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

a hit and a miss

Who: J1 and J2
When: throughout the day (summer vacation!)
where: around the house
what materials: paper and scissors, access to pencilcode.net

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Here are a couple of recent activities/discussions with the kids. More than the content, though, is a reminder that sometimes an activity won't click right away, but they may still come back to it later on their own time.

Multiplication windows

This first activity was a big hit, so thanks to Musings of  Mathematical Mom from which we copied the Many-many game. To play, we needed to make factor cards. While the kids were doing something else nearby, I started making a couple by just cutting slits out of square pieces of paper. Almost immediately, they took over and insisted on making their own. Here are some of their products:

As an example, this is the pattern for 3x5:

While the product isn't beautiful, making the factor cards was fun for them and it really engaged their thinking about the multiplication model. In particular, there were some interesting mini-discussions about which number factors to make and how long to make the slits (if some are too short on one card, they might not get an open window when matched with another card).

In general, our kids respond really well when an investigation is prefaced by making something and that preliminary always leads to deeper questions about the structure around the topic we are exploring.

One other question they raised: how can we use these window cards to do 3 factor multiplication?

How many triangles?

The next activity was initially a miss, but J1 has been keeping it in mind and even asked tried to introduce it to a friend during a skype call tonight. The challenge question in isolation: if you can only make angles that are multiples of 30 degrees, how many different triangles can you make (up to similarity)?

Now, this wasn't how I posed the question to him, so you'll have to forgive a bit of background. J1 has recently spent more time making his own pencilcode programs. One is a simple program that creates 4 buttons that control the turtle (take a play yourself here). The buttons move the turtle forward, backward, turn 30 degrees clockwise, and turn 30 degrees counterclockwise. I don't know what motivated this program, but he had seen at least one other user make a similar version.  The key difference he noticed was that the other program only allowed 90 degree turns.

J1 was talking about how limiting the 90 degree turns are, so I asked him to show me something he could draw using his program that the other program couldn't replicate. He showed me a triangle (30-60-90). Thus, the extension question: how many different triangle shapes can he make with his program?

He played for a couple of minutes, but quickly moved on to something else. I was a bit disappointed, but didn't express anything to him about it. Somewhat to my surprise, it has come back tonight as he is going to sleep and he asked me to remind him to investigate more in the morning.

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).

Bedtime math extensions

Who: J1 and J2
Where: all around the house
When: 6 am

It is summer vacation for J2 and J3, so we have been doing a lot of activities. Unfortunately, there is almost no free time to blog about it.  For now, let me share some of the conversations J1 and I have had about recent Bedtime Math posts.

Doubling Volcanoes

Today, we talked about this post: Instant Island. In particular, we focused on the last part, the "big kids bonus" question: if an island is ¹/₂ mile wide and the width doubles every month, how wide will it be in 4 months?

Okay, forget about the answer for a moment, does the question make sense? First, we asked: if this were true, how long would it take the volcano to stretch around the circumference of the Earth? Using some familiar powers of 2, we figured out it would be slightly less than 16 months. It seemed pretty clear that this didn't make physical sense, since we clearly don't see small islands cover the Earth like that.

We talked about this doubling growth process for a while. What things do we know that work like this? J1 listed:

• bacteria
• people
• computer viruses
• plants

We talked about what is happening: basically, the new "material" is able to reproduce itself, so the more you have, the more productive potential you have.

Does the growth keep going forever? No, otherwise everything would be covered, actually become, the thing that is doubling. At some point, these all run into a limiting factor, food, water, space, etc.

Back to the island: J1 realized that the growing island would hit other landmasses before it went around the Earth. If you consider connecting with Asia to count as "growth" for the island, then there could be moments of extremely fast growth. At the same time, we know that the Eurasian landmass isn't growing with an exponential process, so this connection won't contribute to the further growth of the volcanic island.

Stuffed with lead?

We were scratching our heads about the weights in this Stuffed Animal post. The average weight per stuffed animal assumed in this post is 7 pounds. We had two conversations about this: how much is that in grams and how much do our stuffed animals actually weigh?

First attempt, a scant 27 grams

Hefty Panda is only 281 grams

Massive Diplodocus is about 100 grams lighter

One of our heavy-weights: still under 400 grams

Not a stuffed animal. This is the trickier we had to use to break 1 kg

Suffice it to say, we had no stuffed animals that exceeded 1 kg.

Heavy Pencils

After having an experience with unreasonable weight measurements, Pointy Gorilla helped launch a similar conversation. Actually, it came from misreading! We connected the following comments:

1. Gorilla weighs 300 pounds (we read this as the pencil gorilla)
2. The big kids question implies a certain number of pencils used (under 600)

So, how heavy are those pencils? Given what we know of real pencils, how much would that gorilla actually weigh?

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.

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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.

Another number guessing game (actually 2!)

who: J2
when: evening/bedtime
where: bedroom
what material we used: a 100 board with removable tiles

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We tried out two new number guessing games that J2 really enjoyed. In fact, he misunderstood the rules of the first game and ended up inventing the second one himself.

Multiples

One player is the number master and the other players are guessers. The number master chooses a secret number and then says a clue, a multiple of the chosen number. We marked the clues on the 100 board for reference. After each clue, the other players make a guess about the secret number.

For tonight, I stayed with single digit secret numbers and, generally, just gave clues that were smaller than 50.

Some tips
When the correct answer was guessed, I asked J2 whether that was the only possible secret number compatible with the clues we had so far. That also led us to talk about what information each clue gave and to realize, without deeply exploring, that some clues are very restrictive (give a lot of information) while others don't help at all. It will be interesting for him to explore further to see what the differences are . . .

Other than through relative choice of clues, you can vary the difficulty by giving smaller or larger clues. Also, 2, 5, 3, 9 are relatively easy to recognize as factors, while larger primes are generally harder to spot. Of course, your mileage will vary with the experience of the particular players.

Marking the clues on the number board gives you the chance to see some nice multiples patterns, though this depends on how many clues are required before they guess the secret. If it seems fun, you can fill in the rest of the multiples pattern after the correct guess has been made.

Factors

As a reverse version of the game, give clues that are factors of the chosen number instead of multiples. This was J2's invention and he had a sneaky plan in mind. The first number he chose was prime and then he refused to give any more clues!

Some further explorations

If you play this game, you might enjoy thinking about some related questions:

1. Playing the multiples game, I gave a clue and J2 immediately guessed my secret number. What do you guess about my secret number that time?
2. Is any collection of clues possibly valid for the multiples game? The factors game?
3. Can the number master cheat at the multiples game? That is, if they don't write down their answer, can they change mid-game? Restricted to a 100 board, what sequence of clues will keep their options open the longest?
4. In the factors game, can the number master cheat? Is it easier or harder than for the multiples game? What if the secret can be as large as you want?

Just try some operations: a problem solving phase?

Who: J1
Where: in bed
When: bedtime

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Bedtime math discussions have become something of a treat for J1, available if everything else is done before bedtime. Tonight, we didn't quite meet that deadline, but I made an exception for some mitigating factors. Turns out it was a lucky night for this as the subject was ROBOTS!!! And there were some short videos!!!!! Here's the link so you can play along: bedtime robots.

The chats
This was in reference to the "little kids bonus" question:
J1: 10.  Wait, no... 9 seconds
J0: why did you originally say 10?
J1: Because I just added.
J0: What made you change your mind?
J1: Because you said "faster," so I knew it should take less time.
...

The second snippet was for the "big kids" main question:
J0: <reads the question>
J1: I guess I need to multiple 5 x 30
J0: Why?
J1: Hmm, should it be adding?
J0: Maybe you should draw a picture.
J1: <gets paper and pencil> Okay, how does the robot move? <looks at question> Zigzags and makes 5 turns . . .<draws a zigzag and counts 5 turns, looks at the question again>. Throws after the fifth turn <draws a little picture of an angry bird next to the fifth turn, then starts counting another zigzag path>....ten will be 2 throws.  So, the answer is 6.
J0: How did you get the answer?
J1: There are 3 groups of 10 in 30, so I need 2x 3.
J0: Why 2 x 3?
J1: Because there were two throws every ten turns.
....

Why do I remark on this?
I was really struck that his immediate reaction was to throw numbers from the question into an operations grinder and see what came out. Sure, I have seen this reported by a lot of other people, but this was the first time it seemed so apparent for our kids. I think this underlines the need for us to encourage them to first make sense of what is being presented, to draw pictures or diagrams, to use manipulatives, and to explain why.

Projects: Finale (programming class 18)

Who: Baan Pathomtham Grade 5
Where: at school
when: Monday morning for 2 hours

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This was our last class of the term, so we spent the time working on finishing the projects.
For each project, there are many potential extensions, but everyone has made a complete program (or very, very nearly complete).

Programs can all be found here: Class Projects.

Plan of work today:

• Titus: (1) instructions for the game (2) webcam version?
• Win: (1) correct flow, (2) add fractions?
• Boongie: (1) how do players win? (2) is the penalty condition working the way you want?
• Gun: (1) penalty for hitting the walls, (2) winning condition, (3) finish maze 

How we play math games, an example

Who: J1
Where: dining table
What materials: a pack of playing cards, a 100 board
When: mid-afternoon

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Some conversation snippets that illustrate what we talk about when playing math games.

For background, we were playing our 35 game from class, with some slight modifications. Each player gets 5 cards and can choose what card to play on their turn. After each play, you draw a new card so your hand should always have 5 cards. In this case, I was just picking cards at random from my hand, not choosing the cards to play.

Conversations:
J0: 26 + 3 = 29. Now, what do you need to get to 35 this turn?
J1: 6 <looks at cards and makes a sad face>.
....
....
J0: What's the count?
J1: 21, can you make 35 on this turn?
J0: <looks at his card and thinks> Oh, no, I can't.
J1: How did you know?
J0: I looked at my highest card and added  to 21.
....
....
J1: <playing a card> 11 + 5 = 16
J0: Can I get to 35 on this turn?
J1: <thinking for a bit> No, the highest you can get is 29.
....
....
J1: <playing a card> 30 + 8 is... oh I have to subtract. 30 - 8 is 22
J0: What is the lowest you can get by bouncing back and subtracting?
J1: <thinking for a while, looking at the 100 board> Oh, we could get back to 10.
J0: How do you get to 10?
J1: We start on 23 and play a king (value 13).
....
....

What's the point?

There are two things I was trying to do: get him to explain and verbalize his thinking and explore the structure of the game. For almost all activities, there will be "what is the largest..?" or "what is the smallest ..?" investigations worth pursuing.

What did I miss?

At least two things I could have added to our conversations:

(1) talking more about my own thinking when doing calculations

(2) Asking for or introducing alternative ways to do a particular calculation.

What else do you see? Feel free to criticize (constructively).

Some shapes

We recently had another round of flexagon making. In our attempts to practice making hexahexaflexagons, we ended up with some spare paper with a triangular tiling. These were good for making little hexahedrons and octahedrons:

Ever since reading Christopher Danielson's class activity on the taxonomy of hexagons, I've been on the look-out for natural occurrences of irregular hexagons. In my experience, they are actually pretty rare. Here is one, captured in its native habitat:

Obviously, a close cousin to the regular hexagon

Examing a graph and 35 game (math class 12)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Weekend outing

Our school went to a play over the weekend, somewhat about chickens. Unfortunately, there were no mathematical chickens, but maybe someone will draw one this week?

Dice turn into a graph

The homework last week was to play damult dice with subtraction, targeting a score of exactly 100. We had discussed some observations about the game and had left open the observation that a lot of scores seemed to be multiples of 3. Over the week, I made a little pencilcode program to show the scoring distributions and thought this would be another good focus for the kids to make observations, conjectures, and pose questions. The distribution we discussed today was the max scores distribution, so you can play along here: Scoring histograms.

Once again, this turned into a good conversation:

• Why are some bars red and some blue?
• What is the meaning of the two rows of numbers on the bottom?
• How does this relate to our dice game?
• In the top row of numbers, why is 2 the smallest? Why is 72 the largest? Why is 3 missing?

For homework, we gave everyone a copy of the possible scores histogram and assigned them to write an observation or question about the graph.

35 Game

This is another card game. I'm not sure of the source, but will link back here when I find it.
Material: Pack of standard playing cards (J=11, Q=12, K=13), manipulatives for adding (we used 100 charts) are also helpful.
Set-up:Deal out all cards to the players. In the first version, it doesn't matter if people have different numbers of cards. Players don't look at their cards.
Play:Going clock-wise around the table, the player turns over one card and adds its value to the running total, if that result is less than 35. If the sum is 35, the player scores a point and the running total resets to 0. If the sum would be more than 35, then the player subtracts the value of their card from the total instead of adding.
Example: if we have gotten to a total of 26 and the next player puts down a 9, they would get a point. If they turn over a 10 or face card, then they subtract that value from 26.
Tips:

• When the card value is too large, encourage the kids to state the addition equation anyway. For example, "30+10 = 40"
• Before playing a card, ask what value would take you to 35.
• Occasionally ask for values larger than 13 and ask if there is a card that would let you get to 35 on the next play.
• For a more involved version, deal equal numbers of cards and let player's choose which card to play on their turn.

Homework

1. Write down at least one observation about the damult dice possible scores histogram
2. Play the 35 game with a parent or sibling until someone gets to 5 points and wins