Twist on an old puzzle and our number scavenger hunt

who: J1, J2, J3
when: after lunch, before violin lessons
what did we use: mini-white board and things around our neighborhood

---

Digit Substitution

You've seen these puzzles before: there are triangles, squares, and circles that represent digits in an arithmetic equation and you need to figure out which digit is represented by each shape. During a recent RightStart Math session, J1 got a couple of these. The puzzles were fun, but standard, each with a unique answer. We decided to add two twists.

How many answers?
The first twist was to find how many answers there are to this puzzle. You can see us mid-work, having already identified two choices for the circle and several options for the other two shapes.
The point is that it invites the kids to think carefully about the abstract relationships between the values that the shapes can have.

Build your own
I always love reversing a puzzle and having the kids build their own puzzles. I opened with the challenge to see if they could create a puzzle with no solutions. Initially, they draw the shapes at random and assume there's no solution. However, once we worked through a couple, they started to notice patterns and to carefully plot their strategies for making the puzzles impossible. Also, it led to discussions about what should be allowed as a solution: can a number have a leading zero, is it ok if two shapes represent the same digit, can we allow a shape to represent a negative number or a number larger than 9, etc.  Really cool thoughts that transcend the specific puzzles we suggested to each other.

Our number scavenger hunt

J3 went looking for numbers around our neighbourhood the other day. This was suggested long ago on Kids Quadrant last September, but I didn't really appreciate how good it would be for us until we started. Here are pictures J3 took of our finds from 1 to 4.

One swimming pool (that is soo big)

Two black metal pieces to open the door

Three plastic bottles (prior to us doing a civic duty and helping clean up)

Four light switches

Except the first, all pictures were taken by J3. We also found an office chair that has 5 wheels, but the picture didn't quite capture it.

Playing this game, J3 was really excited to count everything in sight. I was a bit surprised that she seemed to have very little prior expectation about whether any particular group would meet the target we were seeking, so we will make sure to play this again soon.

Home construction and guessing game (5th and 6th grade programming)

Who: Baan Pathomtham 5th and 6th grade classes
Where: at school
When: Monday morning

---

We had our first class of the near year today! This term, we have an hour for each grade; initial impressions are that this will work well.

Fifth Grade

We started the new class slightly differently than last year (notes here). Our agenda for the day:

1. talk briefly about what we will be doing in the class
2. get everyone set up with a computer and account
3. start playing with code based on this worksheet.

What will we be doing?
None of the students seemed to know what we would be doing, so I got to set the agenda:

• telling computers what to do
• playing and making games
• experimenting and exploring
• learning about programming, geometry and algebraic thinking
• creating art

I'm sure there are other things, but this is enough to anticipate now.

Setting up
Pretty mundane, but a reminder for anyone who wants to follow along at home:

1. open a browser
2. go to pencilcode.net, 
3. click the New Account link in the upper right part of the page, enter your username, make up a password, and you are running. 

We experimented with dragging in blocks of code and typing directly. You play (run) your code by pushing the blue triangle button in the middle of the screen. during the class, I showed a couple of examples of typing commands into the test panel, but will flag that more explicitly next time.

Don't forget to save your work! Just hit the handy save button on the top of the page.

Our play
The essential structure of our class is that we give everyone selected code or concepts to explore, usually with some mini-project as an organizing objective. As the kids play, we will flag ideas that someone has discovered or highlight difficult points. During this class, three of these came up:

• computers are dumb. Sure, they can do calculations precisely and quickly, but they only do exactly what you tell them. If you try to tell pencilcode pen pank to set the pen color, the machine won't know what to do, even if your friend could read that and understand you meant pink.
• Order matters. getting dressed and then taking a bath is not the same as taking a bath and getting dressed. Similarly, these two snippets of code don't do the same thing, even though they each have the same lines of code:

First Version	Second Version
fd 100	rt 90
rt 90	fd 100
fd 50	fd 50

• Some angle chat: How many degrees in a circle? What does it mean to turn right (clockwise) 90 degrees? What about left (counterclockwise) 180 degrees?

5th Grade Homework

The homework is to do the third and fourth exercises on the worksheet:

1. make a sailboat
2. draw and code your own picture

Sixth Grade

All four students are with us once again, so we are continuing their studies from last term. We will be reinforcing concepts such as if-statements, for loops, and functions (love, love, love functions!) Of course, there will be a lot of mathematics in the work as a natural part of their programs. The other area we will discuss explicitly is the process of writing developing a program. I'm not dogmatic about a set recipe, so mostly we practice different strategies, tools, and concepts.

Guessing Game

One initial focus will be building and extending a simple guessing game. I gave them this sample output:

Guess my number.
You get 5 guesses.
⇒ 25
Too big!
4 left.
⇒ 10
Too big!
3 left.
⇒ 5
Too small!
2 left.
⇒ 7
Too small!
Last guess!
⇒ 8
Too small!
Game over.
It was 9.

Their challenge: figure out the rules for this game and create a program to implement it.
This is a simple game, but a great review as it necessarily uses if statements and loops. If you want, you can even throw in a function.

Where will we take the guessing game?
We will also follow the ideas that the students have, but here are some of my plans:

• helping game 1: as long as the user makes sensible choices based on the feedback (Too small!, Too big!) then you always let the user win on their last choice. You can choose whether it is possible for the user to win earlier.
• cheating game: you always lose. The computer keeps track of your guesses and then tells you that the real answer was something you didn't choose. Of course, this would be easy for them to do with pencil, paper, and a friend, but how do you code it?
• Visual game: create a way to show, visually, which numbers have been guessed. Some easy choices are list, number line, grid, but I'm sure there are other fancy things we can do.
• Helping game 2: make the hints more detailed. Is the number even or odd, a multiple of 3, a square, etc. This will involve creating a collection of interesting facts about the numbers.
• Pico-Fermi-Bagel/Mastermind: we've written about this type of game in our math class posts. It seems a reasonable project given the general theme of "guessing" games.

6th Grade Homework
Two homework assignments:

1. Everyone is going to choose a project for this term and write a brief description
2. Win is going to review his game from last term and explain to the class how to create a loop that will ask the user to keep guessing. (This opportunity to explain an idea will rotate through the class)

15 piece tangram (more math from trash)

who: J0 and J2
what did I use: a postcard advertising some property for sale

---

Note: when I originally wrote this, J2 and J1 were busy doing something else. By the time I had half composed this post, J2 had noticed what I was doing and started playing with the tangrams again. More evidence that the easiest way to get kids into an activity is just to have it out and available or doing it yourself.

15 piece tangram

In Bangkok, we get a lot of junk mail touting property viewings. One postcard was just the right size and durability to use for a 15 piece tangram set. I would encourage you to make and play with your own 15 piece set (or a 7 piece, if you don't already have one).

In fact, you should do what I didn't do this time: work with your kids to cut apart the original square or have them make their own sets by themselves. Even just following the diagram is a geometry experience as well as an exercise in scaling.  Our card was 15cm on a side, the diagram I used gave dimensions based on a 3 inch side.

I was a little concerned about the dissection of the central square because of the circular curves. I traced out the dissection with a ballpoint pen that made a slight indentation in the card, then traced over that outline with the tip of a very sharp knife. It isn't perfect, but I'm very pleased with the result:

The image adds some hints when we want to reform the basic square. Good or bad?

I was, perhaps, remiss in not providing full credit to the tangram book we are using. here it is, in all its Dover Publications glory, Tangrams 330 Puzzles by Ronald C Read.  Looking over my shoulder, J2 said, "the book actually had 334 puzzles." So, boom, 4 free puzzles!

Platonic Solids Defying Gravity I

Unrelated, J2 made a temporary installation of mathematical art today in the kitchen: Gravity Defying Platonic Solids I. Except for the obvious one, he also took the pictures included here:

Tangrams and math at the market

who: J2 and J1
when: all day (sick kids at home)
what did we use: tangrams

---

A quick-start activity and conversation from our kids' recent home-sick days.

Tangrams

(Note: people like pictures, but I don't like spoilers. I've included some of our tangram pictures at the bottom of this post)

We got a book of tangram puzzles from the grandparents when we were visiting. It was a good catalyst for getting out the nice tangram set that came with our RightStart math kit. While the book gave us some good ideas, the best one was a simple progression we (J2 and I) came up with on our own: make isosceles right triangles with 1, 2, 3, 4, 5, 6, and 7 pieces.

When we did attempt puzzles from the book, we quickly noticed that we never came up with the same solution that the book had. Admittedly, we only did about half a dozen puzzles, but this led to the natural question of how many solutions we could find for our triangle progression. That naturally opens a really interesting discussion about when you should consider two solutions to be the same (rotations, reflections)?

Another path to follow is related to dissections: we had a sense that some solutions are more satisfying than others because they can't be broken into "typical" sub-shapes. Making this idea more precise is difficult, but worth pursuing.

One last path for the triangle progression is to see what solutions are possible simultaneously. There are several ways to specify this, but here is a specific challenge for you:

Let P be a set of positive integers summing to 7. Using the 7 traditional tangram pieces at one time, make isosceles right triangles so that, for each p in P, there is exactly one triangle with p pieces.

Can you find a set P that works?

Math at the market

Someone was nice enough to buy me a bag of passion fruit. For some reason, the price came up: 80 baht for 1 kg (we weighed it, just to confirm). I recalled another market where I had purchased 800 grams for 100 baht. Of course, that leads to instant discussion:

• Which seller has a cheaper price? How do you know?
• How much cheaper is one price than the other? What are sensible ways to compare?
• Why might the prices be different? Different place and time are obvious ones.
• If the two sellers were next to each other in the market at the same time, would people only buy from the cheaper source? Why/why not? What factors complicate this?

Also, if you were paying attention, you will realize that, yes, this is how I thank someone for giving me a gift: lead them along a mathematical conversation!

Some pictures

Avoid this section if you don't want hints about some tangram configurations.

We thought our approach to making a letter "L" shape was better than the one suggested by the book. Both have an annoying triangle tip poking out. Our version otherwise has a common and consistent width on the two legs which the book didn't have.

Simple rectangle. This is an example of something that comes quickly once you figure out the classic 7-piece square.

One of the members of our triangle family and a cousin of the class square. This gives away solutions for 1, 2, 5, and 7 piece triangles, so sorry about that.

The Miller's Puzzle (a multiplication investigation)

who: J1 and J2
what did we use: good old pencil and paper
when: intermittently through the weekend as they recovered from being ill

---

Denise Gaskins (of Let's Play Math, you know, one of my top recommendations for parents) included a game from Dudeney's puzzle book: The Canterbury Puzzles. The game is a good one which we will use in an upcoming class, but I won't steal her thunder on that one, so go subscribe to her newsletter and find out for yourself.

I'm always excited to see a new source and had never heard of Dudeney or the book, so I went to look. My suggestion is to skip the introduction and go right to the puzzles. One of the first for our play is the Miller's puzzle. Nine numbered sacks are arranged as below:

You notice that 7x28 = 196, but 34 x 5 is not 196. Can you swap sacks, keeping the 1-2-3-2-1 arrangement, so that the products on both sides are the same number in the middle?

Bonus: those sacks are heavy, so how can you get a solution swapping as few sacks as possible?

Our initial attempts

First, the kids were intrigued and suspicious: is 7x28 really 196 and is 34x5 really not 196?

196 is, of course, an old friend: the square of 14. J2 was really excited to see it appear in this puzzle and the factorization 7 x 28 was a nice complement to his identification of 14 x 14.

Second, their inherent sense of efficiency kicked in and they wanted to see if they could fix the problem by simply changing the three bags on the right side. In itself, that was a good discussion about:

• What are all of the possibilities?
• How do we know, without multiplying, that 43x5 and 45x3 won't be 196?
• What are 43x5, 35 x 4, 53x4, 45x3, 54x3?
• What parity do we get when multiplying an even and an odd? When checking our answers, we can at least look to see if we got the right parity.

Their third step was to try a couple of other simple products with the numbers 1 to 9. Overall, this part of the play had some good conversation and a lot of multiplication practice.

A way forward

I didn't give solutions, but I did ask some questions to help guide the investigation further:

1. Where could the 1 go?
2. Where could the 5 go?
3. Where could the odd digits go?

Don't just take those questions in turn, think about them, then see if your answers to one question give new insight into the others.

Another Idea

I like to look for calculation short-cuts and encourage you to do the same. That gives rise to another mini-hint: what are the relationships between the following number pairs and how can you use that to simplify your search for a solution:

1&2, 2&4, 3&6, 4&8
1&3, 2&6, 3&9

Also, isn't it sad that this cute sequence isn't a solution: 4 x 54 = 216 = 8 x 27? So close . . .

Finally, once you have found solutions, you have opened a new can of worms: how many sack moves does it take to go from the Miller's original arrangement to your solution? How can you tell that you've found the solution with the least moves?

An unrelated picture

A slightly more related picture

the avocado lives!

who: the avocado tree (our youngest baby?)

---

Well, for all of you who were anxious about the fate of our avocado tree, you can breathe a sigh of relief. The current stats and observations from the 3 Js:

• height (soil to top): 99cm/38 inc 
• Widest leaf 11cm/4.25 inch 
• Longest leaf 26.5cm/10.5in 
• Diameter 51cm Longest branch (excluding leaves) 25cm
• The pit is now black/dark brown/same color as the soil (from J3)

Unfortunately, we are now unable to measure many of our original stats: mass, total length from roots to highest shoot.

Someone, I think J3, even made a little video of the tree:

Sadly, our attempt to grow a sister tree at Grandma's house was unsuccessful, the avocado pit didn't sprout while we were there and the side project had to be abandoned. There was an ambitious little grapefruit seed trying to make progress on its own, so perhaps Grandma or Grandpa will be able to provide an update of that.

All triangles are equilateral: a coordinate geometry exploration

this was my comment on Dan Burfeind's blog. It has been too long since I copied it back here for me to remember all the editing intentions I had, so, I simply offer as-is:

For another great coordinate geometry investigation, you could spring off this numberphile video: All Triangles are Equilateral, which I saw in a post from Mike Lawler. Can the kids use coordinate geometry to investigate where the proof goes wrong? I’d suggest they follow Mike’s son and use a right triangle (even 3-4-5 as in the blog video).

Choosing a right triangle and setting it in the coordinate plane carefully (right angle at the origin, legs along the axes) makes the equations really nice and, I think, forms a useful demonstration of the power of coordinate geometry. If they play with the picture, there are further rewards as they see interesting relationships between triangle side lengths and the lengths of other key line segments.

A further tidbit out of this whole sequence is that you can also have an interesting discussion about what it means for something to be a proof. The numberphile argument looks, smells, and feels like a solid proof, but we know it is wrong. Something subtle was left out, but there are always subtle things left out of real proofs. Furthermore, we start with the feeling that, where two lines intersect is minor, technical and obvious. Then, it is suddenly a lot more difficult when we try to prove it. Then it turns out the original idea was wrong.

A classic progression: “obvious, hard, false.”

Cake is wonderful

who: J1 and J2
where: kitchen
when: afternoon during J3's nap
what material did we use: dish soap, water, a plastic drink bottle, zometool set, and two small blocks of solid CO₂

---

Last weekend, a generous uncle brought two pieces of chocolate cake for the older J's (littlest J can't eat dairy, so has to make due with dark chocolate squares). The cake was nice, but the real delight was in two small pieces of dry ice that we part of the packaging to keep the cake cool. While J2 had his violin lesson, J1 and I looked for good activities to take advantage of this bounty.

We hit on the Crazy Russian Hacker's 8 dry ice activities (video linked below). Of those, we selected the smoke bubbles and the smoke rings since they seemed cool and possible with our resources. These turned out to be really easy and fun activities.

Here are some videos of the smoke rings:

By coincidence, our friend Pongskorn Saipetch was also playing with dry ice recently, you can see his video here. From the size of his piece of dry ice, I have to guess he had a lot of cake!


After the dry ice had melted, we made bubble wands out of zometool and continued to play with the soapy water:

Our video inspiration. While our photos and video didn't turn out this well, the actual experience was at least as much fun as what you see here:

Helping with homework

who: a high-school student in a semi-public venue
what: stuck on a homework question

---

A few weeks ago, J1, J2 and I witnessed a frustrating scene: someone stuck in a corner of a party trying to finish math homework and a parade of adults failing to break through the block. The three of us, briefly, tried to help as well. I'll describe what we observed, and then offer an idea I had when reflecting on the experience.

The Block

The problem was about graphing functions of one variable for a high school algebra class. As far as I could tell, the student felt moderately comfortable with this topic, but hadn't achieved mastery. However, the meta-problem was a perception that the teacher was asking for something very specific, that idiosyncratic definitions were being used, and that the student didn't have the information required to answer the question. To be concrete, the problem/meta-problems were something like:

Problem: Graph the function , show your data set
Meta-problem:

1. my teacher defines "data set" in a particular way
2. A "data set" for graphing a function is the set of characteristics of the function we investigate to understand what the graph looks like (for example, is it symmetric across an axis, does it have a max or a min, etc)
3. I don't have my teacher-supplied "data set" for these functions
4. Thus, I cannot do this work

Strategies that were tried

I basically saw two approaches. The first was for the helper to offer their own interpretation of the questions, including supplying their own definition of "data set." These approaches all failed, I think because the definitions did not match the student's required form of definition and because they were presented without confidence. The latter meant that the student expected these weren't the same "data set" as what their teacher would have supplied and the former confirmed that.  I wonder how it would have continued if the helper had said: "Oh yeah, I remember this. Let's talk about it and you can help me remember what the data sets are for these other functions."

The second approach was to focus on completing as much of the work as possible. For example, "so just graph the functions and do that part, even if you don't know how to include the data sets." Honestly, I don't know why this advice was rejected, but I've seen it before. There seems to be a common aversion to doing an intentionally incomplete job. I guess the message that the incomplete work sends is "I couldn't do everything," while the student would rather send the message "I chose not do bother with this assignment." In other words, non-compliance is better than inability.

Strategies to try in the future

As I said, I've encountered this myself in the past and expect to see it again in the future. Here are my ideas for strategies to try:

1. Say, "I can't help you with this question, but let's play a game related to this part of math." In this case, the game could have been asking each other questions about the functions, say yes/no, with the objective of getting the other person to answer incorrectly or say they don't know.
2. Bluff that I know what the teacher wants but need their help remembering. Along the way, work through the actual concepts together.
3. As a last resort, tell the kid to leave the work and come do something else together. Even a 5 minute break could be enough of an emotional release. Also, my objective sense is that no single piece of work is worth the unproductive battle.

In the first two, at least the student will get to have a conversation about the actual content of the homework and may even learn more than doing the assignment. There will will probably be some lingering confusion about definitions, so those will need to be addressed later.

What do you think?

I would love to hear ideas from other parents. What do you do in these cases?
Also, it would be great to hear from teachers, too. What do you think parents should do when this happens?