# Double U, not W (Age 5)

## The Activities

1. Topic: Subtraction: Book: Taking Away with Tigers by T. Steffora.
2. Topics: Geometry, Spatial Reasoning:  I went to a different part of the kitchen where the kids couldn’t see what I was doing, and using pattern blocks I made a shape.  Then, I described the shape to the kids and they had to make it just from my description.
3. Topics: Games, Addition: We again played the game where we rolled two dice with varying numbers of sides and then either jumped or clapped a number of times equal to the sum of the numbers on the dice.
4. Topic: Reflections: I gave them a picture I drew with reflected versions of every letter hidden inside, and they had to find all the reflected letters.
5. Topic: Graphs: The kids were given a grid with letters on the rows and numbers on the columns, and a set of coordinates grouped into one or more colors.  They had to find those cells and color them the right color (similar to counted cross-stitch).  You can download the materials here.

## How Did It Go?

We had all five kids this week.

#### Taking Away with Tigers

A very simple introduction to subtraction; very easy for some kids but about right for others.  The most interesting part was probably the tiger facts at the end.  One kid’s review was “I didn’t care about that book.”

#### Describing Patterns

The kids were much better at this than I expected.  They got almost every shape that I described, including “3D” shapes with some pieces standing on edge.  One thing that helped was they all could look at what each other was doing; because I was away from the table, I don’t know whether it was one or two kids figuring it out and the others copying.  But from a few times I came over a little early, it looked like many of them were working independently.  They were able to choose shapes based on just the name (including hexagon and trapezoid), to get the right number of each shape, and to follow complex instructions like “Take two white diamonds and put them so each one is touching both the square and the hexagon”.

#### Dice Jumping

They continue to love this activity.  I had a bunch of different kinds of dice; the kids have mostly caught on that they should choose the die with the most number of sides each time (so they can jump as much as possible).  I had two different kids choose one die each, a third choose jump or clap, and a fourth role the dice, which helped keep everyone involved.  The most number of times we had to jump was 35; counting past 20 is kind of slow for them, so most of them jumped 70+ times.

#### Hidden Letters

The kids enjoyed this quite a bit, and only needed help a couple of times.  They understood the idea of covering half of a reflected letter in order to see the original letter.  One note is that although some letters are part of other letters (e.g., F and E), each letter is present by itself, not as part of another letter.  One tricky case was the reflected U, which they thought was a W.

#### Coordinate Drawings

This is the second time we’ve done this; they were quite a bit better than last time.  All of the kids were able to make progress on their own this time, and several were noticeably faster than last time.  Most of the kids still made a mistake or two.  One interesting thing is that the kids had trouble telling what the pictures actually were; for example, one kid couldn’t tell it was a car until I told them, but then when they showed it to their mom, she knew it was a car right away.

# Probability Gambling (Age 7)

## The Activities

1. Topic: Logic, Problem Solving: Book: More Stories to Solve by Shannon. This is a book of 15 short stories with surprise endings you can try to figure out. The kids really loved this, and begged to do more stories.  We read 5 stories, and were able to solve 4 of them by discussing them together.

2. Topic: Logic. We played Logic Links again. This time I had printed out sheets with the formations needed for the harder puzzles.

A solved logic puzzle.

3. Topic: Probability. Probability race: each kid gets two dice, rolls them, adds them up and colors in the corresponding number on the chart. The first number to get rolled 5 times wins the race.

## How did it go?

School is back in session now, so we had all 5 kids. It was a really fun circle with lots of excited contributors, and even some hard thinking.

This time we did puzzles with nine clues.  The first puzzle was easy enough that everyone solved it with few problems.  The second puzzle was harder, and eventually one kid stumbled across the correct arrangement, and the kids used her paper as reference.  This game works pretty well with a large group, though it can be frustrating for the kids who don’t happen to get the solution first.

#### Probability Race

These kids did a bunch of probability races in past years, but at first only my daughter remembered the activity.  As soon as I explained the rules, they started speeding through the races. The kids are MUCH faster now at rolling, adding, and marking their charts.  Again, all the kids enjoyed my commentary about how 7, 8, and 9 are tied on Kid X’s paper, etc.

After we finished around 8 races, we stopped and reviewed the winners.  7 won four times, 3 once, 6 twice, and 9 once.  I asked the kids what number they would pick if they could win a dollar for guessing right.  Most kids said 7.

I asked why 7 had won so many times, and a girl suggested that there are more ways to get 7 than any other number.  She counted up “1+6”, “2+5”, “3+4” to get 7.  Then she figured out there were 3 ways to get 6 also: “1+5”, “2+4”, “3+3”.

At this point I got a piece of paper, and wrote die A and die B at the top, and we made a chart of possible ways to get 7 based on various rolls of die A and B.  We found 6 ways.  In contrast there were only 5 ways to get 6.  The kids were pretty interested in figuring out how many ways to get various number.

One kid said she thought “3+4” and “4+3” should count as the same.  I said that they were different because they were on different dice, but she asked how we could tell the dice apart.  Next time, we should use dice of two different colors.

While we were figuring out the ways to roll 5, I asked what if die A rolls a 6? What does die B need to be to add up to 5?  At this point, one of the kids said that negative numbers are not allowed at her school.  I said no? And she said they never do problems like 5 – 6 at school.  I asked everyone if negative numbers are allowed at Math Circle?  They all said “Yes!”.  I asked, “Can we roll a negative 1 on a die?”, and we all agreed no.

Finally, I let the kids pick 3 numbers. If one of those numbers wins the race, they get a prize from our prize box.  They quickly (and wisely) picked 6, 7, and 8. We had a very dramatic race, ending with 6 winning everything, and lots of cheering.  We picked prizes, and circle was over.

# 2 1/2 Using Only Fives? (Age 7)

## The Activities

1. Topic: Fractions: Book: Fraction Action by D. Adler.
2. Topics: Arithmetic, Order of Operations: First I explained how to use parentheses in arithmetical expressions.  Then, using just the number 5, the four arithmetic operations (+, -, x, /), and parentheses, we made as many different numbers as we could (e.g., (5 + 5 + 5) / 5 = 3).
3. Topics: Maps, Spatial Reasoning: I had a floor plan of the first floor of our house, and using an iPad I took 7 different pictures in different places and directions.  Using an top-view of a person (with a big nose so you could see which direction he was looking), they had to show where I was standing and which way I was facing when I took the picture.  Then, I found a top down sketch of a room with a corresponding perspective drawing from one point in the room; using the paper person to show what perspective to use, the kids had to draw a picture of what the room looked like if you were standing there. Then we compared to the professional version. The original site is here, and a PowerPoint with the two pictures and the paper person is here.

## How Did It Go?

There were only three kids this week.  For whatever reason, the kids were less well-behaved than usual, particularly during the maps activity where there was a lot of (unhelpful) laughing.

#### Fraction Action

The kids paid attention but afterwards complained about the book a little.  Unlike most of the books we’ve been reading lately, it didn’t have a story.  Also, it has a number of hands-on activities that we didn’t actually do; you could spend at least half of circle on this book if you did all the activities, which were pretty good.

#### All Fives

Two of the kids already knew about parentheses, so I didn’t need to spend very much time explaining.  We went through a couple of examples, and then I gave them the 5’s problem.  They got 1 and 2 relatively quickly, and then got a bunch of multiples of 5.  The kids got 5 + 5 + 5 + 5 = 20 pretty early, but I mentioned you could do it with fewer 5’s and one of the kids got (5 X 5) – 5 after a while.  Each kid added their answer to a shared sheet (picture above).  But they had a tough time getting beyond that.  Eventually I said “If you can get 15, what else can you get?”  They didn’t get what I was hinting at, so I asked for each operation what happened when you applied it to 15.  Then they saw you could get 3 by dividing.  When the kid wrote down 5 + 5 + 5 / 5, they put the parentheses as (5 + 5) + (5 / 5).  I asked what this made, and they realized it was 11.  I pointed out that 5 + ((5 + 5) / 5) was also different.  One of the kids decided to write down a random sequence of operations, and with the parentheses they chose it was 5 – (5 X (5 / (5 + 5))), or 2 1/2.  The kids definitely didn’t realize the general patterns yet — for example, after we got 3, they didn’t see they could get 4 by taking 20 and dividing by 5.  So we should revisit this later and try to get more numbers.

#### Maps and Perspective

Unlike the little kids, the big kids were very good at knowing where various pieces of furniture were on the map, and wanted to a draw a bunch of them.  They could also take a point on the map and easily say where it was.  However, it turned out to be pretty hard for them to figure out where a picture was taken.  They could identify items in the picture, find them on the map, and decide that the person had to be nearby.  But what they had trouble doing was taking information about multiple different objects and combining them together to get the final position.  For example, in one picture, you could use the fact that the dinner table was between the camera and the beanbag to figure out where the camera was; but they wanted to put the person between the table and the beanbag.  Orientation of the camera was also somewhat difficult.  When we did the drawing activity, two of the kids drew overhead views similar to the picture I first showed them, while one of the kids drew a picture with approximately the right perspective (see above).

# Math Circle Ice Cream Shoppe (Age 5)

## The Activities

1. Topic: Combinations: Book: The Sundae Scoop by Murphy. picThis is a fun book about kids running an ice cream booth at the school fair, and how many different sundaes they could make.

2. Topic: Combinations: Use ice cream clip art to make all possible sundaes from the book, choosing one ice cream (chocolate or vanilla), one toppping (nuts or sprinkles), and one sauce (hot fudge, caramel).

The Math Ice Cream Shoppe

3. Topic: Maps: I taped up a bunch of red squares around the house, one of which had a star on the back of it.  Then I showed the kids where the starred square was on a map of our house, and they had to go find it.  Next, each kid got a turn to hide the star, and show the spot on the map.

4. Topic: Reflections: The kids reflected half letters to make a phrase. Here are the 2 phrases they did.

5. Topic: Probability: Each kid rolled two dice, added them up, then colored the corresponding square on the chart. The first number to get rolled 6 times wins the race!

## How did it go?

We had 4 kids this week. Most of the kids are starting kindergarten next week, which is bittersweet and exciting!  We’re still planning to meet for circle each week, so we’ll stay in touch.

#### The Ice Cream Shoppe

The kids enjoyed gluing the clip art ice cream to the papers.  We got a lot of repeat sundaes toward the end, but eventually, a couple kids came over to the wall to examine the sundaes we already had, and they found the last two.

#### Map Hide and Go Seek

The kids were amazingly good at reading the map, and only got the first one wrong. They all enjoyed hiding the star for their friends, so each kid took two turns.  All 4 kids seemed to totally understand how to use the map to find the starred paper.

#### Reflections

The kids hadn’t seen this type of reflection much before, so it was a bit tricky, but after a bit of help from me, the all caught on.  We worked together to sound out the two phrases.

#### Probability Race

We did the activity when circle first started, but the kids’ attention span was noticeably shorter last time. This time all 4 kids stayed on task and finished at least one race. They all enjoyed my commentary, e.g. “Oooh, 7 is pulling ahead, but 5 and 8 are close too!”.

One kid completed 4 races very quickly. He could add the dice without counting the dots. The slowest kid was not only counting the dots on the dice to get the total, but also counting along the bottom of the chart to find the right column to color.

After everyone finished, we laid out a bunch of probability races, some from this week, some from previous weeks.  We noticed that 6, 7, and 8 were the most common winners.

I asked if ‘1’ had ever won, and the kids said no.  I asked if anyone had ever rolled a ‘1’.  They claimed they had gotten a one but I pointed out they had to count both dice.  One boy showed me that you could roll two ones, but he agreed that then you would add them to get a 2.  No one could convince me that it was either possible or impossible to roll two dice and get a 1.

# Where Is The Potty?

## The Activities

1. Topics: Numbers, Place Values: Book: Earth Day — Hooray! by S. Murphy.
2. Topics: Numbers, Place Values: Using Base Ten Blocks, we practiced going from Base Ten Block representations of numbers to the numerical form, and vice-versa.
3. Topics: Addition, Subtraction, Games: We played Math Dice Jr.  Each round, you roll 1 12-sided die and 5 6-sided dice (2 of them with only numbers 1-3) and then try to make the number on the 12-sided die using the numbers on the 6-sided dice using addition and subtraction.  We had a small modification where each player could only match one set of dice per round.
4. Topic: Maps: We happen to have architectural drawings from the remodel of our backyard, so we all went outside and found things on the map.  Then, I made an overhead view of the first floor interior, and I had the kids find and draw various pieces of furniture and other things on their maps.

## How Did It Go?

We had 4 kids this week.

#### Earth Day — Hooray!

A nice introduction to place values for 4-digit numbers.

#### Base Ten Blocks

The kids were a bit better at this than the last time we did this, half of them are fairly solid and the other half still need more practice.

#### Math Dice Jr.

The kids were all in pretty different places with this game, ranging from coming up with the answer right away every time to mostly just getting it when the die matched the number exactly.  That’s why I introduced the rule that each person could only get one match each time, and also once each kid got to the finish line they watched until (all but one of) the other kids finished.  This worked out fine, the kids who were finished were still mostly interested in watching.

#### House Maps

This activity also had a pretty big range of abilities, but not in the same order as the dice game.  One of the kids had a pretty decent handle on how to read a map, and when drawing objects got the exact right spot most of the time.  The other kids were much spottier — they could do reasonably well on getting the right room, but within the room had a lot of trouble finding the right spot.  The kids definitely enjoyed running around outside.  They were good at looking at textures (such as the stone-paved part of the backyard), but the spacial layout was much harder.  Any talk of bathrooms was extremely funny.

# Snowflakes in August (Age 7)

## The Activities

1. Topic: Zero, Mathematical Operations: Book: Zero the Hero, by Holub.  This is a very funny book about Zero feeling left out and useless until the Roman Numerals attack. The kids loved the jokes in the book, but it was bit long (took 15 minutes to read).
2. Topic: Logic. We played the game Logic Links.  The kids used Unifix cubes instead of chips, so we would have enough cubes for each kid to solve the puzzle on their own.
3. Topic: Symmetry, Snowflakes. Each kid folded and cut out three snowflakes from this book:

## How did it go?

We had 4 kids this week.  It was a pretty easy circle, with nothing too challenging. The kids all loved the book, and really LOVED the snowflakes.

In this game you take 4 or 5 chips of different colors, and then read a couple clues from a card to figure out what order they belong in. For example, the blue chip is on the far left, or the orange chips are not touching.

We did the puzzles in order, but the early ones were too easy so we started skipping ahead by 4 or 5 puzzles after we solved each one.  The trickiest part of these puzzles was remembering which side is the left vs the right, especially since kids were sitting across the table from each other, so their lefts and rights were reversed.

After 10 or so puzzles, we skipped to puzzle #150, but we had to solve it together because we needed to use the board because the pieces needed to be in a particular circle.  We successfully solved it together, but some kids were disappointed they didn’t get to do it on their own.

#### Snowflakes

Everyone loved these snowflakes, and begged to do more, so we ended up doing more snowflakes and skipping the last activity I had planned.

# Ellemmennopee (Age 5)

## The Activities

1. Topic: Division. Book: Divide and Ride by Murphy. This is good book about 11 friends going to an amusement park.  How many ride cars will they fill? How many empty seats will be left over?

2. Topic: Sorting. Work together to sort the letters A – Z.  How fast can you do it?

The trickiest part of the alphabet: Ellemeno.

3. Topic: Combinations. Find all unique animals for the Math Pet Shop.  We had pictures of three animals: dog, bird, and snake, each with a bow around its neck. The kids had 3 color choices for the body of the animal and 2 choices for the bow. Here are the animal pictures.  After finding the combinations, we sorted the pet shop:

4. Topic: Spatial Reasoning, Tesselations. Cover a half-sheet of paper with wooden pattern blocks, so that no white paper is showing.

A successful strategy.

Another successful strategy.

An unsuccessful strategy.

## How did it go?

All 5 kids attended this week.  It was a very focused and fun circle.

#### Divide and Ride

The kids loved the amusement park theme of this book, and each kid got a chance to put the 11 friends into cars for a different ride. We used plastic pumpkin tokens as the friends.

#### Sorting the Alphabet

We’ve sorted parts of the alphabet the last few weeks. Today I gave them the whole alphabet on foam paper squares, and said they should work together to sort it.  I handed out a few letters to each kid. I also timed them while they worked.

Their strategy was to sing the alphabet, and then each check their letters to see if they had the next letter.  Also, the people with the X, Y, and Z put them at the end immediately.

This strategy worked pretty well since the number of cards each kid had was pretty manageable. However, singing the alphabet proved to be a bit tricky, -especially L-M-N-O-P, which the kids sang very quickly.  They ended up missing N & O initially…skipping from M to P.

They worked together surprisingly well, though at the very end they started to get silly.  They laughed really hard at the end when I found their two mistakes.  O was missing, and V was in the wrong place.

#### Math Circle Pet Shop

The first 6 or so animals were pretty easy…almost anything you picked would be a new.  Then it got tougher…a couple kids got a bit frustrated when they brought me an animal and found that there was already an exact match on the wall.

Then they started looking at the animals already on the wall.  The kids would pick a type of animal they wanted to color, and then a body color.  They could then look at the wall to see what color bow would be new.

Eventually we got 6 puppies, 5 snakes, and 5 birds.  The kids immediately suggested there should be one more snake and one more bird.  Two kids were able to figure out the missing snake and bird.

#### Tesselations

We did this a couple months ago, but most kids had not been able to finish the activity.

Last time one kid tried to cover the whole paper with squares.  Last time it didn’t work because he switched to a different shape halfway through.  This time he came up with the same strategy. Another kid saw what he doing and also started using squares. Unfortunately, there were not enough squares to cover both papers. Eventually the second kid switched to hexagons (because I was using hexagons), and she generously gave all her squares to the first kid, who was able to finish covering the whole paper.  Afterward he said it was a lot of work, like one or two hours to finish.

Two kids covered their whole papers with hexagons, after I started doing that.

My son tried to use a mix of shapes, but that gets tricky quickly, so he eventually got frustrated and just threw a handful of shapes on his paper.

The last kid and I worked together to cover the paper with blue diamonds.

After this, we all picked up the shapes and circle was over!