Photo Math, computer based math, and hand calculations

Who: J2
Where: all over the house
when: after school almost every day last week and all weekend

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Recently, there has been a lot of excitement about the photo math app (on-line community at-large) and hand calculations (just within our house). Is there a place for doing hand calculations and learning standard calculating algorithms when technology has already automated so many mathematical operations and is attacking problems of increasing complexity?

I'm not going to attempt to answer comprehensively or theoretically, I'd just like to make some observations based on J2's explorations this past week.

It started with squares

Who knows why, but J2 was building a sequence of squares one afternoon with our colored tiles. I think he had seen something when we were doing another investigation or had heard me make a remark and wanted to investigate square numbers himself.  Making these squares was a fun and colorful way to do the calculations.

At some point, he realized that he wasn't going to have enough tiles to keep making separate squares and he consolidated into one square which he kept growing. I think this 13x13 is where he stopped that day.

What was he thinking?
He was absorbing the numbers and looking for patterns.  Early on, he realized that it was annoying to keep counting all the tiles to calculate the new square, so he wanted a faster way.  At one point he came to report his progress and explained: "I have 100 tiles in my 10x10 square.  When I make the 11x11, I know it will have 100 + 10 + 11 tiles."

Symbolically, he was recognizing (n+1)^2 = n^2 + n + (n+ 1)

After that, he kept using this relationship to check his results. We also played around with doing the multiplication directly. Whenever we did a multiplication, I would find a way to illustrate the distributive property and usually invoked some number bonds.  Here is one illustrative example, though mostly I just drew diagrams like this on a paper:

Incidentally, I got this design from Mike Lawler's video giving a physical illustration of why the product of two negatives is positive.

Another tool:
For several of the calculations, J2 was using a 100 chart or our 100 board.  He spent a bit of time looking at the board thinking about whether the squares were easy to see on this board.  his intuition was that, somehow, it would be nice to see them as the vertices of growing squares within the grid.

I suggested he build his own 100 spiral and look for patterns along the way.  This is slightly less than halfway (well 40% of the way, to be exact):

While he did this, he noticed three things:
(1) the squares are appearing along diagonals
(2) the even squares are rise moving northeast and the odd squares increase going southwest.
(3) we also get non-square rectangles at 1x2, 2x3, 3x4, 4x5, etc

I have it on good authority that  you can see something interesting with the primes in this configuration (see Ulam's Spiral) but that will have to come later for our J's.

Some further adventures
Along with his hand calculations, J2 started entering his squares into a spreadsheet. This let him explore larger squares than he could multiple right now and well beyond our tile collection. Also, we could explore first and second differences, seeing his old recursive relationship in a new way.

Beyond the squares, he has since done similar things with cubes (constructing physical cubes out of trio blocks, building a table in the spreadsheet, looking at differences), powers of 2, and quartics. Looking at these all together allowed him to start seeing connections around more advanced questions:
- which powers of 2 are squares?
- which cubes are also squares?
- which quartics are squares?
- how fast do the different sequences grow?

A hint of what is to come: before going to sleep last night, J2 mentioned that he wants to talk to me about triangular numbers next . . .

What do I conclude
For J2, the hands-on manipulations and associated hand-calculations are helping him see number patterns more closely and become familiar with a lot of interesting relationships.  This work has provided him with a platform to then engage with more computationally powerful tools. Importantly, he doesn't see it as a binary choice between manual and automated calculation, but is very happy to alternate between the two.

Further reading
For a more thoughtful and comprehensive discussion of the app and the impact on teaching, see Dy/Dan.

Creating new creatures (Programming Class 8)

who: Baan Pathomtham Grade 5
when: Monday morning, bright and early
where: at school

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Sharing our Work
First, we looked at programs we have written:

1. Birthday animation for an uncle
2. Win's flower
3. Spirals: Boongie and Titus.  Remember, this was the inspiration: popSpiral.

For the spirals, we discussed briefly differences in how these were implemented and then flagged the key similarity of the nested loop.

Nested For Loops

What does this code create?

speed 50
for x in [0..200] by 25
  for y in [0..200] by 25
    moveto x, y
    dot 10, rgb(x, y, 0)
ht()

This is going to come back next week when Pooh is leading the class and there will be some other examples of nested for loops.

Creating a new creature
Another block of code for experimenting.  The kids played with this and then started trying to build their own versions with spooky ghosts:

ant = new Sprite
  color: transparent #can use other colors or 
#  width: 500
#  height:50
  
drawon ant
turtle.speed 100
fd 9
pen black
lt 60, 15
lt -60, 15
rt 60, 15
rt -60, 15
lt 75
fd 10
bk 10
rt 150
fd 10
bk 10
lt 75
pu()

dot crimson, 10
bk 9
pd()
lt 90
fd 10
bk 20
fd 10
rt 90
pu()
dot crimson, 8
bk 10
pd()
rt 75
bk 10
fd 10
lt 150
bk 10
fd 10
rt 75
pu()
dot crimson, 12

# sync makes the ant wait until
# the turtle is done.

drawon() sync ant, turtle

ant.pen orange
for [1..5]
  ant.fd 100
  ant.rt 100

Homework
(1) look at the program antGamer (http://jgplay.pencilcode.net/edit/class/antGamer) and try to figure out what it does.  What can you change?
(2) work on your ghost programs

Feedback
Two points I thought were worth flagging from the feedback cards this week:
(1) when asked what they learned, the kids still generally talk about the whole program, not the underlying concept.  This makes me concerned that they don't see the more general idea and won't think of using it when it is appropriate.  Based on this, we are going to spend a bit more time next week talking about how the programs work.

(2) I love this other answer to what I learned: "I learned that there are things I don't know yet."

Math games (class 1)

Who: Baan Pathomtham First and Second grade classes
Where: at school
When: mid-morning

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Today was our opportunity to play some mathematical games with the younger kids at J1's school. It was fun and we played two games that the kids can take home to play with their families for further investigation.

Apologies, no pictures: we were too occupied to take any snaps.

Warming-up
We start each session with some quick question warm-ups.  Going around the class, everyone gets a question in turn. These are meant to be fairly easy, but get them actively engaged.

Examples:
- questions about days of the week e.g., what day comes before Monday?
- months of the year, e.g,. what month comes after December? How many days in June?
- continuing number sequences, e.g., What comes after 19, 20, 21? What about 35, 34, 33? What about 75, 73, 71?
- Skip counting, e.g., counting to 20 by 2 starting at 0.  Counting up by 2 starting at 1.
- What are some ways to make 10 with addition?

Making mathematical observations
After warm-ups, we gave the children a picture and asked them to make mathematical statements or ask mathematical questions about the picture.   The types of statement depend on the picture, but examples include:
- counting specific objects, e.g., There are 5 apples in the picture. My favourite is something like "There are zero footballs in the picture" or another object that is totally unrelated to the picture.
- comparing numbers of objects, e.g,. There are more people than lions.
- number sentences: there are five children and three adults, all together there are 8 people
- comments about shapes

This is an idea we got from Mathematics Mastery.  For younger grades, they have developed a series based on fairly tales that are really good.  Naively, I had thought it would be possible to take random photos off the web, but most just aren't that detailed or there is limited variation.

Game 1: Euclid
Our first game was from Let's Play Math: Euclid's Game. The rules are simple:
- start with a 100 grid (we used 60 for the 1st graders)
- First player chooses a number and draws an X over it
- Second player chooses another number and draws an X over it
- players take turns crossing out numbers that are the difference between two numbers already crossed out.
- last player with a legal move is the winner

We played several rounds on the whiteboard, J0 against the kids and also half the class against the other half. We recommend using contrasting colors to make it easier to see any patterns that emerge.

At the basic level, this is just practice subtracting two digit numbers.  We found that both grades were struggling a bit with this, so the extra practice was useful. For the next level, ask about patterns, during and at the end of the game:
- can you see what pattern of squares we are crossing out?
- how do you know if there are any moves left?
- who do you think is going to win?
- what is the largest number we are going to cross out?
- what is the smallest number we are going to cross out?

On their own, the first grade class started to speculate about whether they could know in advance who would win.

We gave out 100 sheets so the kids could play at home with family and friends.

Look: your very own 100 grid!

Game 2: Don't Make a triangle
This game comes from Math4Love and is called Don't make a triangle.  All  you need to play at home is a pencil and paper.

We actually didn't play the Math4Love version, but instead showed them a simpler variation where players take turns connecting the starting dots and they try to avoid forming a triangle with vertices on the starting dots. Here is a progression of variations and explorations we considered:

- variation 1: students start with 6 dots, take turns connecting pairs. the one to make a completed triangle first loses (a segment drawn by either player counts)
- variation 2 (as suggested in Math4Love): start with 6 dots, students use different colors and only a triangle with all sides their color count
- exploration 1: will it make an interesting game if you are trying to be the first to complete a triangle? Test this for variations 1 and 2
- exploration 2: is there a winning strategy? do you want to be the first or second player?
- exploration 3: try these games and questions with a different number of dots


For future sessions:
There is a lot more we can do to explore the two games we introduced today.  I plan to spend at least part of the next session on a bit more of an investigation into Euclid's Game.  Here are some other ideas we may explore

(1) three chips puzzle

(2) Always truthful/always lying logic puzzles.  there are many of these, here are two examples:
a) Tom always tells the truth, Dick sometimes tells the truth and sometimes lies, Harry always lies. You don't know who is who, but start to ask their names.  Their answers:
- First person: I'm Dick
- Second person: I'm Harry

When you ask the third person, what answer does he give?

b) A family has 2 sons, one who always tells the truth and the other always lies. Their house is next to a Y- intersection.  Walking by there house, you get confused about which direct you need to go (left or right). You go to their house and one son comes to the door. What can you ask that son so that you find out the correct way to go?


(3) Nim variations
- 1-2 Nim
- 1-2-3 Nim
- Nim on 10 frame
- Poison
- points for taking counters and bonus for ending condition

(6) Einstein/elimination puzzles
I did a lot of these around this age, so am curious to see how the kids find them.  I just found this collection, so will see if any seem suitable.

Doing the challenges (programming class 7)

Who: Baan Pathomtham 5th grade class
Where: at school
When: Monday morning, first day after a 1 month break

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Well, the kids hadn't worked on the challenges I posted here, so we worked on them in class.

Homework: work on challenge 3
This challenge is to reproduce the spiral below. They all made some progress and I am keen to see what they can figure out. Having reviewed my own code, I expect they will produce something more elegant than what I wrote.

Notes from today

As usual, they all did well with some finding different tasks easier and others harder. Mainly, I think it was an effective refresh of some things they might have forgotten and helped me assess where they are starting for this term.

Challenge 2: They all did this first.  I think they enjoy replicating a block of code and then experimenting with it. If I hadn't pushed them to move on, they would have been stuck playing with this for the whole session.

 

Challenge 1: All of them found it pretty easy to figure out what code was missing from the two short programs.  Also, they were able to copy and save the code, demonstrating that they haven't forgotten how to move around the system.

Kan did the most experimenting with code and came up with some really interesting stars by increasing the number of iterations on the star loop. For next time, I will ask them about some of these patterns, compare what would happen with a 5 or 6 pointed star and see if we can figure out why they get created.

Challenge 3: They all found this to be hard and have only started making progress. The key point is that they remember how to use for loops, but will need to be reminded of the different ways they can be structured and how they can be put together.

Feedback form

At the end of each class, I will ask them to fill out a short feedback form:

(1) Two things they learned:

They all said they learned how to make a submarine and many mentioned fixing the missing line programs from challenge 1.

Refresh of how to use for loops

Some talked about learning how to debug their programs when they made a mistake

(2) One question they have about the lesson or 1 new question they have because of the lesson

Everyone asked some variation of how to make the spiral, either how do we make it larger as we iterate or how do we change colors?

(3) One piece of feedback about what we did in class today

- Like making the submarine

- I liked working on the spiral

- I don't like doing calculations because they are hard and made me tired

- Don't like making the spiral because it is hard and uses too many techniques

Square One TV (A talking math with your kids experience)

who: whole family
when: dinner (and then at odd times for the next week)
what did we use: internet connected TV

So, you've been reading Talking Math with Your Kids and are eager to kick off some chats? Here's one effortless cheat that has led to many conversations in our house.

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Let's Play Math had a link to a Square One TV skit that reminded me about the show.  Of course, I loved it as a kid and have been pleased to find that I enjoy it just as much now. I had to introduce these to my kids.

This is the one we watched:


Some parts of the presentation haven't really kept up with the times, I admit, and the youtube video quality is low, so I think there's a lot the kids don't catch.  However, it has led to a lot of interesting follow-up discussions and not necessarily about the most typically entertaining parts of the show. For example, when we got to this point, suddenly the kids were quite and focused:

What was so interesting?  A grey screen with an equation and enough space to promise more equations! After this segment played, we paused the video and had a good chat about it.  J2 was particularly excited and explained the patterns he saw.  Over the next several days, he returned to these equations and added steps to the sequence.

The other segment that has stuck with them was the Dirk Niblick cartoon.  J1 has been doing a post mortem analysis of almost every part of the sketch.  The crescendo came yesterday when he wanted to analyze the cost of the car, the percentages and the mistake.  We spent about 2 hours talking about different percent calculations, translating to and from fractions, comparisons.  I'm not sure he completely got it, even at the end, but he continues to be intrigued and brings up the points.

After all of that, I showed them a couple of the SQ1TV songs, like Perfect Squares (to which I linked on my last post) and 8% of My Love.  Some resulting discussions and explorations:

- Perfect Squares caught J2 in a phase of thinking about square numbers and has spurred him to more questions about them (specific squares and patterns between them).  One specific thing I've been doing here is drawing area models for his squaring calculations, so that is getting to be a familiar diagram for him.

- Battle of the bulge sandwiches (also from the very first SQ1TV episode): J1 and J2 wanted to know how many sandwiches could be made if multiple meats and cheeses were allowed.  J1 made a guess it would be about 30 and then they worked on a systematic way to organize the list.

A little question about squares (SQ1TV warm-up)

Remember the perfect squares song from Square One TV? No, then try this link. 

Did you catch the part about 14?  Not a square number, but the save is to add an extra digit to make it 144. Let's call that a square-save. Here's the section of the song, if you want to enjoy the 14 square-save in song.

With that as the inspiration, can you always square-save any positive integer? In other words, can you always add some extra digits to make a perfect square?

Extra credit

What do you make of the following sequence:

1, 5, 6, 2, 23, 8, 27, 9, 3, 10, 34, 11, 37, 12, 39, 4, 42, 43, 14, 45 . . .

What number comes next?

Do you notice any patterns?

Extra Extra Credit

One square-save of 10 is to add a 0, making 100. One hundred, of course, is 10 * 10. Are there any other numbers that can do this, e.g., their own square is a square-save?

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.