Wednesday, July 29, 2015

ABCs of logic puzzles

who: J1
when: while on holiday from school

After a long gap, I had a chance to look at Tanya Khovanova's math blog again recently. She has a nice mix of questions/puzzles, some of which are beyond our kids right now while others are perfect. Yesterday, we talked about a pair of problems involving a trio of puzzling characters: Alice, Bob, and Carl.

My hidden number
In the first puzzle, Carl has a secret number and gives out some clues. This puzzle shares characteristics with the (recently) famous Cheryl's Birthday puzzle. In particular:

  1. There is some common information
  2. There is some private information that the characters in the story have, but we don't have
  3. The characters make comments about whether someone else can solve the puzzle
  4. Being told something you already seem to have known (e.g., "You don't know the answer") actually gives the character enough additional information

I like Tanya's puzzle more than CBP because it is more self-contained and also invites us to a bunch of (elementary) number theory observations in addition to working through the logic.

Here are some highlights of the discussion:

  • Realizing that there are some numbers where it is sufficient to see either the 10s or the 1s digit to reconstruct the whole number (given multiple of 7, less than 100, etc)
  • Realizing that there are some numbers where one person could know the answer, but the other doesn't and that it could be either the person with the tens or the ones digit.
  • Thinking about what it meant to Bob when Alice said that he didn't know the number.
  • Identifying related clusters of multiples of 7 (like {14, 84}, {21, 28, 91, 98}) that helps us see some (slightly) more subtle relationships between numbers we don't normally associate

Where's the party
In the prior puzzle, we could trust everything that Alice, Bob, and Carl said as being true. In our second challenge, where's the party, we now confront a problem where there is always something distorted in their comments.

Once again, we felt there were some parallels with some of the scenario's from Smullyan's Alice in Puzzle Land. You have to play with the statements you are given to extract the useful information.

The key issue in our discussion of this puzzle was the process of going back and forth between "true" numbers and numbers spoken by the characters. This led us to talk about functions, like Alice(t) is the number Alice will say when she is talking about true number t and the inverse functions. J1 called the function inverse operator Undo, so Undo(Bob)(Bob(t))=t and Bob(Undo(Bob)(s)) = s.

Suddenly, J1 had so many questions about these new objects, Alice(), Bob(), Carl(), and their Undo relatives:

  • When are they the same, i.e., Alice(t) = Bob(t)?
  • Which one is larger, for a given true number t?
  • Do we ever have Alice(t) = Undo(Alice)(t)?
  • etc, etc

This was an invitation to make some pictures, a simple graph of the three functions. Here is J1's and then the one we made together:

The pictures then gave us some new things to notice. For example:

  • Carl only says the largest number for a bounded region of true numbers
  • For any true number, Carl never says the smallest number

A call for puzzle extensions/mash-ups 
J1 asked something I wouldn't have considered on my own: are these the same Alice, Bob, and Carl in the two puzzles? If so, does something interesting happen if we combine the distortions of the second puzzle with the basic set-up from the first puzzle? What if Alice and Bob don't know Carl's constant?

Please go forth and consider this version, as well as create new ones of your own. If you need further inspiration, consider this mash-up Cheryl's sweets, from the fantastic Aperiodical crew.

Monday, July 27, 2015

a physical feeling for densities

who: J1
when: after lunch
where: our stairs

Thanks to Mom, our stair number line recently got a (partial) refresh with some snazzy new number posters. This stimulated a new exploration by J1 to study how dense different groups of numbers are. Well, that wasn't how he phrased it . . .

J1's dialogue as he played on the stairs:
Hmm, what if I walk up the multiples of 2? <walks up skipping stairs, no problem here>
Oh, that's pretty easy, but J3 might find it tricky.

What about squares? 1, 4, what's next ...?
Ooh, 9, that's hard. <finds a cheat by wedging his feet against the sides of the stairs so that he isn't touching the stairs in between>
What's next? 16, oh no! <seeing that it is too far to get by himself>

What about primes?
2 <easy>
3 <very easy>
5, 7 <not to bad>
9? No!
11... daddy, please help! <I help lever him up onto 11>
13, easy!
15, no... 17 <slips as he tries a couple of tricks to get to the 17th step>

Okay, now let's try multiples of 4

Sunday, July 26, 2015

counting at the pool

who: J3
where: at the pool

In the spirit of Christopher Danielson's marshmallow post, Tale of Two Conversations, I wanted to flag some simple ways to add some numbers to a toddler's day. None of these are earth-shattering and they aren't hard to do, but they sometimes take a bit of awareness to remember.

Counting up

At the pool recently with J3, I noticed several times that we chose to use numbers when we might have said something else. First, she started jumping in from the side, so we led with a count: "1, 2, 3 jump!"

That's standard and comes easily to most parents (my observation from watching other parents at playgrounds). A math educator once told me that some kids start school thinking that the counting sequence is "1, 2, 3, go" because they hear that formula so often!

One of our additions is to add a little counting song after each jump. I always think of the sesame street count, so we sing after each jump: "1 [later 2, 3, 4, etc] mighty jump, ha ha ha." She got up to 17 before moving on from the jumping game.

Counting down

Later, she did rocket launches from the ladder. Of course, this was a good chance for counting down practice: 5, 4, 3, 2, 1, 0 blast off!

Oh, look at the time!

Wrapping up all of this, we had a series of comments about the time throughout our swimming session. When we first arrived, I read the time to her, with a couple explanatory comments (it was 10:55, so I explained that the long hand on the 11 meant 5*11 = 55 minutes after the hour and the short hand near that 11 meant that it wasn't quite 11 o'clock, so it would be 10). I didn't expect or require her to get all of this detail.

Then, we talked about how long we could stay: 15 or 30 minutes? 30 minutes was the choice, then when would we leave? As the time passed, how much time did we have left? When we got to the agreed leaving time, how much more time did she want?

I found that the conversation about time, in particular, had to be a conscious effort. Since she doesn't have a strong understanding or awareness of time, it would have been just as functional for me to keep all that information to myself rather than explaining it to her. On the other hand, how can the little ones develop their own understanding if they aren't part of these types of discussions?

Thursday, July 23, 2015

tmbg inspired primes, perfect, deficient and abundant numbers

A quick conversation summary, inspired by They Might Be Giants kids song: Seven

J1: why is the only way to subtract 7's by using up all the cake?
J0: they are really hungry because they are primes. They only have 2 proper factors.
J2: hmm, then what about 6? I guess 6 doesn't like cake?
J1: oh, the other numbers like cake, they just aren't as hungry if they have a lot of factors.
J2: what about 144? 144 must really not need to eat much extra!

Sunday, July 19, 2015

more age chat

who: J1
when: bedtime

J1 wanted to return to our discussion about their three ages. In particular, someone had noticed that, prior to the recent birthday, their whole year ages were all primes: 3, 5, and 7. J1's question this evening: will this ever happen again?

First case

We talked through 2 different cases. The first uses 2 year gaps, so AgeY(J3)+2 = AgeY(J2) and AgeY(J2)+ 2 = AgeY(J1). As we worked through examples, J1 quickly saw that it wouldn't work whenever J3's age was even, so we focused on odds. From 5, 7, 9, he realized that we could skip the cases where J3 was 7 or 9, so we looked at 11 and 17. Along the way, we had identified that J1's age was 9 (3x3), then 15 (3x5), then 21 (3x7).  Seemed to be a pattern of always being a multiple of 3...

Second case

Next, we looked at their current gaps, so AgeY(J3)+2 = AgeY(J2) and AgeY(J2)+ 3 = AgeY(J1)
This one was easier, solved by just looking at the parity of the ages.

So, sad news all around: they will never again have all prime ages!

Fractional ages

Next, we talked about the equation Age(J3) + Age(J2) = Age(J1). This works out now when we use whole year ages (or using the floor function, if you prefer). There had been some chatter earlier about fractional ages, so J1 wanted to check that.  These were the estimates J2 had given earlier for their ages:

  • J3: 3 1/4
  • J2: 5 1/2
  • J1: 8
So, does the equation hold? Unfortunately, it doesn't. J3+J2 is 8 3/4. Will it ever hold in the future and when?

We talked for a while and J1 realized that, as time passes, the sum of the sibling's ages increases twice as fast as his age does. That gave him a clue that it won't happen in the future, but did in the past.  So, when was that?

Keeping in mind the pattern he'd recognized about changes, he guessed it was 3/4 of a year ago. Checking through was a good exercise in fractions and confirmed his conjecture.

From that, he wanted to get more precise about when that date was, 3/4 of a year ago, and then started talking about that having been a really special time for the three of them (unrecognized at the time, of course). When he started getting that precise, though, I offered that J2's estimates were a bit off and substituted J3 = 3 1/6, J2 = 5 5/6, and J1 = 8. This gave an even more satisfying conclusion once he worked out this case.

Thursday, July 16, 2015

Freeform tangrams: an imagination game by J1

Who: J1, J2, and J3 (a late appearance)
When: just before bedtime
What did we use: two sets of tangram pieces

J1 came up with a new game this evening. The rules:

  • number of players: 1 or more
  • playing pieces: 1 or more sets of tangram pieces
  • designer: each round, one player takes the role of the designer, putting the tangram pieces into a configuration. This could be a random arrangement or intentional
  • Taking a turn: in a clockwise order starting with the person to the left of the designer, players say what they think the tangram looks like from their perspective. After everyone has taken a turn, the players shift 90 degrees to their left and repeat the cycle, but this time they are looking at the tangram from a different orientation.
  • Once everyone has had a go from all 4 orientations, the next player becomes the designer
  • Advanced play: once someone has said what they think the tangram looks like from their perspective, the experienced players can add a comment to start building a story based on that object.
  • Winner: everyone!

We played with J1, J2, and J0 taking turns as the designer. To give you a feel for it, here is one of our designs from 3 different orientations:

Here are some of the ideas for what we saw: person jumping out of a box, a wheelbarrow, an army officer, a flower, an 8th note in a mirror, a catapult, a road grader, a knife cutting an apple, the sword in the stone, I was surprised that no one said "poop," which just shows how engaging the two of them found this activity. I doubt it would have been as successful if I introduced these rules and asked them to play!

Our own puzzle book

Wednesday, July 15, 2015

Some comparisons (two tmwyk transcripts and a puzzle)

who: J1
when: just before bedtime

the value of being alive

J1: Daddy, now I've got a question for you
J0: Ok?
J1: if I get a new book every time I write 20 pages in my journal, how valuable is each page?
J1: The books are about 150 baht
J0: How much?
J1: let me check, I think the price is on the back cover . . . 169 baht
J0: if you could tell me what I need to calculate, I'll calculate for you
J1: hmm, so 20 pages is 169 baht, I want to know how much one page is, so I need to divide by 20.
J0: do you think it will be more or less than 10?
J1: less than 10
J0; Are you sure? How do you know?
J1: Well, 10 * 20 is 200 which is more than 169
J0: what about 5 baht per page? Is it more or less than that?
J1: More, 5* 20 is half of 10*20, so 100, which is less than 169.
<we figure out that the amount per page is 8.45 baht/page>
J1: That's not very much!
J0: How much did you get for your birthday?
J1: [x] from grandma, [x] from grandpa
J0: well, how much is that per day. Is it more or less than 10?
J1: More than 10
J0: how <interrupted>
J1: how do I know? well . . .10 * 365 is ...
<some discussion of whether he was right, various other estimates of the amount of money per day>
J0: How does that compare with each page of your journal?
J1: More...but what if I include the [present a] and [present b]?
<he estimates how much different presents cost, figures the total, estimates how much that is per day, etc>
J3 (who has been listening all this time): wow, J1, that's a lot of money!

J3 explores bricks

Earlier in the evening, J3 has been building sticks with 1x1x1 TRIO cubes. She made four, all the same length, then handed two to me as drumsticks. I counted the cubes in one (I got 11) and then she counted one of hers (she got 12). I put them side-by-side and we saw they were the same length.

J3: but...daddy, I really counted 12, you are wrong
J0: are you sure they should have the same number.
J3: yes, let's count them again, together
<I point at the cubes and she counts them, 11>
J3: Ok, now I'm going to build a shape and you see if you can make a copy. It will be tricky!

A birthday puzzle

With their current ages expressed as whole years (you know, the way everyone talks about ages, except for mothers of very small children):

  1. What is a number sentence that relates the ages of J1, J2 and J3? Hint, oldest is 8, middle 5, and youngest 3
  2. Will this ever be true again?
  3. Was it ever true in the past?
  4. When/why not?
  5. What about multiplying? Will it ever be the case that AgeY(J1) = AgeY(J2) * AgeY(J3)?
  6. Was this ever true in the past?
  7. When/why not?
Note that there is a complication since they were not all born on the same day, so the difference in their year ages changes depending on the day of the year we are considering.

J2 wanted to investigate more precisely, so he asked to work things out in months. That meant we had to calculate how many months are between them.

Monday, July 13, 2015

We are totally flipping out!

Who: J1, J2, J3
when: dinner time
What did we use: 4 coins

In Sue VanHattum's linkfest post, there was an intriguing puzzle: given four coins in a square, can you call out commands to flip coins so that you are guaranteed to eventually get them all up or down? More precise instructions/rules are in the link.

We saw very quickly that this is an excellent puzzle:
  • complex terminology? Not at all.
  • difficult to find equipment? Nope.
  • low threshold? Yes, Even J3 was able to investigate along with the older ones
  • high ceiling? Yes. Solving the initial puzzle is hard enough, but there are plenty of possible extensions. 
  • Fun and deceptively tricky challenge? YES!
Everyone got to have fun taking a turn as the coin master (following the flipping instructions and rotating the square) and most of us took a turn as the (attempted) solver. Here are our tools:

For most of the conversation, I just listened. The kids naturally focused onidentifying states: what are the possible conditions for the coins, which of these are, for purposes of the puzzle, identical and which are truly different? One other part of the discussion was about moves, again, which are identical and which are different. Of course, the didn't quite use this terminology.

Some extensions

I suggested they try the puzzle with a smaller number of coins. 1 coin, no problem, it always starts solved. 2 coins were pretty easy again. What about 3 coins? Here, there was a little thinking about what the equivalent version would be for 3 coins. In particular, do the coins have this configuration:

and the coin master can only swap the outer two coins, or are they in an equilateral triangle like this,

and rotations are the only transformation?

Well this is math, so there are no right or wrong answers, you just have to try out your different ideas and see what is the most beautiful.

Another simplification we thought about was to eliminate the rotations.

Probabilistic Attack vs Strategic Game
The problem asks for certainty, but let's assume the rotations are done at random. What if  you are happy with a high probability of getting a solved configuration? Is there a different strategy that has a faster expected ending time? Can you say anything about confidence levels (probability of ending at n or fewer moves)?

In contrast, what if the coin master is playing against you, trying to increase the number of moves you take by selecting tricky rotations. Does that alter how you think about the game? How you play, in practice?

See, I told you the ceiling could be pretty high.

Make it bigger
Okay, you've solved the 4 coin puzzle, what about 5 coins? Are there interesting versions for larger numbers of coins?

A humble suggestion

Whatever you do, I encourage you to try the 3 coin triangular version with rotations.

Friday, July 10, 2015

Predestination stories (reading lessons)

who: J1 (guest appearances by J2 and J3)
when: early afternoon
where: in the dining room
what material did we use: pack of sight words, Usborne book of Fairy Tales, Jeffrey Archer collection of short stories.

We have started to be more consistent about the literacy activities we are doing with the kids.


We have found very quick warm-up activities worked really well in the math classes we teach at school, so we thought we'd do the same for our literacy sessions. We are still in the process of collecting appropriately short and fun activities, but here is a short list of things we've done so far:

  • Sight word sentences: using a pack of sight words, draw 2 and then form a sentence with them. The sillier, the better!
  • Sight word story: take 6 or 8 sight words and form a short story with them. 
  • Crazy Sentences: reading the strange sentences that come out of this program which was inspired by a game from Peggy Kaye (whose site looks like a good source for other qiuck games).
  • Talk about a picture: this is a direct translation of one of our math warm-up activities

Intro to predestination (aka Sleeping Beauty)

For the main event, J1 read (or re-read) the first story in our fairy tale book. As he was reading, I wrote out a couple of questions for us to discuss:

  1. What is the location of the story?
  2. Who are the main characters (2-4)?
  3. What is the conflict in the story?
Of course, these are generic questions we can discuss for almost every story he reads. There were several highlights in our discussion.

What happens with the mean fairy?
In our version, the mean fairy only shows up once, explicitly, in the story, to curse Rose. It is implied in the pictures, that she shows up later to introduce the fated spinning wheel. J1 was strongly drawn to this interpretation based on his own narrative sense of closure and connection. We were a little disappointed that she didn't figure in the ending sequence, but more on that later . . .

Why was Florien successful in rescuing Rose?
Prior to his attempt, several other chaps made an effort and were unsuccessful. J1 said that the reason he was successful seemed to have something to do with Rose drawing Florien's picture earlier in the story and Florien dreaming of Rose. Of course, we don't know how many pictures of princes Rose actually drew, how exact the likeness was to Florien, nor whether the unsuccessful princes had dreamed of Rose or not.

We talked about other stories and came up with these suggestions:

  • Maybe Rose and Florien were partners in a prior life (an idea J1 got from the Thai Ramakien)
  • Maybe Florien did something nice for the mean fairy and she granted him the ability to rescue Rose. J1's favorite version: The mean fairy turned into a squirrel to run through the forest. She accidentally got caught in a hunter's trap. Florein found her and bandaged her wounds.
  • Maybe Florien defeated the mean fairy in battle and won the power to rescue Rose. This version was accompanied by some wild jumping around in a simulated sword (Florien) vs lighting blast (mean fairy) battle.
Predestination vs Free Will
I told J1 about two competing theories: predestination and free will and then we went back to the story to see whether/how each theory was represented. There was a very clear winner, with predestination getting all the points:

  • Rose's story told in advance by the curse/blessings of the fairies.
  • Rose predicts Florien saving her by her drawing
  • Florien predicts saving Rose by his dream
  • The other princes fail to save her "just because" (because they weren't fated to do so)
  • The king acts to prevent the foretold curse by destroying spinning wheels in the land, but the fate is inescapable
The last point flagged up a classic element of predestination stories: even if the characters take action to change their fate, the results still end up the same. Often, the action taken to prevent the fate is somehow critical in causing it to happen.

Appointment with Death
By chance, I had just read this short story in a Jeffrey Archer anthology. I got it out and we read it together, then talked more about predestination vs free will.

Some other tidbits from the chat:
- who is Death? Why do people anthropomorphize death like this?
- why does the story seem to suddenly shift to first person, form Death's perspective? Is the story more or less clear written this way?

Daily Journal

J1 and J2 both have small notebooks for writing a daily journal. They usually write something about what happened that day, but are free to write whatever they want. Sometimes, it becomes a short story or even a never-ending story.
We talk about what they wrote and then make a vocabulary list related to their note. Usually, it is formed from words they mis-spelled, but the vocabulary words could be things they spelled correctly that highlight interesting patterns or a word they didn't use that is related to the topic.

What about the little one?

For J3, we have been working on phonics and letter recognition. Each evening, we have a focus letter or sound that we ask her to find as we read bedtime stories together. Also, we have been singing phonics songs, particularly ones from this collection: Jolly phonics

Thursday, July 9, 2015

A magic trick and magical discussion (part 1)

who: J1 and J2
where: in bed
when: just before going to sleep

Another mystery process trick
I found this post Little Math Magic on JD2718's blog. We did something similar at the beginning of the year with calendars (here) and the kids really liked making a choice, doing some calculations that obscure the choice, then seeing if I can figure out their choice, so I expected that they would enjoy this, too. Computationally, it is a bit more challenging for the kids as it involves squaring 2 digit integers.

14 Squared
talking though 14 squared, J2 asking if we could do 10*10 + 4 *4. We talked about why that doesn't work. In fact, J1 raised the example of 11x 11. Since J2 knew that this is 121, they were able to compare with the other "algorithm" and see that 10x10 + 1x1 isn't right.
J1 said that, instead, we could do 10 * 14 + 4 * 14, which J2 then calculated. When he got to 196, he was delighted, since he did recognize that old friend. Also, he mentioned 169 was another familiar square friend: "13 squared, right?"

From 196, we keep the 6 and then square that, getting 36. We keep the 6 again. Finally, we need to multiply this result by our original number, so 14*6. J2 remembered 4*14 was 56 from an earlier calculation, so he just calculated 14x6 = 14x4 + 14x2= 56 + 28 = 84.

At the time, it felt really good to hear them helping each other work through these calculations, especially their thought process checking the possible algorithm for multiplying 2 digit numbers.

Next time
Well, now that they understand the algorithm, we still have to do it as a trick, where they don't tell me their starting number. After that, let's see if they can figure out how it works?

Wednesday, July 8, 2015

Dice mining (grades 2 and 3 math)

Notes for Grade 1 will come later

This week, we continued our run of dice games. This is a simple game we found in Marilyn Burn's About Teaching Mathematics. She calls it Two Dice Sum Game, but I like Dice Miner better (to pair with Dice Farmer, of course). Here's how to play:

  1. Students make a number line with slots for each integer 2 to 12
  2. Everyone gets 11 blocks or other counters (we used unifix cubes which were nice for stacking).
  3. Players put their counters on the numbers, distributing them in whatever way they want. In particular, they can put more than one counter on a number or none on a number.
  4. Everyone takes turns rolling two dice (normal 6-sided). Players can take one counter off their board for the sum of the two dice. For example, if I have 3 counters on 7 and roll 6+1, then I can take one counter off and now have 2 counters remaining on 7. If I had no counters on 7, then I would not take any action.
  5. The first player to have all counters removed is the winner.

Any activity with construction cubes is bound to create opportunities for other mini-conversations:

  • using the cubes as a ruler to make the number lines
  • comparing how many cubes one person has to another by measuring lengths (before we made sure everyone had exactly 11)
  • using the cubes to form a ruler to measure other things (our whiteboard, a table leg, a friend)
  • making a pattern with light and dark colors from the cubes
As to the main game, there is some basic arithmetic practice through all the addition, but the real interest lies in trying to figure out how best to arrange your counters at the start. All the kids had their own hypotheses, but it was interesting to see these highlights:

  • One student realized that 7 was the single most likely result and put all his cubes on 7 (this did not win that round)
  • Most students started with a uniform distribution, one cube on all numbers
  • There was a surprising amount of enthusiasm for 2 and 12, with many students initially playing multiple cubes on those extremes
  • One student tried to be tricky and put cubes half on 6 and 7, presumably planning to take them off if either number came up.
Toward the end of the class, there were two comments that were really interesting. First, one student suggested rolling the dice many times, recording the results, and seeing how often all the numbers came up. This would then inform his strategy for how to distribute the dice.

We asked the students how many times they would have to roll. Would 1 or 2 rolls be sufficient? No, everyone was sure that wasn't enough. What about 25 times? One student pointed out that there are 11 slots, so 25 times is only an average of 2 per slot, so that didn't feel like enough to get a good sense of the "true" answer. Most students were eager to see for themselves, so this became their homework.


  1. test the distribution by rolling two dice 100 times (or more) and tallying how often each number comes up as the sum of the two dice.
  2. Play Dice Miner 5 times with your friends/parents at home. Record your initial starting position for the cubes, how many rolls to take them all off and who won each round.

Tuesday, July 7, 2015

Yin-yang and lasers (programming class)

Grade 5

Yin-yang programs from homework
As before, everyone did a really great job attacking the yin-yang challenge. As with last year, this turned up some interesting questions about using the fill command.

A creative variant from Chun (this is my slight recreation, she edited her own program during the class):

One of Jung's programs, with the other versions faded and tricky:

Classic from Tatia:

Interestingly, Pitchee used a different method to create the same picture. It is worth taking a look at her program in the editor so you can see the animation difference: Pitchee's YinYang.
In particular, this illustrates a method of building up more complicated pictures using simple elements that you already understand.

Inspired by Jung's name last week, I made my own version with a pair of dancing turtles:

Fill investigation
Based on the questions from the homework, we wanted to do some investigation of different fill variations. These are versions we looked at:

  1. What happens if you have a program without a pen command, but you end a series of movements with fill red? The answer we found was that the fill doesn't do anything.
  2. Does it matter where, within a sequence of moves, you order your command pen path? Yes, it matters very much. Movements prior to that command will only position the turtle for the start of the fill.
  3. What if we use pen blue (or some other color) instead of pen path? This still creates a filled shape, but now we get an outline of our chosen pen color.
  4. What if we make a path that doesn't have a clear "inside" and "outside," what will get filled? We made an "N" shape and were all a bit surprised about what got filled. The key point is that the computer will still follow whatever algorithm it has for determining where to fill, but it may not do what you expect. The solution is to control the output by creating cases where you know what the computer will do.

More for loops
Finally, we worked through these:
- look at range program, also looked at see [5..1]
- replicate dandelion picture:

Finish the flower and replicate this star:
Try to use loops for both!

For next time
We will continue working on loops, probably in next term.
- look at rectangle:
- replicate rainbow:

Grade 6

Laser attributes
Our focus today was on creating a new object with various attributes and then getting objects to interact.

In the first investigation, we simply created a new object and then changed the attributes to see what would happen:
We had to do a deeper investigation of the function rgb() to make sure we understood what colors were getting produced. One of the ideas we discovered was that we could make our laser sprite larger so that it is easier to see the color changes, then change the size back once we are satisfied with the color choice.

As with for loops, the attribute investigation was a reminder of the key difference between code lines that are subsidiary to an earlier line (in the loop, a property of the object) and lines that are independent.

Building up our program, we added another element and a simple interaction. Now there is a wall and the laser shoots whenever we click the mouse, but then disappears when it hits the wall. Here is Titus's version:

The final homework is to add a target for the laser, something exciting when the target gets hit, and any other embellishments you like. For inspiration, here is my version, modified a bit during the class:

Wednesday, July 1, 2015

Thai tangrams and pattern blocks (math games grades 1-3)

Grade 1

Homework 100 board
For homework, both students colored their 100 board to emphasize even numbers. They were excited to talk about the pattern the recognized: stripes on the page. This links back and we used to compare with one of the later patterns in our warm-up.

Warm-up patterns
Following our clapping patterns from last time, I thought it would be helpful and fun to add a visual component. I repeated the pattern cubes activity from last year.

For each pattern, I reveal one cube, then ask them to guess what color comes next and why. After we revealed the whole pattern, then we would clap the pattern, using hand claps for one color and lap claps for the other color of the pattern.

For the alternating pattern (in our case, red-pink-red-pink) we compared with their 100 board pictures of evens and odds.

Have blocks, will use them
After going through the patterns, the kids wanted to play with the cubes. They did several impromptu investigations, partly inspired by the fact that they both ended up with different collections of cubes. Here were some questions we discussed:

  • how many block do they each have? Turned out to be 14 and 21.
  • how many blocks are there all together?
  • are there an even or odd number?
  • which color is more?
  • what are some different ways we could group and count?
Dotty six
The kids played dotty 6 with dice. This was something we discussed two weeks ago with the 2nd and 3rd grade classes.

Play dotty 6 (or dotty 10) with their parents and color the multiples of 3 on the 100 board.

Grades 2 and 3

Tangram investigations
After working on creating some animal and letter outlines, the kids were already pretty familiar with the 7 piece tangram set. This time, we added some more structure to the investigations:

  1. sort the pieces by number of sides
  2. Create new triangles from subsets of their 7 piece sets. These can be hard because the kids have the extra complication of needing to decide which pieces to include and which to exclude:
    • Make a triangle from 2 pieces, 4 pieces, 5 pieces
    • Make non-square rectangles of with different numbers of pieces
    • Make a square from 2 pieces, from 4 pieces, from 5
  3. Order the pieces from smallest to largest. This generated a lot of discussion for the shapes that were difficult to directly compare, particularly the square, the middle size triangle, and the parallelogram
    • Which pieces are the same size?
    • What are your reasons for the ordering? How do you know what is larger?
  4. Area calculations: taking the smallest triangle as a unit, what is the area of all the other shapes? What is the area of all the shapes together?
  5. Create squares using different numbers of pieces: 1 - 7. Are there any numbers which don't have a solution? What is the area of each square? Are there different ways to do it, for example, 3-piece squares that have different areas? 
Thai letters
Josh made 4 Thai letter outlines for ก จ อ ช:

The kids have several associated challenges:
  1. find solutions for Josh's outlines
  2. make their own, improved, versions of Josh's letters.
  3. make their own Thai letter, vowel, tone mark, or number
Looking online, we couldn't find any tangram puzzles involving Thai letters, so this is our gift to the rest of the world!