  1 2 2 3 3 3 4  __  __  __  __
      3. 91 82 73 64 __   __   __
      4. 11 22 33 __  __  __  __
      5. 1 1 2 3 5 8 __  __  __
      6. 1 2 4 8 __  __


    How did it go?


We had four kids today and they were all very motivated by wanting to earn a prize in honor of my son’s upcoming birthday. We played the game with 37 hexes, and the kids got unlucky and didn’t find the jewel until they had uncovered 30 hexes. Toward the end I started letting them move 2, 3, or 4 hexes without solving the problems, just to make sure we found the jewel.

All four kids worked hard on the game questions. My son is quite far ahead of his age in calculation and story problems but he did a really good job not telling the other kids the answers. The other kids stayed involved though, and we made sure to work out each answer as a group, using Base Ten blocks or counting on our fingers if necessary. One kid got bored after 30 minutes but didn’t distract the others. Another kid especially enjoyed problems the required counting by 4, 20, or 11. At first he didn’t think he could count by 11s, but quickly he saw the pattern and took the lead.

The fourth kid is the least comfortable with the number line but he got really excited by square numbers and solved all three square problems before anyone else (smallest square above 0, smallest square above 5,  smallest square above 10). We used Base Ten Blocks to do this. I showed the kids how 9 is a square number because you can make a square out of 9 unit cubes, and he then spent some time making other squares out of unit cubes. He also solved this pattern: 1, 2, 2, 3, 3, 3, 4, _, _, _, _ first.

Everyone enjoyed decoding the witch’s code and trying to sound out the trickier words…pasta? pesto? poh-aye-son? Ooohhhh: poison!

The unicorn fraction problems turned out to be tricky. All the kids could follow the instruction: draw a circle and divide it in half. But “Now divide each piece into three pieces” was tricky. Only my son figured out how to divide each half into three equal pieces. The other kids ended up drawing straight lines and getting three very uneven pieces. Most kids also forgot to divide *each* half, so they would get ‘4’ as the answer instead of 6.

We finally uncovered the jewel, and celebrated. Then everyone picked a prize and ran around outside to get rid of their pent up energy. A very successful circle!


A Pride of Fish? (Age 6)

The Activities

  1. Topic: Comparisons: Book: Too Tall Tina by D. Merritt.
  2. Topic: Measurement: I gave each kid a 12″ ruler and asked them to look around the first floor for something 3″ long.  Then I asked 6″, 11″, and 1″.IMG_2065
  3. Topic: Verbal Discussions: I asked the kids a bunch of questions about what you call groups of things: cows (herd); sheep, birds (flock); wolves (pack); flowers (bouquet, bunch, garden); fish (school); geese (gaggle); cats (?); ants (colony); bees (hive, swarm); lions (pride); people (crowd); whales (pod); witches (coven); rabbits (warren); thieves, robbers, musicians (band); soccer players (team); dancers (troupe, company); soldiers (troop, army, legion); girl/boy scouts (troop); kittens, puppies (litter); math students (circle); cards (deck, pack); grapes, bananas (bunch); books (shelf, stack, library); wheat (field); hay (bale); knives (rack); ships (fleet); stars (galaxy, cluster, universe); planets (solar system); sailors (crew); actors (cast).
  4. Topics: Logic, Numbers: I did an activity from  Math Logic & Word Problems, Gr. 1-2, Guess Benny’s Number and Guess Jenny’s Number.  Each had a series of clues that narrowed down to a single number.  We used a 100 Number Board to keep track of which numbers were eliminated.The first puzzle was
    1. The number has two digits.
    2. Both digits are greater than or equal to 5.
    3. The tens digit is greater than the ones digit.
    4. The sum of the digits is 12.

    The second puzzle was

    1. The number has two digits.
    2. Both digits are less than 8.
    3. The ones digit is greater than the tens digit.
    4. The sum of the digits is 10.
    5. The number is even.
  5. Topics: Counting, Games:  Using the 100 board again, we played the following game.  Each turn, a kid rolled a six-sided die.  They could then advance that number of spaces up to 10 times (so if they rolled a 5, they could advance 0, 5, 10, …, 45, 50).  The goal was to get to 100.  The first time they started at 0, but the second time I had them start at 1 since it’s more interesting.

How Did It Go?

We had three kids this week.

Too Tall Tina

Not much math in this book, but loosely ties into the next activity.

Finding Objects

Some of the kids needed some help measuring at first.  One of the kids spent a lot of time measuring different parts of their mom’s body.  They were pretty excited when they found matching things.


The kids weren’t able to think of many of them — e.g., for cows, they only knew herd once I told them.  One of the few that they did get was bees, where our son got both “hive” and “swarm” right away — which is pretty funny, because he’s rather afraid of bees.  They also got band of musicians, circle of math students, pack and deck of cards, and solar system of planets.  One kid guessed “pride” for fish, and then when we got to lions realized that it actually went with lions.  For stars, with some help one of the kids thought of pictures in the sky, but couldn’t remember the word constellation.

Guess Jenny’s Number

This activity was kind of hard for them.  First, they weren’t that familiar with the concept of ones and tens digit.  Second, it’s pretty tricky that you need to cover all the squares that DON’T match.  They kept trying, though, and with some help, they were able to do it.  One neat thing is you get some nice patterns along the way.  Our 8-year-old daughter worked on one of them after circle, and it wasn’t trivial for her either.

Skip Counting

This was a good exercise for skip counting — the game made it a bit more interesting, but mostly it was about practicing skip counting.  Switching to starting at 1 made for a much more interesting game — the first time, two players finished in 3 rolls.  One of the kids realized that once you were on 96, if you rolled a 3, you should stay on 96 because there are more ways to win, which is the most interesting part of this activity.


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).IMG_20160814_175721
  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.IMG_20160814_175308
  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.IMG_20160814_175551

    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.


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).

Unifix Estimating (Age 6)

The Activities

  1. Topic: Estimating. Book: Betcha! by Murphy. Two friends walk around town estimating the number of people, cars, and jelly beans they see.
  2. Topic: Estimating, Counting. Predict how many Unifix cubes can fit in a small bowl. How many Unifix cubes tall are you? How many Unifix cubes tall am I?
  3. Topic: Logic. A little boy rides the elevator alone to and from his 15th floor apartment. Whenever he goes down, he goes all the way down to floor one. Whenever he goes up, he takes the elevator up to the 7th floor, then the stairs up to the 15th. Why?
  4. Topic: Geometry. How many rectangles are in various pictures? How many triangles?
  5. Topic: Spatial Reasoning.  Cover a checkerboard with rectangular tiles that are two squares long. Are some boards impossible to cover? Why?

How did it go?

This week we had four kids, after a couple weeks with just two kids per circle. The kids were all interested in the activity and stayed on task really well.

Unifix Estimating

First we each guessed how many cubes would fit in a cup. Then each kid tried to get as many as possible inside.


The guesses ranged from four to eight. At first everyone fit 9 in their cup (with the lid sealed). But I managed to fit 11 in.  After a lot of trying my son managed to squish in 12 cubes, much to his excitement.


12 cubes!!

Next we guessed how many cubes tall each kid was. We estimated by hold a stick of 10 cubes up to the kid’s body. A taller kid then decided to estimate his height by adding a few to the other kid’s height. The guesses were around 59 – 64 cubes. It was quite challenging to stick together that many unifix cubes, but the kids all stuck with it, and ended up with ~68 cubes per kid. We then guessed that I must be 100 cubes tall. I laid on the floor while kids made a very long unifix pole, and when we counted, it was 90 cubes long


The Boy in the Elevator

I got this story from Math from Age Three to Seven by Zvonkin. A little boy rides the elevator alone. When he goes down from the 15th floor, he goes all the way to the bottom. But when he goes up, he only goes to the 7th floor then walks up the stairs the rest of the way. Why?

The first suggestions were that maybe he wants exercise. Or maybe he doesn’t like the other buttons. At that suggestion, I drew them the buttons to see what they looked like:


I taped them up to the wall. No one had much to say about this, but then I asked one kid what would happen if her little brother pressed the buttons? She said he may be too short. Then another kid suggested maybe the boy was too short to reach the 15, and could only reach up to the 7. And on the way down, he can reach the 1 button easily.

Counting Shapes

In this activity, I showed the kids pictures of shapes I had drawn and we tried to find all triangles or rectangles in the picture.

At first the kids only see four rectangles in a picture like this. But after some looking, they noticed the big rectangle around the outside edge. Then later they noticed the long thin rectanble highlighted in green, and lastly the squareish rectangle in black. All the kids enjoyed this activity.

Tiling Checkerboards

I gave the kids a bunch of tiles that each would cover two squares on a checkerboard. Then I gave them increasingly interesting checkerboards to try to cover.

First they got a 4×4 checkboard which everyone easily covered.

Next was a 5×5 board:


Notice that one square is uncovered. The kids spent several minutes trying to rearrange the tiles to cover the last square. Eventually I suggested that maybe it’s impossible? If so, can you explain why? One kid suggested the tile is the wrong shape. Or maybe you should be allowed to let the square hang off the edge of the checkerboard?

Eventually, my son counted the squares on the board (5 on top, 5 down the side => 25 squares) and he said: “it’s impossible! 25 is odd, and the tiles can only cover an even number”. We checked it out with the other kids and eventually they were convinced.

Next was this board:


My son said it should be possible because there’s an even number. But no one could do it. A couple kids suggested they would need to put the squares diagonally. I asked about the color of the remaining squares? We noticed it was always two white squares left. I asked if one tile can ever cover two white squares? The kids tried it and said no, but were not fully convinced.


The final board

This was the last board. Everyone immediately said it was impossible. One kid pointed out it would be possible if you could overlap the pieces, but no one had a clear explanation of how they were sure it was impossible otherwise.