Count on Pascal (Age 8)

The Activities

  1. Topic: Mathematicians, Pascal’s Triangle, Geometry. Book: Mathematicians Are People, Too, by Reimer. Chapter: “Count on Pascal”.
  2. Topic: Protractors, Angles, Triangles, Quadrilaterals. We used rulers to each draw triangles. Then we measured each angle, and added them together. Next we tried out quadrilaterals and pentagons.
  3. Topic: Pascal’s Triangle. The kids filled in Pascal’s Triangle as far as they could go.IMG_20160501_174446

How did it go?

We had only three kids this week, which is usually very easy to manage. However, this week my daughter was in a very bad mood, and had to be sent out of circle during the story. The other two kids were a bit distracted by her, but did a pretty good job overall.

Count on Pascal

This was the first time I’ve read Mathmeticians Are People, Too. David’s read several chapters in past circles. I skipped a couple pages to keep the chapter shorter, but even so, the kids attention drifted. My daughter kept banging on the table and complaining about wanting a different book…she actually had to be sent out during the story which has never happened before.

The content of the chapter was pretty good, it discussed Pascal’s triangle, and his life. Unlike the ancient mathematicians, Pascal was not murdered for learning the dark arts of mathematics. By 1650, the world was a bit more prepared for math.

Geometry and Protractors

When Pascal was 12 he worked out a proof that the angles of a triangle add up to 180 degrees. Our first activity this week was to use a ruler to draw triangles, and then measure each angle and add them up.  I’m not sure if any kids had used a protractor before. It was a bit tricky to use because you have to orient the protractor correctly or you’ll get the inverse angle measurement.  With help from me, everyone measured and added their angles. We got between 170 and 185 for the sum.

Next we all drew quadrilaterals and measured and added the angles. This all took awhile because it required precision and help. At this point my daughter was tired of angles but the other two wanted to try pentagons.  So I let them continue and had my daughter work on Pascal’s Triangle.

Pascal’s Triangle

My daughter wrote down the first 10 rows of Pascal’s triangle pretty quickly, with no help from me. Eventually one of the other kids also started this activity, but their triangle wasn’t quite correct because they had missed a few of the numbers.

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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.

Laziness is a Virtue (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 5 (John Napier).
  2. Topics: Multiplication, History of Math:  I printed out one set of Napier’s Bones per kid.  I showed the kids how to use them to do multiplication of a large number by a single digit number.  A follow-up that we didn’t do this time is to use the bones for multiplying two large numbers.
  3. Topic: Sorting:  The kids attempted to sort cards with all numbers from 1 to 300.

How Did It Go?

We had four kids this week.

John Napier

The kids liked the (non-mathematical) stories about John Napier and the rooster and John Napier and the pigeons.  We’re now thankfully past the part of the book where the mathematicians all die at the end of the story.

Napier’s Bones

The kids understood how to use the bones fairly quickly.  One of the kids is very good at paper-and-pencil multiplication, so while I could slightly win on large number X single digit multiplication, the other kids couldn’t.  I showed them why it worked (see second picture above).  The place where the bones really shine is multiplying two large numbers, since you can use the same arrangement of the bones for each single digit multiplicand, so we should do that race at some point.

Sorting to 300

The kids decided to use labels, this time making sure to have labels bigger than 200.  It took them 10-15 minutes to make the labels, so next time I’m going to count this time as part of the overall time.  In particular, they didn’t think of simply using the cards that were multiples of 10 as “labels”.  They went with the same strategy of running back and forth even though I tried to get them to come up with something better; our daughter said that she WANTED to run back and forth so she could get some energy out.  I let them go for 17 minutes (which meant that circle went over by ten minutes), and then got 200 of the 300 cards sorted.  One of the kids was feeling tired and decided to first find all the numbers between 1-100 and do those first, since that required less running.  If only more of them were lazy…

Pokemon on Parade (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 4 (Hypatia).  Note: I skipped the ending of this story, only briefly summarizing (see notes below).
  2. Topic: Decision Trees, Addition:  First I reviewed the Pokemon decision tree from last week (a couple kids weren’t there).  Next, each kid “drafted” a team of Pokemon cards.  I split the cards into 5 piles, gave one pile to each kid; they each picked one and passed the pile to the right, and we repeated until each kid had 5 Pokemon.  Then, I gave out prizes based on the attributes of the Pokemon; some were based on the single Pokemon (e.g., most HP), and others were based on the entire team (largest total weight).  You can download the prizes here (the Pikachus are to give out when there are ties).   After all the prizes were awarded, the kids worked together to built a decision tree which identified whose team each Pokemon was on (that is, the labels at the leaves were the kids’ names).  I did the initial few splits until I had 5 different leaves, numbered them 1-5, and then had each kid work on one of the pages.
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How Did It Go?

We had all five kids this week.

Hypatia

I definitely wanted to read this one because it’s one of the few stories about a female mathematician.  Unfortunately, she was killed by a religious mob.  So I stopped about 2 pages from the end of the story and just summarized as “some people didn’t like that she asked questions about the things they believed in, and so they killed her”.  The kids really wanted to know how she died, but I said “No one knows for sure” (which is true, although it was almost certainly quite unpleasant).  Out of the four mathematician stories, three have ended with the murder of the mathematician — so if you decide to read this book to kids, be aware of that.  I checked out the Wikipedia page for Hypatia, and it’s consistent with the book, so they’re not making things worse than they actually were.

Pokemon Round-up

For the review, I had the kids who were there last week show the other 2 how to use the big decision tree from last week.  They had seen decision trees before, so they caught on very quickly.

The drafting went fine, although some kids were confused about the mechanics of the draft (only take 1 from each pile, don’t mix the ones you’ve already picked into the current pile, etc.)  Most kids picked carefully, but one picked randomly.  Overall, most of the kids were pretty excited by the theme.

The prize giving went well, everyone was motivated even when they had to do somewhat complicated sums.  The trickiest part was computing the total height, because it involved feet and inches.  Everyone got some prizes.  One of the kids really wanted to win the prize with the picture of Heracross (a bug with lots of attack).

I asked the kids whether they wanted to work alone or in pairs, 3 said pairs and 2 alone.  I worked with one of the 3.  They all had trouble getting started, but after some help they made pretty good progress on their own.  One kid noted that they could use a split to identify a single Pokemon that wouldn’t have worked earlier but because of an earlier split was now unique.  Some of the kids were drawing Pokemon by the end, but they did manage to finish the entire tree of 25 Pokemon.  They used a few interesting splits that were different from our big tree, including “Is bird?” and “Name starts with ‘s'”.

 

Shappy Valentine Day (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 3 (Archimedes).
  2. Topics: Factors, Logic:  We played the Big Bad Wolf game (idea from youcubed).  It’s really a puzzle, not a game, since the moves of the Wolf are deterministic.  You start with the numbers from 1 to N (we started with 6 and then moved on to 10 and 15).  Each turn, the player picks a number, and then the Wolf gets all the factors of that number.  The Wolf always must get at least one number.  At the end, the Wolf gets all the numbers that are left.  Whoever has the highest total wins.IMG_1825
  3. Topic: Codes:  We finished decoding the little kids’ Valentine’s message from last week.

How Did It Go?

We had four kids this week.

Archimedes

The kids seem fairly interested in this book, but the chapters are a bit too long — they take at least 20 minutes to read.  From now on, I’m going to figure out sections to skip so the book takes no more than 10-15 minutes — probably a bit more than half of each chapter.

Big Bad Wolf Game

The kids are good enough at factors of smallish numbers that they can play this game.  Some of the kids realized you should pick numbers that only give the Big Bad Wolf one number — but they didn’t realize you also want the things you pick to be as large as possible (as far as I know the greedy strategy of picking the largest number that only gives one thing to the Big Bad Wolf isn’t optimal, but it does pretty well).  A couple of the kids got tired of the game after a bit and stopped making progress, but one of the kids who was playing it for the first time really liked it, I heard later that they mentioned it at home.

Decoding Valentines

The kids had trouble getting started decoding the two harder messages.  I helped direct the work, and we were able to decode one of them without too much trouble.  The main challenge with the first of the two was that you couldn’t tell the boundaries between numbers (i.e., 26 vs. 2 and 6).  The other was considerably harder because it was written in a color changing pencil, was hard to read, and the rows of numbers overlapped.  As a result, we thought the first letter of the second line was part of the first line, giving us “Shappy Valentine Day”.  The kids thought this was hilarious.

 

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.

Donkeys and Salt (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 1 (Thales).
  2. Topic: Story Problems: Minute Mysteries 2: More Stories to Solve by T. Witkowski and J. Hirsch, Chapter 1 (Bake Sale).

  3. Topic: Programming: We did the Conditionals with Cards offline activity from Hour of Code.  As a follow up, I dealt three random cards as “YES” and three cards as “NO” cards, and they had to write a conditional expression to separate the two groups of cards.IMG_1800

How Did It Go?

We had all five kids this week.

Mathematicians are People, Too

I was a bit worried they might find this book boring, particularly since the chapters are a bit long, but the Thales chapter held their interest the whole time.  Our daughter immediately noticed that the story of the donkey and the salt was the same as one from a book of Aesop’s Fables she had listened to, and went to get the paper copy.

Minute Mysteries

Once you extract the relevant information, the problem to solve is “There are 180 brownies and cookies, combined.  There are twice as many cookies as brownies.  Brownies sold for 10 cents each, cookies for 5 cents.  How much money did the girls make total?”  The kids were able to solve this problem without any help at all.  One kid made two nice insights: first, that they needed to find a number X so that 3 * X = 180; and second, that the amount of money from cookies = amount of money from brownies, since there are twice as many cookies but they cost half as much.

Conditionals with Cards

We started with the warm-up exercise suggested in the activity, where I went to each kid in turn and then they had to do something different based on what I did.  This was very easy.  They also had very little problem with the main activity: each team takes turns flipping over a card, and then either they or the other team gets points depending on the card.  The first one (“If red, we get a point, otherwise they do”) was very easy; the second (“If red, we get a point, otherwise if <= 5, we get that number of points, otherwise they get a point”) some of them firmly understood and a others were a bit shaky.

The follow-up activity, where they had to write a program to separate two sets of cards, was quite a bit harder.  Two of the kids didn’t make much progress; another had a good initial idea but got stuck; a fourth got correct tests which together could make a correct program but they weren’t nested properly; and a fifth started slow but ended up with a correct program.

Afterwards, I realized that they would probably understand this much better if it were expressed in tree form (similar to the Choose-your-own-adventure activity we did a while ago).  We’ll revisit this activity soon and try again.