Trick or Treat Math (Age 6)

The Activities

1. Topic: Counting. Book: How Many Donkeys? An Arabic Counting Tale, by MacDonald. In this simple book a man can’t figure out if he has 9 or 10 donkeys because he keeps forgetting to count the donkey he is riding. The kids caught on quickly and laughed whenever he got it wrong.
2. Topic: Maps, Spatial Reasoning, Logic: Fill in a map of a treat-or-treating neighborhood based on the following clues. Here is the clipart we used: halloweencharacters.
1. Directly to the West of your house is the Witch’s house.
2. The Zombie house is 2 houses West of the Witch’s house.
3. Olaf’s house is across the street from the Zombie’s house.
4. Elsa’s house is directly South of the Witch’s house.
5. The pumpkin house is directly East of your house.
6. The Spider house is on the very West end of the South side of the street.
7. The Butterfly is scared of the Spider. The Butterfly’s house is on the same side of the street as the Spider’s, but as far away as possible.
8. The Goblin is between the Zombie and the Witch.
9. The skeleton is directly across the street from the Spider.
10. Next to Elsa’s house is a Graveyard that takes up two houses.
11. The Ladybug’s house is right next to the Butterfly’s.
12. The Fairy can fly right across the street to the Ladybug’s house.
13. The Wizard’s house is East of the Fairy’s.
14. Anna’s house is next to Elsa’s house.

The completed puzzle

3. Topic: Estimation, Subtraction. Guess how much candy is in a container. Then put the same candy in a shallower container and guess again. Then count the candy and figure out whose guesses were the closest.

The Candy

4. Topic: Logic. Tape a Halloween character to each kid’s head. Then the kids ask each other yes/no questions to figure out who they are. The hardest part of the game is not telling your friends what is written on their heads.

How did it go?

I wore my witch costume during circle, and I organized it so the kids would get to ‘trick or treat’ after completing each activity from my bucket of small prizes and candy.

Halloween Logic

Each clue was pretty easy for kids, especially after they understood what phrases like “directly West” means. The hardest clue was: “The Spider house is on the very West end of the South side of the street.” Two of the kids figured it out on their own. The other two needed some help from their friends to understand the “south side of the street”.

Candy Estimation

The kids were very excited to see so much candy, especially when I told them that the person who guesses closest would get to trick or treat twice after the activity. Interestingly, the guesses did not get closer after I spread out the candy. Most second guesses were at least as wrong as the first guess. I guessed after the candy was spread out (and I got within 2 of the correct number).

After everyone wrote down their guesses I asked the kids to count the candy. They immediately began discussing counting strategies. They eventually decided to sort the candy by type and then count each type. However, they soon realized that some types had too many pieces to be easily counted, and they didn’t know how to add the results anyway. So they switched to counting each piece of candy as it was thrown back into the tub. Two kids both wanted to throw in candy and everyone ended up missing a bunch of pieces when the two throwers could not coordinate. They came up with 67 pieces, but I counted it again and found 72 pieces.

Halloween Twenty Questions

The kids loved seeing costumes taped to their friends’ heads, especially when one boy got ‘Princess Leia’. I told them at the start that it is very important not to tell your friends what is written on their heads, and the kids did pretty well at this. However, some kids asked questions like “Am I a zombie?” because they saw “Zombie” on their friend’s head. The hardest to get turned out to be superman. The kid knew he was a strong hero who wears red and blue, and has an S, and has a cape, but couldn’t think of superman.  Everyone else figured theirs out eventually (with some hints from me about what questions to ask). Everyone really enjoyed this activity. At the end, we had five minutes left so one of the kids moms played and had to figure out she was a pumpkin. The kids loved hearing her questions and shouting out answers. “Can you eat me?” “Yes, but it’s yucky and too chewy!”

Taking a Finger Walk (Age 6)

The Activities

1. Topic: Puzzles: Book: Taro Gomi’s Playful Puzzles for Little Hands.  We’re most of the way through now, probably one circle left of puzzles.
2. Topic: Logic: I printed sheets with 6 uncolored flowers on one side, and 9 on the other.  There were two puzzles: For 6 flowers, “There are more red flowers than purple, and more yellow flowers than red.  For 9 flowers, “There are more red flowers than purple, more blue than red, and the same number of blue and yellow.”
3. Topic: Spatial Reasoning:  Corey and I built a number of models out of Legos.  The kids each picked a model and had to copy it exactly.  They could pick it up and look at it from any angle.  Each kid copied several models.
4. Topic: Attributes, Games:  We played a couple rounds of Set with just the solid cards.

How Did It Go?

We had four kids this week.  One kid had been gone for a couple months, but now everyone is back from summer trips.  This circle went well, the kids were all interested in all the activities.

Playful Puzzles

We spent quite a bit of time on a puzzle with two kids at either end of a very windy path, with the question “Where will they meet?”  We measured by placing coins from either end.  They also enjoyed a page where you were supposed to “take a walk” with your finger by tracing a path and following various instructions along the way (e.g., “Take a rest here” or “Go around this corner really fast”).  Every kid did it once.

Flower Logic

The kids figured out the answers pretty quickly.  Interestingly, different kids figured out the second one from the first.

Lego Models

Different kids definitely had different skill levels on this one.  One kid breezed through a whole bunch, while others took quite a bit longer.  The trickiest ones were the dinosaur, because it was irregular and a bit complicated internally, and an offset colored square because it was tricky to get the right pattern of blocks on the bottom row (the kid working on it initially had the colors going the inverse rotation).

Set

We’ve played before with this circle, which the kids remembered, but not all kids remembered the rules.  Our son has played a lot, so after he got a few I said he had to let other kids get sets.  I was happy because all the other kids got at least one set on their own.

Unifix Estimating (Age 6)

The Activities

1. Topic: Estimating. Book: Betcha! by Murphy. Two friends walk around town estimating the number of people, cars, and jelly beans they see.
2. Topic: Estimating, Counting. Predict how many Unifix cubes can fit in a small bowl. How many Unifix cubes tall are you? How many Unifix cubes tall am I?
3. Topic: Logic. A little boy rides the elevator alone to and from his 15th floor apartment. Whenever he goes down, he goes all the way down to floor one. Whenever he goes up, he takes the elevator up to the 7th floor, then the stairs up to the 15th. Why?
4. Topic: Geometry. How many rectangles are in various pictures? How many triangles?
5. Topic: Spatial Reasoning.  Cover a checkerboard with rectangular tiles that are two squares long. Are some boards impossible to cover? Why?

How did it go?

This week we had four kids, after a couple weeks with just two kids per circle. The kids were all interested in the activity and stayed on task really well.

Unifix Estimating

First we each guessed how many cubes would fit in a cup. Then each kid tried to get as many as possible inside.

The guesses ranged from four to eight. At first everyone fit 9 in their cup (with the lid sealed). But I managed to fit 11 in.  After a lot of trying my son managed to squish in 12 cubes, much to his excitement.

12 cubes!!

Next we guessed how many cubes tall each kid was. We estimated by hold a stick of 10 cubes up to the kid’s body. A taller kid then decided to estimate his height by adding a few to the other kid’s height. The guesses were around 59 – 64 cubes. It was quite challenging to stick together that many unifix cubes, but the kids all stuck with it, and ended up with ~68 cubes per kid. We then guessed that I must be 100 cubes tall. I laid on the floor while kids made a very long unifix pole, and when we counted, it was 90 cubes long

The Boy in the Elevator

I got this story from Math from Age Three to Seven by Zvonkin. A little boy rides the elevator alone. When he goes down from the 15th floor, he goes all the way to the bottom. But when he goes up, he only goes to the 7th floor then walks up the stairs the rest of the way. Why?

The first suggestions were that maybe he wants exercise. Or maybe he doesn’t like the other buttons. At that suggestion, I drew them the buttons to see what they looked like:

I taped them up to the wall. No one had much to say about this, but then I asked one kid what would happen if her little brother pressed the buttons? She said he may be too short. Then another kid suggested maybe the boy was too short to reach the 15, and could only reach up to the 7. And on the way down, he can reach the 1 button easily.

Counting Shapes

In this activity, I showed the kids pictures of shapes I had drawn and we tried to find all triangles or rectangles in the picture.

At first the kids only see four rectangles in a picture like this. But after some looking, they noticed the big rectangle around the outside edge. Then later they noticed the long thin rectanble highlighted in green, and lastly the squareish rectangle in black. All the kids enjoyed this activity.

Tiling Checkerboards

I gave the kids a bunch of tiles that each would cover two squares on a checkerboard. Then I gave them increasingly interesting checkerboards to try to cover.

First they got a 4×4 checkboard which everyone easily covered.

Next was a 5×5 board:

Notice that one square is uncovered. The kids spent several minutes trying to rearrange the tiles to cover the last square. Eventually I suggested that maybe it’s impossible? If so, can you explain why? One kid suggested the tile is the wrong shape. Or maybe you should be allowed to let the square hang off the edge of the checkerboard?

Eventually, my son counted the squares on the board (5 on top, 5 down the side => 25 squares) and he said: “it’s impossible! 25 is odd, and the tiles can only cover an even number”. We checked it out with the other kids and eventually they were convinced.

My son said it should be possible because there’s an even number. But no one could do it. A couple kids suggested they would need to put the squares diagonally. I asked about the color of the remaining squares? We noticed it was always two white squares left. I asked if one tile can ever cover two white squares? The kids tried it and said no, but were not fully convinced.

The final board

This was the last board. Everyone immediately said it was impossible. One kid pointed out it would be possible if you could overlap the pieces, but no one had a clear explanation of how they were sure it was impossible otherwise.

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.

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.
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.

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.

Is it a keychain? Is it a pony? (Age 8)

The Activities

1. Topic: Logic. Book: True Lies, by Shannon. We read four more chapters from this book, and the kids were begging for more.
2. Topic: Spatial Reasoning, Tangrams. Each kid had a set of tangrams and tried to recreate shapes from a tangram book.
3. Topic: Logic, Decision Trees. I made several sets of objects, and the kids had to write a decision tree that would classify each object into my groupings.

How did it go?

We had four kids this week. The circle was great overall. The Tangram activity was really hard for everyone. The decision trees were easy for some kids, and hard for others.

True Lies

This time the kids had lots of ideas for how to solve the riddles. In fact, the kids solved 2 out of 4 all by themselves, and had good ideas for the others. The best one today was:

A man said “I have a hog so tall that a fellow can’t touch its back if he stretches his hand as high as he can.” His friends went to go see the hog, and it was just a normal size. Was he lying?

One kid said you can’t pat a normal sized hog if you stretch your hand up high, because your hand will be above its back. This turned out to be the correct answer.

Tangrams

We did Tangrams two ways. First, we had some tangram pictures printed out, where the picture was the full tangram size. You could do the tangram by arranging the pieces on top of the picture. This was easy for everyone.

Second, we had printed out a few pages from a Tangram book, where each picture was much smaller than the Tangrams. Only one kid matched any picture from that book. I also failed…it was surprisingly hard.  The kids did give it a good try, but after a few minutes there were lots of comments about how this was too hard, or impossible. After a few more minutes, we stopped, and my daughter said she was so exhausted she could hardly open her eyes.

Decision Trees

I had several stations set up with two or three groups of objects at each station. The task was to make a decision that would classify the objects into the groups I had made. You could ask any yes/no question, as long as you didn’t use the word ‘and’ or ‘or’.

First we did one tree as a group, and then kids cycled around to the different stations, moving one whenever they finished one. Some of the stations were easy, and some were tricky.

The easiest was a station that had one pile with yellow stones in it, and another pile with a mix of blue and green stones.  3 of the 4 kids started their tree with “Is it yellow?” if yes, then pile A, if no, then pile B.  The fourth kid started with “Is it green?”, but then didn’t know how to continue. I suggested looking at the other colors, and the kid switched to “Is it yellow?”

My daughter was very interested in making sure her decision trees used different questions from everyone else’s. At the end of circle we compared trees, and saw the different ways people had solved the same problem.

I watched one girl working on this set of objects. First she said “Oh, is it a keychain?” but then she noticed that there is a keychain in 2A.  She then changed the top question to “Is it a pony?”, and then followed up with “Is it a keychain?” in the “no” branch.

One kid’s decision trees.

Which Dice Did I Roll? (Age 7)

The Activities

1. Topic: Numbers: Book: The Number Devil by H. Enzensberger, Chapter 2.
2. Topics: Probability, Graphs: We revisited the activity from last week where we made two big charts, one for the sum of two six-sided dice, the other for a twelve-sided die.  This time, I wrote a computer program to simulate dice rolling, and ran it until the winning number came up 50 times.  I ran it twice each for 1d12, 2d6, and 3d4 (left to right below), and then copied each result into a chart.  Then, I showed the kids each chart and asked them how they thought I had made each of the charts.  We had only done 1d12 and 2d6 before, so including 3d4 was kind of cheating.
3. Topic: Spatial Reasoning: We continued the activity from last week where one kid made a shape using pattern blocks and then described it to the other kids.  This time I varied whether the kids worked individual, in one big group, or 2 small groups to follow the instructions.
 Original shape Copied shape

How Did It Go?

We had all five kids this week.

The Number Devil

We read most of the 2nd chapter, which talked about zero and how Roman numerals are unwieldy because they don’t have place values. We stopped and talked at a number of places, the kids were pretty involved.

Probability Charts

At first the kids didn’t want to volunteer many opinions. I had them vote on each one, for a while only one or two kids voted each time. After a while, they had grouped into two piles: (1d12, 1d12, 3d4) and (2d6, 2d6, 3d4). They decided 3d4 didn’t fit, and moved it to the other pile. So they got everything as right as possible. Then I told them that I had actually rolled 3d4 for 2 of them, and pointed out how 2 was missing on those charts and how 3 and 12 were so low. They thought it was cheating :).

Describing Shapes

We did 3 different patterns this time, the kids are getting increasingly ambitious in their patterns. The pictures above show one of the results. The top part was described as a flower; you can see that the petals are pointed in the wrong orientation. Also, the relationship between the three trapezoids wasn’t described. The final pattern was quite a bit more complicated, and I needed to help or they would have been way off (due to under specification of directions). They do well when the shapes are totally symmetric, because they can guess what it should look like.