# A Bag Full of Dice (Age 9)

## The Activities

1. Topics: Geometry, Three Dimensional Shapes: Book:  Sir Cumference and the Sword in the Cone by C. Neuschwander.
2. Topics: Geometry, Three Dimensional Shapes:  A while ago we bought 5 full sets of “D&D dice” (4, 6, 8, 10, 12, and 20 sided).  We counted the edges, faces, and vertices for each of these and made a chart like in Sir Cumference, showing that “Faces + Vertices – Edges = 2”.  I also pointed out the dual relationship between 6 & 8 and 12 & 20 sided polyhedra (i.e., 6-sided has 6 faces, 8 vertices, and 12 edges; 8-sided has 8 faces, 6 vertices, and 12 edges; you can switch between the two by putting a vertex in the middle of each face and connecting adjacent vertices).
3. Topic: Numbers: We did What’s the Secret Code? from youcubed.org.  There are some clues about what the secret number is like “The digit in the hundreds place is ¾ the digit in the thousands place.”  There is more than one answer which is cool.
4. Topics: Origami, Geometry: We did Paper Folding from youcubed.org.  There are a number of folding challenges like “Construct a square with exactly ¼ the area of the original square. Convince yourself that it is a square and has ¼ of the area.”

## How Did It Go?

We had four kids this week.  As usual some kids followed along better than others, but most people were engaged for both the dice activity and the paper folding.

#### Sir Cumference and the Sword in the Cone

The kids liked the book, they laughed at quite a few of the math puns.

#### Euler’s Polyhedron Formula

The kids definitely enjoyed making the chart.  They did a pretty good job staying on task (it was easy to get distracted and start rolling the dice).  Counting the edges on some of the dice was fairly tricky but was much easier with good grouping strategies.

#### What’s the Secret Code?

The kids did well on this except that they had trouble with the decimals.  They did find one of the decimal answers, because they knew that .5 = 1/2, but I believe there were other possible decimal answers as well.

#### Paper Folding

The kids solved all the tasks except the last one, which was making a non-diagonal 1/2 area square.  I figured out a pretty complicated way to do it (by transferring the side length of the diagonal answer onto a horizontal edge), they copied what I did but it was pretty tricky (see picture above).

# 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.