Showing posts with label class. Show all posts

Monday, March 6, 2017

Cryptarithmetic puzzles follow-up

I was asked to write a bit about strategies and answers for the puzzles we gave two weeks ago.

BIG + PIG = YUM
Because the digits in YUM are all distinct from BIG and PIG and there are only 7 letters in this puzzle, we should expect there to be many solutions.

The easiest way to get a feel for the puzzle is to start trying values and see what develops. This was part of the idea of using this puzzle as the opening challenge.

As we play with examples, the kids should notice these things that constrain our possible solutions:

B, G, I, M, P, U, Y must all be distinct
We are adding two three digit numbers and the sum is a three digit number
B, P, and Y are all leading digits
The largest sum possible with two numbers 0 to 9 is 18.

Some conclusions:

(a) G is not 0. If it was, then M would also be 0.

(b) B, P, and Y are all not 0. They are leading digits, the rules of our puzzles say they can't be zero.

(c) G + G is at most 18. It may contribute at most one ten to the calculation of U. That will only happen if G is 5 or larger.

(d) I + I is at most 18. Along with a potential ten from G+G, that means we have at most 19 coming from the tens. That will only happen if I is 5 or larger.

(e) B+G is at most 9. If there is an extra hundred coming from the tens digits, B+ G is at most 8.

(f) If I is 9, G must be less than 5. Can you see why?

(g) If G is less than 5, I cannot be 0

After these observations, I'd suggest picking values of G, then seeing what values of I are allowed, then checking what remains for B and P. Because we aren't allowed to have duplicates, we quickly see that our choices are constrained.

For example, if G is 1 or 2, then I is at least 3 and we get the following possible solutions (B and P can be interchanged):

431 + 531 = 962

341 + 641 = 982

351 + 451 = 802

371 + 571 = 942

381 + 581 = 962

132 + 732 = 864

132 + 832 = 964

152 + 652 = 804

152 + 752 = 904

182 + 582 = 764

192 + 392 = 584

192 + 592 = 784

There are some more advanced ideas that could come out of trying to count or list all of the solutions, so I'd encourage people to explore. Even this simple puzzle can be a lot of fun!

CAT + HAT = BAD

The A in BAD is the key part of this puzzle. We can get two cases:

(a) A is 0 and T is 1, 2, 3 or 4

(b) A is 9 and T is 5, 6, 7 or 8.

Again, while there are a lot of solutions (and counting them would be a fun challenge) they are easiest to build up by choosing A (either 0 or 9), then T, then seeing what flexibility is left for C and H. Here are some examples:

301 + 401 = 702

301 + 501 = 802

301 + 601 = 902

302 + 502 = 804

302 + 602 = 904

103 + 403 = 506

395 + 495 = 890

SAD + MAD + DAD = SORRY

This was a puzzle without a solution. In this case, it isn't too hard to see that SORRY has too many digits. The best explanation was given by one student:

The largest three digit number is 999.
If we add three of them, we will at most get 2997.
SORRY has to be bigger than 10,000.
This isn't possible

CURRY + RICE = LUNCH

Unfortunately, this also doesn't have a solution, but the reasoning is more subtle than the previous puzzle.

Here, we can reason as follows:

R cannot be 0 because it is the leading digit in RICE
Because the tens digit of RICE and LUNCH are both C, R must be 9 and we must have Y + E > 10.
This also means R + C + 1 = 10 + C.
That will mean the 100s digit of RICE must be the same as the 100s digit of the sum.
However, the 100s digit of RICE and LUNCH are different.

Too bad, it was such a cute puzzle!

ALAS + LASS + NO + MORE = CASH

This is the most challenging puzzle from this set.

Some things we notice:

There are ten letters (A C E H L M N O R S) and they must all be distinct.
We are adding three 4-digit numbers and a two digit number to produce another 4 digit number.
A, L, N, M and C are leading digits, so they can't be zeros.
The tens and hundreds digits of CASH (S and A) are also involved in the sums for those digits.

Point 4 has a subtle implication, which I'll illustrate with the hundreds digits. Since L + O must be more than 0, but A is the hundreds digit of the sum, we must have some number of thousands carried over. Because A, L and M are all distinct and larger than 0, the smallest their sum can be is 1+2+3. Putting these two observations together, C must be at least 7.

In this case, I find it helpful to put together a table showing possibilities that we have eliminated:

We can see some more restrictions from the fact that A + L + M must be less than 9. That means we have only the following possible triplets (ignoring order):

{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}

One thing we notice is that 1 is in all of these triplets, so either A, L or M must be 1 and none of the other letters can be 1. Another thing we notice is that we don't yet have any way of differentiating A, L, or M, so any ordering of our triplets is possible. That would mean we have 24 cases to consider.

Let's see how we would work through the cases, starting with A = 1, L = 2, M = 3, the first on our list. Now this, happens to be a stroke of luck, as we'll see.

Starting from the thousands digit, we see that this would make C = 7, if there is a single carry from the hundreds. Indeed, we can see that this must be the value (in the case we are testing), as the carry from there could only come from L + O (plus any carry from the tens digit). Since L is at most 5, L + O is at most 14 and any carry from the tens digit must be less than 6.

Now, in the hundreds digit, we have 2 + O + carry from the tens = 10, so O = 8 - carry from tens.
We know there must be at least one carry from the tens, so O is at most 7. Since 7 is already used by C, let's try 6. That means we need to get 2 hundreds carried over from the tens, so we need
A + N + R + carry from ones = 20, or N + R + carry from ones = 19. Since we have already used 6 and 7, the only way this is possible is if N and R are 8 and 9 (in either order) and we are carrying 2 from the ones.

At this point, the case we've worked through has:
121S + 21SS + 86 + 369E = 71SH

We still have to allocate digits 0, 4, and 5. and we know that S + S + 6 + E = 20 + H. Given our remaining digits, the biggest the left hand can be is if S is 5 and E is 4, making 20. The smallest the right hand can be is if H is 0. Fortunately, this makes the equality hold, so we get our final answer:

1255 + 2155 + 86 + 3694 = 7150

Through the process of checking this case, we learned more about how the carry from lower digits is restricted and it would be faster for us to check through remaining cases.
Let me know how many other solutions you find!

LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL +

LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL + LOL = ROFL

There are 71 LOLs, so this is 71 x LOL = ROFL. While this looks daunting, there are some ideas which take us a long way to the solution.

First, ROFL has 4 digits. If L were 2, 71 x LOL would be more than 14,000, so L must be 1. In fact, ROFL is less than 9861, so LOL is smaller than 9871 / 71 which is 139. We can quickly check

101, 121, and 131 and see that 131 works.

71 x 131 = 9301

Tuesday, February 21, 2017

Cryptarithmetic Puzzles for Grades 1 to 4

Inspired by a series of puzzles from Manan Shah, I decided to have the kids play with cryptarithmetic puzzles today. In addition to borrowing some of Manan's puzzles, I also used some from this puzzle page: Brain Fun. I've included some more comments below about the Brain Fun puzzles.

My main concern was whether the puzzles were at the right level. In particular, I was afraid that the puzzles would be too hard. In fact, I tried solving a bunch of them yesterday and actually found myself struggling. I'll ascribe some of that to being tired and sick. However, my intuition was to make some simpler puzzles of my own. In particular, I added:

puzzles that have many solutions: I figured that many solutions would make it easy to find at least one.
a puzzle that "obviously" has no solution. Now, obviously, the word "obviously" is a sneaky one in math, but I was pretty sure the kids could see the problem with this structure.

Grades 1 and 2

For the younger kids, I started with a shape substitution puzzle. This is one our family explored almost 2 years ago: Shape Substitution. I don't recall the original source.

Two reasons why I started with this. First, it has a lot of solutions, but there is an important insight that unlocks those solutions. Second, by using shapes, we can write possible number solutions inside them as we solve or guess-and-check the puzzle. This made it easier for the kids to see the connection that all squares have the same value, etc.

The second puzzle: BIG + PIG = YUM

Really just a warm-up practicing the rules and doing a little bit of checking that we haven't duplicated any numbers.

The third puzzle: CAT + HAT = BAD

Again, lots of solutions, but noticing leads to a good insight.

Fourth puzzle: SAD + MAD + DAD = SORRY

This is a trick puzzle. The kids know that I like to tease them, so they are aware they need to look out for things like this. We discussed this in class and I suggested they give this puzzle to their parents.

Fifth puzzle: CURRY + RICE = LUNCH

When I translated this to Thai, all the kids laughed. I was sneaking a little bit of English practice into the lesson and then they realized that it was worth trying to read all the puzzles, not just solve them.

Sources: I think I made up all of these puzzles (original authors, please correct me if I'm wrong).

Grades 3 and 4

The older kids already had experience with these puzzles. We did refresh their memory a bit with BIG + PIG = YUM

I asked them to give me the rules and explain why those rules made sense. As with most games, I want to communicate that we're doing things for a reason, but those reasons can be challenged. If they think it makes sense to do it a particular way, we're open to their ideas.

Second puzzle: SAD + MAD + DAD = SORRY

Same discussion as for the younger kids. When prompted, this was pretty easy for them to spot, but they weren't naturally attuned to think about whether a puzzle had solutions or how many. This led me to take a vote on all the puzzles at the end to see who thought the puzzles would have 0, 1 or many solutions.

Third puzzle: ALAS + LASS + NO + MORE = CASH

A puzzle from Brain Fun. I think this is one of the easier ones on that page. Again, a bit of English practice.

Fourth puzzle: LOL + LOL + LOL + .... + LOL = ROFL (71 LOLs)

This was from Manan. I think it is one of the easier ones in his collection, but it looks daunting. Turns out none of the kids in the class were familiar with (English) texting short-hand, so my attempt to be cool fell flat.

Fifth puzzle: CURRY + RICE = LUNCH

Again, everyone was delighted when I translated this one. We're in Thailand, after all, so at least one puzzle had to be about food.

The key exercise

The final assignment everyone (all four grades) was given was to make up a puzzle for me to solve. I was thinking it would be nice to have one in Thai, but we decided to keep it in English as further language practice.

Manan wrote a nice post about having kids design their own puzzles. If it goes well, this is actually the activity that ties a lot of the learning messages together: they think about structure, they think about what allows multiple or single solutions, they apply their own aesthetic judgment, they use their knowledge of the operations, they are empowered with an open-ended task that cannot be "wrong."

We'll see how it goes. At the very least, I expect a lot of work for myself when their puzzles come in!

An extra sweetener

Two kids asked if we could use other operations than addition. That prompted me to put this on the table (also from Brain Fun):

DOS x DOS = CUATRO

Brain Fun Problems

The first time I'd seen the Brain Fun problems, I added them to a list and called them "basic" (see this page.) When I actually went to solve them, though, they didn't seem so easy.

Big confession time: I actually looked at some of the solutions. However, I was disturbed to see that the solutions involved extra information that wasn't included as part of the problem statement! For example, in THREE + THREE + FIVE = ELEVEN, the solution assumes that ELEVEN is divisible by 11. This seems to be the case for several of the puzzles involving written out arithmetic:

TWO + TWENTY = TWELVE + TEN (assume 20 divides TWENTY and 12 divides TWELVE, I wasn't clear about whether any divisibility was assumed for TWO and TEN)

I'm not sure if similar assumptions are allowed/required for any of the others.

Maybe I shouldn't complain, since this assumption creates an additional constraint without which there could be further solutions. Perhaps part of the reason it doesn't sit well is aesthetic. In the 3 + 3 + 5 = 11 puzzle, 3 doesn't divide THREE and 5 doesn't divide FIVE.

Lastly, there is a typo in the final puzzle of the Brain Fun page. That puzzle should be
TEN x TEN = FIFTY + FIFTY

Monday, February 13, 2017

NRICH 5 Steps to 50

A quick note about the game we played in first grade today: 5 Steps to 50.

This is an NRICH activity that I've had on my radar for a while. I even made a pencilcode program to explore the activity in reverse. True to their other activities (check them out!!!) 5 steps to 50 requires very little explanation, is accessible to students with limited background, but has depth and richness.

Our lesson outline
I explained the basic activity and did an example at the board. To get my starting value, I had one student roll for the 10s digit and one for the 1s digit. Then we talked through together as we added 10s and 1s.

I then distributed dice and had the kids try 3 rounds. As they worked, I confirmed several rules:

the only operations allowed are +1, -1, +10, -10
we must use exactly five steps (I note that this is ambiguous on the NRICH description, they say "you can then make 5 jumps")
we are allowed to do the operations in any order
we can mix addition and subtraction operations

After everyone had been through 3 rounds, we regrouped to summarize our findings:

Which starting numbers can jump to 50?
Which starting numbers cannot jump to 50?

We helped the kids resolve disagreements and then posed the following:

What is the smallest number that can jump to 50?
What is the largest number that can jump to 50?

For those to challenges, we kept the restriction that the numbers must be possible to generate from 2d6.

Basic level
To engage with the activity, some of the kids just started trying operations without much planning. This quickly reinforced the basic points about addition and place-value and commutativity of addition.

For these kids, it was helpful to ask a couple of prompting questions:

What do you notice? This is a standard that never gets old!
If this path doesn't get to 50, does that mean there is no path to 50?

This second question, particularly, raises the interesting observation that it is easy to show when a number can jump to 50 (just show a path) but to show that no path is possible requires a different type of thinking.

Getting more advanced

The next level of sophistication was really about noticing that the key consideration is the distance to 50. In particular, this identified a symmetry, where n could jump to 50 if 2*50 - n can jump to 50. Of course, the kids didn't phrase this relationship in this way....

The next major step is thinking about a way to systematically write down the paths.

Wednesday, February 1, 2017

Perfect Play for My closest neighbor

Joe Schwartz at Exit10a wrote a fraction comparison post that prompted me to write up more of my experience and thoughts on this game.

Let's find perfect play
This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:

If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?

My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?

How did it go?
In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:

common denominators
common numerators
distance to 1
relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark.

For visualization, drawing on a number line seemed to work best.

I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.

What have I learned?
This game is really effective at distinguishing levels of understanding:
(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.

(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.

(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.

(3) proficient: have at least one consistent strategy they can work through to make a comparison

(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.

What would I do differently?
Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions mean. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.

More locally, for this game in a class of mixed levels, I would

lean toward doing this more as a cooperative puzzle
re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)
I also would consider allowing equivalent fractions to the target as winning plays

ideas for upcoming classes

warm-ups for all

WODB: (1) shapes book (2) wodb.ca

any: Traffic lights/inverse tic tac toe/faces game

good options here, mostly grades 1/2: some games

dots & boxes (maybe with an arithmetic component)

loop-de-loops

Grades 1 and 2

31 Game: could be used by any grade
Integer addition game
jump to 50, chain of changes puzzles

close to 100 game:

Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.

Procedure:

- Deal out 6 cards to each player

- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers. The goal is to get the sum of the two numbers as close to 100 as possible but cannot exceed 100.

3 chips puzzle

Grades 3 and 4

Factor finding game (maybe warm-up?)
Factors and Multiples game
Contig for 3 and 4 (explanation).
Times tic-tac-toe: review for Grades 3 and 4
Fraction war for grade 4 (smallest card is numerator)
Multiplicaton models: worth making for grades 3-4 for solidifying concepts? Associated games
d 2

card on head game

Pico Fermi Bagel

Magic triangle puzzles

damult dice

4 dominoes puzzle

tables and chairs

(1) Dice game perudo

Equipment

- multi-player, 2-5

- Everyone gets the same number of 6 sided dice (full game they get 5, I would start with 3)

- Everyone has a cup to shake and conceal their dice

Basic Play

- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.

- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s.

This bid signifies that the player has 2 (or more dice) showing the value 3.

- the next player has two choices:

call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.
raise, either the number of dice or the value or both get increased

Advanced rules

- Ones are wild, they count as any number toward the target bid

- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round

- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.

remainder jump

we played this game before, but we could give them blank boards and let them create. See the last page here: http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf

https://www.beastacademy.com/store/products/5B/printables/GCF_LCM_Webs.pdf

another good game:

https://saravanderwerf.com/2015/12/13/5x5-most-amazing-just-for-fun-game/

We should still use the pyramid puzzle sometime:

http://mathforlove.com/lesson/pyramid-puzzles/

(1) double digit and double dollar:

We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.

http://mathforlove.com/lesson/double-digit-and-dollar-digit/

(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here are 5 questions).

http://mathforlove.com/lesson/1009569184/

(3) some of these games are promising:

https://danburf.files.wordpress.com/2014/11/tapson-games.pdf

http://mathforlove.com/lesson/pyramid-puzzles/

Tuesday, January 17, 2017

My Closest Neighbor Fraction game

Denise Gaskins recently flagged a post about a good fraction game: My Closest Neighbor. I tried this out in class today.

A pre-test
First, I wasn't sure whether the level of the game would be right for the kids. I was considering it for the 3rd and 4th graders, but had some alternative activities planned in case. To start, I posed the following questions:

Which is closest to one-half: 1/3 or 2/5? The third graders really struggled with this, so I left it alone and went to my plan B games. The fourth graders were all confident on this one.
Which is closest to 3/4: 5/11 or 11/12? This was a challenge for the fourth graders, but I thought it would be ok to play the game.

In our discussion of the second question, we explored two strategies:

making a common denominator
comparing with reference numbers

The common denominator is a bit of a pain, since 11 is prime, though at least we have the fact that 4 is a factor of 12. One student soldiered through this approach, but it was difficult for the other kids to follow.

For the second strategy, we made use of some observations that were more elementary for the kids:

(a) 5/11 < 5/10 = 1/2

(b) 3/4 is halfway between 1/2 and 1

Combining these, it was easy for us to draw a rough number line, place 5/11, 1/2, 3/4, 11/12, 1 and see that 11/12 must be closer.

The game
We played three rounds: target 0, target 1/3 and target 1/2. I think this game was very challenging for the kids. Everyone had to work to figure out the best play from their hand and didn't always make the right (local) choice. For example, whether to choose 5/8 or 5/9 for a 1/2 target.

Once everyone had played, the challenge was still just starting. They had to figure out who was closest. I structured the discussion by helping them figure out which plays were lower than the target and which were higher. For the ones that were lower, they could put them in order and only needed to consider the highest. Then, we worked on the ones higher than the target and got the lowest of those.

In the course of this discussion, we added a third strategy to the ones listed above:

making a common numerator

Summary thoughts

Fraction comparison like this was still too difficult for the kids to make an engaging game. If I were to do it again, I would change to make it more of a puzzling exercise, removing competition and any sense of time pressure.

Once the kids gain a bit more experience, though, I think this game has some nice features. It is particularly good for practicing fraction sense, and the multiple rounds allow some scope for strategic play.

Sunday, August 14, 2016

Block blobs redux

Last December, we played Block Blobs (notes here). This week, we are trying a slightly modified version for two digit multiplication.

The Game

Materials

4d6. Two dice are re-labelled 0, 1, 1, 2, 2, 3 (see notes below)
Graph paper (we are using paper that is roughly 20 cm x 28 cm, lines about 0.5 cm apart)
colored pencils

Taking a turn

Roll all four dice. Form two 2-digit numbers using the standard dice as ones digits. Then, use your color to outline and shade a rectangle in the grid so that:

the side lengths are the 2-digit numbers you formed with the dice
At least one unit of the rectangle's border is on the border of your block blob
Note: the first player on their first turn must have a corner of their rectangle on the vertex at the center of the grid. The second player has a free play on their first turn.
Write down the area of your rectangle

Ending the game

The game ends when one player can't place a rectangle of the required dimensions legally.

When that happens, add up the area of your block blob. Higher value wins.

Notes

"Counting" sides
The side lengths of the rectangles are long enough that counting on the graph paper will be irritating. Instead of counting directly, they can measure the side lengths. For our graph paper, the link between the measurement and the count is nice, since the paper is very nearly 5mm ruled, so they just double the measure. I think this is a really nice measurement and doubling practice, too.

Dice labels

Other labels could be used on the special dice. We chose this arrangement because of the size of the graph paper. Rectangles with sides longer than 40 often won't fit and we think we will even need some single digit rectangles to allow a fun game length. An alternative we are considering is 0, 0, 1, 1, 2, 2.

As an alternative to labeling the dice with new numbers, you could label them with colored dots and give out a mapping table. For example:

blue corresponds to 3
red corresponds to 2
green corresponds to 2
yellow corresponds to 1
black corresponds to 1
white corresponds to 0

This would keep the tens digit dice distinct from the ones digit dice and allow rapid modification if you want to change the allowed tens digits (just tell everyone a new mapping). Alternatively, you could create a mapping using "raw" dice and even allow more strategic flexibility from the players. I have a feeling that this would be confusing to most kids, though.

Reinforcing the distributive law

To facilitate calculating the area and reinforce the distributive law, you might have the students split their rectangles into two (or four or more) pieces and calculate the partial products. You can further decide whether to ask them to split the sides in particular ways or encourage them to find the easiest way to split to help them calculate.

Monday, June 20, 2016

Secret Numbers (Addition Boomerang variants)

We have been enjoying Mathpickle activities in our math games classes recently. This week, the 3rd and 4th graders will be playing with some of the more advanced Addition Boomerang variations.

During our planning, we came up with one extra pointer to tie the activity more closely with multiplication. Also, we had ideas for variations and wanted to record our notes so we can use them again in the future.

Tie with multiplication

In Gord's video explanation, he sometimes records in the center of each loop how many times that loop has been used. We emphasize this and write some related equations to help draw out the connection between the repeated additions in this activity and multiplication.

The first way we do this is by making tally marks inside the loop every time that branch is chosen. At any time, you can pause and write down an equation for the current total in a form

AxN + BxM = Total

where A and B are the values of the loops, N and M are the number of times each loop has been used.

Alternatively, we can show a "completed" round of throws by simply writing the number of passes for each loop in the middle and, again, write out an equation showing the total as the sum of two products.

Secret Number Variations

We start with a basic addition boomerang lay-out, either with 2 or 4 branches, both players (or teams) share a common set of addends and take turns adding on to a common running total. In our variants, the players choose and write down a secret number that helps inform their target for the game:

Version A: players pick a number between 70 and 100. This is their target for the game and they win if the common total hits that value, whether the target is reached on their turn or their opponents turn.
Version B: players pick a number larger than 15. They win if the total hits a multiple of their secret number. For example, if they choose 17 and the running total hits 51 (aka 17 x 3) then they would win. If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.
Version C: players pick any number. They win if the total hits a multiple of their secret number that is larger than 60 (not equal to 60). If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.

Version B is, I think, the most directly playable.

Possible issues
I'm not sure how to deal with the case where both players choose the same secret number.

For Version A, it will be interesting to see what modification kids can find that will deal with the fact that it is very easy to miss any particular target. In the basic game, once the total is larger than your target, there is no hope of recovery. There are several ways to address this. I would be eager to hear any rule sets that kids create and hear about the experiences.

In Version C, I wonder if choosing 2 as the secret number is too strong a move?

Monday, May 30, 2016

Class summaries

Quick notes on the activities for the classes today.

Grades 1 and 2

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

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

There are some natural extension explorations:

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

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

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

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

Grades 3 and 4

We started with the Shapes x Shapes puzzle from NRICH:

We added a couple of extra questions to this challenge:

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

Observations:

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

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

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

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

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

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

Monday, May 23, 2016

Division Dice (math games class)

Who: grades 3 and 4
Where: in school

A dice division game

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

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

Key constraints

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

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

Modifications/Extensions

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

We addressed these by adding two extra rules:

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

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

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

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

Wednesday, May 18, 2016

Factors and division

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

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

Dots & Boxes and Factor Game Mash-up

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

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

Homework

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

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

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

Tuesday, May 17, 2016

Build the chair spatial reasoning (Gr 1 and 2)

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

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

Build a chair

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

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

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

Chairs for bears

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

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

Homework

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

Extension

Building off the three bears activity is a nice extension:

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

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

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

Tuesday, February 16, 2016

Dots and Boxes variation

In grades 1, 2, 3, we played this variation of dots & boxes: Mathify the Squares Game.

I'm enthusiastic about this game, but can't resist a quick comment about the "mathification." Dots & boxes is already a mathematical activity, it doesn't need to be "mathified." This term implies confusing arithmetic and calculation with math, something I've written about elsewhere and, I hope, is clearly not implied by our blog.

In any case, I'll use the shorthand MD&B to refer to this dots and boxes variation.

Notes from playing in class

In class, we first introduced the kids to vanilla Dots & Boxes with a pre-printed grid of dots. We knew that it would be too much to play on a lattice covering the whole A4 sheet, but we thought a quarter of the grid would work. That turned out to be too big and the game started to seem monotonous to the kids as there was too much time spent on the opening (playing on squares that don't yet have any filled sides.)

We rectified this problem in 3rd grade and played on much smaller grids, with sizes between 7x7 lattices (which yield 6x6 squares) and 10x10.

After they were comfortable with the vanilla game, we introduced the product version with dice. In our case, we just had the players take turns and didn't give an extra turn when someone completed a square.

Some rule variations
There are some simple variations depending on how you deal with completed squares:

no extra turn (this was the version we played in class)
player adds another side to a different square with the same value. For example, say the dice are 2 and 4 (product 8) and the player fills a square. They also must add a side to another square with value 8.
player rolls the dice and adds a new side (a full extra turn)
player adds a side to any square (dice and square values ignored)

Of course, you could also make the extra turn optional instead of compulsory. You might also have some ideas about different ways to handle cases where there are no more free sides on squares with the required number value.

Probability questions

Dice games naturally lead to probability questions. Here were two that I really liked, based on scenarios from a recent game play:

What question are we asking?

What is the chance a player will get both the 20 and 36 boxes:

One great answer was 1/2. The reasoning: we are going to play until all boxes are filled and each of us have an equal chance to fill this box, so 1/2. This is not quite right, since the person who is about to roll has an advantage, but I thought it was an interesting interpretation of the question.

A 2x2 square

In this configuration, what is the probability that the next throw will allow the player to complete a box in the 4-30-25-15 zone?

A bigger D&B family

One of the reasons I thought this D&B variation was so cool is because of our games matrix. Whenever we play games, J1 and I talk about some key characteristics of the game, particularly the amount of randomness and the strategic complexity. These are not entirely independent dimensions, since a larger amount of randomness reduces the number or importance of each player decision, thus the strategic depth.

The mechanics of this game gave us some ideas about how to dial up or down the amount of randomness in this family of games. Here is our list of members of this family, roughly ordered from least random to most random:

vanilla dots and boxes: no random element
mash-up with product game: squares are still labeled, but players control the two factors using selectors (like in the product game) instead of using dice. This can be played with different collections of factors and different size boards (including board variations where a product appears multiple times or only a single time, where values are ordered or randomly distributed).
half-way house: one factor is chosen by players moving a selector, the other is determined by a dice roll (either before or after the "free" factor is selected).
MD&B game: as played in class and described in the first link
MD&B game where each factor appears only once. This is a case of dialing up the randomness by reducing the strategic options of the players.

Ideas for other games

I'm excited to see what other games we can modify use the underlying idea from the MD&B variation. To be clear: use numbers to label parts of the game and then constrain the players' actions based on a die roll to involve either the pieces with corresponding labels or board positions with those labels.

Three specific examples:

Hackenbush variation where segments of the picture are numbered. This could nicely incorporate probabilities by putting values of the least likely dice rolls closer to the ground.
Ultimate tic-tac-toe meets the product game: from Art of Math. This is an old post, but I just happened to see it when preparing this post.
Dice chess. Here's the wikipedia article. For some reason, I often forget about this variation, even though it is a nice way to reduce the strategic complexity of vanilla chess for beginners and has some nice links with probability.

If you have some favorites, I would love to hear in the comments!

*Update*
Playing through the MD&B version several times, we came up with these rule variations that are worth your consideration:

Game stops as soon as someone rolls a value that can't be played (alternatives are to let that person roll again or have them pass their turn)
Remove some of the randomness: (a) on your turn, you roll and play a side with the required value, but they opponent also plays a side with that value. As we played it, that means moves (without filled boxes) go: A, B, B, A, A, B, B, etc. (b) When a player fills a box, they can choose to re-roll both, either, or none of the dice for their extra turn.

I think our favorite was a combination of all three of these components. Mixing 2a and b, you have to be careful to keep track of whose turn it is, but it lead helped bring out elements of strategy and more thinking about probability.

*Update 2*
Game phases
Above, we talked about how A4 (or even 1/4 of an A4, which I guess is equivalent to A6) is too big for beginning players of Dots & Boxes. They found the game "boring." J1 and I talked about this experience and it led us to considerations around game phases: opening, middle game, and end game. These are terms we first learned in chess, and we found it useful to contrast the two games.
Here were some observations:

Opening: a lot of choices, not obvious how most of those choices link with "scoring" or the winning objective of the game. At this stage, there seems to be little interaction between the players (there is enough space that most of their actions either don't bring opposing pieces together or there is a lot of open territory).
Middle game: still many playing choices, increasingly direct conflict between players, interim objectives within the game become more clear and there are some chances for plays that either score or more clearly move closer to the overall game objective.
End game: significantly fewer choices for each turn than the other phases, either because there are fewer pieces (chess) or most territory has been claimed. At this stage, players are able to focus on the overall game objective, rather than interim objectives.

What we realized is that the larger playing area for D&B significantly increases the length of the opening. Because this phase is the least connected with capturing boxes, it is the hardest for beginning players to see how their choices ultimate lead to scoring and it is the phase with the most available choices on a turn.

Monday, February 8, 2016

Consecutive capture and Multiplication zones (math games class notes)

The games last week were taken from Acing Math's collection of card games and John Golden's wonderful blog (here's a list of games).

Consecutive Capture

This game comes from John Golden. The idea is simple, but it is a fun game. We used this in the first grade class. We made some slight changes to his rules.
Materials: pack of playing cards, including jokers, a number line labelled -13 to +13
Players: Two to Four (though seems naturally a 2 person game)
In this games, red cards are negatives, jokers are zero, and black cards are positive. Players are dealt a hand, then take turns putting their cards on the number line. Whenever they form three (or more) in a row, they can collect the cards that form the run. The cards they collect from runs count as points toward winning. At the end of each turn, they draw a card to replenish their hand.
If a point on the number line is already covered by a card, a player can add another card with the same value on top. If that subsequently becomes part of a run, the player collecting the run only takes one card for each value.
Variations: as noted, you can play with different numbers of players. When there are multiple cards on a value, you could allow a run collector to take all of the stacked cards. In John's version, he lets black aces take the value of 1 or 14, up to the decision of the player who adds them to the number line (and red aces -1 or -14).

Multiplication zones

This is from Acing Math. For 2nd and 3rd grades, We modified the card values from their rules, keeping aces as 1, J = 11, Q = 12 and removed the kings.

Tuesday, January 19, 2016

Midpoints and Sierpinski Triangles (programming class)

We started the new year with some exercises that refresh several of the concepts we've been covering:

loops
arrays
coordinates

We haven't done very much with coordinates, so this is probably the part that feels the most "new." By this point, loops should feel reasonably familiar and, in the code today, we aren't doing any complicated or nested loops. The use of arrays is still difficult; these objects are a tricky beast for the kids to fully understand.

Midpoint jumps

Our first challenge is to get the turtle to jump halfway from where it is standing to a target point. For this, we recall the getxy() function and use the xy-coordinates for the screen (the home where the turtle starts is (0, 0) in this system). Here's an example:

Sierpinski Triangle

Using this basic code, we can reorganize things a bit to play the Chaos Game and generate an amazing picture. In each iteration, we randomly select one vertex, then jump halfway to that vertex, leave a dot, then continue:

This version already has a nice little twist with the random colors on each new dot. Nice!

Homework

Finish the challenges from the class (including writing explanations of the lines of code we highlighted) and then modify the last program to create your own picture. Here's an example of something I created:

A slightly different version is here: <a href="https://jgplay.pencilcode.net/edit/Math/Fractal3">more Fractals.</a>

Tuesday, November 24, 2015

another race to 100 game

Today's game at the math classes was not particularly well liked, but we are including this note for completeness and future reference.

Race to 100

how many players: 2-5
material: 1d6, 100 board, position markers (the kids made their own out of play-dough)
start: all players start on 1 on the 100 board
turns: each player's takes a separate turn. They roll the dice, then move their piece up the 100 board some multiple of the dice value (up to 10x).
winning: first player to get exactly to 100 wins

This game practices multiplication, skip counting, and factoring. Here are some example questions to stimulate thinking about game strategy:

Would you rather have your piece on 99, 98, or 96?
What about 71 and 70?
If you are on 88, what are your chances of winning on the next roll?

Game reception
The kids found this game fairly easy. In retrospect, perhaps we should have played this game before the Times Square factors game.

Potential extension
The game is nicely suited to analysis by working back from the higher positions and/or analyzing a simpler version of the game. This may be a nice exercise for our programming classes, especially as we have recently been working with arrays.

Wednesday, November 18, 2015

Times square variations (math games classes)

In grades 2 and 3, we have been playing with variations of NCTM's game Times Square, one of their offerings on Calculation Nation. This is one of my favorite multiplication games because, like the puzzle Bojagi, it is fun and multiplication is integral to the game, it isn't just a set of flashcards in disguise.

Here's a basic Times Square board:

The AI doesn't understand edge vs center!

Players take turns moving one of the square windows at the bottom to select two numbers, then get to take possession of the square that is the product of the values the windows are on. In our starting game, the AI moved the first window to 6, I moved the second to 5 and captured 30 (5x6). The AI then moved from 6 to 1 and captured 5 (1x5). On their turn, the player can move either window, but has to capture an open area (you can't duplicate a product you've already captured or take over an area your opponent has previous captured). The first person to get 4 in a row wins.

A pen and paper version
We didn't have (or want) computers for all the kids to play online. Instead, we created a simple paper and pencil version. We made many copies of the board on a piece of paper, with the numbers 1 to 9 at the bottom. We then used small rubber bands (loom band left-overs!) to select the factors and players used colored pencils to claim their territory.

It was an easy, colorful, and fun implementation of the game:

Notice the sad faces where mom/dad won that round?

Noticing the structure
As usual with this kind of activity, there are many possible extensions, with two obvious groups being strategy (how do you win the game) and structure (what do you notice and could change about how the game is set up).

So far, we have been looking at structure. Here are some of the things we discussed relating to the basic board:

What shape is the playing board? How many numbers are on it?
What is the largest number? Why aren't there that number of spaces?
What is the smallest number (positive integer) that isn't on the board? Why?
What numbers are missing from the board?
If we say an integer between 1 and 81, can you tell, without looking, whether it is on the board?

Make it simpler

The next iteration was an exercise in simplifying. Do we need to use all factors 1 to 9? What if we made an easier game with factors 1 to 4? Here is the version we came up with:

Surprise, surprise, we can still make a nicely shaped grid! Now, we aim for 3 in a row, like standard tic-tac-toe, but with a constrain that means we can't always move where we would like. With this simplified version, maybe we can go back to our strategy questions and gain some wisdom that will help for the 1 to 9 version?

Make it more complicated

What if we wanted a harder (calculation) challenge than 1 to 9? Are there other collections of factors that would give us nicely shaped grids? We had them work out creating a grid based on factors 1 to 13.

It was really interesting to see the different strategies that the students took to determining what would go on their boards. Some people tried creating full multiplication tables and then removing duplicates. Other people counted up from one and tested each number as they went along. Some people identified patterns, essentially working with the diagonal and upper half triangle of a multiplication table.

Here's a student, hard at work calculating the 1 to 13 board:

In this case, there are 72 distinct products, so the students also had a choice of making near-square boards that are 8x9 or 9x8. We didn't guide them to these shapes, but it was interesting that no one made a 6x12, 4x18, 3x24, 2x36, or 1x72 shaped board,

For the 8x9 and 9x8 boards, we had them take some time to play on each version. Does play feel different on the two different boards? Is there a different strategy for the two boards? Perhaps you will also experiment with this.

Further exploration

A sequence

How is the sequence 1, 3, 6, 9, 14, 18, 25, 30, 36, 42, 53, 59, 72, 80, 89, 97, 114, 123, 142, 152... related to this game? These are the numbers of distinct products of the integers 1 to n as n grows. What can we say about this sequence? For example, how quickly does it grow with n? Is there a closed form for the nth term?

Can we see anything interesting if, instead of using integers 1 to n, we use a different collection of n integers? For an easy one, try using the first n primes. Maybe using n integers that are in an arithmetic sequence would be interesting?

This simple pencilcode program could get you started on gathering some data: TimesSquareBoards.

For some light, related reading: Number of Integers with a divisor in a given interval (Ford 2008) which was linked on this Math Stackexchange question.

Strategy and Structure

Ok, so we can take factors 1 to n, then create a board that arranges the distinct products into a rectangle. Because we see primes in the sequence above, we know that some of these rectangles are just 1 x p (or p x 1) shaped. Even so, what can we say about winning strategies:

When does the first player have a winning strategy?
When does the second player have a winning strategy?
When does optimal play by both players lead to a tie (like classic tic-tac-toe)?
Are there n for which differently shaped boards have different winning strategies? Is there an n which has 3 differently shaped boards that cover each of the different strategy outcomes (one that is a first player winner, another that is a second player winner, a third that ends in ties?)

In particular, I think it would make for a delightful bar bet if, say, the first player had a winning strategy for the 8x9 board, while the second player has a winning strategy for the 9 x 8 board!

A picture, just for the heck of it

Having nothing to do with any of this, what estimation and math questions do you have about this picture:

Yes, these are gold, but just covered with gold leaf, not solid!

Tuesday, November 3, 2015

Loop-de-loop festival (math games and programming classes)

who: all grades at Baan Pathomtham
when: throughout the school day

First, Apologies! With other obligations, this is over a week late.

Second, I've already written about loop-de-loops here and here. You can find the basic explanations and background there.

In this write-up, I just want to explain how we played with loop-de-loops in the classroom and the reaction of the students. With pictures! Our experiences come in two flavors, based on the two different kinds of classes we were leading:

Math games/exploration, for grades 1-3, notes here
Programming, for grades 5 and 6, notes here

Math games and exploration

In these classes, I started at the front of the class with a small(ish) whiteboard to show them the simple rules. Following Anna Weltman's instruction page, I drew a 2-3-4 loop-de-loop as follows:

Draw a line up the page for 2 units (I marked ticks to provide a reference for 2 units)
Turn the whiteboard clockwise 90 degrees
Draw a line up the page for 3 units
Turn the whiteboard clockwise 90 degrees
Draw a line up the page for 4 units
Pause and ask the kids what they thought I would do next, with a little discussion, then ...
Turn the whiteboard clockwise 90 degrees
Ask them how long a line I should drawwith a little discussion, then ...
Draw a line up the page for 2 units
Ask them, if I continue this 2, rotate, 3, rotate, 4, rotate, 2, rotate, 3, rotate, etc, will I get back to my starting place? After a little debate amongst the kids with opposing views expressed, I turned them loose to try it out on their own graph paper.

For the rest of the class, the kids asked me for more seeds and/or experimented with their own ideas. A couple are worth noting:

3-5-2: This is on Anna's instruction page. The kids found it surprisingly challenging. The issue comes during one step where you end on a pre-existing line, but not at one of the endpoints. That seemed to make it easy for people to lose their place or get confused about what they should do next.
4 number sequences: both closed loops (like 4-7-4-7) and open ones (1-2-3-4) really interested the kids. I have a (mild) reputation for teasing them, so they were somewhat on the look-out for a twist like this.
6 number sequences: they discovered these on their own or had a more experienced friend suggest them.

Why was this a great activity for the kids?
First, mathematically, there are tons of patterns waiting to be discovered, almost all of which are easily accessible and where the kids can set their own direction for exploration. We will write up an example in next post about the math classes.

Second, this shows some important aspects of mathematics that we often forget: it isn't just about calculating and it has a deep aesthetic (artistic) side.

Programming

The basic introduction was similar for the two programming classes. I showed the essential rules, then the kids drew some loop-de-loops on paper. Of course, the natural next step is writing a program to generate the pictures.

After more or less coaching, all the kids wrote a double for loop to draw their loop-de-loops.

Why was this a great activity for the kids?

First, it was a very natural context to use double for loops, including an outer loop where the steps are just repeated exactly and the other where the iterating variable changes as it moves through a list of step sizes.

Second, repeating, exactly, a list of instructions over and over shows off the power of the machine over hand-calculating. In this sense, it was easier to create programs to draw loop-de-loops than to draw them by hand. When doing them manually, almost all of us occasionally lost track of where we were, turned the wrong way, or made a line the wrong length.

Which brings us to: third, we got to use the computers as a tool to support our own investigation of the loop-de-loop patterns. This was because it was so easy to draw so many versions so quickly. One example was comparing the 1-2-4 shape with the 4-1-2 shape and the 4-2-1 shape. Wait for the next post for another example.

Fourth, when writing their programs, all the kids scaled their drawings. For example, in the 1-2-4 shape, some chose to make the step lengths 100-200-400, while others chose 25-50-100, while others made different choices. This gave us a chance to talk about these scaling choices and to introduce an explicit scaling variable. Some of this continued into the next class.

Finally, in the 6th grade class, the use of computers gave them free rein to explore much longer and more complicated step sequences than they could have considered by hand.

Pictures

Oh, right, you just wanted to see pictures. Here you go!