Saturday, February 28, 2015

23 still isn't prime (sometimes)

who: J1 and J2
where: walking around the neighborhood
when: after lunch on a saturday

In response to our earlier discussion that 23 is prime in the integers but not in the rationals, someone on Google Plus (either Curious Cheetah or Paul Hatzer) mentioned that 23 isn't prime in base 9. That led to another good discussion with the kids.

We went through a bunch of bases larger than 3 and discussed whether 23 was prime in that base. A different way of seeing this is that we were looking for primes in the arithmetic series 2b+3:


They were really excited to feel that they had found a prime-making machine. At this point it seemed that a clear pattern had emerged: all primes except when the base is a multiple of 3. I asked if they had any ideas why and they quickly identified that, when the base is a multiple of three, then 3 will divide 2b+3. We went back to some lower bases for which 23 wouldn't be sensible, but we saw that our prime/non-prime pattern still held:


Finally, of course, we had to see the bad news: this sequence doesn't hit primes on every base that isn't a multiple of 3 and, in fact, our earlier exploration had stopped just short of the first counter example:
11No (boo hoo)

Wednesday, February 25, 2015

Build a chair

who: J2
where: our reception room
when: after swimming lessons

NRICH has a simple activity: use unifix cubes to make a table and chairs. Here's the building material we were supposed to use:

We don't have these cubes at home, so we used TRIO blocks instead. These are 1x 1x n cuboids with a top and a bottom that can be connected. There are also sticks that can connect from any side of  and block to any other side of any other block. these sticks are either 3 or 4 units long.

Here is J1's chair:

After building the set, I asked what he thought of the challenge. He had to solve 2 problems in the construction:
(1) How to deal with the corners where he could only have one block connect to the leg
(2) how to connect the middle of the table so it wouldn't fall through the hole.

His answer was to use a two layer table with corner pieces in the two layers arranged orthogonally. For the middle of the table, he has some sticks along the bottom that support the unattached pieces in the middle (you can see a slight depression in the picture.

Finally, since he knows about the unifix cubes from school, we got to talk about how this construction would have been different using that building set. As always, this compare and contrast discussion was really good for getting him to think about the characteristics of the materials he was using..

Tuesday, February 24, 2015

We noticed . . . (math class 11)

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

Warm-up patterns Today, we started skip counting by 4 (first grade) and 6 (second grade). We are extending the damult dice game today and the kids benefit from as much practice as they can get.

What did we notice (dominoes)?

We split the class into pairs (the extra student went with P) with the following instructions: talk with your partner about what you noticed or discovered about the dominoes puzzle, write down one sentence and be prepared to share with the rest of the class. I assigned the partners and intentionally broke up traditional pairs. I made sure to have some boy-girl mixing and also paired students with different levels of historical engagement. Many kids had questions about what types of ideas they were meant to be discussing, so we gave them the following prompts:
  • What were your strategies for solving the puzzle?
  • What did you find for the smallest side sum?
  • What did you find for the largest side sum?
  • How many solutions did you find?
  • How did this compare with the triangle puzzle? What was the same, what was different?
After 5 minutes (longer on request by the first graders who were still writing), we regrouped to discuss.
With this structure, we had a really good conversation. There were a lot of ideas shared and some good debate about each of the points. Some highlights:
  1. Some students claimed to solve the puzzle by randomly ordering the paper dominoes we made last week. They were surprised when I praised this approach as they were expecting to be criticized. We discussed that the whole strategy was actually: make manipulatives, create a possible solution, check whether the solution is valid, reorganize into a different configuration. P emphasized that this is widely valid, if they are working on something and get stuck, one thing to try is to use a tool to help them, not to suffer in confusion.
  2. For each answer we discussed, there were usually multiple ideas. This gave kids an opportunity to talk about what they found and how they found it.
  3. Comparing with the triangle puzzles was a very rich vein for conversation. The contrast helped them identify structure in the puzzle that, otherwise, would have been either too subtle or too obvious to mention. For examples, the dominoes force the numbers to sit in pairs, the square arrangement has 4 sides vs 3 for the triangle, the domino puzzle had 8 numbers vs 9 (or 6) for the triangle puzzles, and there were some repeated numbers in the domino puzzle while the triangle puzzle had distinct digits.

What did we notice (damult dice)?

We repeated the discussion process again, this time based on playing damult dice. Again, this was a really good discussion, covering a range of ideas:

  • the smallest result possible (2= (1+1) x 2)
  • the largest result (72 = (6+6) x 6)
  • strategies to make the largest result with any given roll
  • strategies to make the smallest result with a given roll
  • why do we get so many multiples of 3? (more on this below)

"New" Game: exact damult dice

We turn again to Math4Love's page on Damult Dice. In the comments, there were many suggestions for extensions to the basic game. The one we introduced today was simple: you have to hit the target number of points exactly and you are allowed to subtract the result of your dice roll instead of adding it. The basic calculation remains the same: roll 3 dice, add two, then multiple the sum by the third.

This is a very simple extension, but it makes the game much richer. Gone is the simple strategy of accumulating as many points as you can on each roll, replaced by something a lot more subtle. I'm looking forward to hearing what the kids find.

Food for thought: Multiples of 3

Playing damult dice last week with my own son, we noticed that maximizing the result often gives us a multiple of three. The challenge: why is this the case and how often do multiples of three actually occur?

Monday, February 23, 2015

Divisible by 3

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

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:


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.

Send me a message and projects 3 (programming class 17)

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

Reminder, this is our standard lesson plan for the rest of the term:
  1. New or review concepts
  2. Exercises related to the projects
  3. Project work


Today, we had three new concepts: while loops, objects, and message passing. This was clearly too much for everyone to understand everything, so I focused on the while loops. My reason for introducing the other ideas and short programs was to give Titus some tools for his project.
These exercises and discussions were based on these two programs from the pencilcode guide:
Talking about while gave a natural opportunity to talk about for again. We had two segments to the conversation, first comparing while vs for, then deciding which is best for different scenarios.
  • Do you know how many iterations you want, before you start to loop? If so, use a for loop, if not, while.
  • Eating: while hungry, eat. If you use a for loop, you may either still be hungry or explode!
  • brushing teeth: while mouth feels dirty, brush your teeth.
  • Putting on socks: for [1..2] put on a sock
  • Cleaning vegetables for stir fry: while veggie bowl has food, clean and chop veggies
  • Adding eggs to a cake: for [1..numEggs], crack an egg and add it to your wet ingredients
  • etc 
This was a really good conversation as each kid had a chance to think about the funny outcome if you used a for when while was more sensible.

Shared Memory
We didn't talk extensively about the concept of an object. Like while/for, objects would pair naturally with arrays, but we also haven't talked a lot about them. For now, the kids should understand that the object is something like a chest of drawers with the drawers given their own names.

As our exercise, we extended the pencilcode guide program to use a button and add an extra value to create this program where the turtle's angle of rotation is entered through the input boxes and the distance moved each step is increased by the button:

Message Passing
Like shared memory,we didn't spend a lot of time on this new concept. However, inspired by Kan wondering why the program stopped responding to his button clicks, we slightly modified the guide program to use a while loop instead of the for loop. Our version is here.

Project exercises

Next week, I will focus the practice/exercises/review on concepts we've learned that are being used in the projects.

Projects and homework

As a reminder, I have stored copies of their work up to the start of today's class in this folder: Project Directory. The homework this week:
  • Kan: finish drawing his maze. Next module is to decide what penalty he wants when the turtle runs into the wall. This is the same assignment as last week. Unfortunately, he overwrote his maze drawing program sometime during the week.
  • Boongie. implement the penalty when the turtle runs into the wall, the turtle bounces back to a previous shell in the maze. There are several ways to do this and I'm curious to see his approach.
  • Win: create functions to ask subtraction, multiplication, and division questions. Another area to think about are gradations in how hard the questions are.
  • Titus: Use the shared memory and message passing code ideas we learned about today to check to see if the player chooses a match.

Friday, February 20, 2015

Chinese New Year math game

Who: J1 and J2
Where: dining room floor
When: after lunch

I've been asked to write about a new game designed by two of the J's.

Number of players: 2
Material required: stacks of banknotes with different denominations. We play with Thai notes, which come in 20, 50, 100, 500, and 1000 flavours.
Game play:
  • one player closes her eyes
  • the other player swaps several stacks of notes. when that player is ready, he slaps the ground to signal the non-looking player.
  • the non-looking player slaps a stack of banknotes. they get points equal to the denomination of the notes in the pile they slapped.
  • These points are added to a running total
Winning condition: unclear

Is it fun?
I think it depends on two factors: (a) did you invent the game and (b) how much do you like mixing, sorting, and counting stacks of money? If you answered "yes" to (a) or "very much" to (b), then you are in business.

Other ideas for Chinese New Year

Frankly, I was underwhelmed when I did a google search for math activities around Chinese New Year. Feel free to use these prompts to come up with your own:
  • 12 animals in the zodiac that serve as mascots for each year
  • properties of the lunar calendar
  • differences between the lunar and Gregorian calendars
  • puzzles involving envelopes with different amounts of money in them (or maybe empty!)
  • game theory analysis of how much to put in your envelopes for relatives, including whether to mark your (giver's) name on the envelope.
If you come up with anything you enjoyed, add it to the comments and we'll have something good for next year!

Wednesday, February 18, 2015

5 minute sharing

Who: J1
When: after dinner and before brushing teeth
Where: bedroom

I've talked before about Peter Liljedahl's Numeracy Tasks as explorations in fair sharing. Tonight, J1 and I briefly discussed the cookie question and the cupcake conundrum.

Sharing Cookies

Six cookies, 3 friends (J1, Ji-Ping and Tanya), sharing is easy peasy, right? Well, one mom insists that her child (Ji-Ping) can eat only one cookie, so what do you do?

J1 responding quick: Ji-Ping gets one, Tanya and I each get 2 and a half
J0: Is that fair?
J1: Its fair because he gets to eat as much as he is allowed and then Tanya and I get the same amount.
J0: How will Ji feel if you get so much more?
J1: Well, his mother probably only wants him to eat one because he is going to get more treats at home, like birthday cake, so that's fair. (By chance, today is Ji's sister's birthday!)
J0: Are there any other ways to split?
J1: We could all have one today and save the others for tomorrow. We could ask Ji-Ping's mom to let him have more.
J0: Maybe you could share with other friends?
J1: yeah, and then in the future, they might share with us. At first, I thought you were going to say that the snack was yo-yo bear <laughs>
J0: What if it was yo-yo-bear, how would that change the situation? Each pack has two strands, but what if Ji-Ping was only allowed to eat one?
J1: <thinking> I guess it wouldn't change it.
J0: So, what about your first idea with the cookies?
J1: Oh, Ji-Ping would get one loop and the treasure card, Tanya and I would each get a treasure card and 2 and a half loops.
J0: how does that feel compared to the cookie split?
J1: it seems pretty fair

Sharing cupcakes

Again, we've got three friends and six treats, but this time 4 cupcakes have delicious chocolate frosting and two do not. This time, J1 had a clear sense that the right answer was to cut in equal portions.

J1: well, we each get 2/3rds of a cupcake with no frosting.  And we get two with frosting. Wait, how many had frosting?
J0: 4
J1: I thought it was 6 <laughing>. Hmm, then we get .... and 1/3rd with frosting.
J0: any other ideas about how to split them/
J1: Well, this is fair, we all get the same treats and we don't have any left over so that's got to be best.

A bedtime math confession

Recently, we've been talking about the "fun nightly math" activities on Bedtime Math. Both J1 and J2 enjoy the scenarios and they have fun calculating to answer the questions. Frankly, I'm not in love with the questions as they usually seem a bit artificial to the story and are often just a single arithmetic calculation. However, it is a very handy resource to easily add a couple minutes of number thinking to the end of a day.

Tonight, the 1000-year Rose led to some good diagrams and a fun discussion about very long times (hundreds of years). J1 made a number line to answer the "sky's the limit" question and a labelled array to answer the big kids bonus question.

Hardly a thrilling photo, but some evidence I'm not making all of this up

So, I hereby officially give you permission, nay encouragement, to open the next evening when you don't have time or energy to have a more extensive TMWYK conversation.

Tuesday, February 17, 2015

Dominoes and Damult Dice (math class 10)

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

Warm-up patterns

Today, we started skip counting by 3, 4, and 6. This was because I wanted to play a game involving multiplication and I wasn't sure whether the kids were ready. They are familiar with skip counting and understand the linkage with multiplication, but clearly need more practice to become fully comfortable.

New Game: Damult Dice

The new game comes from Math4Love. Like NRICH, they are a consistent source of good material for our activities. I wanted this to be a more structured game than usual, so we established the following set-up:
  • Class is split into two groups (3 and 4 students)
  • One player in each group rolls three dice
  • that student chooses two dice to add together. The third is for multiplying
  • By skip counting, the team works together to multiply the two numbers.
  • that result is their team score for that round and gets added to their total.
This game worked very well. It was especially encouraging to see them thinking about which combinations would yield the largest result and remembering prior results.

Old homework: The Triangle Puzzle

Homework from last time was to work on the triangle puzzles. I was curious to hear what they found. Nearly everyone found a solution to at least one of the sizes (6 gaps or 9 gaps), so they all got to share something about what they found.
  • For the 3 space triangle, were there any solutions? No one had really thought about this, but during the discussion a couple of second grade students realized that all the numbers had to be the same for the sum of sides to be the same. I let them share their ideas about why.
  • For the 6 space triangle, students found answers with sides summing to 9, 10, 11, and 12. 
  • We talked about why 9 was the smallest (6 has to be linked with numbers at least as big as 1 and 2) and why 12 is the largest (1 has to be linked with numbers at most as large as 5 and 6).
  • For the 9 space triangle, students claimed to find answers with sums 17, 20, 21 and 23. For an extra investigation, they can confirm these and try to find solutions with sums 18, 19, and 22.
For the 9 space triangle, I noticed a relationship between two of the students' answers, so drew these on the board:

Admittedly, I made some adjustments to highlight the relationship. You see that the numbers in corresponding slots add up to 10, right? In a sense, these are complements. If
you look closely, there is one other relationship, but I'll leave that for you to discover.

A dominoes version

I was lucky to find a direct complement to our triangle activity on NRICH: 4 Dom. This time, the challenge is to arrange 4 dominoes into a square with three numbers on each side where the sum on each side is the same. We got the kids to make their own dominoes by cutting out and coloring little strips of paper. That gave them a hands-on tool to explore three questions, all assigned for homework:
  1. Allowing sides to have different sums, what is the smallest sum you can make with these 4 dominoes? How do you know it is the smallest?
  2. Allowing sides to have different sums, what is the largest sum you can make with these 4 dominoes? How do you know it is the largest?
  3. What arrangement makes all the sides have the same sum? What is the sum? How many solutions are there?


We gave them four pieces of homework. This seems like a lot, but the game is very short and most of the kids answered several of the dominoes questions already in class:

  1. Damult dice: play the game with someone in your family, first to 200 wins. For each roll, write down the equation you are calculating, for example (6+1) x 2 = 14
  2. Dominoes puzzle: answer the three questions listed above. In short, what is the smallest sum that a side of the square could have, what is the largest, what sum and arrangement works so that all sides are the same?

Congratulations for getting this far. Here is something pretty for your efforts. Please post in the comments any mathematical ideas this picture gives you!

Monday, February 16, 2015

Functions review and projects 2 (programming class 16)

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

Reminder, this is our standard lesson plan for the rest of the term:
  1. New or review concepts: For loops, functions, if statements. Often, this will be integrated with the exercises.
  2. Exercises related to the projects
  3. Project work


The new concept today was the random function. We played with three snippets of code to understand how this works and, of course, keep practicing for loops:

  1. Generate 4 random integers between 1 and 10 (inclusive)
  2. Modify the previous program to multiple your random numbers by 10
  3. Change the first program to generate numbers in [10, 100]
We had a good discussion about these programs: first identifying the ranges for possible values, then drawing Venn diagrams and talking about intersections and subsets. This opportunity to talk about additional math concepts is one of the extra dividends from the programming class.

Using the random function has many applications for games. In the projects this term, I expect that Kan and Boongie will use it to help create new mazes each time the program is run, Win will use it to generate a new set of math questions, and Titus will use it to shuffle the picture cards.

For our review, I wanted to reinforce the ideas related to functions. In one of our last sessions, we realized that the kids didn't really understand function definitions, calls, and variables. This time, i gave them three code snippets:

  • Define hypotenuseStep1(a, b) = a+b
  • Define hypotenuseStep2(a,b) = a*a + b*b
  • Define hypotenuseFinal(a,b) = sqrt(a*a+b*b)
I wanted to see if they could figure out how to call the function and whether they would do anything with the output. After many interesting attempts, we went through the details again and explained how the components work. In retrospect, I should have given them a function that would generate visual output once called. As it was, they needed to get two things right (call the function correctly and write the output) in order to get positive feedback.

Project Exercises

As noted above, the exercises on random have several potential applications in the projects. The other exercise today was most directly related to Win's project. We combined for loops, random, and write statements to loop through some math statements.

Here are some ideas for future project exercises, particularly using functions:
(1) reset/restart the game
(2) B: draw random gaps in circles for maze
(3) Gan: maybe randomize drawing of maze, depends on how he plans to structure the maze
(4) T: check to see if chosen cards match the next picture symbol

Projects and homework

The students are all making reasonable progress on their projects. I have stored copies of their work up to the start of today's class in this folder: Project Directory. The homework this week:

  • Kan: finish drawing his maze. Next module is to decide what penalty he wants when the turtle runs into the wall.
  • Boongie. implement the penalty when the turtle runs into the wall, the turtle bounces back to a previous shell in the maze. There are several ways to do this and I'm curious to see his approach.
  • Win: create a looping function so that they get a second chance for an incorrect answer. Next module is to think about gradations in how hard the questions are.
  • Titus: arrange the memory cards into a grid, randomize each time the game is played, if possible. Next module is to think about how to check to see if the player chooses a match. 

Sunday, February 15, 2015

2048 vs 2584

who: J1 and J2
Where: online
When: before and during violin lessons

I don't have many games on my phone, but 2048 is there. It nicely fits J2's love of powers of 2, but J1 also really enjoys it. I will admit that I play a lot more than I think I should.

Assuming that you know the game, what do you make of this board:

Hmm, 2 and 8 are familiar, but 1, 3, 13, and 610?!

Searching recently for something related to the Fibonacci sequence, I found 2584, the Fibonacci sequence version of 2048. Thinking about it briefly, you will see why the sequence fits so nicely into this game structure. Of course, this was bound to be a favourite for the younger J's, too.

A bit of compare and contrast

J1 and J2 played back-to-back games, one round each, swapping in between. Then we talked about how the games compared. Because this was interspersed with other activities, J1 and I talked without J2 and then later got J2's opinions, but J1 waited to hear his thoughts before interjecting. Here were some snippets:

  • J0: Which one is harder? 
  • J1/J2: Fibonacci is harder. 
  • J0; what does that mean, "harder"? Is it harder to play each step or harder to keep going in the game?
  • J1/J2: Harder to play each step. For powers of 2, you just match up the number. For Fibonacci, you have to think about which numbers can combine together. 
  • J0: which one do you think is harder to keep going?
  • J1: the Fibonacci one is easier because each number can combine with two others. Like 3 can combine with 2 or 5, 5 can combine with 3 or 8, 8 can combine with 5 or 13, 13 can combine with 8 or 21.
  • ---------------
  • J0: are the games similar in anyway?
  • J2: yes, both are on a 4 x 4 grid
  • J1: yes, both have sliding number tiles that get added together
  • ---------------
  • J1: why do you win when you get 2584?
  • J0: is it the closest Fibonacci number to 2048?
  • J2: no, 1597 is closer
  • J0: Let's see, what power of 2 is 2048?
  • J2: 14
  • J0: is that correct?
  • J2: .... 11
  • J0: is 2584 the 11th Fibonacci number?
  • J2: let's count them
  • <together>: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584. That's 18 (or maybe we should call it 17 for purposes of the game?)
  • J2: Oh, we knew it had to be more than 12 because 144 is the 12th (which is their favourite Fibonacci number right now because it is also 12 squared)

Load bearing tangents

who: J1, J2, and J3
when: at dinner
where: in the dining room

So, can you get any useful math out of Peppa Pig? I was indulging the munchkins in an episode when we came to this scene (at 3m12s):

Obviously this is just meant to be silly, but the pseudo-mathematical nonsense irked me. An alternative like "I derive solutions to equations" would have sounded nearly as complex to the target audience, would have been a (nearly) sensible job, and would have fit the quadratic formula on his whiteboard.

Anyway, one of the little ones asked: "what's a load bearing tangent?" I told them the whole thing seemed silly to me, but I understand "tangent" and "load bearing." This lead us into drawing a bunch of pictures:
- a circle with a tangent line kissing a single point
- a smooth curve with a tangent line that intersects the curve at another point
- a triangle, on which we tried to find the tangent to a non-vertex point and then a vertex point.
- two circles tangent to each other (externally)

Next we talked about "load bearing" as "carrying a weight." They understood that pretty easily because they'd spent part of the evening earlier hanging off my arms.

To round it all out, they proceeded to spend the rest of the time before bed running around, pointing at things and people, and shouting out, "there's a load bearing tangent!"

Oh, and if you think I'm only a curmudgeon when it comes to math, I'll admit this grammar nonsense also grates on my ears (at 2m25s):

Monday, February 9, 2015

Puzzling triangles and Pico/Fermi/Bagel (math games class 9)

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

Warm-up patterns

Some students requested we talk about the Fibonacci sequence, so I decided to start with some number patterns. If I were going to do these again, I would stop after the second and third terms and ask for a range of different ideas about how the patterns might continue.  Once we have 4 or 5 terms, though, it becomes pretty difficult to identify a non-obvious pattern that fits the available data.

The point we want to communicate is that there are many possible answers and the only condition is that they should fit the existing data.

Patterns we used today:
  • 0, 1, 2, 3, 4, 5.... universally extrapolated as counting up by 1
  • 3, 10, 17, 24, 31, .... best fit was counting up by 7 starting with 3
  • 18, 14, 10, 6, 2, -2, -6, ... Subtracting 4, starting with 18
  • 1, 2, 4, 8, 16, .... powers of 2 or doubling the prior element of the sequence
  • 1, 1, 2, 3, 5, 8, 13, .... Fibonacci sequence
In the first grade class, we had an interesting conversation about possible continuations from 18, 14.
One idea was 18, 14, 12 which we explored a bit more and came up with the following:

  • 18, 14, 12, 12, 14, 18, 24 .... the difference of the differences is +2
  • 18, 14, 12, 12, 12, 12, 12, 12.... hits a floor and stays at 12
  • 18, 14, 12, 11, 10.5, 10.25, .... the differences are getting halved at each step

Triangle Puzzles

Mathematics Mastery sent a Christmas card that inspired this exploration. As a basic set-up, you have a triangle with circles on the vertices and some number of circles on the sides. You have to place distinct digits in the circles so that the sum of the numbers on a side is the same for each side. You can see the pictures for triangles with 0 extra circles (only the circles on the 3 vertices), 1 extra on each side and, if  you look closely, 2 extra on each side:

The kids had a lot of fun with this puzzle. Here are some notes:
  • 0 extra circles/3 total circles: Is it possible with three distinct numbers? If not, why not? Can you convince your friends you are right?
  • 1 extra circle on each side/6 total circles: fill in with the integers 1 to 6. How many solutions can you find? How do you know you have found them all? Given a collection of 6 distinct integers, can you always find an arrangement that works? If not, are there any conditions that must be satisfied? Are there any conditions which are sufficient?
  • 2 extra circles on each side/9 total circles: similar questions to the 1 extra circle
  • non-equilateral triangles: with n-circles, when can you fit the numbers 1 to n into the triangle to meet the condition of equal side sums?
The non-equilateral version was suggested by Prim. She challenged me to work on putting 1 to 7 into a triangle with 7 circles: 3 on the vertices, 2 extra on one side, and one extra on two sides.

If you get tired of triangles, of course, you can explore square figures, polygons with more sides, and stars (put circles on the points and internal intersections).


This game was also a student request. I've talked about this game before (here), so won't bother to repeat the rules. Today, we did two rounds with 3 digit numbers, then let the kids play in pairs for a couple minutes. For playing in class, we found this a good game because:

  1. It was fun!
  2. Everyone could guess a number and stay involved in the game
  3. Students got a chance to explain their thinking and try to reason logically through the evidence
We also experimented with having one teacher score guesses for different target numbers for individual students. This was workable, but it significantly reduces our ability to have a deeper conversation with the students about what they have learned from each clue and why they are choosing a particular guess.


The kids are assigned more exploration for the triangle puzzles:
  • Find multiple solutions to the 6 space triangle
  • Find a solution to the 9 space triangle
  • Look for solutions to the 3 space triangle and talk about what you find

Sunday, February 8, 2015

Magic Cards (programming 15, projects 1)

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

Today, we began to focus on the projects for this term. These are the kids project ideas:
  • Gun and Boongie are both making mazes
  • Win is making a math worksheet game to quiz the younger students in arithmetic
  • Titus is making a version of the memory match game
We plan to follow this agenda for the rest of the term:
  1. New or review concepts
  2. Exercises related to the projects
  3. Project work


Today, we integrated this activity with the project exercises. For the future sessions, the three concepts we want to keep discussing are:
  • For loops: the kids are close to mastering these
  • Functions: still some confusion about the essential features, function calls and arguments
  • If statements: introduced this week, but will take more discussion.

Project Exercises

These exercises have several objectives. First, I want to help the students with their projects. Each exercise has at least one idea that can be used directly, or with small modification, in someone's project.

Second, I wanted them to really understand the concepts in the exercises. This group does best when they have a mix of experimentation, discussion, and explanation. Short snippets of code serve very well to catalyze these. It is also very useful to have them work on some material in common. In particular, during the experimentation, it is nearly guaranteed that at least one of the students will do something that nicely illustrates a critical aspect of the idea they are learning.

Third, I wanted to continue to make sure that everyone would have exposure to the interesting ideas in each project.

Here are student examples of the three exercises:
Clicking done by Boongie

If...Else done by Win

Forever and Turn done by Titus

Project Work

Boongie focused his time on drawing his maze. Gun tried to figure out a way to block the turtle from going through the barriers once he has drawn his maze. Win expanded the if-else pattern from the exercises, introducing variables to his write statements and random values. Titus added mini pictures to the outside of his memory game playing area.


For the most part, each student has different homework based on their project.
  • Gun and Boongie: draw the outline of the mazes
  • Win: work on how to respond when the player enters an incorrect answer and think about how to loop for multiple questions.
  • Titus: fill out the pictures on the boundary circle