A Triangular Circle (Age 8)

The Activities

  1. Topic: Logic: Book: Still More Stories to Solve, by G. Shannon.  We read a few stories picking up where we left off last time.
  2. Topics: Geometry, Proofs:  I started by exploring the properties of angles resulting from a line A crossing two other parallel lines B and C.  Each kid drew a picture and measured the angles, and noticed that the “Z” angles are the same, as well as the angles “translated” along line A.  I told them that this is a property of parallel lines, and then we used it to prove that the sum of the angles of a triangle is 180 degrees.IMG_1909
  3. Topics: Patterns, Sequences:  I made a very large Pascal’s triangle (20 rows) on a sheet of poster board.  The kids looked at it and noticed some patterns (symmetry, rise then fall across a row, counting numbers, triangular numbers), I pointed out the “dog-leg” property, and then we covered all the even numbers with pennies, resulting in the pattern below (which is a discrete approximation of Serpinski’s triangle, a famous fractal).

How Did It Go?


The hardest part of this was getting the abstract idea that this would hold for ANY picture, not just the one they drew.  This was particularly true for the triangular proof — it’s tricky to keep track of the difference between the name of the angle (e.g., A) and the number of degrees in its measurement.  Once you get the idea, mathematicians often gloss over this point and use the same variable to apply to both, but for the kids this was confusing.  Some of the kids understood the triangle proof, I think.

Pascal’s Triangle

The kids did a pretty good discovering the various properties.  I gave them some hints for the up/down property, but the only two things I directly suggested were the dog-leg property and covering the even numbers.  The pattern is really cool!


Lots of Bunnies (Age 7)

The Activities

  1. Topics: Fibonacci, Sequences: Book: The Number Devil by H. Enzensberger, night 6, which covers Fibonacci numbers.  We explored a number of the properties of Fibonacci numbers (such as that fact that the squares of two consecutive Fibonacci numbers is also a Fibonacci number), and drew “Bunny diagrams” to see how Fibonacci could arise in nature.IMG_1726
  2. Topics: Graphs, Coordinates: The kids each did another worksheet from worksheetworks.com.  This time the coordinates included negative values.
  3. Topics: Games, Addition: We played Clumsy Thief, which involves finding pairs of numbers that add to $100.

How Did It Go?

We had only 3 kids this week.


The kids were able to write out the Fibonacci sequence until about 1000.  The kids varied in how interested they were in the various properties; one kid was “bored”.  One of the kids really understood the bunnies diagram and did quite a few rows.

Coordinate Pictures

As usual, the kids enjoyed doing the pictures.  One of them had brought back their homework from a previous week (another coordinate picture) and got a prize.  The negative coordinates confused them for a while, but still the actual difficulty is knowing whether the first coordinate is horizontal or vertical.  By the end they were doing pretty well, and they all finished one picture.

Clumsy Thief

Two of the kids liked the game, the other wasn’t so sure; I think this corresponded roughly with how quickly they were able to find the pairs that added up to $100.

JPM (Jumps Per Minute) — Age 5

The Activities

  1. Topic: Time: Book: The Case of the Missing Zebra Stripes, “Monkey with a Minute”.
  2. Topic: Time: We figured out how many times we could do various things (draw triangles, jump) in one minute.  We counted to 60 seconds using “1-Mississippi”, and watched the second hand on an analog watch for one full rotation.  We also tried to stand on one leg for a minute.  Finally, we timed 4 different sand hourglasses (1 minute, 2 minutes, 3 minutes, 5 minutes).IMG_1709
  3. Topics: Attribute Blocks, Sequences: Each kid made a chain of 10 attribute blocks, varying only one attribute per step.  Then they had to make it into a loop.  Finally, the whole group worked together to make a loop out of all 60 blocks.
  4. Topics: Geometry, Optimization: We revisited the activity with pastures and animals.  The rules are only one type of animal per pasture.  Each 1×1 square can hold two animals, and each equilateral triangle can hold one animal.  The goal is to hold a certain number of animals using as few fences as possible.  Initially, each kid had their own set of animals, but at the end the kids worked together to make one farm for a bunch of animals.

How Did It Go?

All five kids attended this week.

Monkey with a Minute

This was only a couple of pages, but it gave the idea for the next activity.

How Much in a Minute?

I used an iPad to do all the timing, they all enjoyed watching the digital display.  As usual they loved jumping — but one minute is a pretty long time to jump with such high energy!  Only one of the kids managed to stand on one leg without touching their other foot for a full minute.  The hourglasses turned out to vary in accuracy: 1:05, 2:01, 2:45, and 4:55.

Attribute Block Sequences

Most of the kids have a pretty good handle on this now, but many still make mistakes.  The kids are more inclined to vary shapes than any other attribute, followed by color — so there were lots of long sequences of thick shapes, for example.  After they had gotten a full loop using most of the shapes (which they could do mostly by themselves, with a few mistakes), they weren’t that great at finding places to add the few remaining shapes.

Fencing Pastures

All the kids understood the rule about only one kind of animal per pasture, and most understood the rules for how many animals there could be per pasture (although they’re not that great at actually counting up larger pastures).  They were less consistent about trying to use as few pastures as possible; and none of them really understood that you could save fences by putting all animals of the same type close together.  In the group activity, we ended up with one region built by a couple of the kids that was pretty efficient, and then a few outlying pastures — initially there had been a bunch of large unused pastures connecting the outlying pastures, but the kids realized they could take away all those fences.

How To Make a Square Number

The Activities

  1. Topic: Fractions: Book: Fraction Action by L. Leedy.
  2. Topic: Scale: This was a repeat from last week, where we had some line drawings on graph paper and the kids had to scale them up by a factor of 2. This week, the drawings were a bit harder, and for the kids who were better at this, I had them scale up by a factor of 3 instead.
  3. Topics: Square Numbers, Sequences, Proofs:  I introduced and explored the fact that the sum of the first n odd numbers is n^2.
  4. Topic: Origami: I handed out origami instructions for an origami horse to each kid, with the goal that they would do it from the instructions without me showing them how to do it.

How Did It Go?

We had 4 kids this week.  This circle went pretty well, although our daughter did get frustrated and had to leave during the origami activity.

Fraction Action

The kids enjoyed this book quite a bit.  It wasn’t too complicated, but there was a slightly harder question at the end of each section that tested them a bit.

Upscaling Drawings

We did this in the previous circle with only two kids.  They liked it so much we decided to do it again.  The two kids who hadn’t done it before picked it up pretty easily.  Most of the figures weren’t too hard for them, but all the kids had an off-by-one error at least once.  The bow-and-arrow and the crown (not pictured) turned out to be the hardest.  I think the reason the bow-and-arrow was hard is that I didn’t draw a line for the “string”, which meant that in order to draw a nice curve, it was best to count out the top and bottom of the bow before you started drawing, which wasn’t what they normally did.  The crown was the only picture that had a slope that wasn’t 45 degrees, and this was pretty hard for them.  Again, the best solution is to find the two endpoints and connect them, but this wasn’t the way they had been doing the easier ones.

Sum of Odd Numbers

I started by having the kids make squares out of Base Ten Blocks for all side lengths from 1 to 10.  I kept track of the area (number of blocks) for each side length in a two-column table.  One of the kids has been practicing multiplication a lot, and already knew all the answers, but they were all still willing to arrange the blocks into squares.  All the kids liked the table and most of them made their own copy.  Next, I asked what they did when they were looking at the Fibonacci sequence, but no one remembered.  So I started writing down the successive differences, and they saw the pattern after the first few and completed it.  Next I made a table of 1, 1 + 3 + 5, 1 + 3 + 5 + 7, …, with the sum in the other column, to show that the squares were the sums of the first k odd numbers.  I was hoping they would be a bit excited by this “coincidence”, but not so much.  Next I made a 3 x 3 square of blue blocks, and asked how many I needed to add to make a 4 x 4 square.  I showed how you could add 3 on each of 2 sides, plus an additional 1, to complete the square.  Most of the kids were distracted by this point, but one of the kids learned the pattern and could answer questions like “How many do you need to add to an 11×11 square to make a 12×12 square?” without help.  Finally, I was hoping to teach them the trick that the sum of the odd numbers up to n was (n + 1) / 2 squared — but everyone just wanted to add up the numbers manually, they weren’t excited about a trick (and didn’t really understand it).  So in the end, there was some progress, but we didn’t succeed in proving anything.


The goal this time was to have them follow the instructions on their own, but they turned out mostly not to be ready.  One of the kids has been practicing quite a bit, and made a lot of progress on their own, but the rest needed lots of help.  Also, I accidentally picked one that required cutting, and the words on the instructions were very small — which was particularly a problem on the step that said “Repeat steps 4-7”.  Another challenge was knowing when to fold only one layer vs. several layers.  Yet another issue was that several steps showed multiple overlapping folds, and you needed to know that you were supposed to do the folds one at a time, unfolding after each.  We should try this again with a slightly easier model.

Which Princess has the Most Toes?

The Activities

  1. Topic: Doubles, Addition: Doubles Fun on the Farm by Freese.
  2. Topic: Shapes, Charts: We have a bag with an assortment of Attribute Block shapes.  There are two activities.
    1. Put your hand in a bag, find a shape.  Then, without looking say what size, shape, and color 🙂 it is.  Next, I specify a size and shape, and each kid needs to find it without looking.  Then the kid puts it in the right box of the shape/color chart.

      The shape and color chart.

  3. Topic: Sequences, Transitivity: 
    1. This is from “Math from Three to Seven” by Zvonkin. To go to work, I drive to the train station, get on the train, get on a bus, then walk to my desk.  How do I get home?
    2. A plane is faster than a car, car is faster than bike. Is a plane faster than a bike?
    3. Jim is taller than Fred. Fred is taller than Susan. Who is taller, Susan or Jim?
  4. Topic: Logic: Some stole the princess’s jewel. I gave the kids a lineup of 4 clipart characters, and also 4 name labels. The kids have to figure out which name goes with each character so they can identify who stole the jewel.
    1. 4 girls, different heights.
      1. The person with the red hat stole the jewel.
      2. Anne is the shortest person.
      3. Dinah is the tallest.
      4. Cara  is taller than Betsy.

        The Girls Lineup

    2. 4 boys with different amounts of hair.
      1. The thief is bald.
      2. Cody is has less hair than Ben.
      3. Cody has less hair than Alex.
      4. Cody has more hair than Dan.

        The Boys Lineup

How did it go?

We had 4 kids at circle this week.

Doubles on the Farm

This is a simple book where two kids see farm animals and do 1+1 = 2, 2 + 2 = 4, and so on. Once we got to 3+3=6, one kid predicted that next would be 4+4=8. Then the kids all said next would be 5+5. Before turning the page, I asked how much that was. The kids thought for a bit, and then one suggested it would be 10.

When we got to 7+7, the kids asked me to stop showing the answers, so they could figure it out first. I checked the book and realized there were only 3 more sums, but 4 kids, so I asked if someone would be ok with not getting another turn.  Everyone protested.  Then I said I would do them all, but they said that wasn’t fair. One kid suggested that the parents could take turns answering, but another kid pointed out that one parent wouldn’t get a turn.  I said, “Do you think that parent will cry?” And all the kids laughed, although they agreed that they would cry if they didn’t get a turn.  So three parents double checked the last sums.

Shapes and Charts

I explained that we need to feel a shape and say what it is without looking into the bag.  I had two bags so two kids could go at the same time.  Most of the kids were pretty good at this. Next I asked 2 kids to pull out a circle, however, Kid #3 said that hexagons were her favorite so she insisted on getting a hexagon.  Kid #2 didn’t understand what I meant and pulled out a triangle.

Then I asked Kid #1 and #4 to get out a circle, which they did successfully. After that, we did another round, and all the kids did better.

Then I got a chart that had square, triangle, rectangle, circle across the top and red, yellow, blue down the side.  The empty space in the top right corner was crossed out.  The kids immediately wanted to know what could go in the empty space I said nothing went there.  I then realized that the chart did not have any hexagons (I forgot about them), so we decided to put the hexagons into the empty square.

I asked each pair of kids to take out a shape and then put it in the chart.  Kid #1 got a red triangle and she wanted to put it right on top of the triangle outline.  I showed how it needed to line up with the red patch of color, but she was unconvinced.

Kid #4 put her yellow circle on the chart with no problem.

Kid #3 got a hexagon (intentionally) and so she put it in the crossed out space.

Then the kids kept taking turns placing shapes.  Kid #4 and #2 were the only ones who really seemed to understand the chart, so we should do more chart work in the future.

At the end, the kids started to get rather wild. I settled them down by saying that I needed their help solving a mystery.  Someone stole a princess’s jewels and they needed to follow clues to figure out who.

Who stole the jewels?

Each problem started with 4 clip art suspects and 4 names written on paper.  The kids’ reading ability varies from pre-K level to fluent.  I read the clues to the kids, and they had to work together to figure which name went with each person.

On the first problem (with the girls of different heights), I told the kids that the jewel thief was the girl with the red hat.  Then I said we had to find the person’s name so we could tell the police.

My first clue was that Anne is the shortest person.  Anne is the one with the red hat, so they already knew who stole the jewel…oops! But we decided to assign the other names anyway.  Kid #4 assigned Anne to the shortest person, but Kid #2 wanted to change it and give the shortest name to the shortest person.  Next I said Dinah is the tallest. Last I said Cara is taller than Betsy and Kid #4 was able to correctly assign the names.  We all agreed Anne was the thief and we should tell the police.

The next problem was the lineup of boys with different amounts of hair.  This one was much harder.  I didn’t tell them the theif was bald, I saved it to the end.  I said all the clues, but no one could deduce anything.  Finally someone assigned Cody to the hairiest person.  I said let’s see if that possible.  I said Cody has less hair than Ben, so who could Ben be? There was no one.  So I asked if there was someone else who could be Cody?  They said maybe he was second hairiest.  So we put Ben as the hairiest but had nowhere for Alex.  Then we scooted Cody down to 3rd hairiest and put Ben and Alex ahead (no one noticed or cared that we couldn’t distinguish A&B).  Finally we put Dan down for the bald person.

Then I went around the circle having each kid verify one clue, and they all fit.  So I said…ok, the theif is bald.  And they excitedly said that Dan took the jewel. I said we’d tell the police and that I hoped Dan would give the jewels back. I said maybe the princess could just ask for the jewel and then call the police if Dan says he won’t give it back.

The quietest kid in circle was a bit lost during this activity because the other kids were super loud.

Sequences and Transitivity

“To go to work, I drive to the train station, get on the train, get on a bus, then walk to my desk.  How do I get home?”

Kid #1 shouted, you do it backwards! Then she correclty explained it, except for forgetting the original order.

The kids asked me if that is really how I go to work.  I said I walk to the garage, ride my bike to my son’s school, ride my bike to work, and then walk to my desk.  I asked how I get home.  Everyone yelled “Backwards!” but had some trouble actually saying the backward order.

“A plane is faster than a car, a car is faster than a bike. Is a plane faster than a bike?”

I asked this one to Kid #2, and who correctly said a plane is faster, but it wasn’t clear if it was just because everyone already knows planes are fast, or because of the clues.

“Jim is taller than Fred. Fred is taller than Susan. Who is taller, Susan or Jim?”

I asked this one to Kid #1 who correctly said that Jim is taller.

“Aurora has more jewels than Cinderella and Cinderella has more than Belle, who has the most.”

“Belle read more books than Aurora and Aurora read more books than Ariel, who read the most? The fewest?”

Everyone was shouting answers, mostly correct.  I asked if they thought Ariel liked to read, and I said I hoped so.  One kid said the stepsisters did not like to read.

Next I asked the only boy at circle today what he wanted a puzzle about… Trains? Thomas? Spiderman? After thinking for a few moments and looking over at all the girls, he said he liked princesses. Very sweet of him 🙂  So I said “Belle has more toes than Cinderella, Cinderella has more toes than Snow White. Who has the most toes?” The kids all laughed at this one, and he answered correctly.

I asked if anyone could think of a princess who really had no toes.  One kid said Ariel, because she is a mermaid.

After circle my daughter wanted to play the jewel thief game more and said it was her favorite part of circle.