The Parallelogram and the Pendulum (Age 8)

The Activities

  1. Topics: Logic:  Still More Stories to Solve, by G. Shannon.  We read and discussed the first two stories.
  2. Topics: Spatial Reasoning, Tangrams:  We did the same set of tangrams from a few weeks ago (letters, numbers, and things from Cinderella).
  3. Topics: Physics, Experiments:  Inspired by the Galileo chapter of Mathematicians are People Too from a few weeks ago, I hung a makeshift pendulum from the ceiling — a roll of tape suspended from an 8′ thread, hanging from a sticky hook attached to the ceiling.  I had pre-marked the 2′, 4′, 6′, and 8′ points away from the center of the roll of tape.  We released the pendulum twice at each length, varying the height that we released it at, and timed how long it took to go 20 swings.IMG_1888
  4. Topics: Sorting, Patterns:  We have a card game called Blink — basically a racing version of Uno.  Each card has some number of symbols 1-5, one of six colors, and one of six shapes.  There are 180 possible combinations, but only 60 cards in the deck.  After we figured out there should be 180, I asked the kids to find out which ones are missing.

How Did It Go?

We sat on the floor this week to make room for the pendulum; this tends to make them a little crazier since they can easily roll around on the floor.

Still More Stories to Solve

I wasn’t crazy about the first puzzle, but the second one, about two brothers having a contest to see whose horse would get somewhere LAST, was nice.  The kids figured it out with some hints.

Tangrams

Corey and I discovered that I’m better at Tangrams than she is :), so unlike last time, where Corey AND the kids were stuck I was able to help them solve the puzzles.  The main thing I tried to teach them was to figure out where the big triangles go first; it’ll be interesting to see if next time we do Tangrams they remember this.

Timing a Pendulum

As you can see from the chart above, we had really reproducible results.  I believe we were actually only counting 19 swings (we started on 1 as we let go and then stopped when we said 20, when we should have let it swing again).  Anyway, I had incorrectly remembered from physics long ago that the time was linearly proportional to the length of the pendulum, so I was initially worried about the timings we were getting — but once they were all in, it become obvious (to me, not the kids) that the time is proportional to the square root of the length.  I asked how long the pendulum should be to get 15 seconds; and also, how long would a 32′ pendulum take.  They were comfortable assuming a linear relationship, but when I pointed out that 8′ was four times 2′ while 60 s was only two times 30 s, they couldn’t really use that information — one kid did guess 1/2 foot for the 15 seconds question, but they didn’t stick to their answer so I think it was just a guess.

One thing that worked out well is that the pieces of tape served as resting spots for the thread so that it would stay at the right length (I didn’t cut the string, we had looped it over the hook like a pulley).  If I hadn’t had the tape sticking out, it would have been hard to maintain a constant length.

Blink

I only had time to explain the problem and figure out how many cards there should be before circle ended.  We’ll probably do the main activity next week.

Who Gets to Jump the Most? (Age 6)

The Activities

  1. Topic: Infinity, Even and Odd. Book: The Cat in Numberland ,Chapter 4, by Ekelund. In this chapter, Mr Hildebrand wants exactly one half of the numbers to go visit the letters. But how do we take one half of infinity? Later, the hotel is half empty because all the odds have gone. How can the evens move down so there are no lonely, empty rooms? We acted this part out with a diagram.
  2. Topic: Addition. The kids took turns throwing bean bags at a number chart on the floor. Then they used based-10 blocks to add up the three numbers. Once all the kids had their sum, we all jumped together, and each kid stopped jumping when we got to their sum. This means the kid with the highest sum gets to jump the longest.Which, believe me, was considered quite a high reward 🙂IMG_20160424_174531
  3. Topic: Primes and Composites. We worked to prove that various numbers between 1 and 100 were prime or composite. To prove something is composite, you should make a rectangle out of that many cubes. Proving something is prime is much harder, since you have to convince me that it is not possible to make a rectangle out of the blocks.

 

How did it go?

We had four kids this week. Everyone *loved* jumping, but my son had to sit out a couple activities because he was bored/wild/frustrated. Everyone else was pretty excited, especially about the beanbag addition.

Hotel Infinity

There were several interesting discussions this week. The hotel owners, the Hildebrands, decide that half of the numbers should go to visit their friends for one week. How do we make sure exactly half of all numbers go? One kid had a great suggestion: send the negative numbers away, and keep the positives. I was very impressed by this idea, but in this book, there are no negative numbers. Someone also suggested sending the numbers away, 50 at a time. I pointed out that this would definitely not be half of the number. And, we had yet another discussion about whether there is a biggest number. I think I (again) convinced them that we can always make a higher number, so there is no biggest.

The next part of the book is about the even numbers being lonely because the odd rooms are empty. Zero figures out that each even number can divide itself in two, and move into that room. We made a diagram of the hotel, and worked to divide the evens in half to find their new room number.IMG_20160424_174658

Three of the kids figured out the new room number by making two equal rows of base 10 blocks. For example, above we see that ’14’ will go to Room 7. Then the kid would write the number in the correct number, and cross it off the list. I gave out numbers in a random order, so it would not be obvious which room each belonged in.

My son got pretty impatient with this, because he could divide the numbers in his head, and just wanted to write them all into the hotel. After 2 or 3 numbers for each kid, we filled in the rest of the hotel by counting by twos and filling in the empty.

Bean Bag Addition

Everyone was excited to throw the bean bags at the chart. All four kids were able to add three numbers between 1 and 30 together, using base ten blocks. There were a few adding mistakes, but these were easily corrected.

After each kid added up their number we discussed who had the highest and lowest numbers, and then we all started jumping. When we got to each kid’s number, they had to stop jumping. At the end, it was just me and the kid with the highest number who were jumping.  Everyone absolutely loved the jumping part, and desperately wanted to get the highest number. The highest number of the day was 64, and all the kids knew exactly who had gotten it.

This caused some problems for my son, who kept wanting to redo his throws to try to get higher numbers. Also, the adding was too easy for him. Initially I told him to multiply his three numbers, but his first set was 5 * 12 * 12. He knew 12*12 = 144, but 144 * 5 was pretty hard for him. This frustrated him, and eventually he decided he wanted to add his numbers like everyone else was doing.

We played 3 rounds of this game, and the last round we used 5 bean bags instead of 3. The kids all cheered when I asked if they wanted 5.

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Primes and Composites

Last week David and the kids tested which number 1 – 14 were prime. This week one girl was able to explain that a number is prime if you cannot make a rectangle out of that number cubes (unless one side is just one block wide).

This week, we explored some of the numbers above 14. We tracked our progress on the 100 board, using a blue square to indicate prime numbers and red to indicate composites. I just told them which were prime this time, but made them prove the composites by making a rectangle.

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The kids started to ask to do very large numbers like 99. Unfortunately it takes forever to count out 99 cubes. In a future week I might do it ahead of time, and let each kid try one big number.

Eggs and Boxes (Age 6)

Age 6

The Activities

  1. Topics: Numbers, Codes, Algebra:  The Cat in Numberland, Chapter 3, by I. Ekeland.  In this chapter the letters come to visit the numbers, and we learn about letter/number ciphers and letters standing in for numbers.
  2. Topic: Algebra:  I made problems of the form “X + 3 = 5” using unit cubes from Base Ten Blocks and a small cardboard box.  I.e., I would secretly put 2 blocks into the box and close it, put 3 blocks next to it, and then say “There are 5 blocks total, how many are in the box?”
  3. Topic: Primes:  I introduced the idea of primes using Base Ten Blocks: a number N is a prime if the only rectangle you can make using N blocks is 1 X N.  I gave different numbers to each kid and had them figure out whether it was prime or not.
  4. Topics: Combinations, Combinatorics:  I printed a bunch of “Easter eggs” with a top and bottom section.  Using five different colors of crayons, I asked the kids to make as many different eggs as they could, coloring each section in solid colors (not stripes/dots/etc.).  I taped each one to the wall (stacking repeats).IMG_1886

How Did It Go?

We had four kids this week.

Cat In Numberland

The algebra in this chapter is tricky because it includes addition, subtraction, multiplication, and division; most of the kids don’t know multiplication or division yet.

Box Algebra

This worked pretty well.  The kids understood what was going on right away, and they were always excited when I opened the box and dumped out the blocks inside to see if their guess was correct.  At the end they made a problem for me, which was something like “X + 3 = 39” (of course, they used as many blocks as they could).

Rectangle Primes

We did up to about 14.  I kept track of each result.  The only odd composite number <= 14 is 9, so for the most part they just needed to check a 2 row rectangle.  Proving something is prime is tricky, of course, and whenever a kid said that something was prime, I always asked them “did you check 3-wide”?  Whoever had 9 didn’t initially check 3×3.

Easter Eggs

The kids were really into this activity and worked very hard to get all the combinations.  They got all 10 two-color combinations pretty quickly and without help (first two rows in picture above) — but there was no pattern to which color was on top vs. bottom.  Then one of the kids realized that you could flip the colors.  They quickly got 6 more, but the next 3 took them a lot longer to find, and I had to help them find the last one.  This got them to 20, but they didn’t think of having the same color on top and bottom.  I suggested it to them and they quickly made the last 5.  Then I rearranged them so that there were same color tops along the rows and same color bottoms along the columns.  I realized afterwards that I should have made this chart before I gave them the hint about same color top/bottom, because then there would have been gaps and I could ask them what went in the gaps.

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Merge Sorting 224 Numbers (Age 8)

The Activities

  1. Topic: Logic. Book: True Lies, by Shannon. We did stories 14 – 18 today, finishing the book. Again, everyone loved the stories, and there were lots of lively discussions and theories about how the apparent lies could actually be true.
  2. Topic: Gravity, Physics. We discussed the results of dropping things off our balcony last week, and whether Galileo was right or wrong when he said everything falls at the same speed.

    IMG_20160417_175418

    Which will hit first? The marker or the dinosaur?

  3. Topic: Proofs. We investigated various number properties, and tried to ‘prove’ them using blue blocks as a way to explain graphically:
    1. odd + odd = even
    2. even + even = even
    3. even + odd = odd
    4. even * even = even
    5. odd * even = even
      IMG_20160417_175216

      Visual illustration that even * odd = even.

       

  4. Topic: Sorting, Merge Sort. We tried out the sorting strategy we had used in the Easter egg sorting activity: each kid sorted their own stack of numbers, and the merged all the stacks together at the end.

How did it go?

We had all five kids this week, and it was a very successful circle. My daughter was very tired from activities earlier in the day, so she caused problems at various points, grabbing away materials or getting off task. Otherwise, everyone was generally engaged.

True Lies

The kids really do love this book, and everyone offered lots of interesting theories this time. For one story, I liked our theory better than the one in the book. In “Pockets” a tour guide bets a rich visitor: “I bet I have more money in my pocket than you have”. The visitor pulls out his fat wallet and then says “I accept”, and he loses the bet.

The kids came up with the idea that because the rich man said “I accept” after he had already taken his money out of his pocket, that meant he had zero dollars left in his pocket, so he lost.  The book says the answer is that the guide meant “I have more money in my pocket than you have in my pocket”.

Gravity Followup

One kid was very clearly able to recall the story of Galileo dropping objects off the tower of Pisa to prove that everything falls at the same speed.  Everyone also remembered that the Kleenex did not fall at the same speed as the coin, when we dropped them off the balcony last week. One kid was able to explain that the air caught the Kleenex and made it drift. We then tried out a few different dropping tests, in the room (not off the balcony this time). Everyone generally correctly predicted when things would hit at the same time or not. However, one kid initially said the heavy bottle of lotion would fall faster than the coin, but she changed her vote after everyone else said they’d hit at the same time.

Even * Even = Even

Next I got out the blue unit cubes. Everyone groaned and said they didn’t like this material.  I said that we were going to the same activity the little circle did last week, and that they had done when they were 5 or 6.  This made them more interested. One girl said “Are we doing primes?”…I was glad she remembered that activity from long ago.

I started by asking whether an even + even number will be even or odd? The kids thought of a couple examples, and then confidently stated that it will be even. One girl was able to explain using the blocks, that each even number could be made of a 2 equal rows of cubes. When you add them together, you know the resulting two rows will also be the same length, so the result is even.

Next another kid demonstrated (with a bit of help), that odd + odd = even. They explained that the two extra cubes from the odd numbers would combine into another pair, yielding an even result.

At this point all the kids had caught on, and many different people want to explain how odd + even = odd.

Next I asked about even * even?  First we tested it by trying out a few concrete examples: 2 * 4, 6 * 4, 8 * 2. They were all even. Is it always true? The kids showed that if you have a bunch of sets of even numbers, then when you combine them, the result must also be even.  This also worked for odd * even.  An odd number of even numbers can still be combined to an even number, with no left over cubes.

Merge Sorting 224 Numbers

I asked the kids about the strategy we used in the Easter sorting activity. Everyone remembered how each kid had sorted their own numbers, then made a sorted pile and combined them together into one line. I said we would do the same again, and see if it seemed useful.

I handed each kid a pile of about 20 – 30 numbers, and asked them to sort it. They all begged for more numbers, so I handed out a few more. It was quite fascinating to see the individual strategies they used. A couple kids laid out all the cards around them, so they could see everything. Then they picked them up in order. If they missed a card they had different ways of adding it in: one kid paged through her stack until she found where the number belonged. The other kid dumped all the numbers off the stack in a random pile until they found the number’s place, and then had to sort them all back.

The slower strategies involved lining the numbers up in order, and sliding down the numbers to make space for the next picked up number. The line management took a lot longer than keeping a stack.

One girl actually got a bit stuck in the individual sort. I couldn’t tell what her strategy was, but eventually I helped her by separating out the numbers < 100, 100..200, and 200+. She took by far the longest, so I started handing out extra numbers to the other kids who had already finished. The two kids who kept stacks of numbers were able to quickly add in the new numbers. The ones who used lines took longer as they shifted their lines around.

Eventually everyone finished the individual, and carefully stacked up their cards with the smallest ones on top. Then the merge sort started. The kids were great at working together, especially in the beginning before they got tired. Their stacks ended up having 224 numbers in them, randomly distributed between 1 – 300. This meant there were gaps in the sort order. The gaps really slowed down the sort because the kids weren’t that good and determining that there was a gap, and which number to put down next. Even so, merging 224 numbers took less than 10 minutes, and the individual sort also took about 10 minutes, so that’s reasonably fast for the first try.

 

 

 

Bombs Away! (Age 8)

The Activities

  1. Topics: History of Math, Mathematicians:  Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians (Volume One) by L. Reimer and W. Reimer, Chapter 6 (Galileo).
  2. Topics: Order of Operations, Parentheses:  First, I introduced how parentheses work in simple arithmetic expressions involving +, -, and x.  Then, I had them practice evaluating expressions with parentheses with 3 or 4 numbers.  Finally, I gave them an expression without parentheses and asked them to figure out all the different possible results by adding parentheses in different places.

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    One kid already was quite familiar with parentheses

  3.  Topics: Gravity, Experiments:  Inspired by the story of Galileo, we dropped various items off our 2nd-story balcony and saw which landed first.IMG_20160410_173800

How Did It Go?

We had all five kids this week.

Galileo

The kids were pretty interested, as usual.  There were three interesting possible activities in this chapter: dropping things, pendulums, and telescopes.  We’ll probably do pendulums and telescopes in future circles.  I mentioned early in the chapter that we would be dropping things later in circle, and they got distracted because they found the stuff we were going to drop.

Parentheses

The kids were at very different places initially.  Two knew about parentheses, and one of those was really good at evaluating expressions with parentheses (unfortunately, they also kept saying how easy it was).  They also varied a lot in how long it took the others to understand how parentheses worked.  One of the kids didn’t want to try, but then when I helped them specifically they got it.  Another kept saying they didn’t understand; I helped them run through an evaluation a couple times, and I think they got it; but then they had trouble with the next part.  Part of the reason I wanted to do this activity was because the kids have had trouble with braces in programming; and indeed, parentheses are a rather challenging concept.  The idea of evaluating “inside-out” doesn’t make sense without more explanation, and the idea that you should find an expression that is bounded by parentheses and doesn’t contain any other parentheses and replace it by the result is rather tricky.

When we got to the part where the kids needed to add parentheses, most of the progress was made by the kid who was the best at evaluating.  However, for the 4-number case we did, one of the other kids found the final possibility.  I realized a useful way to think about it is picking two adjacent numbers and replacing them with the result of applying the operation between them.  This suggests a different notation, which is drawing non-overlapping circles instead of parentheses (really, that’s what parentheses represent).  I think this would help the kids understand what’s going on better.

Dropping Things

We did this activity in combination with the Age 6 circle.

We took turns having kids drop things (same kid held both items and dropped at the same time).  The others stood below.  As it turns out, non-breakable household items really don’t all fall at the same rate — anything with significant surface area will be noticeably slower.  So, there are two possible strategies: 1) arrange to have all the “well-behaved” objects first, so you can establish that weight and size of, say, spherical objects doesn’t affect the speed, before moving on to the tricker items, or 2) do things in a more or less random order and let the kids try to figure it out.  Both have advantages and disadvantages — in some ways, #2 is more scientific, but #1 might give them better intuition.  I did #2 — and the result was that at least some of the kids came away saying that heavier things fall faster.  However, at least one of the kids really got the idea that surface area matters, and could even explain why a book falls faster held on edge than flat: because it “torpedoes” through the air if held on edge.

One amusing thing is that one of the items was a bag of (somewhat stale) hot dog buns left over from earlier in the week, and while we were managing the experiment, a couple of the kids started eating the buns.

Even + Even = ? (Age 6)

The Activities

  1. Topic: Infinity. Book: The Cat in Numberland, Chapter 2, by Ekelund. In this chapter, all the numbers are living in the infinite hotel,when along comes Zero, who also wants a room. How can they fit him in?
  2. Topic: Proofs, Even and Odd Numbers. Define even and odd numbers. Then play with blue blocks adding even + even, and see if the result is even or odd.  Will the result always be even? What happens if you add odd plus odd?IMG_20160410_174413
  3. Topic: Sorting. Sort the number tiles 1 – 100 on the number board.IMG_20160410_170852
  4. Topic: Gravity. We took turns dropping objects off the balcony. Which object will hit the ground first?
    IMG_20160410_173800

    A balled-up tissue vs. a flat tissue

     

 

How did it go?

We had only three younger kids this week, so it was a pretty easy circle. The big kids were reading a book Galileo, who experimented with gravity by dropping items off the Leaning Tower of Pisa, so David wanted to drop stuff off our balcony. I figured that’d be pretty distracting to the little kids, so we decided to do it as a joint activity.

Cat in Numberland

This is one of my favorite books…it really emphasizes the wonder of infinity, and that some questions are not easy to answer.  In this chapter, the number Zero needs a room in the hotel. Ultimately, all the numbers decide to move up one room: One will stay in Room 2, Two in Room 3, etc. This leaves Room 1 empty, so Zero can move in. Finally the numbers each re-label their rooms to match themselves.

The kids were very interested in the book. One girl said she was worried that some very big number would no longer have a room, because they all moved up one room. I said that that big number could stay in the room for the number one bigger.

After the book, I said that the number Negative One knocked on the door and wanted a room. What can we do? At first the kids thought Negative One would have to sleep outside, or build a separate hotel, but then someone proposed moving the numbers up one room again. This seemed to work.

Even and Odds

First I asked the kids what is an even number? Kids started naming numbers: 4, 6, 12. My son started counting by 2s.  I then got out unit blocks and gave each kid a handful. I asked if that was an even or odd number of cubes? Some kids answered by counting the blocks and then knowing that 22 is even.  I showed the kids that instead, you could make two rows of blocks, and if the rows were the same length then number is even.

Even after this explanation, so kids wanted to instead count the cubes, but I kept demonstrating that what I really cared about was not what the number is, but whether it is odd or even. I showed that I could tell this without knowing how many cubes there were.

Next we tried a few addition problems with cubes: four + six, is it even or odd? 12 + 14? Is every even + even always even? Why?

The kids had some intuition about why even + even would be even, but it was hard to put it in words. One girl said something like “same number + same number means the answer will have the same too”.

Next we explored odd + odd, and found that it seemed to always be even. Why? One kid suggested that you could combine the two left over cubes to make even rows.

Sorting

We had a few minutes left before it was time to drop things off the balcony, so I got out the 1 – 100 board, and the number tiles. I seeded the board with four or five numbers, and then the three kids worked together to put in the rest of the numbers. They are much better at this than when they started circle. In the beginning it took them forever to sort the tiles on the side of the board that shows the numbers. This time it took about 7 minutes to sort the tiles on the side with no number labels. All three were able to place ’78’ in the right place by finding the ’71’ and counting up.

Dropping Stuff off the Balcony

See David’s blog entry for a full description, but this was fun and wild as you might expect. The older kids expected everything to drop at the same speed because they had read the book about Galileo. The little kids had no expectations, but really enjoyed having stuff thrown off the balcony, and seeing which would hit first.

Everyone enjoyed how slowly the tissue fell.

My Friend’s Mom Is Always Right (Age 6)

The Activities

  1. Topics: Numbers, Large Numbers: The Cat in Numberland, Chapter 1, by I. Ekeland.
  2. Topic: Large Numbers: First, I asked the kids to come up with numbers with increasing sizes (first 1 digit, then 2 digits, …).  After they stopped being able to come up with things, I wrote powers of 10 up to a quintillion.  I also wrote a googol and had them help me count the zeroes.  Next, each kid wrote down the biggest number they could on a sheet of paper.  I asked them which kid’s number was largest, and then I asked if it was the largest possible number.  Then I guided them through a proof by contradiction that there is no largest number.
  3. Topic: Programming: We revisited the Dance Programming activity.  First I called out commands (Down, Up, Jump, Spin) and the kids did them.  Then I taught them a few programs I had made that used the four commands plus “Do X times {… }” loops and function calls.  Next I had them each say a 3-command function for us all to do.  Finally, I asked each kid to write down a named function, with the rule that if you used a “Do X” loop X was at most 5.  Then I made a program that called all their functions and a couple of the kids did it.
    IMG_1864
  4. Topic: Programming: We revisited the Cargo Bots activity.  I made a new version of the board, download here.  First, we moved a single block from one end to the other; then two blocks; and finally, they needed to move two blocks of different colors from one end to other so that the final ordering was the same (i.e., if red is on top initially, it should be on top at the end as well).
    IMG_1865

How Did It Go?

We had four kids this week.  This circle went pretty well, we had some good discussions and some good problem solving.  The kids sometimes thank us after circle(usually prompted by parents) , this time one of them said “Thanks for giving circle to me.”

The Cat in Numberland

This was a popular book with the older circle.  There was a part about how not all numbers can play division together — two of the four kids got the idea.  When it got to the part about infinity, several of the kids already knew something about infinity.  One kid said “My friend says that her mom is always right, and her mom says infinity is a number.”

Large Numbers

Most of the kids had trouble coming up with a number above 1000.  Even the kids that knew larger numbers like 1,000,000 couldn’t make arbitrary seven digit numbers.  The kids enjoyed counting out 100 zeroes when I was writing a googol.  The kids varied in their approaches to writing large numbers.  Some did 1 followed by lots of zeroes, others wrote somewhat random sequences.  I wrote a large number as well — I started with 9’s for obvious reasons, but I discovered that 9 is pretty slow to write, and so I switched to 1’s which are really fast to write.

Next, I asked them which number was biggest; I’m pretty sure it was mine, but one of the kids had filled up their page with larger numbers, so they picked that one.  I asked whether it was the biggest number.  3 of the kids said no, one said yes.  I asked whether they could make a bigger number.  They didn’t really come up with adding 1, but they did suggest adding more digits, including adding on someone else’s sheet of numbers.  Then I said the idea of proof by contradiction (assume the opposite, find a problem), and said “Suppose you took all paper in the world and filled it with numbers.  Could you make a bigger number?”  After a bit, one of the kids said “You could cut down more trees and make more paper so you could add more numbers.”  So I think some of them got the idea.  Since they had suggested taping together sheets, I taped together all the sheets that we had made and laid it on the floor.  For the rest of circle, when a kid finished their work and was waiting on someone else, they asked to go over and add more numbers to their sheets.

Dance Programming

For an activity involving jumping up and down, the kids payed pretty good attention.  They all understood sequences of instructions, and I think they understood loops and functions.  When it came time to write a program, only one used a loop; the loop they wrote was a copy of one of my functions with a different number.  It’s a good thing I limited X to 5, because they immediately said “I wanted to do 100.”  No one figured out the weakness in my problem specification — nested loops.

Cargo Bots

One of the kids missed the first time we did this but caught up fairly quickly.  All the kids were able to solve the first two problems, at different speeds.  A common mistake on the two-block problem was forgetting to move the crane back to the beginning for the second block.  The two-color two-block problem was way harder.  One of the kids came up with some interesting rule-breaking solutions, such as using their other hand.  After a while I gave them a hint by making a program that flipped the stack onto the middle square of the track.  From this, one of the kids (same one who came up with the alternate solutions) realized if they repeated this again, they would solve the problem.  They had some bugs along the way, which I demonstrated by tracing through their program, and they were able to keep fixing things until they got a correct solution (with a few unneeded instructions).  Another kid understood the solution and tried to copy what the first kid had done and got pretty close.  The other kids were still far off and hadn’t gotten the idea of double flipping.