Sorting and Unsorting (Age 7)

The Activities

1. Topic: Units. Book: Dinosaur Deals by Murphy. In this book, a boy wants to get a T-Rex trading card. He finds a girl who will trade it for 3 Allosaurus cards, but the boy only has 1 Allosaurus. How can he get the T-Rex.

2. Topic: Sorting, Teamwork. We repeated the sorting activity from a few weeks ago, sorting the cards 1 – 104. This time we started by discussing possible strategies based on what went well last week. Then I timed the kids to see how fast their sort was, so we can try to get faster in the future.

3. Topic: Programming. We played the game Robot Turtles, with a few rules changes to make it more cooperative (in past circles some kids have gotten upset if their turtle falls behind):

  • All turtles are trying to get to the same jewel.
  • Turtles can walk on top of each other.
  • Each person gets only one ‘laser’ card, so sometimes you have to work together to rescue a friend trapped behind ice blocks.

 

How did it go?

Sorting

First we discussed what worked well last time, and what strategies we could use this time. One kid said that sorting goes slowly when people hold cards in their hands (they spend a lot of time rummaging through the cards), so someone proposed laying the cards out on the ground so everyone can see.  Then another kid suggested sorting the cards into groups of 10 at the start (1-9, 10-19, etc), then doing the big sort. I made labels for each group of ten cards, and put them around the table. Then I gave each kid 1/4th of the deck, and started the timer.

Sorting into decades went very smoothly, everyone was working together, and in parallel. There were some mistakes, for example someone misread 72 as 27, but overall progress was quick.

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Sorting into piles of 10

Next, two of the kids took the 1-9 pile and started sorting it into the final spot on the ground. The other two kids picked up some random piles. One kid laid out the 100s, 70s, and 90s on the ground, but ended up mixing them all together. The other kid took just the forties, and laid them out in order 40 -49. Then he picked up the 50s and laid them out in order under that 40s.

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Sorting the 40s and 50s (soon to be undone by a friend)

Meanwhile the other two kids got up to the 40s. One of them came over and scooped up the row of 40s, completely mixing them up, and then resorted the cards into the final positions. This wasn’t too too slow, but seemed suboptimal 🙂

The sorting really slowed down once we got to the mixed up 80s, 90s, and 100s. Three kids had a bunch of random cards in their hands, and one kid was distractedly counting the already sorted cards.  I pointed out that it seemed slow, and that people were holding cards, so they laid the cards down and eventually finished.

The final time was 16:30, which is not terrible, but definitely can be improved.

Afterward, I asked the kids what went well, and what could have gone better. One kid said it would be better if I didn’t take out some cards. At the start of the activity I randomly pulled out 7 cards from the deck, and told the kids. In practice this speeds up the sort a lot, because they don’t get stuck trying to find one card forever, and just move on, assuming that it must be one of the removed cards.

In this discussion, I demonstrated how one kid had sorted the 40s, and then the work was lost when the friend scooped them up. The kids then suggested picking the cards up in order. We tried this, and found that it was indeed quicker to lay down a sorted pile of ten cards than a shuffled one.

I also pointed out that the beginning was really fast because everyone was able to help at once, but no one had any strong ideas about how to make the full search parallelizable.

Robot Turtles

Most of the kids had played this before. Some groaned for some reason, when they saw it, but everyone seemed excited. There was a bit of extra energy left over from sorting, so this was a wild 10 minutes, but we did finish a couple puzzles. The tricky parts were that kids wanted to move their turtles while laying down their programming cards, and also, they would mix up the two turning directions without noticing. But overall they were much better at this than I expected. Their favorite part was using the lasers to rescue their friends.

IMG_20170312_183325

 

Valentines Jeopardy! (Age 9)

The Activities

  1. Topic: Money. Book: The Story of Money by Maestro. This book traces the history of money from the earliest people to present day. We read until the Lydians invented the first coins. Both kids were really interested in this book, and didn’t want to stop reading. We had various interesting discussions, for example: what would happen if someone needed a blanket, but the blanket maker didn’t want any eggs.
  2. Topic: Story problems, coordinates, money, combinations. Valentines Jeopardy. We had 4 categories with 5 questions in each category. The questions were worth 100 – 500 points, with the higher point values being harder.  Our categories were “Broken Hearts”, “Time for Love”, “Map of My Heart”, and “Valentines Store”. Here are all the questions and answers.

Valentines Shop

Valentines Shop
Stickers…………12 for $2
Toys…………….5 for $3
Cards…………..25 for $4

Each Valentine is made of 1 card, 1 toy, and 1 sticker.

100: How much do 3 Valentines cost?
200: How much do 11 Valentines cost?
300: How much do 25 Valentines cost?
400: How much to 26 Valentines cost?
500: How much do 100 Valentines cost?

Time for Love

100: Katie sang a love song to Alex. She started singing at 5:22AM, and sang for 1 hour and 34 minutes. What time did she stop singing?
200: Fluffy bunny loved carrots so much she hopped around the garden with joy. Each hop was 2 feet long. She hopped 10 times per minute for 6 minutes. How far did she hop?
300: Luke has been waiting for Valentines day since December 8th. How many days did he have to wait?
400: Sam loves candy hearts. A pack contains 30 hearts, and it takes Sam 3 minutes to each one pack. How long does it take same to eat 5 hearts?
500: Corey loves numbers. She started at 5 and counted by fives for 30 minutes. She said one number every 2 seconds. What number did she end on?

Broken Hearts

100: You have 2 colors. How many ways can you color in a heart split into two sections?
200: You have 4 colors. How many ways can you color in a heart split into two sections?
300: You have 4 colors, and each heart has to use two different colors. How many ways can you color a heart split into two sections?
400: You have 2 colors. How many ways can you color in a heart split into 5 sections?
500: You have 3 colors. Each heart much use each color. How many ways can you color a heart split into 3 sections?

Map of My Heart

What word do the letters at the given coordinates spell? Starting at 300, the words are scrambled.img_20170212_181129

100: (7, 17) (11, 19) (3, 12) (16, 10)
200: (20, 1) (16, 10) (8, 5) (18, 3) (2, 20)
300: (2, 2) (8, 5) (8, 2) (9, 11) (6, 21)
400:
(16, 10) (4, 6) (18, 7) (7, 17) (9, 11) (13, 1) (15, 5) (8, 5) (2, 20) (16, 10)
500: (19, 13) (5, 10) (2, 2) (18, 7) (6, 21)

How did it go?

We only had two kids in the circle, which was unlucky, since competitive activities like jeopardy usually go better if you have teams. Otherwise there can be too much pressure on individual kids. My daughter had an especially hard time with the competition aspect, especially after she fell behind early. She started ripping up all the materials and crying in between questions, but refused my attempts to turn the activity into group problem solving instead of a competition. Here’s the room after the activity was done. Notice all the ripped up paper bits strewn around.

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Ultimately my daughter came back from a 1400 to 100 deficit, to win 2000 to 1900. The other kid was a great sport throughout the activity. She answered 7 questions correctly, compared to 6 from my daughter, but the point value was a bit lower.

The questions were just about the right difficulty. They had to work hard for the 500s.

Time for Love: they missed the 300 and the 500. They were close on the 300, but pretty far away from being able to solve the 500.

Valentines Shop: My daughter solved the 100 – 400, but could not compute the 500 (how many 12s make 100?). The other girl was uncomfortable with this category, even though I worked through each problem right afterward to show how it goes. I think she felt overwhelmed by having to compute how many packs you need to buy for each of 3 objects.

Map of my Heart: The other girl solved 100 – 400 very quickly. She was able to guess the Valentines words from just a couple coordinates. For the 100, she guessed the answer was “LOVE” after seeing the L and that the word was four letters long. The 300 was scrambled (CANDY), and it took both girls a while to figure it out. The 400 went quickly, guessed before all letters were searched.  My daughter got the 500 (CUPID), which was the trickiest word to unscramble.

Broken Hearts: I thought this wouldn’t be that hard, but neither girl knew how to compute color combinations through multiplication. They wanted to enumerate the colors. They only answered the 300 correctly. This was because I had enumerated the 16 options for the 200, and my daughter realized she just needed to remove the double color choices to get the 300. (12).

At the end of circle all the kids got a chocolate covered strawberry that me and my daughter made this afternoon.

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Happy girl, before tragical jeopardy.

Trick or Treat Math (Age 6)

The Activities

  1. Topic: Counting. Book: How Many Donkeys? An Arabic Counting Tale, by MacDonald. In this simple book a man can’t figure out if he has 9 or 10 donkeys because he keeps forgetting to count the donkey he is riding. The kids caught on quickly and laughed whenever he got it wrong.
  2. Topic: Maps, Spatial Reasoning, Logic: Fill in a map of a treat-or-treating neighborhood based on the following clues. Here is the clipart we used: halloweencharacters.
    1. Directly to the West of your house is the Witch’s house.
    2. The Zombie house is 2 houses West of the Witch’s house.
    3. Olaf’s house is across the street from the Zombie’s house.
    4. Elsa’s house is directly South of the Witch’s house.
    5. The pumpkin house is directly East of your house.
    6. The Spider house is on the very West end of the South side of the street.
    7. The Butterfly is scared of the Spider. The Butterfly’s house is on the same side of the street as the Spider’s, but as far away as possible.
    8. The Goblin is between the Zombie and the Witch.
    9. The skeleton is directly across the street from the Spider.
    10. Next to Elsa’s house is a Graveyard that takes up two houses.
    11. The Ladybug’s house is right next to the Butterfly’s.
    12. The Fairy can fly right across the street to the Ladybug’s house.
    13. The Wizard’s house is East of the Fairy’s.
    14. Anna’s house is next to Elsa’s house.
      img_20161030_173822

      The completed puzzle

      3. Topic: Estimation, Subtraction. Guess how much candy is in a container. Then put the same candy in a shallower container and guess again. Then count the candy and figure out whose guesses were the closest.

      img_20161030_180434

      The Candy

      4. Topic: Logic. Tape a Halloween character to each kid’s head. Then the kids ask each other yes/no questions to figure out who they are. The hardest part of the game is not telling your friends what is written on their heads.

      How did it go?

      I wore my witch costume during circle, and I organized it so the kids would get to ‘trick or treat’ after completing each activity from my bucket of small prizes and candy.

      Halloween Logic

      Each clue was pretty easy for kids, especially after they understood what phrases like “directly West” means. The hardest clue was: “The Spider house is on the very West end of the South side of the street.” Two of the kids figured it out on their own. The other two needed some help from their friends to understand the “south side of the street”.

Candy Estimation

The kids were very excited to see so much candy, especially when I told them that the person who guesses closest would get to trick or treat twice after the activity. Interestingly, the guesses did not get closer after I spread out the candy. Most second guesses were at least as wrong as the first guess. I guessed after the candy was spread out (and I got within 2 of the correct number).

After everyone wrote down their guesses I asked the kids to count the candy. They immediately began discussing counting strategies. They eventually decided to sort the candy by type and then count each type. However, they soon realized that some types had too many pieces to be easily counted, and they didn’t know how to add the results anyway. So they switched to counting each piece of candy as it was thrown back into the tub. Two kids both wanted to throw in candy and everyone ended up missing a bunch of pieces when the two throwers could not coordinate. They came up with 67 pieces, but I counted it again and found 72 pieces.

Halloween Twenty Questions

The kids loved seeing costumes taped to their friends’ heads, especially when one boy got ‘Princess Leia’. I told them at the start that it is very important not to tell your friends what is written on their heads, and the kids did pretty well at this. However, some kids asked questions like “Am I a zombie?” because they saw “Zombie” on their friend’s head. The hardest to get turned out to be superman. The kid knew he was a strong hero who wears red and blue, and has an S, and has a cape, but couldn’t think of superman.  Everyone else figured theirs out eventually (with some hints from me about what questions to ask). Everyone really enjoyed this activity. At the end, we had five minutes left so one of the kids moms played and had to figure out she was a pumpkin. The kids loved hearing her questions and shouting out answers. “Can you eat me?” “Yes, but it’s yucky and too chewy!”

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:

img_20161016_173856

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.

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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:IMG_20160905_173605.jpg

 

 

 

 

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!

 

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
    IMG_20160814_175344

    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.