# Captain Invincible and the Magformers (Age 5)

## The Activities

1. Topic: Geometry:  Book: Captain Invincible and the Space Shapes by S. Murphy.
2. Topic: Geometry:  Each kid made a cube, rectangular prism, pyramid, and tetrahedron using Magformers.
3. Topics: Numbers, Games:  We played the higher/lower guessing game from 1-100, with each kid taking turns having the secret number.  Rules summary: each of the other kids, in turn, guessed a number and the kid with the secret number said whether the secret number was higher or lower.  There was no guess limit, but we didn’t use any visual aides to keep track of the information so far so it wasn’t trivial to win.
4. Topics: Geometry, Tangrams:  I had 6 not-too-hard to-scale tangram diagrams.  The kids each solved as many individually as they could.
5. Topic: Numbers:  We practiced making moderate size numbers (e.g. 63) using Base Ten Blocks (each kid made their own).  I also made a number and asked the kids what it was.  Finally, each kid made a challenge number for me to count, using as many blocks as they wanted.

## How Did It Go?

We had three kids this week.  This circle went very well, all the kids paid attention the whole time.

#### Captain Invincible

Not very complicated mathematically but with interesting pictures.

#### Magformer Solids

Except for our son who has played with Magformers quite a bit, it took them a bit of time to get the hang of it.  But once they started, all the kids were able to make each of the shapes.

#### Higher/Lower

The kids have been getting much better at number sense; I had them tell me the secret number each time so I could check their answers, but I hardly ever had to correct the answer of the secret holder.  On the other hand, their guessing sophistication varied quite a bit.  One kid could easily solve 1-100 puzzles on their own, keeping track of the bounds so far.  Another kid often, but not always, made good guesses.  The final kid frequently guessed numbers outside of the current range, often far outside.  For 1-100, these guesses didn’t distract the other kids too much.  After we had done two full rounds of secret numbers (6 games total), I did a secret number between 1-1000.  One of the kids was capable of winning had they been playing on their own; but this time, the non-useful guesses kept “resetting” the current progress, and they weren’t ever able to win.  One of the kids kept guess numbers between 100 and 200, even after they knew the answer was between 700 and 800.

#### Tangrams

I was afraid this was going to be terribly hard and the kids would give up; but actually, they were pretty good at it!  I often had to give a small hint, but generally only one hint per puzzle.  They each finished at least 4 of the 6 puzzles.  They also enjoyed this quite a bit, so we’ll probably do it again soon.

#### Base Ten Numbers

This also was better than previous times, although it’s still a bit tricky switching from counting by 10’s on the rods to counting by 1’s on the cubes.  The challenge problems for me ranged from 69 to 382.

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# Magic Hexaflexagons (Age 7)

## The Activities

1. Topic: Counting, Place Values: Book: The Blast Off Kid, by Driscoll. Last week I read this to the little kids.  This week the big kids heard it. They also really enjoyed this story, and could do subtraction, like, if you need 10,000 wrappers and you have 6,593, how many more do you need?

2. Topic: Probability: I had a 4-sided die, a 6-sided die, and an 8-sided die. One kid would randomly choose a die, and the roll the die and tell the other three kids what number they rolled.  The three guessers could then guess which die the roller had, or ask the roller to roll again.

The Three Dice

3. Topic: Geometry, Origami: I helped the kids each make a “Hexaflexagon”, which is a really interesting hexagon that can be “flexed” to have 3 sides.  See the excellent Youtube Video by Vi Hart here.  This site has the template and instructions I used to make the hexaflexagons.

 The 3 sides of a hexaflexagon.

4. Topic: Partial Ordering, Logic: I told the kids some facts about animals competing in a race.  The facts did not give enough information to tell which animal finished in which place, but we tried to draw graphs (similar to the building block graphs we made in past circles) to represent the information we did know.

One kid’s partial ordering graph.

## How did it go?

We had 4 kids at circle this week. It was a fun circle, except that my daughter was extremely tired, and kept crying throughout circle. After circle we had a discussion about this, where she said she did not want to leave circle to calm down because then she’d miss something.  But I said she would have to leave if she couldn’t stop disrupting circle because it was preventing other kids from learning.

#### Dice Guessing

The kids immediately realized that if someone ever rolls an ‘8’, then they must have the 8 sided die.  However, the kids were initially willing to guess after just one roll, so if someone rolls a ‘4’, some kids wanted to guess it must be the 4-sided die. After a few rounds of this, my daughter started to assert that we should keep rolling several times before guessing.  By the end, they had the roller roll the 6-sided die about 6 times before deciding it was safe to guess.

#### Hexaflexagons

All the kids loooooved this activity! It was not easy to get the hexaflexagons set up for each kid, though they were pretty patient about waiting while I helped others with the tricky steps.  Once the hexaflexagons were built, I had the kids color both visible sides, each side a different color. The kids were all suitably amazed when I flexed their hexagon, and made a new, white side appear.  Then they colored the white side a third color, and had quite a lot of fun switching the shape around. They all enjoyed showing their parents the trick after circle.  One kid made a special point of telling me how much she loved the hexaflexagon.

#### Partial Ordering Races.

We worked through three races:

1. The Dog beat the Cat. The Cat beat the Bird. The Snake beat the Dog.

2. The Fish beat the Dolphin.  The Fish beat the Duck. The Fish beat the Frog.

3. The Fly beat the Ant.  The Bee beat the Ant.  The Bee beat the Worm. The Spider beat the Fly.

At first, a couple of the kids were confused by the premise, but eventually they all got it, and agreed that the animals finished in this order: snake, dog, cat, bird.

For the second race, at first the kids wanted to say that the order was fish, dolphin, duck, but then I pointed out we couldn’t tell whether the dolphin beat the duck or not. Eventually we ended up with a picture that showed the fish first, and the other three animals in a tie.

For the third race, the kids were starting to understand what was going on, but it was a much harder example. We were moving along pretty well until my daughter pointed out that we didn’t know if the Bee beat the Fly, or even if the Worm beat the Fly.  We all agreed that the Ant, Worm, and Fly could not have won.

This was a more interesting and trickier activity than I expected, and I’d like to do more with it in the future.  Maybe next time I’ll make squares of paper with the animals’ names on it, and we can arrange the squares according to the information. That would make it easier to adjust things as you learn new facts.

# Guessing Games (Age 7)

## The Activities

1. Topics: Geometry, Perimeter: Book: Chickens on the Move by P. Pollack and M. Belviso.
2. Topic: Binary Search: I printed out some charts with letters A-Z and corresponding numbers between 1 and 99, increasing from A to Z, but with a different pattern of numbers on each chart.  The kids took turns being the “puzzle master” and being the guessers — the goal was to find a particular number (which I specified each time) in as few guesses as possible.
3. Topic: Probability: Same as an activity the Age 5 circle did last week.  I had two dice, one with twenty sides (1-20) and another with ten (0-9).  Putting the twenty-sided first, if you roll both of them you get a number between 10 and 209.  First game was, write down a number, then I roll the dice, if your number is higher, you win.  Next was, if you’re lower you win.  Finally, the interesting game is you pick one number, I roll 5 times, and you win if at least one roll is lower than your number and at least one is higher.  We also played variants where you needed, say, at least 2 higher and at least 6 lower out of 10 rolls.
4. Topic: Probability: Each kid had a bag with 10 colored blocks, split between red and green (different number per kid).  They could pull blocks out one at a time, putting them back after each draw, as many times as they wanted.  When they were done, they would guess how many green and how many red were in their bag.
5. Topics: Multiplication, Commutivity: We took the product 3 * 5 * 7 and evaluated it 3 different ways — (3 * 5) * 7, (3 * 7) * 5, and (5 * 7) * 3).  Then I showed them an informal proof of commutivity, using a 3-dimensional figure made of blocks (changing the order of multiplications corresponds to rotating the figure).

## How Did It Go?

We only had two kids this week.  As usual with small circles, both kids were very attentive.  Several of the activities could be viewed as competitions; and indeed the kids did compete.

#### Chickens on the Move

A light book that introduces the concept of perimeter.

#### Binary Search

The kids really liked being the puzzle master.  We used a whiteboard where we started with each letter on the board, and then erased the letter and replaced it with the number after they guessed.  They picked this up really fast.  They didn’t quite do binary search, they guessed where in the interval the next number would be based on its relationship to the endpoints; but it’s not clear that binary search was the best strategy anyway, since the numbers were somewhat uniformly distributed (I did have some fairly significant skew in some of the charts, but since there were only 99 possible values for the 26 boxes, you couldn’t get in too much trouble by not splitting evenly each time).  They got every number in 3-5 guesses.

#### Higher/Lower

They also were pretty good at this activity.  One kid chose a really low number on the first round, probably because they were confused about the rules; but after that, they stuck with 209 or 208 for higher (they thought for a while they had to choose a number that was possible; after I said it didn’t, they chose 1 for the next round).  When I said the rules for the final game, they said right away that they wanted to be in the middle.  However, they chose ~ 50 for the middle at first.  Later they realized 100 was the middle.  Since they got this so quickly, I added advanced rounds where they needed, say, 2 higher and 6 lower.  We used colored blocks to keep track of each result (you got a green if your number was higher, red if lower).  Both of the kids got the general idea, but one of the kids chose better numbers than the other.  I also played a couple times; as it happened, even though my numbers were only slightly better than theirs, it made the difference between winning on losing (not that likely that this would have happened).

#### Bag of Blocks

One of the kids stopped after 15 pulls, 3 red and 12 green.  They guessed 3 red and 7 green, when the answer was actually 1 red and 9 green.  The other kid stopped after 30 pulls (they chose to make it a multiple of ten), got 21 red and 9 green, and guessed 8 red and 2 green (actual answer was 7 red and 3 green — although one of them got dropped on the ground along the way, so there were only 6 red and 3 green).  So there’s still plenty to learn here — they only have a general sense of ratios; and they don’t have a strong sense that more pulls = better results.

#### Commutivity of Multiplication

They sort of already knew that you can do multiplication in any order, although calculating 3 * 5 * 7 was hard enough that I think they were still a bit surprised it came out right.  The proof by 3D figure seemed to work for at least one of them.

# A Secret Message For Dad (Age 5)

## The Activities

1. Topic: Counting, Base Ten. Book: The Blast Off Kid by Driscoll.  This is a nice book about a kid collecting 10000 wrappers to win a prize. It shows how to group wrappers into groups of 10, then 100, then 1000 to make counting easier. We used Base 10 Blocks to follow along with the book.

2. Topic: Codes, Reading. We printed out clip art pictures of things like dragons and pencils. Then I gave kids coded words, using a simple number-to-letter code. The kid decoded the message, then matched the word with one of the pictures. Here are the pictures, coded words, and key.

3. Topic: Codes. This time we used the reverse key to encode a secret Father’s Day message.  The message was “I LOV YOU DAD” (spelled out by the kids). Once the message was written, I gave the kids envelopes to put the key and message, and decorate.

4. Topic: Counting, Addition. We rolled various dice and then jumped or clapped that many times. This time each kid had a choice: roll a 20-sided die two times and add the result, or roll a die that had 00, 10, 20, 30, up to 90 one time.

My son, jumping for joy!

## How did it go?

#### The Blast Off Kid

The kids really enjoyed this book. They were excited by the big numbers, and the plot about a kid collecting wrappers to go to Space Camp. We used Base 10 blocks to count along.

#### Father’s Day Codes

All the kids liked decoding the words and matching them up with the pictures. Most of the kids needed help sounding out the words.  Two kids really work quickly on this, but the other two also participated.

When we started writing coded messages for the dads, two of the kids made progress on their own, though they needed help from me to keep track of which letter they were on.  My son started writing a coded message even before I wrote out “I LOV YOU DAD”.  It turned out he was writing random numbers down. When David decoded the message it said: “SRTN JTVA MBWD VIDY BPST”.  My son claimed this was what he had wanted to write.

#### Jumping for Joy

The kids were all very excited to play this again. The first couple kids chose to roll the 20-sided die and add the result.  After the roll, they got to decide whether we would clap or jump that many times. I always begged them to pick claps, and they enjoyed breaking my heart by making us jump.  My son saw big numbers on the other die, so he decided to roll it, and he got a 10.  He was very disappointed.

The next time a kid picked that die, he actually rolled 0.  I asked “Jumps or Claps?” He picked jumps.  We all stood up, and I pretended I was about to jump…and then I said “All done!” Everyone laughed.

Finally some rolled the big die again and got 70.  Luckily, she picked claps!  My son kept playing this game even after circle ended.

# Boys Are People, Of Course! (Age 5)

## The Activities

1. Topic: Symmetry: Book: Seeing Symmetry by L. Leedy.
2. Topics: Graphs, Drawing, Geometrical Drawing: I made some fairly simple line drawings on graph paper, each on a separate small sheet of paper.  I gave each kid a sheet of graph paper, and they needed to (exactly) copy the drawing.  Once they finished, they would get a new one to copy.
3. Topic: Programming:  We used our usual parent programming set-up, with two new commands, “Pick Up” and “Drop” (instruction cards here and here).  The task was command the robot to transfer 3 books from the island in our kitchen to the counter top (so you needed to pick up the book, turn around, go two steps, put down, etc.).
4. Topic: Sets:  We discussed whether there were more boys or more people in the world.
5. Topics: Probability, Numbers:  We have some “percentile dice” — a set of two ten-sided dice, one of which has 10/20/30/… and the other with 1/2/3/…  If you roll both, you get a number from 0 to 99.  We played three variants of the same game.  First, each kid wrote down any number, and then we rolled the dice.  If your number was higher, you won.  After we played this several times, we switched to the same game, but trying to get lower.  Finally, we played a variant where you chose a single number, and then we rolled 5 times.  You won if there was at least one time your number was higher and at least one time your number was lower.

## How Did It Go?

All 5 kids were there this week.

#### Seeing Symmetry

This book explained linear and rotational symmetry.  The kids were pretty good at drawing lines of symmetry on an object by the end of the book.  One of the kids didn’t like the book and wanted to move on to the next activity.

#### Copying Drawings

The kids had greatly varying degrees of success on this task.  It was not easy for any of them.  One common issue was following the gridlines.  Some kids got the hang of this, but even they sometimes just stopped following the gridlines; while other kids had more trouble.  Many of the kids’ first instinct was to free-hand copy the graph.  Diagonals were a big challenge; some kids drew curves instead of diagonal lines, they had more trouble counting the lengths of the other lines, etc.  Our son got pretty frustrated when he made mistakes.  All of the kids did complete at least one successful drawing; many of them finished several, some of them very nice.

#### Parent Programming

As usual with parent programming, some of the kids got distracted some of the time, since it’s a group activity with only one output.  But all of them participated at least some.  In the end, they had a program that successfully moved two of the three books.  As expected, it was GREAT whenever the robot dropped the book on the floor — so much so that one of the kids started trying to sabotage the program so it would keep falling on the floor (they did know how to successfully sabotage, at least…).  I let them add instructions at “run time” for a while, but then had them start over and check whether the instructions they had were correct (not surprisingly, they weren’t, because not all of the verbal instructions to the robot were faithfully recorded).  All in all this was reasonably successful, and in particular no one tried to make a nonsense program on the side, which was a common problem with the older kids’ circle.

#### Boys vs People

As opposed to the square vs. quadrilateral, this was easy for them, they had no doubt that there were more people (“boys and girls are both people”).  I asked whether there were more boys or girls in the world, only one kid had a thought about this and said there were the same.  I asked about insects vs. people and they said more people.

#### Higher/Lower Dice

First we practiced reading the percentile dice.  Three of the kids could do it fairly well, the other two couldn’t.  One of the three often swapped the tens and ones.  Most, but not all, of the kids could reliably answer questions like “is 71 higher or lower than 19”.  Next we played the game where you wanted your number to be higher.  The initial numbers were 10, 40, 100000, 1000, and 101.  10 and 40 lost.  At this point, we started rotating who was doing the rolling, so each kid got a turn, and I guessed as well (foolishly). In the next game, 10 switched to 20, the others switched to at least 100 (including at least one number > 1 million, which that kid thought was 1000), and I guessed 1.  The final round, the 20 may have switched to 40, I can’t remember; I guessed 0.  The kids noticed that I lost, but they didn’t think my guess was hilarious — which suggests they haven’t fully grasped this game yet.  Then we switched to the smaller game.  Everyone chose 1 or 2, and stayed there (except me, I always chose big numbers).  Finally, I explained the higher/lower game.  One of the kids was very concerned and didn’t know what to pick.  Another kid tried to get me to let them choose two numbers (I said no).  The chosen numbers were all big, the smallest somewhere around 100; and they all lost.  The next round, two of the kids stayed with really big numbers, the others switched to 19, 31, and 50; those three won.  The final game, the two kids still stayed with big numbers, and the others switched to 2, 10, 11 — everyone lost.  I realized partway through that it’s a great help if you have tokens to give out every time someone is higher/lower.  That way it’s really easy to keep track of what everyone has gotten so far.  Clearly some of the kids realized you want something that has some numbers below and some above, but not all of them, and the ones who did still weren’t choosing numbers in the middle all the time.

# The Sorting World Record! (Age 7)

## The Activities

1. Topic: Trading, Equivalent Values. Dinosaur Deals by Murphy.

2. Topic: Logic, Indirect Reasoning. I told the kids an interesting story about a poor girl, a prince, and a mean king, and asked them to explain the ending.

3. Topic: Sorting. Sort cards with the numbers 1 – 104 on them.

4. Topic: Graphs, Partial Ordering. Build structures out of numbered Keva blocks. Then build a diagram describing the build order of the structure.

A block struture with its diagram.

5. Topic: Attributes. The kids played the game Set.  A ‘set’ is a group of three cards where each attribute (number, shape, color, shading), is either the same or different on all 3 cards. For example, the top 2 middle card and the 3rd card on the bottom are a set because:

• Each card has exactly 1 shape.
• Each card is a different color.
• Each card is an oval.
• Each card has different shading.

Set

## How did it go?

#### Life or Death?

Here’s the story I told the kids (adapted from Math from Three to Seven by Zvonkin).

A poor girl and a prince fell in love. The King did not want them to marry, but he said he’d give them one chance.  He would put the word “Life” on one card, and “Death” on another, and put both cards in a hat. If the girl drew out “Life”, then she could marry the prince. If she drew out “Death”, then she would be killed. The girl would draw a card out at noon in front of the whole Kingdom.

The king really did not want her to marry the prince, so he told his advisor that he was planning to put two cards that both said “Death” in the hat. The prince heard the King talking, and warned the girl.  The girl was very scared, but then she came up with a plan.

At noon the next day, she drew a card out of the hat, and then…she ate it!

Why did she eat it?

The kids suggested various things like, “Maybe the king will make new cards”, or “Maybe the girl can put two “life” cards in”.

Eventually with some clues they realized that the townspeople could look at the leftover card to tell what the girl had drawn out.  The leftover card will say “Death”, so the girl must have eaten “Life”. The king did not want to admit his trick, so the girl got to marry the prince.

#### Sorting

Two weeks ago I the big kids circle set a new record for sorting.  104 cards in 12:30 seconds.  There were only two kids at that circle, and this time we had 3 kids.  I challenged the kids to beat the previous best time. Before they started, my daughter explained the strategy they had used last time, and the kids agreed to use that strategy again this time.

Each kids would take a stack of cards and then put each card where it should go, leaving gaps whereever there are missing numbers.  The kids started working right away, and made really good progress.  Toward then end, one girl ended up picking up all the cards and sorting them herself.  I said “It seems like you’ve really slowed down now…”  My daughter said “She has all the cards!”  I said she should ask, and the other girl immediately handed cards to each other kids.

In the end they finished in 8:30, destroying the previous record!

#### Construction Graphs

My daughter had done this in the small circle two weeks ago, and immediately started building structures and drawing her own graphs.  I worked through a sample with the other two kids, who quickly got the basic idea. Then we took turns building structures and drawing graphs. We also started with a graph, and then built the corresponding structure.

It was still possible to confuse the kids by building complex structures, so it’s worth doing this again.

#### Set

The kids were all excited to play Set, especially my daughter. She was quite a bit faster than the other two, so I made her slow down or let them have a turn, so they got a chance.

# 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.

#### Origami

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.