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.

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.

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.

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.

Happy girl, before tragical jeopardy.

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.
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!

Saving My Allowance (Age 6)

The Activities

1. Topic: Money, Negative Numbers: Less Than Zero by Murphy. In this book, Perry the Penguin wants to save 9 clams to buy a scooter, but ends up borrowing money from his friends and going negative. The kids really enjoyed this book, and eagerly followed along with the graphs in the book.
2. Topic: Money. I ran a pretend store with items of different prices. The kids got \$3 each as allowance, and could buy something or save their money. I kept track of each kid’s total on an allowance chart. The kids worked with me to add their new allowance and subtract the money they spent.
3. Topic: Advanced Money. The adding and subtracting in the store activity were too easy for one kid, so I gave him a harder problem to work on. Otherwise he couldn’t stop himself from answering for everyone else.  His problem was: If you have \$10, and each pack of cards costs \$0.75, how many packs can you buy?Topic: Sorting. The kids worked together to sort the cards 1 – 100. They were trying to beat their previous record of 19 minutes.

How did it go?

We had all five kids this week. Everyone was very attentive, and enjoyed all the activities.

Shopping

I set out a ‘store’ with a bunch of small toys I found in my daughter’s room. I was very clear that the kids did not get to keep the toys they bought (my daughter would kill me!).

First I gave each kid \$3 allowance, then asked if anyone wanted to buy anything? Three of the five kids bought something the first round. The other two kids saved their money.

No one tried to go negative, even though the book we had read was about a penguin spending more money than he had.

Next I put out some more toys in the store, and had the kids add their new allowance (\$3) to the amount they currently had. Then we had a few more rounds of buying and getting allowance.

One kid decided to save all his money every time. He said there were so many things he wanted to buy, that he couldn’t pick any. He did enjoy having more money than anyone else.

The kids have gotten much better at adding since circle started last year. But several still needed help adding 3 to \$15, for example. I showed them how to count up 3: 16, 17, 18.

Subtraction is much harder for everyone. Some kids could do problems like 5 – 3, but I helped other kids by showing it on my fingers.

One kid had no problem adding and subtracting, but was having a problem with telling other people the answers before they had a chance.  I gave him a different problem, which he worked out on the floor. If you have \$10, and each pack of cards costs \$0.75, how many packs can you buy? He solved this by repeatedly subtracting 75 from 1000. At the end he counted up how many subtractions he had done, and found you could buy 13 packs, with 25 cents left over.

Sorting

Everyone was very excited to try sorting again. We discussed strategy beforehand, and said that it had worked better when all the cards were out on the floor. Then we started sorting, and only 2 of the 5 kids laid out their cards :-).  This time there were some new strategies: several kids started collecting all cards of one decade, for example, one girl had all the nineties. A boy had the seventies and the eighties.  The girl with the nineties actually sorted them in her hands, but they got mixed up again when she handed them off to the other kids.

Ultimately, the kids beat their old time by 4 minutes, and finished all 100 cards in just 15 minutes. If we follow up on the new bucket sorting strategy, we should be able to go faster.

How Much For The Whaleshark? (Age 6)

The Activities

1. Topic: Money: Book: Follow the Money! by L. Leedy.
2. Topics: Money, Addition, Subtraction: I ran a “store” where the kids could (pretend) buy various small toys.  Each round, I gave each kid 4 or 5 “dollars” (play money), and then they chose something to buy and payed me for it.  Some things cost more than \$5, so if they wanted to buy them they had to buy something cheap so they’d have enough next round.  We did three rounds, then reset.  The second run through, I paid them using \$5 bills so they had to make change.  The third run through, I had them keep track on a “ledger” in addition to paying me with physical play money.
3. Topics: Counting, Sorting: We repeated the activity where the kids sorted dominoes according to the sum of the spots.  I gave them 6 dominoes to start, and gave them 3 more at a time when they finished.  I went up to dominoes with 9 spots on each side.  One of the kids has learned his times tables, so he sorted by product instead.  After we used up all the dominoes, I had the kids sort the entire set (by sum).

How Did It Go?

We had all five kids this week.

This book didn’t have a ton of math, but it did show making change a number of times.  It also introduced the various dominations and was a nice introduction to the next activity.

Play Store

The kids got the hang of buying things quickly.  However, they weren’t very good at making change — I think several of the kids didn’t ever quite understand what it meant.  Also, some of them can’t do subtraction, so they also couldn’t do it directly.  That is, if you have a 5 dollar bill and need to pay \$3, either you can make change into ones and pay 3 of them, or you can subtract 3 and get back 2.  Some of the kids didn’t understand either; some understood the change method, and some understood both.  The ledger was also a challenge for many of them, both because they didn’t know subtraction and they didn’t understand how it related to paying for things.  A few got it, but at best the others understood the mechanics but not the meaning.  So we can definitely explore more both the ideas of making change and keeping track of quantities using a running tally.

After we finished the money activity, I had each kid make up a story about a simple math problem.  For example, I would write “3 + 2” and they would say “There were 3 birds sitting on a branch, and then 2 more came, so there were 5 birds”.  Some of the kids closely followed earlier stories, but we got several different types of stories by the end.

Sorting Dominoes

The kids were pretty good at this.  All of them were able to finish on their own, with only a few mistakes here and there.  They even handled the 9 spot dominoes pretty well.  For the group sorting, it took them a while to get organized, but once they had at least one domino in each position they made faster progress.  Not everyone participated the whole time but they all contributed.

Arranging Angles (Age 7)

The Activities

Corey was gone this week, so two parents graciously volunteered to lead circle this week.

1. Topic: Money:  Sold!  A Math Adventure by N. Zimelman and B. Barnard.
2. Topic: Geometry: This activity explored the fact that the sum of the angles of a triangle is 180 degrees, and a (convex) quadrilateral 360 degrees.  Starting with various triangles and quadrilaterals, the kids cut off the corners and then arranged them next to each other.  They also tried to measure the angles using a protractor.
3. Topic: Building: Using our collection of Magformers, the kids tried to build as tall a tower as possible.

How Did It Go?

There were some challenges this circle.

The goal was to get them to discover that the sum of the angles was 180 or 360 degrees.  Apparently, this didn’t work out too well, and the kids ended up spending a lot of time cutting without much discovery.  Afterwards, several of them said (repeatedly) that “they didn’t know what the point was.”  This is a very nice activity, so we’re going to try it again later; the parents who led it suggested that it might work better if the shapes were smaller so it took less time to cut them out.

Tower Building

The two parents had a contest with the kids to see who could build a taller tower.  Unfortunately, one of the kids knocked over the parents’ tower partway through.  Tower building with Magformers works pretty well, but at a certain point the towers tend to collapse in on themselves.

Pompeii Money (Age 5)

The Activities

1. Topic: Lines, Shapes. The Case of the Missing Zebra Stripes by Time-Life Books. (Page 40 – 47). This section was about lines: parallel, intersecting, shapes.
2. Topic: Money, Addition. I showed the kids glass beads that I said were money from a different country.  Green beads were worth \$3, red beads were \$2, and yellow beads were \$1. First I handed out small handfuls of money, and helped the kids add it up.  Next we figured out how much more money each kid needed to have \$20. Finally, I opened up a Math Circle store where the kids could spend their money on tiny toys.

My kids playing ‘store’ after circle.

3. Topic: Attributes. I taught the kids Set.  We only used the solid colored cards. First I explained what a Set is: a set of three cards where all three cards either match or are different for each attribute.  Next we looked at two cards and figured out what 3rd card would make a set.  Finally we played Set, laying out 9 cards at a time. Each kid would raise their hand if they thought they saw a Set.

Each row is a Set.

How did it go?

Pompeii Money

The kids all said they had seen money before.  I told them that we would play with pretend money from a different country.  I asked what our country should be called? One boy immediately proposed “Bumpitup” which he said was a country that had been destroyed by a volcano.  I asked if he meant “Pompeii”, and he said yes.

I showed the kids the values of the different colors of money, then handed a few beads to each kid.  The kids really varied in their ability to add up the money.  Two of the kids just wanted to count the beads, not add up the dollar amount.  They seemed to understand what was going on, but needed one-on-one help to add 3+3, etc.  Another kid could pretty much do it on their own, with just a little help.  My son could quickly add it all up himself.

We did two rounds of adding small groups of money.  Everyone soon understood that greens were the best, since they were worth three.  Next I worked with the kids to add money so that each had \$20.

I asked “What’s the fun thing about money?” One kid said the fun was that you can use money to buy more money.  I then opened the store, which had small stickers and toys available for \$2, \$5, or \$10.  The kids took turns picking an item and paying me the correct amount (often needing help).

Set

My son had played Set before, but the other kids had not.  We practiced finding Sets, and then eventually played a couple rounds. If one kid fell behind, I let that kid have extra time (and clues) to find the next Set.  All 4 kids caught on by the end of circle, and were excited to play.