# A Triangular Circle (Age 8)

## The Activities

1. Topic: Logic: Book: Still More Stories to Solve, by G. Shannon.  We read a few stories picking up where we left off last time.
2. Topics: Geometry, Proofs:  I started by exploring the properties of angles resulting from a line A crossing two other parallel lines B and C.  Each kid drew a picture and measured the angles, and noticed that the “Z” angles are the same, as well as the angles “translated” along line A.  I told them that this is a property of parallel lines, and then we used it to prove that the sum of the angles of a triangle is 180 degrees.
3. Topics: Patterns, Sequences:  I made a very large Pascal’s triangle (20 rows) on a sheet of poster board.  The kids looked at it and noticed some patterns (symmetry, rise then fall across a row, counting numbers, triangular numbers), I pointed out the “dog-leg” property, and then we covered all the even numbers with pennies, resulting in the pattern below (which is a discrete approximation of Serpinski’s triangle, a famous fractal).

## How Did It Go?

#### Angles

The hardest part of this was getting the abstract idea that this would hold for ANY picture, not just the one they drew.  This was particularly true for the triangular proof — it’s tricky to keep track of the difference between the name of the angle (e.g., A) and the number of degrees in its measurement.  Once you get the idea, mathematicians often gloss over this point and use the same variable to apply to both, but for the kids this was confusing.  Some of the kids understood the triangle proof, I think.

#### Pascal’s Triangle

The kids did a pretty good discovering the various properties.  I gave them some hints for the up/down property, but the only two things I directly suggested were the dog-leg property and covering the even numbers.  The pattern is really cool!

# Building Dice (Age 6)

## The Activities

1. Topic: Logic, Puzzles. Book: Playful Puzzles for Little Hands, by Taro Gomi. This is a really cute book with lovely illustrations. The puzzles cover mazes, dexterity, counting, logic, subtraction, and find-the-differences.  We did about 10 pages in 15 minutes.
2. Topic: Cubes, Spatial Reasoning. I gave each kid a cube pattern and a die. I drew six dots on one of the sides of the cube, and asked the kids to copy the die onto the cube so it would be exactly the same. Then we cut out the cube and glued it together to make a die.
3. Topic: Probability.
1. I gave the kids some colored stones and a bag. They put 10 stones in the bag that were a mix of red and green (e.g. 9 red and 1 green).  Then I repeatedly drew out a stone, and replaced it in the bag, trying to figure out how many of each color the bag contained. I made silly guesses along the way, to entertain the kids.
2. I divided the kids into two teams which each had a bag. The bags each contained 10 stones, but one bag had more red stones than the others.  One team member repeatedly drew out one stone and then put it back in the bag. The other team member kept track of how many reds vs. greens were drawn out. In the end the two teams presented their chart, and then tried to guess which bag had more red stones.

The bag with 10 stones.

## How did it go?

All five kids came to circle this week. There was lots of extra energy because we had a picnic afterward, so the kids were excited, and there were lots of younger siblings playing in a nearby room. Even so, everyone generally paid attention, except for my son who drew pictures instead of participating the in the team probability activity.

#### Building a Die

The kids were surprisingly good at this. Once I drew the six side on their cube printout, they could generally figure out where the 1 should go (with a few mistakes), and then most of them even figured out where to put the remaining sides, with only a bit of help from me. Everyone was quite good at cutting out the shape, except my son who got frustrated and cut his in half…I gave him the one I had been cutting out.

Another parent helped with gluing the cubes together since the kids all needed help at the same time.

Overall, I was impressed by their spatial abilities.

#### Probability Trials

First I had one kid put 10 stones in the bag without telling me the colors. The kids LOVED this.  Then I started drawing and replacing one stone at a time. First I drew a green stone, so I said: “Oh, I think they’re all green!”.  The kids giggled. Then I drew a red stone, and said “Oh no, I think they’re all red!”. The kids corrected me, pointing out I just drawn a green stone the time before. Then I started drawing more, keeping track of how many more reds than green I drew out.  After 20 or so trial I made a final guess. This is probably not enough trials, because I never actually got it correct.

We let 3 different kids put stones in the bag for me, but then the kids started to lose interest, so I told the last two kids that they got to be captains of the next activity. This really pleased them.  Being captain meant they got to pick their teammate, and decide whether they wanted to be the note taker or the one who drew out the stones. One captain decided to be the stone person, and the other wanted notetaker, so both positions were apparently desirable.

Four kids were actually pretty efficient at drawing out stones and making tally marks to count them.  My son was on a team of three, and decided to draw pictures instead of participate. I let him because he wasn’t distracting anyone else, and we were running out of time.

After about 30 draws, I called the kids back to the table.  One team had drawn 33 times and gotten 16 greens and 17 reds.  The other team had drawn 25 times and gotten 12 greens and 13 reds.  The kids decided that the team with 33 draws must have more reds — because they had drawn red 17 times and 17 > 13.  After circle, David pointed out that I should have then asked which bag had more greens? and complain if they again said the team that had drawn out 33 times.  But I just let it go because I didn’t think of it at the time.

In reality one bag had 3 reds and the other had 5 reds, but 25 – 35 draws were not enough to distinguish the two cases.  So we’ll have to repeat again, and give more time for repeated draws.

# Misguided Record-Keepers (Age 6)

## The Activities

1. Topic: Puzzles: Book: Taro Gomi’s Playful Puzzles for Little Hands by T. Gomi.  We did the first 10-15 puzzles.
2. Topic: Probability:  I had a cloth bag and two colors of glass beads, red and green.  Secretly, I chose four total beads and put them in the bag (3 red and 1 green).  The kids took turns drawing out one bead at a time, checking its color, and putting it back.  After the kids had done this a couple times each, I had them guess how many green and red there were (I told them there were four total).  We did that again, with 4 green this time.  Finally, I took another bag, filled bag A with 8 green and 2 red, bag B with 2 green and 8 red, and had them try to figure out (using the same draw one and put back) which bag had more green beads.
3. Topic: Geometry: I downloaded some cube net diagrams from the internet (a cube net is an “unfolded” cube).  I had one full sheet for each kid, each kid getting one of two diagrams.  I showed them how to cut out and fold the cube, and then they each made their own (I helped them assemble it once they had put glue on the tabs).  After that, I took mine (which wasn’t glued) and wrote a letter on each face.  Then I asked “What letter is on the opposite side from the A?”  After they answered, I folded it up and we checked.  After we did a few different questions, I switched to the miniature cube diagrams from page 3 of the PowerPoint.  I wrote letters again and asked the same kind of questions, and then checked by folding up the cube.

## How Did It Go?

We had four kids this week.

#### Playful Puzzles

These activities went really well, the kids were all into it.  Different kids were good at different puzzles — one kid knew right and left, another is good at counting, another was the fastest at finding things in a picture.

#### Drawing With Replacement

The kids were better than expected at not looking in the bag — when we did this with the older circle, people tried to cheat.  The first time they drew 9 reds and 1 green.  Two of the kids thought there would be 2 greens and 2 reds; the other two thought 1 green and 3 reds.  I played the fool and said there could be 4 reds, but they were onto me.  In retrospect, I should have gone farther and had them make a puzzle for me and then I could make a bad but not impossible guess (e.g., “I think there are more reds than greens because the first thing I drew was red” or guessing more red than green after drawing more green than red).   The second time there were no reds, and everyone guessed correctly (they drew about 10 times).

Then I introduced two bags, asking which had more greens.  They took a bit to get going, but then one kid took charge of one bag and another the second.  I had given them paper to keep track, but didn’t tell them what to keep track of.  They ended up all making a chart to track the TOTAL number of reds and greens across both bags, which obviously doesn’t help much.  The bags were skewed enough that the kids in charge of each bag could tell that their own bag was mostly one color, so in the end they all guessed the right answer.  But the record keepers had no idea without listening to the bag holders :).

#### Cube Nets

The kids were pretty good at cutting out their diagrams.  They were decent at folding as well.  I helped assemble because that part is pretty tricky.

The first time I asked them a “what’s on the opposite side” question, they had no clue.  But they quickly noticed how the line of four squares formed a ring around the cube with the two other flaps on opposite sides, so by the 2nd or 3rd question they were all correct.  And when I switched to a new diagram with the same backbone of four consecutive squares plus a slightly different arrangement of the opposite flaps, they still got it right.   However, once I switched to one where the opposing flaps were at opposite ends (a sort of 2 shape), they got it wrong.  And I didn’t have a chance to get to a diagram that was a zigzag, with no backbone.

# The Case of the Missing Blink Cards (Age 8)

## The Activities

1. Topic: Measurement. Book: Measuring Penny by Leedy. Lisa gets a homework assignment to measure something in a variety of ways. She decides to measure her dog Penny.
2. Topic: Sorting, Patterns, Charts. We have a card game called Blink, which has cards of six different colors, with six different shapes, and five different numbers (1 – 5). Here are some sample cards.

We calculated that there should be 6 * 6 * 5 = 180 unique cards. However, the Blink deck only contains 60 cards.  I asked the kids to figure out which cards are missing, and if there’s any pattern.

3. Topic: Protractors, Measurement, Triangles. Each kid got a protractor, and  triangle I had drawn before circle. We measured each angle, and then added them up to see what we got.

## How did it go?

We had 4 kids this week. Overall, it was a fun, focused circle.

#### Measuring Penny

We had read this book over a year ago. Some kids remembered it, but 3 out of 4 kids wanted to hear it again. This time I took several different breaks to discuss the book. For example, when they talked about ‘nonstandard units’, I measured one of the kids’ hair in number of “Corey Hands”. We also measured everyone’s ears in centimeter. The kids had a great time with the book, and stayed focused and interested.

#### Missing Blink Cards

We had told the kids about the missing cards several weeks ago, and they all remembered that there are supposed be 180 cards.  We recalculated it again, just to be sure.  Then I asked if we could figure out which cards are missing?  I started them off by sorting through the cards and showing that there were only two cards that had red lightning bolts on them.

The kids took over from there. At first they just randomly picked a color and shape, and looked for the matching cards. Soon this became unmanageable, so one of the kids suggested moving to the floor, and making a separate row for each color, and use columns for the shapes. This resulted in the following chart:

I then asked the kids several questions about the cards, which were easy to answer with this chart. How many of each color are there? 10.  How many of each shape? 10.

How many of each number? This one was trickier because the chart is not sorted by number. One kid wanted to rearrange the chart, but instead we went row by row looking for each number. We found there are 12 of each. We also found that for each color, there are two of each number.  During this time, two kids counted the attributes, and two kids were keeping notes.

One kid’s notes.

Next I asked how many cards are missing from each row? The kids looked at their chart, and said two are missing from each row. We then calculated there must be 12 missing cards total. But that would make only 12 + 60 = 72 cards, not 180, like we calculated.

I should have asked the kids where the other cards were, but instead I just showed them how to update the chart to sort by number too.  So we had a row for each color, and a column for each number/shape combination. Two kids helped me fix the chart:

The new chart.

Then we counted the missing cards in each row. This time we found there are 20 cards missing in each. 20 * 6 = 120 + 60 = 180!

#### Triangles and Protractors

I handed out several big triangles I had drawn with sharpie before circle. The kids used protractors to measure the three angles, and then add them up.

Using the protractor was still challenging for the kids, but they all made progress when I helped them. The kids added angles up to 180 four or five separate times. We also got 182  and 183 a couple times. Weirdly, when I did it myself, I got 188 for a triangle…Not sure why.  But the kids actually noticed that it was near 180 all the time, so we may be almost ready for the proof that they must always be 180.

My daughter measuring a triangle.

# Mother’s Day Origami (Age 8)

## The Activities

1. Topics: Combinatorics, Combinations:  Anno’s Three Little Pigs by M. Anno.
2. Topic: Origami:  To celebrate Mother’s Day, we made two different models: a double heart from Essential Origami and a rose from Origami Made Easy.

## How Did It Go?

We had four kids this week.

#### Three Little Pigs

We spent some time looking at and understanding the pictures.  We’ve read this book before, and while I still don’t think they fully understood it, they understood a lot more than last time.

#### Origami

The double heart model was fairly tricky and they needed some help.  But it’s a pretty cool model!

# 91 Is Not Prime (Age 6)

## The Activities

1. Topic: Addition: Book: Mall Mania by S. Murphy.
2. Topic: Primes: As a followup to last week, I made bags of 65, 91, and 95 unit cubes, gave one bag to each of the three kids, and asked them to prove that those numbers weren’t prime by making them into a rectangle.
3. Topic: Logic: We did about ten puzzles from Logic Links, numbers 1-8, 50, and 51.  To make a set of pieces for each kid I used Unifix cubes and printed-out boards.  These puzzles have clues like “There is a blue cube directly to the left of the orange cube.” and you have to figure out the position of all the cubes.

## How Did It Go?

We had three kids this week.  Two of them were somewhat out of sorts, so the group was less attentive than usual.

#### Mall Mania

This book has a bunch of different interesting adding strategies, so it would be a good lead-in to an addition activity.

#### Large Composites

Kid 65 recalled that you could get to 65 counting by 5’s, and tried that out right away successfully.  Kid 91 tried a very long (and spread out) 2-wide rectangle.  Kid 95 decided to use the hundred plate as a guide and tried out a 10-wide rectangle.  When 2-wide didn’t work, kid 91 didn’t want to try anything else.  I mentioned that you could check things quickly by skip counting and seeing whether you got your number.  We skip counted 3, 5, and finally 7, but kid 91 wasn’t interested in checking whether you could make a 7-wide rectangle.  Kid 95 worked slowly and eventually found that 10-wide didn’t work.  Kid 91 had noticed that you got 95 counting by 5’s, but kid 95 didn’t see how that helped.  I showed them that 95 did work with a 5-wide rectangle.  Kid 95 had suggested doing the prime rectangles activity with larger numbers the previous week, so it was surprising that the kids weren’t a bit more interested in this activity — might have just been a one-off problem, but it does take quite a while for them to make rectangles with this many cubes.

Even with the L and R printed on the boards, understanding what “The blue block is directly to the left of the red block” means was challenging.  In particular, you really have to pay attention to the order the blocks are mentioned.  Besides that, the kids were good at following the directions individually, and decent at combining all the clues to get the final answer.  However, they definitely aren’t good at abstracting what the clues imply — for example, one of the clues was “There are 3 red cubes.  One of the red cubes only touches red cubes” which means that there must be an L shape of red cubes in the corner.  One of the kids got tired of the puzzles and said they were bored and wanted to stop.  Again, might be a one-off, we’ll have to see how it goes next time.

# 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.

## 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.