If you are a first-time visitor to this blog, you may want to start by reading About the Math Circle followed by the Circle 2 Archive. You can also use the Age 5, Age 6, and Age 7 tags to find activities for a particular age group.
- 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).
- 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.
- 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.
- 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.
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.
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?
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.
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.
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.
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.
- Topic: Measurement: Book: Taro Gomi’s Playful Puzzles for Little Hands. We only did a few puzzles towards the end of the book, most of them involved measurement and were pretty hard!
- Topics: Geometry, Graphs: I made a set of Valentine’s Day themed arrow direction drawings, downloadable here. The rules are, using graph paper (ideally with fairly small squares), you start at a vertex and have one of 8 directions and a distance. I introduced something new this time, which is some of the instructions were in red, which means you moved your pencil but didn’t draw a line.
- Topics: Counting, Graphs: I gave each kid a box of the kind of candy hearts that have messages like “Be Mine” or “Sweet On You” printed on them. Each kid sorted their box by heart color, and then we made a combined graph with how many there were of each color. Then we found as many distinct hearts (message + color) as we could.
How Did It Go?
We had all five kids this week. It was a high energy circle, partly because of candy and partly because four of the five kids had just been to Cirque du Soleil. We spent five minutes at the beginning of circle so each kid who had been to the circus could say their favorite part, and then we got through the rest of circle without any mention of the circus!
One of the problems asked which of a bunch of hats was the shortest and tallest — we tried to find some kind of trick (e.g., number of stripes), but in the end all we could figure out was the measure. Similarly, the next page had two different colored poles cut into pieces and asked which pole (when put together) was longest, which seemed really hard as well.
The kids did pretty well on these, but there was a pretty big spread in ability. Most of the kids made a small mistake from time to time, usually either going the wrong distance or not doing a diagonal at 45 degrees. One kid was noticeably better, going faster and without mistakes. I was worried the red instructions (pick up your pencil) would be confusing, but they understood it easily.
I was originally planning to have them sort by message and make a graph that way, but when we opened the boxes, it turned out that the printing quality on the hearts is really bad — probably at least 1/3rd of them have missing or unreadable messages. Also, it turned out there are a TON of different messages (“Be Happy”, “Nuts 4 U”, …) — we counted 30 different ones — which would have made it hard to make a graph. So we did color instead. And then there was another surprise — there were FAR more oranges than anything else — 3 times as many as most of the other colors! And it was consistent across boxes as well. Seems like a pretty solid result that I’d expect to hold up across many boxes. The kids were pretty excited to find all the different messages and laughed every time we found a new one. The kids were also REALLY excited to eat some of the hearts, but as far as I know they listened to me and didn’t eat any until the end (they got three each).
- 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.
- 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.
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?
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)
(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.
- Topics: Geometry, Three Dimensional Shapes: Book: Sir Cumference and the Sword in the Cone by C. Neuschwander.
- Topics: Geometry, Three Dimensional Shapes: A while ago we bought 5 full sets of “D&D dice” (4, 6, 8, 10, 12, and 20 sided). We counted the edges, faces, and vertices for each of these and made a chart like in Sir Cumference, showing that “Faces + Vertices – Edges = 2”. I also pointed out the dual relationship between 6 & 8 and 12 & 20 sided polyhedra (i.e., 6-sided has 6 faces, 8 vertices, and 12 edges; 8-sided has 8 faces, 6 vertices, and 12 edges; you can switch between the two by putting a vertex in the middle of each face and connecting adjacent vertices).
- Topic: Numbers: We did What’s the Secret Code? from youcubed.org. There are some clues about what the secret number is like “The digit in the hundreds place is ¾ the digit in the thousands place.” There is more than one answer which is cool.
- Topics: Origami, Geometry: We did Paper Folding from youcubed.org. There are a number of folding challenges like “Construct a square with exactly ¼ the area of the original square. Convince yourself that it is a square and has ¼ of the area.”
How Did It Go?
We had four kids this week. As usual some kids followed along better than others, but most people were engaged for both the dice activity and the paper folding.
Sir Cumference and the Sword in the Cone
The kids liked the book, they laughed at quite a few of the math puns.
Euler’s Polyhedron Formula
The kids definitely enjoyed making the chart. They did a pretty good job staying on task (it was easy to get distracted and start rolling the dice). Counting the edges on some of the dice was fairly tricky but was much easier with good grouping strategies.
What’s the Secret Code?
The kids did well on this except that they had trouble with the decimals. They did find one of the decimal answers, because they knew that .5 = 1/2, but I believe there were other possible decimal answers as well.
The kids solved all the tasks except the last one, which was making a non-diagonal 1/2 area square. I figured out a pretty complicated way to do it (by transferring the side length of the diagonal answer onto a horizontal edge), they copied what I did but it was pretty tricky (see picture above).
- Topic: Probability: Book: A Very Improbable Story by E. Einhorn.
- Topic: Probability: First, I secretly put 2 red and 8 blue stones into a small drawstring bag. Each kid took turns pulling one stone out, looking at it, and then putting it back. The question was, are there more reds or blues? I repeated it with 4 red / 6 blue, and also 5 red / 5 blue. Finally, I made two bags, one with 10 red / 10 blue, and the other with 11 red / 9 blue, divided the kids into two teams, and asked them to figure out which bag had more reds. I gave the kids paper and pencil and they decided to make charts to keep track of the results.
- Topics: Numbers, Sorting: I had about 20 different numbers on squares of paper, 0, 1, 3, 4, 6, 8, 12, 13, 100, 105, 1001, 1052, 1053, 1000000, -5, and -100. First, I handed each kid one number and asked them to sort themselves. We did this several times, starting simple and then using some of the trickier numbers. Then, instead of handing them the numbers, I taped a number to each kids’ back, and without telling each other what the numbers were, they needed to sort themselves. We did this a few times as well.
- Topics: Tangrams, Geometry: I gave each kid six different tangram puzzles. For the kids who finished earlier, I had them work on the letter “A” from Tangrams: 330 Puzzles.
How Did It Go?
We had four kids this week. It was a good circle, a few of the kids got a little antsy when we were discussing the results of the bag counting, but otherwise they were all engaged the whole time.
A Very Improbable Story
The kids liked the cat on the head :).
The kids immediately grasped the idea of looking for whichever color came out more often. Not surprisingly, they were overconfident — once, after only 3 draws one kid concluded red was the winner and dumped out the bag, only to find out that there were 5 of each.
For the team activity, one of the teams delegated one person to pull the stones and the other to record, while the other was taking turns drawing out stones. The former strategy was about 2x faster, so I suggested the other team use it as well. It was very interesting to see the two charts (pictured above). One was a standard tally chart, except with 6 instead of 5 in each group. For the other, the kid started by writing a bunch of numbers, and then checking them off as stones were pulled out of the bag. The results came out pretty nicely — exactly 50% for the 10/10 bag, and 55.6% for the 11/9 bag (expected 55%). However, the kids were a bit confused by the fact that team 1 had counts of 15 red and 15 blue vs. 30 red and 24 blue for team 2 — at one point, one kid concluded that team 2 had more reds AND blues. In fact, the only way I got them to conclude that team 2 had more reds was to ask them to guess what was in each bag. Their guess for team 1 was 10/10, while their guess for team 2 was “6 more reds than blues” (not coincidentally, they had drawn red out 6 times more than blue). I asked them how many reds there would be if there were 6 more reds than blues, and 20 total — this was actually quite hard for them and I had to help them a lot (the initial guess, 16, didn’t work). Of course, 13/7 doesn’t match their observed results. So, there’s clearly a lot more them to learn for the fine shades of probability!
This activity was pretty easy for them, even with the numbers taped to their backs. They had a lot of fun, particularly when I gave them negative numbers or really big numbers. They did a great job not telling each other — the closest they came was saying one kid’s number was really low (when it was -100).
This group has done these puzzles before, but that wasn’t an issue, they didn’t remember the solutions. They were better than last time, but the puzzles still definitely weren’t trivial. The bonus puzzle is much harder because it wasn’t to scale, but they made a good effort and made progress.
- Topics: Puzzles, Arithmetic: Book: Edgar Allan Poe’s Pie: Math Puzzlers in Classic Poems by J. Patrick Lewis. We read 5 or 6 of the poems and they solved the math puzzle. For some of the poems, I found the original version and read it to them first.
- Topics: Logic, Hard Problems: You have available an unlimited number of airplanes. Each airplane can hold 12 units of fuel, and the airplanes can refuel each other in midair. Each unit of fuel lets an airplane go 1000 miles. Airplanes can only land at the starting line — if they run out of fuel anywhere else they crash. I asked the kids to try to get as far from the starting line as possible without having any planes crash. I created a powerpoint with planes and distance track as a visual aid — the planes show the fuel units and the kids could fill in the units in pencil as they simulated their solution.
- Topics: Counting, Factors: We did the Robot Stepper activity from youcubed.org. I made a square grid of the numbers from 1-100 for the kids to fill in, and gave each kid a different starting number and number of steps. After each kid had done several different charts, we looked at them as a group to see what kind of patterns we could find.
How Did It Go?
We had all five kids this week.
Edgar Allan Poe’s Pie
The kids liked the puzzles and did a pretty good job listening and trying to solve them. However, they weren’t very interested in hearing the original poems, some of them said they were boring or “Why are we doing this?” I was surprised because I thought they might like the change of pace.
As is often the case on this kind of problem, a couple of the kids tried hard and the rest were distracted most of the time. They all liked the planes — one kid was even grabbing other kids’ planes :(. One of the kids made quite a bit of progress. I gave the kids a way to get to 7 using 2 planes (they both move 4 spaces, one plane gives 2 fuel and returns home, other plane has enough to get to 7 and then back home); the one kid quickly figured out you can get to 8 using 2 planes, and kept improving until they got a plane to distance 12 and back (using 5 or 6 planes, can’t remember). Framing the problem as “How far can you get?” rather than “Can you get to X?” was good, I think, because it took the pressure off.
Everyone was into making the charts. One kid made a couple mistakes, decided to X out the mistakes, and then decided to go ahead and X out every skipped square. All the kids noticed patterns as they were coloring, and often stopped actually counting and just used the pattern instead. The best insight on this problem was one kid was able to explain why stepping by 9 created a backwards diagonal (going down adds 10, going to the left subtracts 1). Unfortunately the kids weren’t super interested at the end when we laid out all the diagrams and analyzed them, but maybe it’s just because circle was almost over at that point.