Robots, Planes, and Pie (Age 8)

The Activities

  1. 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.
  2. 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.
  3. 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.
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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.

Long-Range Airplanes

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.

Robot Stepper

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.

Shappy Valentine Day (Age 8)

The Activities

  1. Topics: History of Math, Mathematicians:  Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians (Volume One) by L. Reimer and W. Reimer, Chapter 3 (Archimedes).
  2. Topics: Factors, Logic:  We played the Big Bad Wolf game (idea from youcubed).  It’s really a puzzle, not a game, since the moves of the Wolf are deterministic.  You start with the numbers from 1 to N (we started with 6 and then moved on to 10 and 15).  Each turn, the player picks a number, and then the Wolf gets all the factors of that number.  The Wolf always must get at least one number.  At the end, the Wolf gets all the numbers that are left.  Whoever has the highest total wins.IMG_1825
  3. Topic: Codes:  We finished decoding the little kids’ Valentine’s message from last week.

How Did It Go?

We had four kids this week.

Archimedes

The kids seem fairly interested in this book, but the chapters are a bit too long — they take at least 20 minutes to read.  From now on, I’m going to figure out sections to skip so the book takes no more than 10-15 minutes — probably a bit more than half of each chapter.

Big Bad Wolf Game

The kids are good enough at factors of smallish numbers that they can play this game.  Some of the kids realized you should pick numbers that only give the Big Bad Wolf one number — but they didn’t realize you also want the things you pick to be as large as possible (as far as I know the greedy strategy of picking the largest number that only gives one thing to the Big Bad Wolf isn’t optimal, but it does pretty well).  A couple of the kids got tired of the game after a bit and stopped making progress, but one of the kids who was playing it for the first time really liked it, I heard later that they mentioned it at home.

Decoding Valentines

The kids had trouble getting started decoding the two harder messages.  I helped direct the work, and we were able to decode one of them without too much trouble.  The main challenge with the first of the two was that you couldn’t tell the boundaries between numbers (i.e., 26 vs. 2 and 6).  The other was considerably harder because it was written in a color changing pencil, was hard to read, and the rows of numbers overlapped.  As a result, we thought the first letter of the second line was part of the first line, giving us “Shappy Valentine Day”.  The kids thought this was hilarious.

 

Spilled Water and Furry Houses

The Activities

  1. Topic: Factors: Book: One Hundred Hungry Ants by E. Pinczes.
  2. Topics: Sets, Attributes: I had a number of groups of four clip-art pictures, download here.  I asked the kids to divide them into two groups based on some attribute (e.g., girls vs. boys).  For each set, we came up with as many ways to group the pictures as possible.
  3. Topics: Geometry, Graphs, Map Coloring: I had several different similar pictures of flowers, with clearly defined regions.  The task was to color each flower using as few colors as possible, subject to the rule that no two adjacent regions could have the same color (this is often referred to as the map coloring problem).  Some of the flowers could be done using 3 colors, others required 4.  You can download the pictures here.
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  4. Topics: Measurement, Conservation of Volume:  I gathered a number of different household containers, such as drinking glasses, tupperware, a champagne glass, etc.  The goal was to compare the relative volumes of each of these containers by pouring water from container to container.
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  5. Topics: Games, Tower of Hanoi:  We have enough sets of Tower of Hanoi for each kid to have one.  I started them with three rings each and then we tried the four ring version.
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How Did It Go?

We had 4 kids this week.

One Hundred Ants

The kids listened intently to this book and noticed that the other animals were stealing the food while the ants were busy making rows.  I didn’t do any follow up discussion about the different ways to make 100.

Clipart Classification

The kids had plenty of interesting ways to group together the different pictures.  They noticed shoes, whether or not the characters were stick figures or not, hair color, boys/girls, and more.  Most of the time they were easily able to explain why they grouped them the way the did.  The fairy tale one (Red Riding Hood, Gingerbread Man, gingerbread house, and three bears) was the most interesting.  They got gingerbread/not-gingerbread right away.  The next thing proposed was red riding hood + gingerbread house, with the explanation that red riding hood could live in the house while the three bears could eat the gingerbread man.  This may have been spurred by one of the kids having already made a mash-up story using all the cards.  I made a silly one where I put the bears with the gingerbread man and said that those two were furry.  They disagreed, but the explanation one of them gave was that the inside of the house could be furry :).

Map Coloring

The kids made rather different amounts of progress on this one.  One kid understood the rules, and I helped them get through all 4 of the pictures.  Another mostly understood the rules but kept accidentally coloring adjacent regions the same color.  Another wanted to keep coloring the first picture with different color schemes.  And the final one wanted to watch everyone else do theirs instead of working on their own.  For the one who made it through all 4, I asked what the difference between the first and last one was (6-petals vs. 5-petals), but their answer was mostly “3 colors worked on this one but not that one.”

Relative Volumes

I decided that I would do all the pouring, which in theory could have made it less engaging; as it turned out, the problem was that the kids got TOO excited (my son had to have a timeout partway through because he was dumping water out of the cookie tray I was using to catch any spilled water).  My measurement method was to fill one of the things and then pour it into the other.  They understood the idea that if it overflowed, then the first was bigger.  They were super excited whenever it overflowed, as you might expect.

The kids don’t really understand transitivity, and as a result they didn’t realize that we could come up with a total order over all the items.  A particularly good example of this is that initially, most of the kids thought the champagne glass would be bigger than a drinking glass, because the champagne glass was taller; and later on, many of them still thought the champagne glass would hold more than an even bigger drinking glass.  They certainly gravitated towards preferring the taller thing.  They greatly underestimated the volume of a 5-6″ square, 2″ tall tupperware — they initially thought it was the smallest but it was actually one of the biggest.  At one point, one kid said that the big drinking glass would hold about twice as much as the champagne glass, because about half the champagne glass was just stem — a nice observation (although actually the drinking glass was way more than that, because it was also much wider).

Tower of Hanoi

The kids got the rules in a reasonable time; there were a couple of things they tried: holding one ring off to the side while they used their other hand to move another ring; and moving a whole pile of rings at a time.  I started with a 3-ring puzzle, and no one could solve it — no one came up with the hard idea of moving the little ring on top of the 2nd smallest in order to make space for the big one to move.  Interestingly, several of them got the (seemingly equally hard) idea of then needing to move the little one to the empty spot so they could move the middle ring onto the big one.  By the end, most of them had successfully demonstrated the solution to the 3-ring puzzle.  The 4-ring puzzle was quite a bit harder; several kids made it to the point that they had moved the 2nd biggest ring (but not the biggest) — one kid did solve the puzzle but with help from me at several points.  So they definitely have lots of room to improve.

Plastic Leaves vs. Poker Chips

The Acitivities

  1. Topic: Counting: Book: Hands Down by Dahl
  2. Topic: Tesselations: Make a tesselation out of the following shapes, using Shape Pattern Blocks.
    1. Hexagons, skinny diamonds, squares.
    2. Hexagons, triangles, diamonds.
    3. Hexagons, skinny diamonds, triangles.
    4. Hexagons, diamonds, trapezoids.
    5. Triangles, trapezoids, diamonds.
    6. Diamonds, squares.
    7. Hexagons, triangles.
    8. Skinny triangles, squares, diamonds.

      My daughter tries to cover the paper rectangle using hexagons, skinny diamonds and squares.

  3. Topic: Multiplication: Have the numbers 1 – 100 on the wall. Assign each kid a number, X, and a color of marker. The kid then counts by ‘X’, and colors in a box above the numbers they get.  For example, one kid will be assigned ‘5’, and color in a segment of 5, 10, 15, 20, etc.
    1. Which numbers have the most colored in boxes? (most factors). The least?
    2. When counting by ‘X’ how many numbers did you color in from 1 – 100?
    3. Counting by 4 always hits the counting by 2 numbers. Why?

      Each kid used a different colored marker to color in the number factors. For example, 2 used light blue, so 2, 4, 6, 8, have a light blue section.

      My daughter colors in her numbers.

  4. Topic: Weight:  Divide the kids into pairs. Each pair gets a different unit we’ll use to weigh different objects in the Pan Balance. The units are poker chips, letter tiles, and plastic leaves.  Have 4 objects that we will weigh. Each pair should guess how many of their units will weigh the same as the object.  Then they use the pan balance to get the exact answer, and fill in the answer on a chart.
    1. Which object weighed the most?
    2. Which unit of measure weighed the most?
    3. How many letter tiles does it take to equal 1 poker chip?  How many leaves?

      Plastic leaves in the Pan Balance.

 Preparation

For the tesselations, I tried out the shapes to find combinations that could tesselate nicely. For the factors, I printed out the candlestick numbers so that each kid could color in a sector for the numbers they count.  For the pan balance activity, I picked out the units of measure, and the objects we would weigh.

How did it go?

This circle was much wilder than usual.  We had all 6 kids, and it was tough to keep them all engaged.

Multiplication / Factors

First I gave assigned each kid a number, starting with 2.  All the kids were quite excited to do this activity.  I gave them 10×10 grids where they would mark their numbers. After I checked their work, they could go color on the candles on the wall.  Many of the kids noticed patterns while doing this.  The kid with ‘2’ saw that she had to color every other column. The kid with ‘3’ noticed diagonal lines in the 10×10 grid.

The kids flew through the low numbers, but it started to get hard when we did counting by 12 and 13. The kid with ’13’ got pretty frustrated.  The kid with ’12’ did ok, but ended up being off by one row.  One kid cried because someone else had the pink marker she wanted.  At that point, I decided we had colored enough numbers.

We went over to look at the candles. The kids all agreed that counting by 2s would color the most candles.  I asked them how many you would color when counting by 2s and they started to count the blue lines, but then it got confusing because the ‘2’ kid had accidentally switched from evens to odds around 47.  This confused the counters so they didn’t end up with an answer.

We kept the candles up on the wall for a month after circle, and my 3-year-old son LOVED the candles. He would constantly count from 1-100 while pointing at the candles.

Weighing

This activity did not go as well as I’d hoped. I divided the kids into 3 teams of 2, and each team was supposed to guess how many of their objects would weigh the same as my big object. Then each team got to use the pan balance to find the answer.  However, weighing took a long time, and the other kids got restless.  Also, some kids played around during the weighing. I ended up cutting this short and moving on to tesselations.

Tesselations

I had prepared bags with construction paper rectangles that should be covered by the shapes in the bag. No duplicate shapes could touch.

Half of the kids worked hard on this one, and half played around.  Some kids finished several bags of shapes, while other kids spent time swapping shapes with their neighbors.  One of the kids mentioned how she did *not* want to work on a tricky one.  Thankfully, this sentiment has been very rare in circle, and I hope it will stay that way.