Castle Logix (product review)

This is in the spirit of Chris Danielson's product reviews at Talking Math with Your Kids.

Who: J2
Where: dining room
When: after lunch
What: Castle Logix block set.

---

Ok, this is Castle Logix:

There are 4 cuboids with holes in their sides and pictures on other sides, three different length cylinders with cones topped by spheres on one end, and the cylinders fit into the holes in the cuboids.

J2 has found two new uses:
1. Rhythm sticks
2. Combinatorics challenge

Rhythm sticks

Really, this is just a fancy way to say that he started beating them together. Or maybe I did? Anyway, he found it a good instrument for listening to Suzuki violin pieces and practicing the rhythm along with the video:

Of course, you can bash together any two things, so what makes these blocks so perfect?
They are a good size, relatively large, but still comfortable for a small 5 year old to hold. They are a bit heavy, so you have to commit to each beat and can't be halfhearted about the game. Most importantly, the holes seem to amplify the sound and make a very satisfying clack. Oh, they are also sturdy enough to take the abuse and seem unaffected.

Conclusion: Two thumbs up (but keep those thumbs on the outside when you are bashing)!

Note: mommy was not around at this time. Your experience may vary depending on who is present during play and time of day. . .

Combinatorics challenge

This is something we are just starting, but the basic questions are:

• How many ways can you put together the Castle Logix pieces?
• What do we even mean by "put together" anyway?
• When do two configurations count as the same?

I will report back as we work through these questions.

Pseudosphere Hat and our Robot begins (some arts and crafts)

Who: J1
When: around lunchtime
where: on the floor

---

Today, we tried making a couple of things. First up, was a pseudosphere.  The inspiration for this is a really nice post from Daniel Walsh: Sudo Make me a Pseudosphere. By all means go to the original post as the pictures, animations, and video he posted are far better than what we managed, but it was still a fun conversation about shapes, angles, slope, and fractions.

The process is easy:

1. cut out a bunch of equally sized circular discs
2. cut the discs into different sized (different angle) sectors
3. make all the pieces into cones
4. stack them from shallowest to steepest

Daniel mentioned something about calculating the optimal sizes, but I didn't really know what he meant. We went for a child's dinner plate for our circular template and cut sectors in multiples of 45 degrees.

Only the finest used newsprint for us!

One good question came up along the way: if I cut out a larger angle, will the cone we make be steeper or flatter?

Pseudosphere taking shape

There was another point where I'd cut out a 135 degree sector and J1 said: that's 1/3.  When he measured very roughly, it did seem to be a third, so I asked him to try it more precisely. He had a sudden realization when he saw the 90 degree remainder.

The cone of our dreams!

We went one step farther and permanently attached all the cones together, then re-purposed the whole thing to create 2015's fashion must-have item: the pseudo-rocket pseudosphere hat:

Starting our Robot

Our other activity is really the beginning of a longer project.. J1 has been talking about making a robot and we are starting to work on some of the main functions. Of course, he is really excited about camera eyes and a laser cannon, which we'll get to eventually (will we?) For now, I have some ideas about how to get the robot to walk.

My plan is to connect a basic rotating electric motor we have, so that leaves us solving an old problem: how to convert rotational motion into straight-line motion. Of course, wasteful methods are easy, but we want our robot to have the maximally powerful stride. For now, we are investigating multiple linkages, in the footsteps of Chebyshev.

Below is a first test: three linked arms:

1. Arm 1 has a fixed end and is intended to rotate in a circle
2. Arm 2 has one end fixed to the rotating end of arm 1. The other end of arm 2 is the motion we want to examine
3. Arm 3 has one end fixed and the other end attached to the mid-point of arm 2. This constrains that midpoint to travel along a circular arc

You can see our ultra-high tech implementation below, using card paper as the arms, a large cardboard box as the base, nails (into the base) to create fixed points, nails point up to create hinge joints, and extra bits of cardboard to cover the point ends of our hinges and past muster with  the health-and-safety inspectors:

The action of the multiple hinges is pretty wild. J1, J2, and I enjoyed cranking arm 1 and watching arm 2 fiddle around. Carefully holding a pencil in place, we managed to draw he path of arm 2's free end. It is the rounded wedge that looks very close to a circular quadrant.

If you want to see some great animations of multi-hinge contraptions, check out the animations at Mathematical Etudes. I'd be delighted if we could get close to this one.

Avocado update

I'm pleased to announce that another family member, D, has started sprouting her own avocado pit and, apparently, has made this into a race.  When told the news, J1 and J2 immediately started guessing what type of sabotage techniques would be employed by D. I think this says more about them than her.

Total mass: 67 grams
Length from root tip to sprout top: 21 cm
Length from pit to sprout top: 12.3 cm

Constructions and calculations (more polydrons)

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

---

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.

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

---

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).

Number sense, calculating, and stuff

Apologies, this post is a bit of a grab-bag of different things we have been doing recently. They are a mix of activities across each of the age groups, done at different times of the day and different levels of engagement, though most of the one-on-one time with J1 comes just before bedtime.

Number Sense: J3
We have made a conscious effort to surround the children with numbers and are often looking for good ways to present concepts in a different format, especially one that has physical objects and some relationship to their body.  Meals are often rich with opportunity, especially if you don't mind a little bit of playing with the food.  Here, J3 has matched up 10 almonds with each of her fingers, counting as we made the layout , then counting again as the almonds got put in a little cup, then counting again as she ate them.

Similar ideas:

• counting stairs going up and down
• counting musical beats as you listen to a song
• weighing food, weighing each other
• estimations (amount, length, weight)

Snakes and Ladders, err, Dinosaurs and Dinosaurs
I've mentioned before and will repeat frequently that I enjoy playing games, but snakes-and-ladders format aren't really games since you can't actually do anything. For now, though, the kids like them and the provide an opportunity for two things: (1) asking interesting questions about structure and (2) introduction to probability.  Along the way, there is practice calculating, but this is just small addition problems and not especially rich.

Tonight, J1 and I played a dinosaur version, you go down if you land on a land-dinosaur head square and up if you land on a designated square associated with a flying reptile:

We made one big modification to the game: playing with two dice instead of one.

The main point of interest is hearing J1 calculate dice sums, which square he is jumping to, and analyzing the size of the boost (or drop) from hitting the bonus squares and the penalty squares. It fun to talk about the probabilities of hitting the bonus and penalty squares, particularly because he asked all the questions and did most of the talking.

Simple starter cues, if you want them:

• is it possible to get to x from where you are now?
• how many squares away are you from y?
• How many ways are there to roll a 5?

Toys

Really, all of this post is just leading up to praise for some toys the kids really like and that can soothe parental anxiety about "stimulating development."  Frankly, I don't really care, since I enjoy playing with these, too!

Rainbow Loom

Extremely popular in J1's school group and even J2 can weave a nice design on his own.

They also discovered that youtube has useful how-to videos and they even worked together!

So, what are the (future) mathematical benefits:

- pattern building and recognition

- building a mental model of how the structure fits together

- understanding algorithms (especially with repeated loops)

Snap Circuits

A fantastic kit with electronic components that all snap into a grid-circuit board for easy assemble and disassembly.

 There is a book of projects which show circuit layouts and give some commentary about what you should be examining to understand the operation of the circuit.  Usually, they introduce each new component in a show piece of its own so the kids can build a sense of function.

I do have one complaint about the kits, though: a lot of the projects depend on using an integrated circuit.  These are black-boxes, so it misses a bit of the fun of building everything up from truly elementary components.  For now, that's certainly fine for our kids level of sophistication since it is fast to build a fully functional circuit. We do miss out on really understanding what is going on inside the box, though.

Trains

Really just an excuse to post a picture that reminds me how much fun J3 had playing with one of our toy train sets: