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.

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.

Taking Mr. Men too seriously

Who: J2 and guest appearance from G1 (grandpa)
Where: in bed
When; at bedtime

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There are a bunch of ways to take the Mr. Men too seriously, and I'm not even talking about this.

Reading out loud
We recently got the full set of Mr Men and Little Miss books.  J2 has especially enjoyed reading them and has his own routines for extracting the books that will be read each night, then collecting the books at the end and flipping the first book for the next night upside in the box.

He really seems to enjoy these books, whether we are reading to him, he is reading to us, or he is reading to his sister.

As he was reading to us tonight (Mr Impossible!) I was wondering about how to be a good listener when the Js are reading. A quick scan of literacy sites suggests that it is both easier to get this right and easier to mess it up than I had thought.

Mainly, I think you need to have the right attitude and, like so much of parenting, the answer here is to be playful and focus on enjoyment.  Choose books, talk about them, help with the reading, let the kids struggle, but all to a degree that it is fun for you and them.

More specifically, the 5 finger test: as the child is reading, have them hold up one finger whenever they encounter a word they don't know/can't read. If you have a full hand up, then the book is too hard for them.

Two nice references, I found are Trevor Cairney's Blog and a New South Wales schools brochure, if you want to pursue this further.

Estimation
When I started writing this post, it was only a reading note, but you know that I'm bound to see a math activity, exploration, or discussion in anything. There were a bunch of counting opportunities in Mr Strong, then we hit this picture:

So, what is the mass of the water in the barn carried by Mr Strong? We had to investigate.
To be honest, I was more interested than the little one who was absorbed in the story, so I'll leave you to come to your own conclusions about how much water there was.

Further exploration: how much pressure does Mr Strong exert on the ground when he walks?
Further further exploration: what happens to soil under that much pressure?

Attitudes

Ok, so further, further explorations about the Mr Men books:

- Is Mr Men vs Little Miss sexist?
- Does the whole series reinforce a fixed mindset?

Does it matter which order?

Who: J1 and J2
When: after lunch
where: reception room floor
what did we use: paper and d6

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At some point, when the three of us were playing a game, J1 had written down 2 + 3 x 2. I asked them what it meant and each had a different interpretation, one said to add 2+3 and then double it, the other said to calculate 2 groups of 3 and then add 2 (I don't remember which child had which order).
My question to them: does it matter which operation we do first?

Of course it does, and they quickly calculated 10 (addition first) and 8 (multiplication first). My next question, will it always be different? Let's investigate . . .

So, we got a 6-sided dice and rolled three times, calculating (a+b) x c and a+ (bxc). We did this about 5 times, keeping a (messy) record of our dice values and calculations.  They both helped each other with the multiplications.

Now, time for finding patterns.  They immediately started talking about
- when are they the same?
- which is always larger (or at least as big)?

Finally, when we agreed that the order usually mattered, I showed them parentheses and explained I liked to use them to avoid confusion. I know the internet code of honor requires me to have a passionate opinion about order of operations, but, at this point, I'd rather teach my kids to over communicate and avoid confusion rather than emphasize conventions.

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.

Some recent musings (Talking Math with your Kids)

who: J1
when: at dinner, about to enter the evening rush to get everyone in bed
what material: paper and pencil (rainbow crayons optional)

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A couple of recent stories to illustrate the habits we read on Talking Math with Your Kids. Sometimes these are easy to get right and, occasionally, a bit harder.

During a break in a conversation we were having, and unconnected to the earlier topic, J1 made a comment: "equals is different from plus, minus, times, and dividing." I thought this was a really interesting concept for him and wanted the subsequent discussion to live up to it. Things I considered saying or asking included:

1. The operations seem to take two things and act on them, while equals doesn't change anything
2. How are they different?
3. Can you draw something to show me?
4. Are they different in other ways?
5. Are they similar somehow?
6. Are there other things similar to = (having < and > in mind)?
7. Talk about operations as functions, taking arguments and delivering an ouput
8. Talk about comparisons as boolean operators

From this list, I thought 7 and 8 were to complicated and would be confusing at this stage. I was reluctant to say what I thought he had in mind (1) and was hoping he could formulate an explanation, so I went with the other questions. He wasn't sure how to articulate what he was thinking, we got distracted, and the conversation didn't really develop.

In retrospect, I think I should have committed more to the idea of asking him to draw what he meant. I'm really not sure what he would have drawn, but it would have become something we could both reference and build from.

In that vein, we had a really fun chat about his other idea: what if the Roman number system had a symbol for 0? In the page below, his new symbol is the green figure on the middle right (a trident on an inverted v with both an x and an equal sign through the middle).

Along the way, we talked about multiplication by 10 in arabic numerals (which you can see he partially gets) and we considered whether his new roman 0 would make multiplication easier in that system. In the first instance, it didn't seem to help that much, so he added more symbols and more rules, making the whole thing increasingly complex. You can see the further two new symbols RA (a sun with 8 rays) and RB (a satellite dish) in the upper left of the page (example X RA V RB = 50.)

Eventually, I started writing down examples in his new system and asking him:

• Does this make sense in your system (e.g., is this an allowed thing to write down, an intelligible combination of symbols)?
• What does this mean in arabic numerals?

It was a lot of fun to go back and forth as he tried to work out what the rules of his new system were. I don't think we figured out how to make the Romans into championship multipliers, like these guys, but he got a chance to invent a new mathematical system and investigate it.

Math Attitudes: Vaccination Booster Shot

Tracy Zager at Becoming the Math Teacher You Wish You'd Had wrote a recent post about her daughter that gave me an epiphany. Go read this story (I don't ever . . .) and then come back.

What I realized is that misconceptions about being good at math are so widespread and entrenched, it isn't enough for us to model good attitudes. At some point, they will meet a teacher or classmate (or teachers and classmates) who get it wrong.  To prepare for that event, we need a proactive vaccination for our kids.

In Tracy's post, she models some useful ingredients:
- explicitly discuss other people's feelings about math as a subject and being good at math
- talk explicitly about fixed and growth mindsets.
- talk about what it means to be good at math
- role play scenarios
- give them positive mantras (both about math and about reactions to other people's comments)
- acknowledge the emotions involved and make sure they have a safe place away from the spotlight to sort through what they really think and feel

Some other ideas:
- talk about examples where well respected people got something wrong or struggled (there should be a list of these somewhere . . . I will link if I find a good reference)
- label this peer pressure
- talk about your own experiences with peer pressure
- identify a helper/defender (someone who is likely to be nearby and can be counted on to have the right attitude)

Of course, a lot of this applies to peer pressure more broadly.