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 .... 1.5.....no......1 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 bedtimemath.org the next evening when you don't have time or energy to have a more extensive TMWYK conversation.

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?

Homework

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!

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

New/Review

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. 

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):

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).

Pico/Fermi/Bagel

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.

Homework

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

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

New/Review

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.

Homework

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