What is 8?

I've had a chance to spend more time doing math with the kids again and am hoping to write up our activities more consistently.  Let's see how this works out!

Graham Fletcher created a set of  Progressions videos for various elementary school themes. J3 and I recently went back to his page and found he had a new(er than we knew) progression on early number and counting.  Even for this simple topic, the video highlights some points we hadn't considered explicitly, for example distinguishing producers (of a number) and counters. Also, the cardinality point that smaller natural numbers are nested within larger numbers wasn't something we had talked about, but we soon realized it was part of many examples in how we understand numbers.

With that as inspiration, J3 and I decided to search for a range of examples of a single number, we chose 8, in different forms.  There is at least one obvious version we're missing.

Add a comment (with picture, if you can) to show other forms of the number 8!

Marking 8 on the 100 board, an easy place to start:

8 beads on the abacus shows the relationships 3+5 = 8 and 10-2 = 8 (also 100- 92 = 8)

8 can hide in plain sight. Without labeling the three lengths, it would have been hard to recognize the longer one as 8 cm and, for you at home, impossible to know without reference to show the scale.

It happened that, within the precision of our scale, two chocolate wrapped chocolate bars were 8 oz (2x3.5 oz of chocolate + about half an ounce of wrapping for each):

8 cups of water ended up being a lot, so this version unintentionally revealed a relationship 4 + 2 + 2 = 8

Though I'm not sure I can articulate why or show supporting research, I feel it is very valuable to build experience with physical models of numbers to create familiarity and intuition about what they are/mean. In particular, I hope this helped J3 anchor the importance of units of measure and scale in the interpretation of numbers.

Finally, this construction has nothing to do with the number 8 (or does it???)

Divisible by 3

who: J2
when: at breakfast
where: at the dining table

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We were recently talking about divisibility rules and J2 was particularly excited about testing large numbers for divisibility by 3. He asked me to provide a large one to see if his mother could figure out whether it was a multiple of 3. This was the resulting conversation:

J0: Let's try 12345678987654321
P: I think it is a multiple of 3
J2: <excitedly> It is!
J0: how do you know?
J2: well, it is a multiple of 111 and 111 is 37 x 3.
P: <stunned/surprised> How do you know it is a multiple of 111?
<conversation continues>

Of course, we parents expected to hear that he added the digits and saw that the digit sum was multiple of three. Unfortunately, that test is just a trick for him right now. He doesn't really understand how it works or why. His own approach wasn't a trick, but it was a bit lucky.

Our friend 12345678987654321
This is actually a number J2 has seen before. In reading the Number Devil, we played with a pattern:

1x1	=	1
11x11	=	121
111x111	=	12321
1,111x1,111	=	1234321
11,111x11,111	=	123454321
111,111x111,111	=	12345654321
1,111,111x1,111,111	=	1234567654321
11,111,111x11,111,111	=	123456787654321
111,111,111x111,111,111	=	12345678987654321

We spent a lot of time talking about these, checking them, and extending the pattern to see what would happen next. From here, it isn't so hard to see that 111 is a factor of 111,111,111 and, of course, everyone knows that 111 is 37 x 3. Right?

Some food for thought

• Where does that sum of digits test for divisibility by 3 actually come from?
• Does it work for any other numbers (hint: yes!)?
• Wait, doesn't this mean that order of digits doesn't matter for those factors?
• What is the divisibility test for 11 and why is it almost as cool as the one for 3?
• What is the divisibility test for 7 and how does the name "primitive root" help explain how messy it is?

Unrelated Pictures

J1's first zometool project, showing the duality of the dodecahedron and icosahedron!

J1 tackles the NRICH table and chairs challenge, with TRIO blocks. More on this later?

A funny looking plant with a pretty flower that caught J1's eye.

Goody bags (fair sharing)

Who: J1
When: before bedtime
Where: bedroom

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Several months ago, I wrote a post (here) about various sharing problems and the difficult question: "What is fair?" I got distracted until I found some ready-made examples from Peter Liljedahl. J1 and I discussed his Goody Bags scenario last night.

Though it seems simple or childish, I found this activity stimulated a rather deep conversation and was really effective. I highly recommend other parents try it.

In a nutshell, you have these 40 gifts to distribute among 5 friends and want to make the distribution fair:

J1 cut out all of the pictures, spent a couple minutes putting them in piles, then we arranged them and talked about whether the allocation was fair.  Each rough column below is a goody bag:

When discussing, we didn't start with them all in a grid, but worked a column at a time, starting from the right.  that means we had two goody bags out and compared them against each other.  J1 naturally came to the arrangement that put common items side-by-side and he was careful to leave spaces where one bag got something that was missing in the other.

As we discussed, he was pretty clear that the common items weren't relevant to deciding whether the two goody bags were fair, we only needed to look at the differences. When talking about comparing unlike objects, he tended to focus on how permanent they would be and compared the difference baskets rather than individual objects within them.  For example, the two rightmost bags differ by +tatoo+sticker - kitkat - gel pen. He judged this fair because the kitkat wouldn't last long, the gel pen was nearly permanent while enjoyment of the the tatoo and sticker would both be moderately lived.

At each step, we compared the new goody bag to the single preceeding goody bag that was most similar.  Eventually, J1 proclaimed that he was satisfied with the distribution and didn't really have a preference between any of the goody bags.

We talked about some of the characteristics of his division:

• when there were 5 of some item, everyone got one
• when there were more than 5 of some item, everyone got at least one
• everyone got the same total number of items.
• no one got more than 1 duplicate

Next, we talked about how he had divided the items. He explained that he had simply done it randomly, by which he meant that he took all the items and dealt them out in a circle. I asked whether it would work well if we tried it again, so he did, but made a very important change.  This time, he dealt the items in a line and gave two items to the bags on the ends in each pass. If that's hard to picture, call the bags A, B, C, D, E and then he is giving goodies to the bags in this order:
A- B - C - D - E - E - D - C - B - A - A - B - C - D - E - E - D - C - B - A (etc).

This process gave us a very different result (not pictured) where most of the previous characteristics were violated:

• A and E got a lot of duplicates
• Often, someone wouldn't get an item, even if there were 5 or more

Everyone still got the same total number, but it was hard to call this split "fair."

We had a good discussion about why the results were different. This included a really interesting idea: if the process used to divide the goodies was "fair," then maybe the ending distribution is fair even if it doesn't look right.  Essentially, he was defending randomness and arguing that everyone had an equal chance to get any particular goody, so the opportunity was fair. This was a pretty sophisticated idea that I hadn't expected to hear. I think our experiment of dealing goodies in a circle vs the line method also demonstrated that apparent randomness can be thwarted with subtle systematic biases, another rather deep idea.

We moved on to discuss whether he had any personal preferences among the items. He ranked them as below, most favored on the left and least on the right. Items that are stacked are considered equal and there is some sense of spacing.

Lastly, he revealed that he had 3 strategies for attacking this challenge, to be employed sequentially, if needed:

1. deal cards randomly
2. swap items if they seemed to be blatantly unfair
3. give up on the whole project and just put all his favorites in one goody bag and day-dream that he got to keep it.

J1's closing comment
When we were all done, I told J1 I was glad we had finally done this sharing challenge. He looked up and said: "oh, this is what you meant! When you said we should do the sharing challenge the other day, I thought you were going to make us share our actual toys, that's why I didn't want to do it."

J2's observation
J2 and I talked a little about this investigation. His solution was simple:

• first row goes to bag A
• second row goes to bag B
• third row goes to bag C
• fourth row goes to bag D
• fifth row goes to bag E
• then the remaining three rows allocated to each bag (in order) by column

He declared this fair because everyone got the same number of items and was entirely unperturbed by the idea that, for example, someone would get 5 potato-head keychains. So, maybe the conversation won't be deep with every child. Or, this indicates stages of maturity in thinking about what fairness is?

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.

Counting with 6 hands

Please read for the *questions* below as I'd love to have your thoughts in the comments.

Who: Baan Pathomtham 1st grade class (J1's class)
Where: at school
When: 2 hours Tuesday morning

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We (P and J0) got a chance to spend the morning talking about subtraction with J1 and his classmates. It was an opportunity to see some differences between talking math one-on-one (or one-on-two) and a larger group.  Here are a couple of tidbits from the discussion.

Practice with poker chips
Of course all children need to be familiar with the standard gambling implements: dice, cards, and poker chips. The first two are already well known, so we did an activity with poker chips this time. How would you count all the chips in the picture?  Well, what if "you" were actually a group of 3 first graders?

Here's the strategy one group implemented, spontaneously, as far as I could tell:
(1) divide the chips into equal piles for each child
(2) count the remainder in the center of the pile (in this case, one chip, so the count was trivial and done without an explicit effort)
(3) take turns putting one new chip into the pile
(4) all count together as the new chips are added

Actually, this was their second strategy. At first, it was a free-for-all with all three trying to count all the chips and messing up each others division between counted and uncounted chips. 

Note: counting the chips was just accidental to the activity we were doing, so this shared counting strategy was just a cool thing we noticed along the way. If it had been more central, I would have talked with them about the equal piles at the start (which links with multiplication) and why there was a remainder (which links to the division algorithm).

Enthusiasm
Most importantly, the kids were all really excited and enjoyed working on math. They liked asking mathematical questions about a picture we presented, had fun doing calculations, trying new modeling tasks, and playing the mathematical game.

This confirms, once again, that enthusiasm and curiousity are things we (usually) kill during the educational process.  Not at our school!

Explaining
Given their enthusiasm and apparent facility with the calculations, I was surprised that they struggled to explain their calculating strategies. I can't tell if this is a language issue, if the calculations they were asked to describe are so ingrained that they don't consciously think about them, or if they don't really understand what they are doing.

My key take-away: I will focus a lot more of my discussion time on getting the J's to talk about how they calculated something, see if they can draw a picture, and see if they can explain using a concrete object.

Extensions
As preparation, P and I talked about 3 models of subtraction: taking away, differences, and counting back. P made two comments:
(1) Word problems are harder than straight calculations (said while we were discussing what types of problems to use to have the kids investigate the three models)
(2) "Counting back is such a waste, I always knew the answer through another method and had to artificially demonstrate counting back."

Word problems seemed, to me, the natural way to motivate using a particular model for subtraction. For three quick examples:

• You started with 5 cookies and ate 3, how many are left? This is taking away, obviously.
• Don has 27 poker chips and Tanya has 13. Who has more and how many more do they have? Differences, naturally.
• Walking along a straight line, you go forward 6 meters and then back 2 meters, how far are you from your starting point? Counting back suits this one.

*Question* is this the wrong way to use alternative models? Is it necessary to force them to use "unnatural" models to demonstrate proficiency (for example, using take-away to resolve the differences question)? Does this create difficulties for problems involving alternative missing values in the same types of questions (i.e., you started with 10 cakes and now have 3, how many did you give away?)

Counting back is the same as the movement model, which I called "forward movement" in my post on addition models. This model leads nicely and really easily to emphasizing the role of 0, negative numbers, and subtraction of negatives. Looking a bit farther ahead, it links with vector addition by just extending our operation to more dimensions. Taken along another path (ha, the puns!) it can be used for modular arithmetic (replace directed movement on a straight line with directed movement on a circle).

Addition, it's just making a bigger collection, right?

Who: J1 (also guest appearance from J2)
When: at dinner
What materials did we use: our mouths (just talkin') and then a bunch of miscellaneous items later

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J1: [Friend X] isn't good at multiplication.
M: That's not surprising. Has he started learning multiplication in school?
J1: Actually, he's not so good at addition either. [Friends Y and Z] also don't really understand addition.
D: Great, so maybe you can help them. Is there a picture you can draw that shows how you think about addition? For example, 4+3?
J2: (looks at his fingers for a moment) that's 7! (exclamation, not factorial)
D: So what does that mean, when you add?
J1: (making a big gesture with his arms) Gathering things together, collecting them.
D: Are there other models of addition that you know? Other things you do where you need to add?
J2: counting, then counting some more
. . . tbc

So, how many models of addition have your kids seen? How many are there? Let me know what you think in the comments.

Repeated Counting

For a hobbyist counter, like J3, this activity is an excuse to count up to 20.  Twenty? Well
5 fingers + 5 almonds + 5 fingers + 5 almonds = 10 fingers + 10 almonds = 20 etwas.

My argument is that the fingers and almonds aren't really being combined, they are all serving as objects for counting, then counting some more.

Grouping Together
On the 4 different stems, we see 1 longan, 2 longan, 3 longan, and 4 longan. How many ลําไย are there all together?

I'm calling this distinct from repeated counting because the 10 ลําไย do make a coherent collection all together in a fruit bowl, while their separation into 4 subsets probably won't be relevant for their future destiny and, in fact wasn't relevant to why they came to our table (they were all bundled together in a cluster).

Forward Movement
You've seen this game before: roll the dice and move your fierce dinosaur closer to the finish. A number line gives you a similar model, with the benefit of fractional steps, but I liked the fact that the motion is only sequential by convention here.



Combining Mass


A lucky one as the measurement error doesn't destroy our perfect whole number addition here.

Combining Volume
Yes, it was necessary to pour out 1/4 cup + 1/2 cup = 3/4 cup of chocolate milk in the course of this investigation!



As with the mass model, note that this is imprecise and the experimenter must decide from the context what gap can be tolerated between the theoretical and empirical results. This is especially critical for valued substances like chocolate milk!

(Musical) Time
One quarter note plus four eighth notes get played for a time lasting one measure, in 3/4 time.  Or, if you want to be measure independent, four eighth notes together last the same time one half note.

Silly
This is a bluebird of happiness. It is not a model of addition . . .  or is it?