Class summaries

Quick notes on the activities for the classes today.

Grades 1 and 2

Mobius loops
We are splitting up games with other activities. This week, we are introducing several geometry explorations. First up are Mobius strip activities, nicely shown in this Matt Parker second favorite shape:

1. cylindrical loop: using 2 colored pencils, draw a line along the center of the loop on the inside and the outside. We can see that there are two sides, no big surprise. Cut along one of the lines and the loop splits into two new loops
2. Mobius strip: give the paper a twist. Now, draw along the middle of the paper and see that there is only one side. Now, cut along this line and see what happens. Repeat this, drawing another line along the center of the new strip. Do you have one side or two? Again, cut along the central line. What do you get?
3. Two connected loops: tape two,  untwisted loops, together in perpendicular directions. Now cut along the center lines of each loop. What do you get?
4. Two connected Mobius strips: tape together two mobius strips and cut along their center lines. What results now? Did everyone get the same result?

There are some natural extension explorations:

• try these with more twists (as per Matt's video)
• keep cutting the center lines
• Connect a Mobius strip and an untwisted loop (half-way step between 3 and 4). Now, cut along the center lines. What happens?
• Inspired by the thinner and thinner loops, kids can explore ways to cut paper so that they get longer and longer strips or loops

Note: these activities can be even more rewarding when something goes wrong. For example, what if there isn't enough tape connected the ends of the loops? These mishaps make everyone pause and consider more carefully what is actually happening.

Also, in the class, we only had time for the first two make-and-cuts, then demonstrated the two connected loops.

Punch (fold and cut)
All this cutting fits nicely with our second exploration: the punch activities from Joel David Hamkins' post punch, fold, and cut from Joel David Hamkins.

Grades 3 and 4

We started with the Shapes x Shapes puzzle from NRICH:

We added a couple of extra questions to this challenge:

• Before completing the puzzle, which numbers do they think are excluded? Why?
• Make extra equations that allow us to include those missing numbers. Are they easier to incorporate using multiplication or addition equations? What about equations that combine multiplication and addition?

Observations: 

Once again, this appears to be a very simple activity, but gave us a lot to talk about. In particular, it was very helpful for highlighting a lot of misconceptions and gaps in understanding. Examples:

• "identity" relationships were still unclear: 1 x n = n, 0 x n = 0
• Several students thought the first equation would be 4x4x4 = 12 (confusing multiplication and addition) 

 

Division Dice move to Cards
For our core activity, we are extending the Division Dice game. This time, we use playing cards, A through 10, instead of dice rolls to generate the random components of their equations. In this case, the aces are wild and can be any number from 2 to 10. When they form a multi=digit number with a 10, the 10 counts as two digits. For example, 3, 5, 10 could form 105 ÷ 3.

With cards instead of dice, we lack the natural move of flipping the dice to the opposite side which we used to make sure all throws could give us whole number divisions. In this version, we allow division with remainder. However, the twist is that the remainder becomes points for the opponent.
For example, if I draw 3, 3, 5, I can form 53 ÷ 3 to score 17 points for myself, but the opponent gets 2 points.

Question: Are there cases where the best play is not to form the largest possible number divided by the smallest number?

An interesting game variant: swap the scoring so that the active player scores the remainder and their opponent scores the quotient.

Improv Math and Division Dice follow-up

We had a really good experience playing Division Dice, the game that we introduced a couple of posts ago.  Mainly, I want to illustrate something fun that came out of really listening and paying attention to what the kids are doing and saying. I like to think of this as "improv math," as a way to credit my improv comedy experiences for heightening my awareness of how important this is.

Division Dice for number sense

I was really pleased about the quality of thinking stimulated by the game. We played with the most loose rules (1s are wild, the components of the 2 digit value can be flipped to their 7s complement). That gave a lot of opportunity for the kids to think through options to (a) make whole number divisions and (b) maximize values.

For example, rolling 3, 4, 6:

• what are the allowed groupings that give a whole number division? Remember, in the 2 digit number, we can use any of the values 1, 3, 4, 6, and it is possible for us to use two 3s or two 4s in our calculation.
• What is the highest scoring choice?

Division Dice for arithmetic exercises

As a way to create virtual worksheets, this game is mediocre. The basic structure means that students are never dividing by a divisor larger than 6. This leave out a lot of fact families. However, because the kids are trying to maximize their scores, they quickly realize that they can almost always get away with division by 2, occasionally must divide by 3, and rarely get stuck dividing by 4 or 5. I haven't yet seen a case in a live game where division by 6 was necessary.

Fun exploration: what scenarios will require division by 6?

Using playing cards or other dice shapes allows us to extend the possible values and reduce the likelihood of dividing by 2 or 3. However, it also increases the number of cases that don't have a whole number division relationship. We are thinking about ways to incorporate division with remainder and will try out a variant tomorrow.

Improv Extension

Playing at home, the 3, 4, 6, case led J1 to consider: how do 63 ÷ 3 and 64 ÷ 4 compare?
As he contemplated that, I realized that we had a nice sequence of multiples, meaning all of these are whole numbers:

There were several cool things for J1 to observe here:

• 4 of the 6 quotients end in 1
• The quotients are all decreasing
• The drops between successive quotients are themselves decreasing
• the dividends are equal to the divisors + 60

We pursued this in two ways:
Extension 1: what if we add something else to the dividends?
We tried three versions.

1. starting with 60 and adding 6 at each step
2. Starting with 60 and adding 60 at each step.
3. starting wit 1 and adding 7 at each step

You can see our notes mid-discussion below:

Later, when J2 was also involved, I offered them another sequence: starting with 66 and adding 6 for each increment:

66 ÷ 1

 72 ÷ 2 

78 ÷ 3

84 ÷ 4

90 ÷ 5

96 ÷ 6

120 ÷ 10

132 ÷ 12

150 ÷ 15

180 ÷ 20

240 ÷ 30

420 ÷ 60

3660 ÷ 600

36060 ÷ 6000

We're breaking the rule about the dividends being multiples of the divisors, but the last two calculations are still easy and nicely illustrate the limiting behavior.

Extension 2: can we find other chains of whole number division equations?
We started this by thinking more simply: for chains shorter than 6. For example, what are the smallest K, L, M, N larger than 1 such that all of the following are whole numbers:

K ÷ 1

 (K+1) ÷ 2 

L ÷ 1

(L+1) ÷ 2

(L+2) ÷ 3

M ÷ 1

(M+1) ÷ 2

(M+2) ÷ 3

(M+3) ÷ 4

N ÷ 1

(N+1) ÷ 2

(N+2) ÷ 3

(N+3) ÷ 4

(N+4) ÷ 5

After getting the shorter cases under our belt, we then went for a chain of length 7. J2 worked by himself for a while, then came back and announced that no chain with dividends smaller than 100 would work.  He went away and then came back quickly with the idea that maybe we could add 7! to each divisor.

Happiness Habits

This may or may not be related to math ....

Discovered an old note listing three habits for happiness. I don't recall the source, some lecture recorded on youtube, I think.

1. 3 gratitudes: write down three things for which you are grateful
2. Positive memory journal: write about one positive experience from the last 24 hours that you would like to remember
3. Thank you: at the start of the day, write a thank  you note to someone in your social support network. Yes, this means you have to interact with them enough to have something to thank them for! 

Logic Puzzle collection and Dropping Phones

Recently, Mathbabe put out a request for riddles. There are some good links in the comments:

• The Riddler on FiveThirtyEight
• Logic Puzzles on Math is Fun
• Alex Bellos at The Guardian
• Riddles.com (naturally)
• Puzzling.stackexchange.com (is there a stack exchange for everything now?!)
• Authors cited: Dudeney, Gardner, Polya, Smullyan, Vaderlind. 

We've played with puzzles from almost all of these sources in the past, but this was a good opportunity to put together a nice list.

The one that stood out to me was on FiveThirtyEight. That's a site I read frequently, especially during this US election season ... and I was totally unaware of The Riddler feature. We kicked off with the oldest puzzle from their archive: Best way to drop a smartphone.

Breaking stuff

Right away, J1 and J2 loved the theme and were into questions about whether they could somehow keep the phones, if they weren't broken, or utterly destroy them, if they were broken. We talked about setting an upper bound with a very simple strategy of starting on the 1st floor and working up each floor. That's not a great answer, but it got them into modifications and improving strategies.

Through the conversation, it was interesting to see them start with the idea that the drops for one of the phones would be de minimis and could be ignored, but then start to pay attention to that aspect. Also, they had to grapple with the idea of balancing the number of drops that would be required in different cases.

While we didn't get to the optimal strategy, but the kids managed to get a version that, at worst, would take 19 drops for the 100 story building. Their intuition was based around taking the square root of 100. They could see that this probably wasn't the best answer, since there were still cases that, at worst, would take 10 drops and others that, at worst, would take 19.

Can you do better?

Smaller before bigger

The puzzle page poses the same challenge for a 1000 story building. However, we found something interesting when working on the version for a 10 story building: there is  strategy where all worst cases take the same number of drops, but it is not the optimal strategy!

It is a little hard to write about this without disclosing the strategy, but here's a hint:

• 4 + 3 + 2 + 1 = 10 and 5 + 4 = 9
• We are always allowed to assume that the phones will break if dropped from the top floor of the building

This led to another extension: what size buildings will have the same issues as for a 10 story building?

Probability comes in

Another extension is to think about the expected number of drops required and strategies that minimize this. Crucially, this extension introduces the idea about our prior beliefs about the sturdiness of the phones: where do we think the phones are likely to break, what is our confidence?

We didn't pursue this extension very far, but it did lead to some interesting conversations about terminal velocities. For those who want to follow that thread, this (other) stack exchange thread might suit you: How to figure out height to achieve terminal velocity.

Labyrinth cool game mechanic, but weak game

Background

Labyrinth, the board game, has a sliding tiles mechanic that I really like. Sixteen tiles are fixed on the board and 33 (7x7 - 16 + 1) loose tiles are available to slide through the frame created by the fixed tiles.

This picture shows the board and J3 punching out the loose tiles:

Here, we've started to put some of the loose tiles into the board:

You can see that each tile has a picture that includes a segment of a maze and, possibly, an item. Paths through the labyrinth depend on how the tiles are arranged. Along the open rows and columns, loose tiles can slide up/down or left/right. This is also where the extra loose tile comes in: sliding a column creates an open space for the extra tile and forces another one off the board.

My problem

We don't find the game very compelling, so I am looking for ideas about how to make use of the underlying structure in a new game. We have done some investigations about arrangements of the loose tiles (can we create a maze without any inaccessible tiles? What do we get when we minimize/maximize the number of inaccessible tiles?) However, I feel like there is a lot more that should be possible.

What are your ideas?

An unrelated project about birds' wing shapes from J2:

Division Dice (math games class)

Who: grades 3 and 4
Where: in school

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A dice division game

We created a simple dice game to practice division. Here's a description of the basic element of play:

1. roll three dice: for example, 3, 4, 5
2. group two of them into a two digit number: for example, 45
3. Divide the two digit number by the remaining single digit: for example, 45 / 3 = 9
4. This value is your score for the round
5. First player to 200 or more points wins (we used 100 for the initial game)

Key constraints

• You can only score points if the single digit is a factor of the two digit number (remainder must be 0)
• Where there are multiple options, the player can choose the combination that gives them the maximum score

This pencilcode program (see code) analyzes this basic game structure, identifying how often there will be no legal scoring arrangement and showing a histogram of the largest scores.

Modifications/Extensions

I wasn't satisfied with three elements of this game: (a) any time a 1 occurs, the division calculation is too easy, (b) too many combinations don't allow a score (about 15%) and (c) there aren't many decisions for the students to make (just six combinations to investigate).

We addressed these by adding two extra rules:

• 1 is a wild that must be replaced by a value from 2 to 6 (cannot be left as a 1)
• On your turn, you can flip the over the dice in the two digit number. For example, a 6 can be flipped to a 1, 5 to 2, 4 to 3, etc.

The first point removes the division by 1 cases, the second one allows more choices and reduces the number of non-scoring cases.

*UPDATE*
Allowing the dice flip and wild 1s seems to make the game too loose. Instead we dropped the wild 1s rule and added these two:

• Division by 1 is not allowed in the game
• If you roll triple 1, re-roll

Build the chair (part 2)

In class, we played with the NRICH Chairs and Tables activity (our outline here). We came up with an extension that we explored at home: making a sequence with smaller and larger chairs.

Kick-off

I knew that J2 had incorrectly counted the cubes in the NRICH sample chair, so I kicked-off by asking him to show me how he counted them. This was a surprising, and unintentional, kick-off. His method was to decompose the chair into three sections: seat, back, and legs. This upper left 2/3rd of this picture show how he determined the number of cubes in each section.

First Sequence

The previous picture also shows notes about how J2 thought of making the chair bigger or smaller. His idea was to keep the same size seat, but make the legs and back of the chair longer or shorter. You can see our drawing of back for the two next smaller chairs.

Along the way, we looked at the total number of cubes in each chair. A simple pattern jumped out to him: each step is a difference of 6 cubes. He quickly realized this was because each leg required one more cube (4) and there are two on the sides of the back.

I asked him how many cubes would be in the 10th chair. When I asked him to explain his thinking, he said, well, going backwards, the 0th chair should have 10 cubes, each step is +6, so I need 10 + 6x10.

From algebra back to geometry

His idea of the 0th chair really excited me as this was one of the ideas I had been hoping we would uncover. We talked about how this chair would look (a 3x3 seat with one cube in the middle of a side as the back). This was not something we would naturally have created when asked to build a chair.

What we had done is gone through a sequence translate from geometry to algebra, naturally extend the algebra to a new case, translate the new algebraic case back to geometry.

Second Sequence

One other delight in this activity was that J2's sequence was not one I had in mind when outlining the activity. Of course,  wanted to share my version, as well.

I followed his decomposition into seat, back, and legs. See if you can understand my notes and picture how my chairs are growing through the sequence. Chair D₃ is the starting example from NRICH.

I asked J1 to fill in the D₁ chair to see if he got the pattern.

Not linear

J2 noticed that, this time, the gaps between cube totals were not the same, but the second difference is constant.

To round up the discussion, we wrote down an equation for the nth chair and calculated how many cubes would be in the 10th chair. Finally, we tried our trick of extending to the 0th chair. 

This time, we realized that there could be something interesting in the 1st chair, too. See if you can build a version of our D₁ chair, using whatever your favorite building material might be.

Math Make-overs

Robert Kaplinsky wrote a post about open-middle vs open-end problems that got me thinking.
The punch-line is that there are simple make-over tricks that you can use to convert almost any problem into the type you need, whether closed- or open- (or half-open) middle, closed- or open-ended.

Note: some of this transformation thinking is clearly inspired by Dan Meyer's "remove the information" method.

I think this post is, mostly, intelligible without reading Robert's note, but everything will make more sense if you have particular problems (and problem-statements) in mind.

Closed vs open middle

Closed-middle problems tell the student how to answer the question, either directing them to a specific strategy or scaffolding a specific approach through explicit interim steps. Open middle problems leave it to the student to find (or select) their own strategy.

Perhaps there is a half-open-half-closed-middle where students are given a menu of strategies?

I conjecture that note that any problem can be made into either type. Specifically, the ones in Robert's post all can, including calculating 475/25. The main technique to change a closed-middle to an open middle is to remove any guidance about the strategy the students use to answer the question. Going in reverse is also possible (tell them what strategy to use, scaffold interim steps) but there are already too many closed-middle problem presentations, no?

Closed vs open-ended

Is there one right answer/end result regardless of which strategy is used? If so, then it is closed-end.

Again, almost any problem topic could be either closed or open-ended, depending on whether it has:
(a) how many interpretations are possible
(b) how much data is supplied and whether that data is consistent.

It is easy to be surprised by problems with multiple interpretations where we only expected one. Recently, Marilyn Burns wrote a post about fractions that seems to have one right answer, but actually depends on how we define our reference unit. Popular probability questions seem ripe for interpretation-based disagreements.

If the question has just the right amount of data or all the data is consistent, then it will have one common ending result and is closed-end. If there isn't enough data or "too much" (some data is inconsistent), then it suddenly becomes open-ended since students have to use other ways to fill in the gaps or make choices between inconsistencies.

In Robert's post, there is a question about hybrid cars that is open-ended because there are some assumptions the students need to make and there are elements in the data that aren't consistent, so choices need to be made. This could become closed-end by adding more data (removing the need for student assumptions) and doctoring the data to make it all consistent.

In contrast, his example of In-and-Out burgers could become open-ended by taking away data or making the data inconsistent. My favorite way to blow open the end would be to go way beyond the assumptions of simple extrapolation: let's order a burger with 1,000,000 patties! How would they price that?

By the way, you might think some questions simply can't become open-end. For example, calculate 475/25. Is it possible to transform this one?

Well...we didn't specify which base we are in. In base 10, of course, we get 19. In base 8, though, the answer is 17 with remainder 2! I'm sure cheeky students out there could find other innovative ways to interpret the question, if given a chance.

Which is better

Sure, we can change the problems from one form to another, but which is the better type of question? Open-middle, open-ended, of course!

Just kidding. All of them have their place and it depends on what you are doing and why. Though it conflicts with my personal preference, I even see a place for closed-closed questions where you want the kids to practice a particular strategy (closed-middle) and you need the consistency of a closed-end answer to quickly check that everyone has gotten to the same result.

Open-middle, closed-end can be good for generating discussions that compare and contrast strategies. That's logical, since the strategies are where you would (should) find differences from these problems. However, aren't some of the kids thinking: "we all got to the same place, I don't care that someone else took a different route." That was something I often thought as a student.

When you have enough time, I think open-open questions create really rich discussions that include strategy comparison. Our typical conversation is something like:

• Hmm, we got different answers!
• What did you do?
• I used method A and data Z
• Oh, I used method B and data Y
• If we used method A with data Y, what would happen? Is that even possible? Why or why not? Would we get the same as (A, Z)?
• If we used method B with data Z, would we get the same answer as (B, Y)? etc etc

You can see I have a bias for open-open questions. One last reason: kids get a lot of closed-end questions already, so I don't feel that I need to add more.

Factors and division

who: grades 3 and 4 at Baan Pathomtham
where: in school

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Sorry about the lack of pictures. This is a short and sweet note.

Dots & Boxes and Factor Game Mash-up

To start the year, we played a version of dots & boxes that integrates the factor game (here is one example). This is based on the game template from Mathified Squares Game that we used last year.
Instead of using dice to determine where each player can play, we introduce factors 1 to 6 at the bottom of the page and selectors.

As with the basic factor game, this version creates multiplication and division. This is the point we want to draw out for the game.

Homework

Play the game at home and write down 15 division equations that come up in the course of play.

*UPDATE* Having now played through this game fully, I really like this structure. Using the factor selectors drives some interesting thinking about common factors, especially during the middle and end-game phases.

We did find that it starts a bit slowly as players can make moves on distant parts of the board and decisions don't have clear connections to capturing squares. For the young kids, we recommend just pushing past that stage. For older kids, that can be an interesting (and difficult) strategic analysis.

Build the chair spatial reasoning (Gr 1 and 2)

who: Baan Pathomtham grades 1 and 2
where: in school

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Here in Thailand, summer is over and we are back to school! We are kicking off the math games and exploration class today with an activity from NRich (chairs and tables) that has a surprising depth. Also, keep your eyes opened for the hidden reasons why we are starting with this activity.

Build a chair

The starting directive is simple: use unifix cubes to make a chair. Here's an example, from NRich:

To start, we ask the kids to get 15 cubes each. What does 15 mean? How do they know they've got 15? Do they think they will need more or less than 15 to make a chair?

After they have built their chairs, how many did they need? If it was less than 15, how many are left over? If it was more than 15, how many more did they need to add?

Chairs for bears

Once we've all got one chair, can we make two more for the three bears from Goldilocks and the 3 Bears? We need a small one for baby bear, a medium sized one for mama bear, and a large one for papa bear.

As the final construction challenge, we ask them to make a table sized to accompany their original chair.

Homework

1. Based on the pictures above:  (a) find out how many cubes would be needed to build these shapes, (b) draw a 2d perspective of one of the shapes from one direction.
2. For those who have construction sets at home, try making chairs of different sizes. What things were similar to using the cubes at school, what was different?

Extension

Building off the three bears activity is a nice extension:

• What is the smallest chair we could make? How many cubes do you use?
• How would you make the next larger chair? The next chair larger than that? How many cubes are used for those?
• What about the tenth chair in this sequence? What would it look like? How many cubes would we use to make it?
• Same questions for the 100th chair?
• What is an equation for the number of cubes in the nth chair?

I'd note that these are challenging questions which go well beyond first and second grade. Also, there is no single correct answer, particularly as different students will have different ideas about what is required to be a chair or how the form should grow through the sequence.

Note: these questions follow the thinking of Fawn Nguyen's Visual Patterns.

Hive at the beach (quickie)

Have been at the beach for the last couple of days before school starts. We got to play with one of my recent impulse buys: Hive (carbon).

Chance for discovery

As an experiment, I started by laying out the pieces with the blank sides up. One at a time, I asked the J's to come in and tell me what they noticed. For each of them, at some point, there was a moment when they tipped or turned a piece over and discovered the insects. Their reactions were a real delight and they realized it was special feeling, so quickly helped rearrange the pieces and get another sibling so that they could have the same experience.

Making patterns

Having started with the blank sides of the pieces, it felt natural to the kids to sometimes play with these tiles just to make patterns. Here is one example:

Playing the game

Of course, there is also delight in playing the game itself:

From this detail, you can see that J3 (playing black pieces) is following an unorthodox strategy. At this stage, she is really learning the rules and sees the whole activity as a strange way to play together to create unusual patterns and combinations of the bugs:

Reversed inequality

Recently, Mike Lawler posted a challenge base on the first question from the recent European Girls Math Olympiad (side note: what is "European" about this contest now that a US team participates?)

I enjoyed thinking about this problem and wanted to come up with something related to do with our kids. One way to see this problem is that it appears to reverse the direction of the inequality between the arithmetic mean and geometric mean. My version for younger students involves exploring this inequality first.

Discovering the Arithmetic Mean-Geometric Mean Inequality

I had J1 and J2 select two dice from our pound-o-dice. Initially, J1 chose a d20 and d6 while J2 chose a d12 and d6. I asked them to make a table with 6 columns and 7 rows (we ended up adding more, so probably better to ask for 8 columns and 10 rows). I had them label the columns:

1. A
2. B
3. AxA +  BxB
4. 2xAxB

They rolled their two dice and put the values into columns A and B, the calculated the other two columns as labeled.

J1's first results were 16 and 5. After filling in the rest of that row, he paused and then switched to 2d4. This wasn't a problem for our investigation, but he did miss some calculating practice with more difficult seed numbers.

After filling in 6 rows of data, I asked them what they noticed.

Here were some of the observations:

• when A and B are the same, our calculated values are the same
• when A and B are different, our calculated values are different
• When A and B differ by 1, the calculated values differ by 1 (also vice versa)
• AxA +  BxB > 2xAxB
• We are only using positive integers

J2 really got into the spirit and asked for some more suggested columns. I told him to add:

AxA - BxB and (A-B) x (A-B). J1 also added the square of the difference.

With that, they noticed two more things:

• AxA + BxB was larger than (A-B)x (A-B)
• AxA + Bx B = 2x Ax B + (A-B)x(A-B)

You can see that they also added some negative numbers on J1's sheet and challenged one of these observations.

A picture proof

To finalize our exploration of the inequality, we used tiles to build intuition for a picture proof of this identity:

AxA + Bx B = 2x Ax B + (A-B)x(A-B)

This is one of the pictures we liked the most. In this case, A = 5 and B = 2. Along the left side and the bottom are rectangles AxB (the green + yellow and the blue + yellow regions). These two rectangles overlap in a BxB square (the yellow region) which they also highlighted with the 4-color square. The remaining red square is (A-B)x(A-B).

Side note: In our discussion, I was delighted that J2 would correct me whenever I slipped and said "greater than" instead of "greater than or equal to" when discussing the inequality.

Two quick algebra extensions

For kids who have already done a little algebra, proving the identity using the distributive law should be fairly straightforward. The other little extension is to show that our expression A²+B²≥ 2AB is equivalent to the arithmetic mean-geometric mean inequality.