The Bird and the Bikes (Age 9)

The Activities

  1. Topic: Money. Book: The Story of Money by Maestro.  We continued where we left off last time, and got as far as early money in the Americas.  My favorite part was the discussion of why paper money caught on better in China than in Europe (the government was more stable in China).
  2. Topics: Algebra, Arithmetic:  I wrote down the equation (5789 + 1286) x 549 = 3,884,175.  I used my phone to compute the right hand side.  Then I asked them a series of questions: What is the answer if you change 5789 to 5790 (the answer increases by 549)?  What if you change 549 to 550?  What if you change 549 to 1098?  Each time, using my phone, I checked that you got the same answer by evaluating directly vs. evaluating incrementally.IMG_2535
  3. Topic: Probability: I attempted to teach the kids how to flip a coin properly, and then each kid (and me) spent 5-10 minutes flipping coins and writing down the sequence.  Then, I asked several questions: “Do you expect more heads or tails?  Is heads more likely after you’ve just gotten three tails in a row?  Is heads-tails more likely than heads-heads?”  For each one, we counted in our sequences to see whether the results matched the kids’ intuitions.
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  4. Topic: Logic: I drew a picture of two bicycles riding toward each other at 5 mph, starting 10 miles apart, and asked them how long before the bicycles met.  Then, I added a bird flying at 20 mph back and forth between the bicycles, turning around and going back whenever is met a bicycle, and asked how far the bird flew before the bikes met.IMG_2533

How Did It Go?

We had four kids this week.  It was a pretty good circle, some distractions as always but a lot of good thinking as well.

The Story of Money

This book has been going well, our daughter was able to explain the China vs. Europe paper money difference later that day when we were talking about circle.

Incremental Algebra

The kids did quite well on this activity, many of them were comfortable with parentheses and they didn’t have much problem getting the right answers.  By the end they were getting confident enough they thought the checking was a waste of time.

Coin Flip Sequences

When I started asking questions, I got some answers like “more heads after 3 tails in a row” — but I was surprised that after hearing each others’ answers they quickly converged to 50/50 no matter what.  So they seem to have a decent grasp on the idea of independence of coin flips.  Flipping was kind of hard for them, they really wanted to move their whole hands instead of just their thumbs.  For this reason the results were slightly suspect.  And of course this is a probability exercise so the results never come out perfect.  But by the end I felt pretty confident that some of the kids understood the idea of evaluating probabilities by counting occurrences from a sequence of trials (i.e., statistics).  The trickiest part was that if you have a sequence of, say, 5 heads in a row, and you’re counting outcomes after 3 heads in a row, you use this sequence 3 times (first 3, second 3, and final 3).

The Bird and the Bikes

One of the kids got the clever answer to the full question almost immediately.  Partly this was because I made it easier by asking the bikes only question first.  But still, I was impressed.  We also started computing the “brute force” way where we figured out how far the bird flew before meeting the first bike (8 miles).  The kids did okay at this too even though it’s a bit tricky.

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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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A Jellyfish Doesn’t Weigh More Than Me! (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 2 (Pythagoras).
  2. Topic: Algebra:  Using a bunch of colored glass beads, we did some simple algebra problems.  I had a chart with boxes numbered 1-8, and then by placing a stone in one of the boxes I could indicate how much that color was worth.  I started with some problems where all the values were known, and asked which pile was worth more.  For example, if Green = 3, Yellow = 2, and Blue = 5, which is bigger, 3G + 2Y or 3B?  Then I made it a bit harder, e.g., 13G + 2Y vs. 13G + B — the goal is to introduce them to canceling equal quantities from both sides.  Next, I changed the problems so that one (or sometimes two) of the values are unknown, and I give them two piles with equal values, and they have to figure out one of the unknowns.  For example, if G = 2 and Y = 3, and G + B = 3Y, how much is B?  Harder, if G = 1 and Y = 4, and G + B = Y + 2B, how much is Blue?  Even harder, if Y = 3, and G + 4Y = G + 2B, how much is Blue?
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  3. Topic: Decision Trees:  We continued the activity from last week.  First, we did a couple more tracing exercises with playing cards and more complicated trees (Tree 4 and Tree 5 from last week’s post).  Next, we did more tree building using random selections of cards, but this time, instead of using playing cards, we used Pokemon TCG cards.  Besides being an interesting theme, these cards are nice because they have a lot of different attributes (color, HP, damage, weight, size, and more).

How Did It Go?

We had all five kids this week

Pythagoras

Just like last time, the kids were pretty interested in this story.  It took about 15 minutes to read — I tried stopping partway through, but they insisted I finish the chapter.  There wasn’t a great deal of math content — but there were good lead-ins to either the Pythagorean theorem or Platonic solids.  Pythagoras definitely had an interesting life, essentially starting a cult…

Bead Algebra

I started with one shared problem for the whole table, but that didn’t work well at all — one or two kids were engaged, but the rest started drawing, writing their names in different styles, etc.  I switched to having two groups each doing the same problem, which helped, but I think I should have had a copy for each kid.  Also, I set up the problems each time — I probably should have had the kids make the problems themselves.

One of the kids (Kid A) already knew how to solve algebra problems such as 5X + 3 = 28, subtracting and then dividing.  I didn’t ever write down the problems in this way, but a different kid (Kid B) wrote one of the earlier problems down like this, and then Kid A immediately solved it.  However, Kid A wasn’t able to generalize to some of the other patterns.

For the initial problems with only given quantities, the kids quickly realized you could ignore an equal number of the same color stones from both sides — although they quickly fell back to calculation if it got more complicated.  Later, when there were unknowns, they weren’t able to apply this idea without help.  Perhaps the hardest problem was B=7, 4B + Y = 4Y + B — one of the kids solved it with the intuition that it would only work if B and Y were equal.  I pointed out the idea of grouping one B and one Y on each side, which some of them understood, but I’m not sure they could apply it.

Pokemon Trees

Tracing went well, one of the kids had been gone last week but picked it up quickly.  The kids sorted quite a few cards in a short period of time.  They also noticed when certain letters happened less frequently, and made some inferences about the output.  The most interesting part of this section was that the kids thought the cards smelled bad, calling them stinky cheese cards (mostly they smelled like plastic).

The Pokemon theme definitely interested them.  They had varying degrees of success writing down a tree.  They all understood how to make a tree, but some of them had trouble making a tree that matched the cards I dealt them.  The kid who missed last week didn’t understand that they weren’t supposed to rearrange the cards — their tree is the second picture above, which mentions A, B, and C even though I dealt only A and B piles.  Some kids used the different attributes more effectively, finding patterns in the HP, for example.  One of the kids used a range 175 lbs – 251 lbs in order to pick the middle two out of four blue Pokemon.  One kid thought it was funny that Tentacool, a jellyfish pokemon, weighed 100 lbs, thus the blog post title.  One of the kids finished two trees quickly (picture 1 above), each time sorting a bunch of other Pokemon after building the tree.

Optimized Cargo Bots (Age 7)

The Acitivites

1. Topic: Algebra. Book: Safari Park by Murphy.  5 kids each get 20 tickets to spend on amusement park rides. Work out simple algebra problems to figure out how many tickets each kid has left.

2. Topic: Algebra. I made several worksheets with simple algebra problems. I divided them by difficulty into three levels. Level One was problems like: 3 + X = 30. A Level Two problem was 3 * X = 15.  Level Three problems were 2 * X + 5 = 25.  Here are the problem worksheets.

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3. Topic: Programming. We continued with Cargo Bot programming problems this week.  Move a stack of 3 blocks from square A to square F. Then ask the kids if there are any repeated sections, and introduce sub-programs. What’s the shortest program that can solve the problem?

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How did it go?

We had three kids this week. It was a quiet, focused circle.  My daughter has been crying a lot during circle recently (much to her disappointment), and she did much better this time.  What a relief! She loves circle, but gets very upset if it doesn’t go how she expects.

Algebra

We did algebra problems about six months ago, and the kids are much, much better at it now. After each kid finished a worksheet, they could either move up to a harder problems, or choose to do another sheet of the same level.

My daughter decided to do two Level One worksheets before moving up. The other two kids moved up a level after each worksheet.  One girl sped through the problems, finishing her Level 3 sheet with essentially no help before the others had finished Level 2.  She worked on the other Level 3, and also the easier levels while the other two kids finished.

Two problems on Level 3 were: X * 0 = 0, and X / X = 1.   Each kid put down an answer for X, e.g. X = 3.  I asked if they were sure X was 3, and they said yes, 3 works.  I asked if X could be anything else, and eventually they realized that X could be any number.  I suggested putting a question mark down in those boxes.

Cargo Bots

There’s an App called Cargo Bots about a machine moving boxes around a warehouse. We used pen and paper to solve Cargo-Bot-like problems.  There are three commands: Left (L), Right (R), and Drop (D) which picks up or drops a box.

The first problem was easier than last week’s question…they just had to move a stack of identical boxes from square A to square F.  Two of the kids immediately wrote out correct (long) programs to do this. The third kid complained that the program would be very long.  I checked her progress, and it turned out she had written code to move the whole stack of blocks from square A to square B.  She was then going to move them all to square C.  I suggested she could move them directly to Square F.

Meanwhile, my daughter noticed repetitive parts of her program, so she replaced the strings of “L, L, L, L, L” with: “Do L 5 Times”.  She then proudly announced that her program was only 11 lines long now.  This was a great insight, but I had been planning to do subprograms with a different syntax.  My daughter was rather upset when I tried to explain that the robot arm didn’t understand the command “D L 5 Times.”  After a minute of discussion, she finally calmed down and we moved on.

Our subprogram syntax was to write a ‘1’ off to the side. Subprogram ‘1’ could contain any basic commands.  For example: 1 = “L L L L L”.  Then we could erase any string of 5 Ls in our main program with ‘1’.  To count the length of a program, you count each line of the main program, plus all the lines in the subprograms.

I then assigned a new task: Move 3 blocks from box C to box A, and try to make the shortest possible program.  Here’s my daughter’s solution:

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The other two kids took a bit longer to solve this, but made good progress on their own.  One girl had two subprograms, which added up to 12 lines.  My daughter helped her try to get down to 9 lines.  In the course of the help, the girls realized that it was impossible to do it in 9 lines, and that my solution had a bug…I was missing the final ‘D’ in the main program.  This made them very happy.

Math Jeopardy

The Activities

1. Topic: Multiplication: Book: Mulitplying Menace: The Revenge of Rumplestiltskin by Calvert.

2. Topic: Many. Math Jeopardy.  Here are all the questions and pictures you need to play this.

We divided the kids into two teams and played a variation of Jeopardy. The categories were Multiplication, Estimation, Patterns, Tangrams, and Algebra.  The first team to write down their answer and raise their hand got to guess. If the first team was wrong the second team got 2.5 minutes to answer.  If they were wrong, then the first team got one last chance to guess. This way the teams are never just waiting for someone to answer.

My daughter working on a Tangram question.

My daughter working on a Tangram question.

The Jeopardy Board

The Jeopardy Board

How did it go?

We had 4 kids this week. The younger circle was cancelled because many of the kids were out of town, so my son was the score keeper for the big kids circle.

Rumplestiltskin

This book has a lot of story, and little bit of multiplication mixed in.  All the kids were really into it, and my daughter asked if she could have it in her room at bedtime.

Jeopardy

We divided the kids into two teams of two, and explained the rules.  I think none of the kids had ever played trivia games before, so they didn’t know some basic strategy: for example, if the first team guesses wrong, the second team should take plenty of time before answering, to be sure to get it right.

Team 1 started out by getting pretty far ahead. This is mainly because one kid was really fast on all the multiplication problems, answering all of them except the 500 point one. No one got that one…it was 101 * 37.  Team 1 tried to do it by writing down 101 thirty-seven times, but they ran out of time. Team 2 tried to do it using base 10 blocks, making 37 piles of 101 each.  I really thought Team 2 might realized that 37 one hundred squares makes 3700, but they didn’t.

Both teams did very well on Algebra, with several kids being very close when the right answer was given.  They didn’t get to the 500 question.

Estimating was hard for the kids. The teams solved the first two by counting each object.  From the 300 onward they tried to estimate, but were never close enough to score points.

The 100 Tangram was pretty hard because the kids assumed our Tangram pictures would be to-scale. We had to give a couple hints for that one.  The kids did much better on the 200, 300, and 400, but ultimately the one kid from Team 2 solved all the Tangrams.

Patterns was a very close category, with multiple kids figuring out what the pattern was, but Team 1 was faster at writing down the answer. We only did the 100, 200, and 300.  The 300 was the hardest: Nov, Oct, Sept, Aug.  Team 2 guessed that the next three would be Sept, Oct, Nov.  Team 1 realized that it should be the months backward, but ended up guessing July, April, March.

Ultimately Team 1 by a score of 1900 to 1600.  Everyone was a good sport, though my daughter had started to get upset when Team 1 was pretty far ahead at the start (because they did all the multiplication problems early).

Overall this was very fun and motivating for the kids, and we’ll have to do it again!

A Octopus Kissed A Ninja

The Activities

  1. Topic: Addition: Book: Math Man by T. Daniels.
  2. Topic: Programming: We did two types of programming this week.  First, we reviewed how loops work, and practiced assigning a variable to itself inside a loop (X = X + 1).  Second, we introduced choosing a random noun/verb/number by drawing a word from a bag — a random version of Ask_a_Friend.  Download the programs here.
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  3. Topic: Algebra: I attempted to show the kids how to solve algebra problems involving addition and simple multiplication using Base Ten Blocks.  For example, in “X + 17 = 68”, you can first make 68, divide it into a pile of 17 and a pile of 41, and then conclude that X is 41.
  4. Topic: Reflections: A while ago I drew a picture that had every uppercase letter hidden in it, duplicated and reflected.  E.g., reflecting “B” around the vertical line gives you a butterfly.  You can download the picture here.
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How Did It Go?

All six kids attended.

Math Man

The kids liked the part where Math Man adds a whole bunch of prices in his head.

Programming

Last week, many of the kids had trouble with loops that had more than one statement inside; and also with self assignment like “X = X + 1”.  We walked through a program with both of these features: “Box_X = 1; Do 5 times { Print Box_X; Box_X = Box_X + 2 }”.  I think by the end of it most of the kids understood what was going on; we’ll try another in a couple weeks and see how it goes.  Our daughter already had a firm grasp of these two concepts, so I gave her some harder problems to work on (second page of PowerPoint above).  She got the first two but got stuck on the nested loops.

As expected, the kids really liked the programs with random nouns/verbs/numbers.  For the most part, they had no problems, but one thing that turned out to be very hard/confusing was “Do Box_X times {Print “really”}”.  Almost all the kids wanted to print the number; and one of the kids ended up print “2 2 really really”.  So clearly they need more practice using variables in control flow rather than just printing them.

Algebra

One issue with this activity is that some of the kids can solve problems like “X + 17 = 63” in their head, but I had everyone work through it methodically.  My approach was a bit wrong, I think; I gave everyone a paper square with “X” on it, and had them make 17 and 63.  Then, I said they should split 63 into two parts, 17 and what’s left, and from that, they can conclude that X is 46.  I think it would have been better to simply have them make 63, and then break it into 17 and 46 and conclude that X = 46.  The problem is, I don’t think they grasped the idea that the two sides were the same, so the extra 17 didn’t really help.  Another problem is that it’s a bit hard to see how to extend this to X – 17 = 63; you need to make a 17 and a 63, put them next to each other, see that if you subtract 17 from this new number, you get 17, so the sum must be the answer.

There’s a different approach as exemplified in the very nice iPad game Dragonbox Algebra which I may try next time I do this activity.  The focus there is much more on the fact that both sides have to be the same.  I also like how they use pictures of monsters instead of numbers, at least at first, which helps emphasize the nature of “subtracting from both sides”.  I think Dragonbox Algebra still gets something wrong, which is that instead of opening the box and revealing what’s inside, the box instead eats the other side, which doesn’t make much sense.  So my ideal version is to have some objects hidden inside the box, emphasize that both sides have the same objects, have them isolate the box, and then reveal it to show that it has the same thing as what’s on the other side.

Reflection Picture

The kids really enjoyed doing this again, working together they were able to find all the letters.