the avocado lives!

who: the avocado tree (our youngest baby?)

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Well, for all of you who were anxious about the fate of our avocado tree, you can breathe a sigh of relief. The current stats and observations from the 3 Js:

• height (soil to top): 99cm/38 inc 
• Widest leaf 11cm/4.25 inch 
• Longest leaf 26.5cm/10.5in 
• Diameter 51cm Longest branch (excluding leaves) 25cm
• The pit is now black/dark brown/same color as the soil (from J3)

Unfortunately, we are now unable to measure many of our original stats: mass, total length from roots to highest shoot.

Someone, I think J3, even made a little video of the tree:

Sadly, our attempt to grow a sister tree at Grandma's house was unsuccessful, the avocado pit didn't sprout while we were there and the side project had to be abandoned. There was an ambitious little grapefruit seed trying to make progress on its own, so perhaps Grandma or Grandpa will be able to provide an update of that.

Leave things around

Who: J2
What: an exponential fractal bug

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As part of our summer vacation, we are doing activities from the Moebius Noodles Multiplication Explorers course. I may write later about the activities themselves, but first wanted to share an observation about getting involvement. The question is: how do we get the kids to participate in an educational activity we want them to do?

Leave it lying around
Again and again, I get the most mileage from just making materials available and around. In this case, I made a tree fractal bug from the suggestions in the ME course. While I was thinking (out loud) about what to draw and how the pattern would work, the kids expressed very little interest. Undeterred, I made my own picture and left it lying around the house.

Today, J2 picked it up and started asking me questions about it. Of course, he also wanted to fill in the bug bodies:

This naturally led to a bunch of mathematical questions: how many bugs are there (at each different level), how many body segments are there to be colored, how many eyes, if we added a new layer...

A technical note
So, should we leave lots of things lying around the house? Yes and no. What we have found is that our kids respond best to an environment that is nearly pristine, with a couple of items around to catch their attention. When the environment is too cluttered, everything gets ignored in the background. On the other hand, a perfectly tidy and sterile environment also isn't optimal. It is simply much easier to respond to something that is already present than to proactively seek out a particular activity, building set, etc.

Apologies to the avocado and sums of cubes

Who: J3 and J2
Where: side of the house
When: this afternoon

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Sums of Cubes

Cathy O'Neil (THE Mathbabe) flagged a proof-by-picture of Nichomachus's theorem yesterday and suggested it would be a fun discussion with kids. J2 and I started exploring it today. My current favourite introduction is to say: "a friend thought you might be interested to explore ..." In this case, to explore patterns from adding up perfect cubes.

From past conversations, his natural inclination was to start with 1³, then 1³+2³, etc and look for patterns. Initially, he confused 2³ and 2⁴, so we clarified that and he embarked on a bunch of calculating. I kept notes for him. To give an easy extra term for his pattern seeking, I started with zero cubed.

First conjecture
We built our table to this level:

03	=	0
03+13	=	1
03+13+23	=	9
03+13+23+33	=	36

At that point, J2 noticed we had squares and guessed that we were going to get every other square. Two nice conjectures, one of which already wasn't quite true, but that was more obviously clear with the next term:

03	=	0
03+13	=	1
03+13+23	=	9
03+13+23+33	=	36
03+13+23+33+43	=	100
03+13+23+33+43+53	=	225

J0: so, are we still getting squares?
J2: ... yes. That's 15 squared
J0: hmm, shall we write that down?
J2: yes, daddy. write down 0 = 0², 1 = 1², 9 = ³ (etc)
J0: ok, so:

03	=	0	=	02
03+13	=	1	=	12
03+13+23	=	9	=	32
03+13+23+33	=	36	=	62
03+13+23+33+43	=	100	=	102
03+13+23+33+43+53	=	225	=	152

J2: hey, those are triangular numbers!

More testing
For the rest of the conversation, he talked about what he expected the next terms would be, then he did the calculations to check. He didn't remember 6 cubed or 7 cubed, so we had diversions to talk about strategies to calculate them. At the end, he was very excited to see that the conjecture was still working, our cubes were adding up to squares of triangular numbers.

To be continued
Frankly, I think it will be a while before he can attack the wallet proof on his own (or with my minimal guidance). In the next couple of weeks, if we have access to some blocks construction sets, though, I'm hoping we can work together to actually do these transformations, rearranging the cubes into squared triangular numbers and vice versa. I expect even this will be a bit difficult, but it should be fun!

The avocado

Our original avocado project was supposed to run for a whole year, with the kids making observations periodically and tracking the progress. The plant (and kids) have grown well during the last several months. Unfortunately, I am pessimistic about the future prospects of our plant as we enter the hot season. We'll see at the next update.

How tall is it? Shoulder height

Key observation: these new leaves are very shiny

The high chair for learning inequalities (also, a broken calculator)

who: J2 and J3
when: at lunch
where: local Japanese restaurant

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Who is taller

While eating lunch today, we found a good excuse to talk about (mathematical) inequality. Next to our table were two spare chairs, a kid high chair and a standard adult chair. The natural questions:

• if J2 sits in the high chair and J3 in the adult chair, who will be higher? 
• Are you sure and why do you think so? 
• What if you switch with J2 in the high chair and J3 in the adult chair? 
• How confident are you of the answer now?

In the course of the conversation, they talked about who is taller standing (J2) and which chair has a higher seat. It made intuitive sense to them that the taller person in the higher seat would end up higher. Still, it was good to test:

For the question about switching seats, they weren't sure, but thought J2 would still be taller (he was). Finally, I asked J2 if this would always be the case: if he sat on a lower seat, would he still be taller? After a minute's reflection, he said it could be either of them. Could they happen to end up the same height?  Also, yes!

With these simple props, it ended up being a surprisingly good conversation.

A broken calculator

After reading Mike Lawler's post about of Dan Finkel's Broken Calculator puzzle, I had to share it with J2. He was asleep at the time, so I made my own in pencilcode (a souped up version here). This morning, after breakfast, I showed it to J2, gave him the back story. We briefly talked about square roots to remind him, and then he was hooked.

You can see his current progress here, working toward finding a way to get every integer from 0 to 109:

Mike's post and videos are very good, so I only want to make a couple points to complement his discussion:

1. Playing with the calculator first made the problem much more accessible. For J2, it helped him see that the +5 and +7 buttons could only make the value larger. It also helped him recognize that he needed square numbers for his square root and to strategize about how to make them. Finally, it led him to discover the trick for making 1.
2. Making other numbers than 2 became a very natural extension that he asked on his own. At first, he started recording (or having me record) the numbers he had made on a paper, then I added the table to our program to keep track automatically.
3. He had fun the rest of the day asking other people, mostly his mother, if they could figure out how to make 2.
4. It was also very easy to extend this by asking about other combinations than +5 and +7. We played with a +6 and +7 version that is, conceptually the same, but practically much more difficult since you lose the ones-digit preservation.

For anyone who wants to sneak in some calculation practice, this served that purpose, too. Why, you might ask? Even though he could always see an answer by pressing the button, there was a cost if he pressed the wrong one because then he would have to go through his sequence again. As a result, he would pre-calculate each operation to make sure it was taking him along the right path.

Finally, this same framework could be used easily with other operations. In particular, for kids who aren't yet ready for square roots, the reduction button could be division (e.g., divide by 4) or even subtraction (e.g., subtract 19).

Load bearing tangents

who: J1, J2, and J3
when: at dinner
where: in the dining room

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So, can you get any useful math out of Peppa Pig? I was indulging the munchkins in an episode when we came to this scene (at 3m12s):

Obviously this is just meant to be silly, but the pseudo-mathematical nonsense irked me. An alternative like "I derive solutions to equations" would have sounded nearly as complex to the target audience, would have been a (nearly) sensible job, and would have fit the quadratic formula on his whiteboard.

Anyway, one of the little ones asked: "what's a load bearing tangent?" I told them the whole thing seemed silly to me, but I understand "tangent" and "load bearing." This lead us into drawing a bunch of pictures:
- a circle with a tangent line kissing a single point
- a smooth curve with a tangent line that intersects the curve at another point
- a triangle, on which we tried to find the tangent to a non-vertex point and then a vertex point.
- two circles tangent to each other (externally)

Next we talked about "load bearing" as "carrying a weight." They understood that pretty easily because they'd spent part of the evening earlier hanging off my arms.

To round it all out, they proceeded to spend the rest of the time before bed running around, pointing at things and people, and shouting out, "there's a load bearing tangent!"

Oh, and if you think I'm only a curmudgeon when it comes to math, I'll admit this grammar nonsense also grates on my ears (at 2m25s):

Stairway to (number) heaven

Who: J1, J2, J3
Where: our stairs
When: permanently (ha!)

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J2's school has a set of stairs used by the kids with numbers posted on each step. I finally realized we could copy this at home. It has been great fun for each of them, in slightly different ways:

• J3 feels like these are "her" numbers. Everyone else has told her we made them for her and the sense of ownership makes her pay a lot of attention to them.
• J2 really likes having numbers and shapes posted around the house, so he is the one who is encouraging us to continue to the top floor.
• J1 likes taking J3 on a tour of the numbers and admiring the pictures. He also likes coming up with new things to add (like writing a related number expression)
• They all like talking about the ones they made and their contribution to the project

I have three recommendations for you replicating:

1. Start with zero
2. Be open to a lot of interesting number and shape conversations while making the cards.
3. Do it! if you don't have stairs, make it into a horizontal number line on the walls. 

J2 almost finished decorating the 12 card

Cards in situ, ready to be enjoyed

Detail of the 8 Card: such craftsmanship!

Secondary Roots appear (observational botany 3)

Who: All J1 and J2
Where: at the dining table
When: after dinner (observations from 28 December/day 27)

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More progress on the avocado sprout. We had to transfer the pit as our previous glass bowl was too shallow and started to restrict the growth of the main root. Pictures courtesy of J1 this time.

Overview

Close-up of the split

Root view

Attention has drifted?

Observations
J1

• if we measure from outside the pit, the sprout is about 1 inch tall
• the sprout outside of the pit is light green with some red dots
• off of the main sprout, there are 6 tiny sprouts that look like little thorns, 4 above the pit and 2 inside
• there are also some black lines on the pit (J0 note: these are from where we cut the original avocado)
• the main root is flexible, the sprout only flexes a bit along the crack of the pit
• the secondary roots are more flexible than the main root.

J2

1. the avocado sprout looks like J1's nose (no one else agreed with this observation)
2. the part of the sprout inside the pit is white
3. the root has 5 pooplish (J0: what is this? J2: it is a little cm, daddy)
4. the toothpick are 5 peebolo (

Measurements:
the whole rig: 550 g
the rig less the pit: 486 g
from which we conclude that the pit plus 3 toothpicks: 64 g (unchanged from last measurement)

length of stem above the pit: 3.2cm (doesn't directly compare with our previous measurements)
length of main root below the pit: 2.9 cm (almost 1 cm longer than last time we measured)

Note: 3 days later on the 31st, the lengths were 6.5 cm above the pit and 14.5 cm total length. There was a striking and obvious growth spurt.

3 little number devils

Who: J1, J2, and J3, mostly engaged on separate activities (J0 and P supporting roles). Note: All activities were done together, so there is cross-talk and listening, even if I only talk about one major protagonist in each activity.
When: throughout the day (no school, so all are at home the whole day)
Where: mainly in our reception room

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School is out and new year festivities are all around us. Lest you think we have been (mathematically) idle, here are some notes of how we've been keeping busy recently.

J3's 3rd counting challenge

We have a little duck sorting game, given to us years ago a by a cousin. Put the ducks on the blue escher-stream and then take turns trying to collect all of  your tribe with common belly markings.

How many ducks are there? How do you count them when they are "swimming" on their stream?

For J3, this is quite a challenge to count all the ducks when they are in motion. She is still at the stage of counting individual objects (in contrast to recognizing clusters) and then she isn't able to keep track of which have already been counted.

Since there are a nice number of ducks, we take turns grouping them in various different ways and talking about the shapes on their bellies.

Arranging and Folding

You may have noticed I created a new page of Upcoming Activities. This is where I keep notes for things I want to remember to do with the kids. True to the promise of that page, J2 and I looked at regular polyhedra and folding. These were inspired by these posts: 3d-2d and Nets&Decorations.

Tetrahedra
Starting simple is always good.  So what arrangements of 4 equilateral triangles are there? Which of these fold to a tetrahedron?

J2 found three ways to arrange 4 equilateral triangles into a contiguous polygon. Two, on the right, fold into tetrahedra while the one on the left doesn't (but is useful for making a square pyramid).

I asked him how we could tell that the three arrangements were really different. Of course, this is a strange question because we can clearly see that they aren't the same, but I persisted and asked how we can be sure that no rotation, translation, or flip will get them to be the same. We talked about this for a while and eventually came up with two ideas:

1. For each triangle in our arrangement, how many other triangles connect to it? For the triangle, we have 1-1-1-3 while the other two have 1-1-2-2. This let us distinguish the triangular arrangement, at least.
2. How many sides does our polygon have? The three arrangements have 3, 4, and 6 sides, respectively. This was strong enough to distinguish all of them.

In the course of discussing how many sides, someone said that one arrangement had 13 sides. I asked them to figure out why that couldn't possibly be correct (4 triangles have 12 sides when they are separate, putting them together can only reduce the number of sides).

Hexominoes/Cubes
Ok, done with tetrahedra, we moved on to cubes. What arrangements of 6 squares fold to a cube? Which don't?

We identified the longest line of squares in our arrangement as an interesting characteristic to classify and called it the "spine." We had 6-spines (just one), 5-spines, 4-spines, 3-spines, and 2-spines. For example, the picture below shows a 2-spine that does fold into a cube

It was fun seeing when J2 would realize that an arrangement did or didn't make a cube and hear his reaction. We talked a bit about how we would know whether we had tested all of the arrangements, but I won't claim we were comprehensive in this exploration. He did develop one hypothesis about 4-spines:

If both "tentacles" of a 4-spine are on opposite sides of the spine, it can fold into a cube. If they are on the same side, it cannot.

Pictures below are our cubable hexominoes and the ones that just don't work out:

At the end, I tried to rearrange things and start talking about pentominoes, but this exploration was already as long enough and he was ready to move on to something else.

How fast do Fibonacci numbers grow?

Both J1 and J2 have recently been introduced to Fibonacci numbers, powers of 2, squares, and cubes. Writing down the Fibonacci sequence, J1 said: "these are growing really fast!" I asked J1 and J2: "do they grow faster than squares?"

This led to a discussion about what my question meant. J2 pointed out that, at the beginning, the squares are growing faster. Very detail oriented he pointed to the first two Fibonacci terms (1, 1) and said "they aren't even growing at all." J1 pointed farther down the sequence and said it looked like they were much larger at some point.

Here, J2 went off to do something else while J1 and I continued talking about the sequences relative to each other. I built a simple spreadsheet and then asked what we should do to compare. Some discussion later, we decided to add the ratio of the sequences and the difference. Through both measures, we saw that, indeed, the Fibonacci sequence becomes much larger than the squares. One little observation he really liked was seeing that the twelfth Fibonacci number (144) is also the 12th square, so the sequences are equal at that point.

J1 then suggested other sequences we could compare: multiples of 100, powers of 2, powers of 3, powers of 10. We did a little to play around with powers of bases between 1 and 2, but we didn't quite get to reveal the magic of the golden ratio.  At least not this time . . .

Number Devil

J1 and I, with occasional visits from J2, have been reading The Number Devil together just before he goes to sleep. Frankly, this was a book toward which I was only lukewarm. Mainly, I wasn't sure about how our kids would take the introduction about nightmares, the relationship between the number devil and Robert, and the negative comments about the math class and math teacher. As it turns out, all of these things are fine, either not taken too seriously or accepted as proof that Robert is a bona fide little boy.

On that last point, J1 was much more attuned to the fact that Robert is supposed to be 12 years old. When we got to a point in the story where the Number Devil asks Robert when he was born (answer: 1986), J1 immediately spotted something was wrong. He didn't know right away how old Robert was, but he had a sense, perhaps from knowing roughly when the 5th graders in his school were born? We spent a bit of time calculating how old Robert would really be in 2014 and then talked about what happened with his age in the story.

Otherwise, I'm finding that most of the math in the book is at just the right level. Mostly, we are reading about things that J1 and J2 have already encountered and they enjoy seeing a slightly different spin on these topics (including silly names for them). We do most of the calculations along with the characters and generally have a grand time.

Sprouting! (observational botany 2)

Who: All Js
Where: at the dining table and in front of our house
When: after lunch (observations from 19 December/day 18)

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Avocado

Our avocado pit has made some progress.



J1's observations

• the root has sprouted
• maybe it is actually upside down and the avocado is confused?
• the pit split in the middle.
• there is a sprout in the middle
• some of the pit has peeled off
• the exposed pit is a bit more rough than when we examined it last week. it feels like bumpy wax.

J2's observations

1. Maybe it is actually upside down and the avocado is confused?
2. the avocado pit looks like it is going to poop on us (referring to the emerging root)
3. the exposed pit is smooth
4. the calculator is 9.5 cm long

J3's observations <made at dinner, had been napping while the older two discussed>

• There's a plant!
• These are floss, we use it to clean our teeth (gesturing to show how)
• Oh, some of the *this* fell off (noticing that dried skin from the pit was in the bottom of the water bowl from when one of the older two peeled it off and dropped it in).  

We also had a discussion about the division of the pit into two halves. When we started, there wasn't any clear indication that it would cleave along this line. We were wondering if there was some mechanism to prevent it from cleaving where we placed a toothpick, if the splitting location is random, or if there is a clear place it will split. I guess we add these to our curiosity list.

Measurements
3 toothpicks: 0 gr on our scale, indicating that they are less than 1/2 gram
The avocado pit and 3 toothpicks together was 64 grams (J2 noted this is 8 x 8)

The sprout in the middle of the pit and extending down was 4.5 cm long.
J0's nose is 6 cm long
J1's nose is 4.5 cm long, so the same as the sprout.
From end to end, the pit and root sprout are 7 cm.
The root protrusion is 2 cm.

When we started, we estimated the pit's mass 56 grams, (J2 remarks, 56 = 7.4833147 x 7.4833147)

J1 measured me with the tape measure and proclaimed that I have 101 kg of fat. His reasoning: some 

distance around was 101 cm and he assumed I must be 1 kg per cm of perimeter along that slice.

J2 added, the following.

√6 = 2.4494897

√4.5 = 2.1213203
I asked J2 if these square roots had any meaning, in the context of our seed. He said, "no, it is just for fun."

Introducing: Orange seeds

Last sunday (13 December) we planted some orange seeds. Vaguely following these instructions, we used two methods:

1. Planting in soil, keeping the soil covered and most: no developments yet
2. Planting in a pool of water with some soil and dried leaves: interesting developments this week

Our "interesting" case this week

So, what happened with the soaking seeds? Here are our observation notes:

• Oh, the seeds are sprouting!
• Hmm, the sprouts seem to be wiggling!
• Those aren't sprouts, they are larvae, probably mosquito larvae
• Ooh, it smells like cows. It stinks

We poured it out on the street in front of our house, in the sun, and watched the water dry out. We observed the larvae moving around in the small puddle of mud as it dried and talked about what they needed to survive. J2 noticed that there was a storm drain a meter away from the puddle and asked what would happen if they went down the drain. Then we talked about whether the could get to the drain (having to cross a meter of dry ground) and how they could know that there was a safe destination on the other side. We made conjectures about their senses and ability to communicate:

• probably cannot see/no eyes
• probably cannot talk, but we guess the do have a mouth to eat
• not sure about ears
• cannot read or write
• no ability to communicate with adult mosquitos, ants, humans, or other creatures

In summary, we concluded that their knowledge is restricted to the limited part of the universe that their limited senses can observe directly. Can you see the editorializing?

As you might expect, fire was introduced at some point in the conversation, we ended up burning a handful of dry leaves and some paper scraps. As you do, you know.

A christmas eve mystery

Who: J3
Where: at school
When: over 2 weeks

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This is actually something being done at school, but it matches our seed growing at home very nicely.
They did a couple of experiments to test the effect of different conditions on plant growth:

• with and without water
• in different potting media (soil, sand, rocks)
• with and without sunlight.

Here is the picture of the plants without (left) and with sunlight:

Sorry about the blur, this was my only shot through a swarm of excited toddlers

Thus, the mystery: why did the plants grown in the dark grow so much taller? Add your hypothesis in the comments!


This is a special day for our family: Grandpa G's birthday.  So, in the spirit of celebration and birthday wishes, we send some powers of 2 (and square relationships):

Sometimes 6s got to get a bit crazy, right?

Observational botany (step 1)

Who: All Js
Where: at the dining table (for future reference, this post has notes from 1 december 2014)
When: 5 minutes a day, before dinner

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The author of Five Triangles made a suggestion somewhere (maybe his/her other blog?) that a great science activity is to plant a seed and make observations of the developing plant for a year. We are starting this with an avocado pit.

The pencil is a stand-in until our dental hygiene
catches up to our scientific zeal

J1's observations

• The pencil smells like okra
• It's red gray
• It feels like my hair
• I think it is 5 cm long
• I estimate the mass is 1trn grams

J2's observations

• The avocado pit smells like okra
• It feels like your poop (J0:"My poop or your's?" "Your's daddy")
• It is 36 feet (J0:"Long, tall, or wide?" "Every dimension")
• It weighs 1000 pooplizes (J0: "what is 1 pooplize?" "The mass of all humans on the earth put together.")

As you can see, someone wasn't really taking this seriously

J3's observations

• This is floss (pointing) and this is floss (pointing again) and a pencil (pointing for a third time).
• It is not symmetrical
• One side is round and the other is pointy (indicating the side down in the water as round and the end pointing up as pointy)
• It is smooth
• it has no smell

J3 also asked for the pencil I was using to take notes, then drew some scribbles on the page and said she was drawing avocado pits.

Measurements
We made the following measures of mass:

• Empty bowl: 107 gr or 3 3/4 oz
• Pencil: 4 gr or 1/8 oz
• Dry pit+bowl+2 toothpicks+pencil: 167 gr or 5 7/8 oz
• water added: 133gr 

By implication, the pit and 2 toothpicks is 56 grams.

References:
A general procedure for growing your own avocado tree (don't expect to eat the fruit, though);
http://www.californiaavocado.com/grow-your-own-avocado-tree/

As usual, wikipedia has something useful to say (my emphasis added):

Usually, avocados are grown from pits indoors. This is often done by removing the pit from a ripe, unrefrigerated avocado. The pit is then stabbed with three or four toothpicks, about one-third of the way up. The pit is placed in a jar or vase containing tepid water. It should split in four to six weeks and yield roots and a sprout. If there is no change by this time, the avocado pit is discarded. Once the stem has grown a few inches, it is placed in a pot with soil. It should be watered every few days. Avocados have been known to grow large, so owners must be ready to repot the plant several times.

In other news
Doesn't the icosidodecahedron look oddly, asymmetrically misshapen?

Constructions and calculations (more polydrons)

who: J2
when: almost all day saturday
where: reception floor

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I've mentioned polydrons before as a favourite construction toy.  Well, my favourite construction toy. Today, the kids spent most of the day building, breaking down, rebuilding, and investigating various creations.  J2 was really the leader of this activity, so most of the discussion focused on his investigations.

Building
First, he has been diligently making all of the example constructions included in the bucket.  He managed the icosahedron, but then we hit the stellated dodecahedron.  Mostly working on his own, but even with my help, we somehow got stuck on a squished version:

He made some interesting comments and questions along the way:
- we need 60 small equilateral triangles
- we should put them together into 5-triangle pentagonal tents
- should the 6-triangle vertices be squished in, out or something else?

He even outsourced collecting the triangles and making some of the pentagonal panels to his older brother, in a brilliant stroke of project management.

We felt that there was something wrong with our approach to the 6-triangle vertices, so he separately put 6 together and examined how the hexagon could flex, comparing it to the possibilities (more limited) for 5, 4 or 3 triangle vertices.

Next on his list was the cuboctahedron. I noticed in his process that he put together two halves first (3 squares and 4 triangles per half) and then put those together.  He ended up with the shape on the left:

At that point, I asked him to compare what he had built with the picture from the instructions: what is the same, what's different? He focused on the components for the faces, pointing out that both his model and picture had square faces and triangular faces. Then he noticed that the cuboctahedron has alternating faces, while he sometimes had triangles sharing edges with triangles and squares sharing an edge with another square.

He was about to break his model and rebuild it, but I suggested that he build a different one so we could compare them more easily.  Again, he built the two halves and then connected them.  We talked about the following things for each of the two figures:

- is the top face parallel to the floor (I called it flat)?

- is the top face always/never parallel to the floor?

- could you cut it in half with a single straight cut? Are there any straight cuts that will just hit edges?

Finally, I reminded him of the half pieces he had built in the middle of his construction.  If he put them together so one square was matched with a triangle in the other half, could any of the squares get matched with another square? Alternatively, if he started by matching a triangle with a triangle, what would happen?

We looked at the faces with edges on the open half and identified this sequence: S-T-S-T-S-T

Actually, it is a repeating cycle as you keep looping around.

If you similarly label the other open half, it is easy to see that you only have two choices for matching: either S gets matched to S or S to T.  Once you make that choice, you either get S-S/T-T shared edges all the way around, or S-T edges all the way around.

To round out our investigation of this shape, I asked why he though it was called a cuboctahedron?  What does it have to do with a cube or an octahedron? He was ready to move on, so I didn't really push on this, but will return to the nice wikipedia page showing it as a rectified cube/octahedron.

Imagining/Planning

At a later point in the day, I noticed him looking at the configurations for other polydron tubs and then he started explaining which of his favourite constructions are possible or not with the available materials:

Smashing

Polydron constructions usually shatter when dropped on a hard floor.  This fact can be used for good or evil. I take no credit for it, but today all three were in a mood to playfully and cooperatively destroy their creations.  Since we had been building a lot of different shapes, they got to investigate which ones broke more easily and which ones broke more completely, dropping once or multiple times and from different heights.

My only contribution in helping to encourage the positive tone and to ask them to attend to different aspects of their investigation:

- "Really interesting.  Did you expect that to happen?"

- "Why do you think X breaks more easily that Y?"

- "Are the angles at the edges sharp or flat?"

- "Did you use solid or skeleton pieces for the faces? Which are heavier? Are the edges they make stronger/weaker/same?"

- "does it matter whether you drop on a vertex an edge or a face?"

Finally, when J1 asked: "daddy, do you know the answers?" I could truthfully answer "no, so we will have to keep investigating together" and everyone was pleased with that.

Math at the beach

Who: J2
When: vacation time
Where: at the beach resort!

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With late October here, we figured that people in higher northern latitudes would enjoy hearing about our recent trip to the beach. Of course, you don't want to hear about sand, sun or seafood, you just come here for the educational tidbits, so I won't bore you.

Instead, let me offer a little example that shows a math discussion can come up anywhere.  What do you notice about this picture?  What does the 2 or 5 or 7 year old next to you think?

Here are some of the points our kids discussed:
- How many floors are there in the hotel? Led by J2.
- What should the numbering be? Led by J1.
- How many floors are there between M and the floor labelled 8? What about the floor labelled 15? J1 and J2.
- Is 7 the same as 17? Led by J3

Also, since the elevator had a glass back, the kids were able to connect higher number floors with a powerful visual of being physically higher.

Even safety indications can be a fruitful source:

We talked about our mass compared to the posted capacity (to be polite, not when mommy or grandpa or non-family members were in the elevator), how many people were there, and how to compare the 17 person limit with the 1150 kg limit.

Finally, you see that this pic was from elevator number 4.  At each floor, there was an indicator to show where each elevator was and which direction it was moving.  Based on that, we played a game to guess which of the 4 elevators would come to our floor first and, of course, I got them to talk about the thinking behind their guesses.

So, a fun trip to the sea, even if I did get a bit too sunburned.

Counting challenge (revisited)

Who: J3
When: at breakfast
Where: the dining table

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J3 and I had an opportunity to revisit the advanced counting challenge described in our post here: How many cows?

The first time, I kept turning the cup as she counted: 1 ... 2 ... 3 ... 4 ... 5 ... 6 ... finished!
Somehow, this time she had a sense that there were a finite number and that she didn't need to count any more.

I asked, so, how many cows are there and held out the cup to her.  Her reply: 1 ... 2... buckle my shoe, then giggled and ran away.

Did she actually count and know there were only two, or was she just amused with herself and felt like adding the nursery rhyme line instead of counting further? I don't know.

How many animals (domino counting challenges)

who: J3, J2, and J1, but the real questions here are for older children
when: morning with J3, evening with the older ones
where: family room
what material did we use: a set of animal dominoes

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When J3 counts, she often says "eleven" after "six" instead of "seven." I set out to show her the difference between 7 and 11.

First, we've got 7 dominoes, I put the blank sides up so that the pictures wouldn't distract us from counting them all as a consistent collection:

Next to those, I arranged a group of 11 (again, all blank sides up):

By this point, she wasn't interested.  I think she found some take-apart cars that needed to be investigated. I, however, was interested in two follow-up questions:

1. how many dominoes are in this set?
2. how many different animal types are there on these dominoes?

The challenge of the first question is coming up with techniques that don't require counting the remaining dominoes.  I have one strategy in mind (other than pure guesstimating) that I think is too advanced for J2 and J1.

For the second question, I think even a direct count requires a thoughtful strategy (though some simple ones are available).  What direct counting strategies can you suggest?

Below, I'll show you how many dominoes are in the set, but I'm still only going to show you the blank faces.  Is that enough information to figure out how many animal types?

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I'll give you the easy way to figure out how many dominoes are in the set

Now, can you answer the second question: how many animal types are used in this domino set?

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Advanced challenge:
Do the same thing with a pack of Spot It cards.

Tools for 2 year olds

Who: J3
When: early afternoon
Where: all over the house
What did we use: assorted play and real tools

Today, J3 was engaged in a serious construction project.  Putting together and taking apart this airplane:

Ours is fun, but the nuts and bolts don't levitate like this

One main step, left out of the manufacturers instructions, was to check whether there were any hidden bolts or screws in my head that could be loosened with the toy power drill. She kept mentioning various pieces that were coming out, but I hope we got them all back in.

Frankly, I love toys like this.  Obviously, there is a lot of counting along the way (do we have all the parts, do we have the right amount of each part), often there is matching (nuts and bolts, usually), and a lot of shapes to discuss and compare.  When we play together, I ask them to describe how things fit together, to encourage them to visualize the connections and to plan how they are going to sequence the construction.

Then, we moved on to the real tool box.

We did two main activities with the real tools. First, J3 found a small flashlight and explored the size of the lit spot she could make.  She had a lot of fun testing her theories about how to make it larger or smaller.  I tried asking if she thought the light would be smaller on her than on my because I'm bigger, but she'd already had enough experience shining it on large walls to realize that didn't matter.

Our second activity was an exploration of the wrenches (few) and screwdrivers (many) that we've collected.  She had near perfect results separating flat and Phillips heads:

That's quite a relief, given how prominently this task figures in so many standardized tests these days!
We arranged them by size, with some interesting discussion about the short 1/4 flat head:

Tool on the bottom: smaller because it is shorter or larger because it is fatter?

A question I posed that was lot on J3 (for now) and I'll try again with the older ones: if that screwdriver is 1/4 made in the USA, where was the other 3/4 made? Also, the longer one says "1/8 made in the USA." Why did they make different amounts of each tool in different places?

I'm afraid, but fully willing to admit, that this behavior shows that a I have a full-blown case of dad humor.

the measure is 27

Who: J3 (also something for 13+)
Where: at home on the reception floor
When: just after breakfast

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A quick picture to show some standard activities in our home.

When in doubt (i.e., too tired to think creatively), I reach for the trio blocks and polydrons and start putting them together. Inevitably, the children will join and take over the activity. We had our tape measure lying around, so J3 started measuring our creations and Ms Rabbit.  After putting the tape measure up to something and looking carefully at the numbers, she would proclaim: "27." She did this several times, each time announcing the same length: 27.

I guess this is similar to her lack of 1-1 correspondence when she's counting: just a developmental step she hasn't yet taken.

A further exploration
Did you notice the star-shaped polydron construction?  It is a cube with the faces replaced with square pyramids. Though it is pretty obvious, I was delighted when we realized that square faces in our constructions could be replaced with 4 triangles arranged as a square pyramid and equilateral triangles could be replaced by 3 sides of a tetrahedron.  Here's an NRICH exploration I found when trying to determine the name of our construction (the cube with pyramids instead of faces).

How many cows?

Who: J3 (2 yr old)
When: at breakfast and again at lunch
Where: the dining table
What did we use: a cup with a repeated picture on it

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Look closely at the cow (cows?) on the cup below. I've taken the picture from 4 orientations, so you can see all around the cup. How many cows are there?

I discussed this with J3 the other day.  She pointed to the cow and counted, then I turned the cup until she saw another cow, pointed at it and counted.  We kept doing this until she had counted up to 5.  I asked her to try counting them again, and again I stopped turning when we got to 5. I asked how many cows are on the cup and she said 5. We did it again, but this time I kept turning until she counted 6.  We did it again up to 10. Finally, I kept turning until her counting got very unorthodox (mid-teens) and with large skips and also going back to smaller numbers.

Later in the day, I put a piece of tape next to one of the cows (this is actually when I took the pictures above). I repeated the same turn-and-count with J3 (I think up to 10).  Then I drew her attention to the tape and repeated the counting.  Same result.

As I typed this, she looked over my shoulder, pointed to the pictures and proceeded to count up to 5 cows.

We do a lot of counting objects, moving them between piles, associating them with fingers, comparing which is larger, so I already knew that her one-to-one association of objects and the count is shaky. Still, I was surprised that she was willing to accept an arbitrary number of cows on the cup.

Challenge
Did you notice the white plus signs on the cup? How many are there?

If you didn't notice them, perhaps this was an invisible gorilla experience.

If you did notice and also counted them, particularly if you distinguished large and small ones for a more detailed analysis, then you can be an honorary member of our family!

Zero...One...Two...Three... (Counting)

Who: J3 (and a little bit of J2)
When: at times you want to help teach counting (e.g., all the time)
Where: anywhere there's stuff to count
What we use: whatever is available, fingers if that's all we've got

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Many of you have heard me say this before: "there are three kinds of mathematicians, those who can count and those who can't."

I never get tired of that (why?), so I should take the opportunity to publicly apologize to everyone who has (or will hear) me say it many times. Part of what lies behind the joke is how fundamental the basic counting skill seems to be for everything else in our standard math education curriculum. I've heard a high school-focused educator claim that many kids struggling at her level are really dealing with the lack of a firm grasp of one-to-one correspondence (a sub-skill for counting). If you really want to, you can see this skill highlighted as a foundation block in the US Common Core Standards: base skill level in counting and cardinality,

So, what do we do? Basically, we play these 4 simple games (from Amy at Kids Quadrant) almost all the time until the kids are counting fluently. There are only some small points I can add to Amy's great post:

1. start with 0. I make two balled fists and wiggle them when I say zero, unless I'm counting hands in which case I just say 0. The point is to make sure they realize 0 is also a number.
2. be silly: this is a game for them kids, so feel free to make silly sounds and gestures.
3. try to find things they can grab and move around as they count. My intuition is that the more of their body involved and the bigger the motion, the more they will remember.
4. (optional) try counting in other languages. If you don't care which language, Chinese and Thai are good choices because of the logical naming system they employ (I think other Asian languages are similar).

Here's another really interesting post from KidsQuadrant outlining the skills behind simple counting: here. What I want you to take away: even though counting seems easy, even obvious to you, be relaxed about how long it takes to click for your kids and keep enjoying it as a repeated game.

Combinatorics

When I tell the joke and say "count," I'm internally thinking about combinatorics.  I have long felt a bit weak in this area, with anxiety that my counts were either leaving out cases or double counting somewhere. That's the real reason I like this joke so much, because the self-deprecation has a meaningful kernel of truth.

Math party (and flexagons!)

Who: Baan Pathomtham First Grade Class+J3
When: 9am - 2pm
Where: our house
What will we used: see below
Why: oh why, oh why? (actually, it was fun!)

First, thanks to all the kids for being so friendly and polite.  Thank you for leaving everything tidy when you left, though that was under a mother's supervision and probably not surprising.  What did amaze me was when Tanya stopped everyone from rushing to eat a snack and you all went right away to collect the toys/games/crafts.

Second, did any of our plan survive reality? Yes, actually some of the things we prepared went well:

• Passports: the kids enjoyed having their activities noted and getting stamps in the passports
• Flexagons: everyone got to play a bit and learned about flexing the hexaflexagons and tetraflexagons. A couple even made their own hexaflexagons and someone decorated a blank I'd left around.
• Pizza: as usual, kids enjoyed assembling their pizzas and were astoundingly patient while they baked.

• Origami stars: some kids were interested in making these

In truth, though, this was simply a group that wanted to play games and could nearly have been left alone the whole day with a selection.  As it was, they played a lot of Uno, managed part of a game of Settlers of Catan, and played some assorted other games: Sum Swamp, Walk the Plank, Spot It!, and Squares

Some lessons (for me):

• Though fairly small, our play room is large enough to host two distinct stationary activities, but they have to be child-selected to be sufficiently engaging for the two groups to remain intact
• Origami (including flexagons) for this age probably needs a smaller group (one-on-two likely works). The kids have the skills required, but either they found it to concentrate, I found it too hard to concentrate, or a friend would suddenly come over and try to take control of the project.
• Activities requiring a meaningful amount of preparatory instructions either need to make sure the whole group is listening first, or have the instructions delivered by one of the children.
• Kids need more encouragement taking things apart (see Flexagons, below)

Below are a sample of the passports, the flexagons, and folded stars from today.

Flexagons
I promised to write up our flexagon experience. After the party today, I'm even more enthusiastic about this activity. There are three reasons why I strongly suggest you start playing with them today:

1. Cheap and easy to make
This is great because, if you break them, then just make more! Since even learning how to flex the shape takes some investigation and practice, there's a danger of tearing a flexagon. That should be encouraged! Cut them open to see how they are folded, force them if you can't see how to flex it. Taking things apart is a great habit/skill.

I think the kids today were overly anxious about their investigation. Perhaps this was the other side to the coin of them being so polite? In any case, this is something I would actively seek to encourage in the future.

2. Mix Art and Math
A blank flexagon isn't much fun, it has to be decorated to make the mystery really come out. For people with a traditional conception of mathematics, this may seem odd: art as a tool to explore the mathematical structure? Yes, yes!

So, if your child doesn't like art, this is a back-door into drawing some patterns or pictures.  If they love art, then this is a back-door into equilateral triangles, angles, rhombus, hexagon, how many sides a piece of paper has, state diagrams, etc.

3. You probably don't know much about flexagons
That means you can let the child lead this activity and just let it develop. That's great because they can be the teacher or you can be equal partners.

Alternatively, you might get excited and have your own questions about flexagons. In this case, this is a chance for you to investigate and for your kids to see you investigating. Do you try to figure it out on your own, draw diagrams, build other models, dissect specimens, experiment, watch videos on youtube, look at wiki pages, ask a friend, talk with your child, all of these things? A great opportunity to show your child how you explore something that interests you.

My flexagon recipe:
Only steps 1 through 6 are really required
1. Watch the Vi Hart video: http://www.youtube.com/watch?v=VIVIegSt81k
2. Make a trihexaflexagon by using tape to complete the loop
3. Color it
4. Flex it and think about how it works
5. Take it apart (this is why we used tape)
6. Make a bunch of additional trihexaflexagons, use different ways to make the series of triangles (folding, compass, 30-60-90 drafting triangle, printed template, etc)
7. draw more pictures and patterns
8. Try making a hexahexaflexagon (again, using the basic comments from the video)
9. Look at some templates online, including Tri-tetraflexagon, Flexagon Portal, More templates and more
10. watch the second part of the Vi Hart flexagon series: http://youtu.be/paQ10POrZh8
11. watch this third Vi Hart flexagon video: http://youtu.be/AmN0YyaTD60