Time for Tessellations (Age 6)

The Activities

  1. Topics: Logic, Puzzles:  Book: Playful Puzzles for Little Hands, by Taro Gomi.  This is the third time we’ve done puzzles from this book.
  2. Topics: Tesselations, Geometry, Patterns:  Each kid made a square-based tessellation by starting with a 3 inch square of poster-board, drawing an inset on two adjacent sides, cutting it out, and taping to the opposite side.  They each kid filled a 8.5 x 11 sheet of paper with the tessellation and colored it.
    IMG_1923

    A rather ambitious tessellation

    How Did It Go?

We had three kids this week.  We took it pretty easy this week after the month long break.

Playful Puzzles

This continues to be a great book, lots of interesting fine details in the “What’s different?” puzzles.

Tessellations

I started by showing them how to make a tessellation and how to trace it.  Two of the kids made fairly simple tessellations, one made a very complicated one.  I cut them all out and taped them.  The two kids with easier tessellations started getting distracted and chatting part way through, so all three kids ended up nearly finishing.  Two of the kids picked rainbow colors.

 

 

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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.

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:

IMG_20150802_175018

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.

A successful strategy.

Another successful strategy.

Another successful strategy.

An unsuccessful 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!

A Ruler is 30 Centimeters Long

The Activities

  1. Topic: Dimensions: Book: A House For Birdie, by Murphy.
  2. Topic: Tesselations, Shapes: Free play with Pattern Blocks. After making some free-form shapes, I asked the kids to completely cover a half-sheet of construction paper, with no gaps.

    My son has covered the piece of paper entirely.

    My son has covered the piece of paper entirely.

  3. Topic: Measurement: Book: What is Big Compared To Me, by Harris.
  4. Topic: Measurement: I gave each kid a worksheet with various centimeter measurements on it: 1,2,4,7,9,11, etc. Then I dumped a big box of random toys out.  The kids used a ruler to find toys of each length, and then worked with a grown-up to write the toy name in the right place.
    A pile of toys!

    A pile of toys!

    One kid's chart.

    One kid’s chart.

How did it go?

I led the younger circle this week, and 4 kids attended.  It was a nice, calm circle today.  The kids were all engaged by the activities.

Tesselations

The kids all loved playing with the pattern blocks. Two of the kids used the blocks to draw objects like a car or a rocketship.  Another kid built a stack of hexagons out of the blocks. While they played, I started making a snowflake out of pattern blocks.  Immediately my son started copying my snowflake.  Another kid decided to help me instead of building her own shapes.

We played this way for 5 minutes or so, and then I convinced the kids to destroy their designs by first taking a picture of each of them with their creation.  My son took some serious convincing, especially since he was already frustrated because it was hard to copy my design.

Next I handed each kid a half-sheet of construction paper, and asked the kids to use the blocks to completely cover the paper.  Two of the kids started laying down random connected shapes. One kid started building a fancy star shape on her paper. The final kid used only squares to make a column covering the right and left side of his paper.

The kids continued to work on their tesselations for about 10 minutes. At that point my son had finished covering his paper.  All the kids admired his work, which was nice 🙂  The other kids had more work to do, so I let them have a couple more minutes before taking a photo of their work, and cleaning up.

Measurement

Two dads joined me to help with this activity. The kids are pretty willing to sound out words on their own, but I thought having a couple more parent to help with the writing would help.

The kids were very interested in the pile of toys, but they also did a pretty good job focusing on measuring the toys instead of playing. In fact, all 4 kids completely finished their chart!

The kids started to get a sense for the size of a centimeter.  They learned that a 1 centimeter toy is very small compared to a 24 centimeter toy.  The biggest length on the chart is 30 centimeters.  Three of the kids found a Barbie doll that was 30 centimeters long.  The fourth kid decided to measure a ruler, which is 30 centimeters!

1 Trillion Easter Eggs

The Activities

  1. Topic: Ratios: Book: Beanstalk: The Measure Of A Giant by A. McCallum.
  2. Topics: Combinations, Combinatorics:  I had printouts of eggs with the top and bottom separated by a line.  First, I gave them 3 colored markers and asked “If you have 3 colors, how many ways can you color the eggs?” (Answer: 9).  Then I asked about 5 colors, without actually giving them 5 colors, to see if they could figure out the pattern without actually doing the coloring.IMG_1328
  3. Topic: Multiplication: I had a bag of plastic Easter eggs with a slip of paper with a number from 1-9 inside each one.  At first, each kid drew out two eggs and had to figure out the product of the two.  After a few rounds of this, they started drawing out three eggs and multiplying them all together.
  4. Topic: Tesselations: Using pattern blocks, we worked together to make this pattern:IMG_1329

How Did It Go?

Four kids attended this week.  Everything went pretty smoothly this week.  At the end of circle, all the big kids went and voted on the birthday party activity in the little kids circle.

Beanstalk: The Measure of a Giant

This book was about ratios and was a good level for the kids.

Egg Coloring

When I asked about 3 colors, several of the kids immediately began coloring, and quickly found all the combinations.  They clearly have gotten better at looking what’s missing rather than just trying random combinations.  I asked how many there would be with 5 colors, the most popular answer was 15 (5 * 3).  Then I arranged the eggs into a grid (as shown above) with same color tops in the rows and same color bottoms in the columns.  I asked them questions about how many were there if you only had 2 colors and 1 color, and arranged the eggs in expanding “rings” to show what gets added each time you add a new color (this suggests another activity, proving that n^2 = sum of first n odd numbers).  I also pointed out how each column and row corresponded to a bottom/top color.  Finally, I asked what shape the chart was for each of 1, 2, 3 colors, and how many eggs on each side; at this point one of the kids saw that the pattern was to make an n x n square.  When I asked about 10 colors, I still got two kids saying 3 * 10, one saying 10 * 10, and one saying “I don’t know.”  After a bit more discussion they all decided it was 10 * 10.  I also asked 1000 colors and 1 million colors just for fun.

Egg Multiplication

A couple of the kids have already memorized their multiplication tables, so the two number multiplication was very easy for them.  However, three numbers was much more complicated.  In particular, they definitely don’t have the idea of multiplying place by place.  One of the largest multiplications was 9 * 9 * 6, quickly reduced to 81 * 6.  I gave them lots of hints for what 80 * 6 was: first I asked what 8 * 6 was (immediately got 48), and then using Base Ten Blocks, I said “If 6 * 8 unit blocks is 48 unit blocks, what is 6 * 8 ten blocks?”  No one ever realized the answer was 48 ten blocks — someone eventually added 80 6 times in their head to get the answer.  They were able to say that 48 ten blocks was 480 once I pointed out that 6 * 8 ten blocks is 48 ten blocks.

Tesselation

I had come up with this pattern when experimenting (playing) with the blocks one day, and thought it was both pretty and somewhat challenging.  At first, they didn’t get the rules and had trouble expanding the pattern.  But once I pointed out that the yellow was surrounded by blues and whites, and the green was surrounded by blues and whites, after some practice they were able to continue expanding the pattern.

Princess of an Alien World

I led the older circle this week. 4 kids attended.

The Activities

1. Topic: Multiplication: Book: Amanda Bean’s Amazing Dream, by Neuschwander. 2. Topic: Proofs: Prove that after every odd number comes an even number. Then prove that even + even = even.

A visual proof that Even + Even = Even

A visual proof that Even + Even = Even

3. Topic: Permutations: You are astronauts who have landed on an alien planet. The aliens do not like you.  However, if you learn their language their feeling will improve. Each alien word is made of 4 letters: ABEK. Every permutation is a legal word. How many alien words can you find?  You start as an enemy. If you learn 9 words, you become a visitor. If you learn 16 words, you’re a friend. 20 = Lord/Lady. 24 = Prince/Princess.

The clip art chart showing how the aliens feel as you learn more alien words.

The clip art chart showing how the aliens feel as you learn more alien words.

A close-up of some of the alien words we discovered.

A close-up of some of the alien words we discovered.

4. Topic: Tesselations: Make a tesselation pattern out of a square.  A tesselation is a shape that can cover a plain with no gaps.  You can make a tesselation pattern piece out of a square by cutting a shape into one side, and taping it to the opposite side. This also works with diamonds and hexagons.

My daughter's pattern piece, made from a square.

My daughter’s pattern piece, made from a square.

My daughter's completed tesselation.

My daughter’s completed tesselation.

How did it go?

This was another review circle where we repeated activities we have done in the last 2 years.  The kids were very excited to see the alien clip art on the wall, and they all remembered (and enjoyed) making tesselations.

The Amanda Bean Multiplication Book

This book is about Amanda who loves to count. She learns in the book that it is important to learn multiplication facts so she can count faster.  All  6 kids in circle go to different schools. One girl’s school is working on multiplication memorization, so at various times during the book she tried to remember a multiplication fact like 8×4. It turns out the other kids are actually faster at counting 8 four times, so I’m not sure if the message of the book was fully convincing 🙂

Even Number Proofs

I started by asking the kids if 2 is even or odd.  They all shouted “Even!”. 3? “Odd!” 4? “Even!”  I got up to about 12 before one girl said, “It’s a pattern! Even then odd!”. I then asked if they could prove that after every even number comes an odd number. I put 2 equal piles of cubes in front of me and said it was an even number. One of the kids then said, “If you add one more cube it will be odd because you can put it in either pile, because they won’t be equal”.  Another kid joined in, “If you add one more, then it will be even again!”. Next I asked them if you add two even numbers is the result even or odd?  They all said even.  I asked them to prove it.  Again I started with an even number of cubes, divided into two equal piles.  Then I gave them another even number of cubes in two equal piles.  One kid then said the result must be even because you can put the same number of cubes in each of the original number’s piles, and you know you can do it because the number is even. I then asked what about odd + odd? A couple kids immediately said odd, but then everyone thought a bit and said even.  We did a couple examples and saw that looked right. However, the kids got a bit restless before they could explain how to prove it.

Alien Permutations

I showed them the clip art chart of alien words. They were very excited by the different pictures and all wanted to get a high alien title, though one kid pointed out that she doesn’t care about princesses 🙂 They all started randomly looking for permutations. As they found new ones, they got to add it to the chart on the wall.  We got to about 21 numbers before it became pretty hard to find new entries.  At that pointed I asked how many they had found that started with A.  We counted and found 4.  There were 6 that started with K, 6 for E and 5 for B.  One kid then said we must be missing 2 that start with A and 1 that starts with B. They started searching for these missing words, and soon found them all!  This was a significant improvement in search strategies from when we last did this activity. At that time they got stuck around 18 or 19.  The kids were all happy to have reached the highest alien title of Prince/Princess.

Tesselations

The kids all remembered tesselations, and were all very impressed by my sample tesselations.  They jumped right in cutting pieces from two sides of their squares. They had to wait a couple minutes while I taped the pieces to the other sides, one by one, but as soon as their patterns were ready they started tracing.  Several kids stayed late so they could start coloring in their tesselations.  This really is a fun activity.

Terrific Tesselations

The Activities

  1. Topic: Infinity: Book: The Cat in Numberland by Ekeland, Chapter 2.
  2. Topic: Prime numbers: The kids try to discover which numbers are prime vs composite. We defined prime as any number where it is impossible to make a rectangle out of that many cubes.  The kids then worked on numbers 1 – 70 trying to make rectangles and crossing off numbers that were not prime.
  3. Topic: Tesselations: A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Each kid made a tesselation pattern piece by cutting shapes out of two sides of a square and taping the cut-out pieces to the opposite side. Then we repeatedly traced the pattern piece until it covered a big sheet of paper, and colored in the tesselation.

    A tesselation pattern made out of poster board.

    Two mathematical artists.

    Preparation

    This circle required hardly any preparation.  I had posterboard, tape, crayons, and scissors out for the tesselation activity.  For the prime numbers proofs, I got out our set of Base Ten Blocks.

How did it go?

The kids caught on to the prime number activity really quickly. They were allowed to choose any number between 1 and 70. If they could show me a rectangle or square made out of that many cubes, then we know the number is not prime, and the kid got to color in the number’s square on our wall chart.  At first the kids randomly picked numbers, but soon they figured out some patterns.  One girl started working on all the even numbers, making 2 by X rectangles.  Another girl figured out all the multiples of 10s would work if you have 10 to a side.  Several of the kids struggled to find square and rectangles…it took some time before they realized that it is better to try something than to do nothing.  We didn’t finish all the numbers 1 – 70. We’ll come back to this another time.

The Tesselation activity was a big success.  It was a bit hectic at first, when the kids were cutting out their patterns, because I had to carefully tape the pieces to the opposite side.  But once the pattern pieces were complete, it went great.  All the kids were able to trace their pattern pieces, and fill the paper with pretty tesselations.

Plastic Leaves vs. Poker Chips

The Acitivities

  1. Topic: Counting: Book: Hands Down by Dahl
  2. Topic: Tesselations: Make a tesselation out of the following shapes, using Shape Pattern Blocks.
    1. Hexagons, skinny diamonds, squares.
    2. Hexagons, triangles, diamonds.
    3. Hexagons, skinny diamonds, triangles.
    4. Hexagons, diamonds, trapezoids.
    5. Triangles, trapezoids, diamonds.
    6. Diamonds, squares.
    7. Hexagons, triangles.
    8. Skinny triangles, squares, diamonds.

      My daughter tries to cover the paper rectangle using hexagons, skinny diamonds and squares.

  3. Topic: Multiplication: Have the numbers 1 – 100 on the wall. Assign each kid a number, X, and a color of marker. The kid then counts by ‘X’, and colors in a box above the numbers they get.  For example, one kid will be assigned ‘5’, and color in a segment of 5, 10, 15, 20, etc.
    1. Which numbers have the most colored in boxes? (most factors). The least?
    2. When counting by ‘X’ how many numbers did you color in from 1 – 100?
    3. Counting by 4 always hits the counting by 2 numbers. Why?

      Each kid used a different colored marker to color in the number factors. For example, 2 used light blue, so 2, 4, 6, 8, have a light blue section.

      My daughter colors in her numbers.

  4. Topic: Weight:  Divide the kids into pairs. Each pair gets a different unit we’ll use to weigh different objects in the Pan Balance. The units are poker chips, letter tiles, and plastic leaves.  Have 4 objects that we will weigh. Each pair should guess how many of their units will weigh the same as the object.  Then they use the pan balance to get the exact answer, and fill in the answer on a chart.
    1. Which object weighed the most?
    2. Which unit of measure weighed the most?
    3. How many letter tiles does it take to equal 1 poker chip?  How many leaves?

      Plastic leaves in the Pan Balance.

 Preparation

For the tesselations, I tried out the shapes to find combinations that could tesselate nicely. For the factors, I printed out the candlestick numbers so that each kid could color in a sector for the numbers they count.  For the pan balance activity, I picked out the units of measure, and the objects we would weigh.

How did it go?

This circle was much wilder than usual.  We had all 6 kids, and it was tough to keep them all engaged.

Multiplication / Factors

First I gave assigned each kid a number, starting with 2.  All the kids were quite excited to do this activity.  I gave them 10×10 grids where they would mark their numbers. After I checked their work, they could go color on the candles on the wall.  Many of the kids noticed patterns while doing this.  The kid with ‘2’ saw that she had to color every other column. The kid with ‘3’ noticed diagonal lines in the 10×10 grid.

The kids flew through the low numbers, but it started to get hard when we did counting by 12 and 13. The kid with ’13’ got pretty frustrated.  The kid with ’12’ did ok, but ended up being off by one row.  One kid cried because someone else had the pink marker she wanted.  At that point, I decided we had colored enough numbers.

We went over to look at the candles. The kids all agreed that counting by 2s would color the most candles.  I asked them how many you would color when counting by 2s and they started to count the blue lines, but then it got confusing because the ‘2’ kid had accidentally switched from evens to odds around 47.  This confused the counters so they didn’t end up with an answer.

We kept the candles up on the wall for a month after circle, and my 3-year-old son LOVED the candles. He would constantly count from 1-100 while pointing at the candles.

Weighing

This activity did not go as well as I’d hoped. I divided the kids into 3 teams of 2, and each team was supposed to guess how many of their objects would weigh the same as my big object. Then each team got to use the pan balance to find the answer.  However, weighing took a long time, and the other kids got restless.  Also, some kids played around during the weighing. I ended up cutting this short and moving on to tesselations.

Tesselations

I had prepared bags with construction paper rectangles that should be covered by the shapes in the bag. No duplicate shapes could touch.

Half of the kids worked hard on this one, and half played around.  Some kids finished several bags of shapes, while other kids spent time swapping shapes with their neighbors.  One of the kids mentioned how she did *not* want to work on a tricky one.  Thankfully, this sentiment has been very rare in circle, and I hope it will stay that way.