What is a pattern?

This week in Kindergarten our focus was on patterns.  I began by asking the kids whether they knew what a pattern was.  All said “yes” in unison.  So I called on someone to tell me.  “It’s when you have one color and then another color, and then the one color, and then another.”  Does it have to be color?  The student said that it does.  But other students disagreed.  “It could also be shapes,” a second student said.  Everyone agreed that you could alternate shapes instead of colors.  Anything else?  No, just colors and shapes.

I called on a kid and asked her to continue the pattern: woof, meow, woof, meow.  Not surprisingly, she had no trouble continuing.  The next kid was asked to continue the pattern clap, stomp, stomp, clap, stomp, stomp…  Again, this was pretty easy.  We continued with a few more variations on the theme, mostly using sounds and physical motions.  Sometimes the students would get confused, but for the most part they were able to continue.

Next, we moved on to doing some patterns on paper, and I was a bit surprised to see that these were much harder for the kids.  The “patterns” were no harder than what we had done with the sounds, mostly ABAB, ABCABC, and ABBABB, but half of the kids had to say the names of the objects out loud to spot the pattern and figure out what should come next.

I began day 2 by asking the students whether a pattern always has to repeat.  I was not surprised to hear a unanimous “yes”.  I then asked the kids one by one to continue the following patterns: 1, 2, 3, 4…;   2, 4, 6, 8…;   10, 20, 30,…;   7, 6, 5, 4.  They had no trouble with any of these.

Then came the most fun part: letting kids make patterns of their own using snap cubes.


We finished off the week with tessellations.  We read Emily Grosvenor’s book Tessellation!   The children loved the pictures of tessellating nature.  This one was voted as the favorite:


Another image that they explored for a very long time, even before we started reading the story, was this one:


They said that it looked like a beehive and also like flowers (a hexagon and all the ones that touch it make a flower).  They also wanted to count all the hexagons and had a discussion about whether it is easier to count them left to right or top to bottom.

Next week I plan on having them look for patterns on the big hundreds chart that is hanging in their classroom.  I am also looking for other ideas.  How would you explore patterns with Kindergartners?





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As many opinions as there are shapes: fun with “Which one doesn’t belong – A shapes book”

Last week, the main theme of both my Kindergarten and my second grade math classes was geometry.  We did a number of activities that explored various shapes and their properties, but the biggest hit in both classes was Which one doesn’t belong: A shapes book by Christopher Danielson.

Each page of the book consists of four shapes and a repeating question: which one does not belong?   The answer is also always the same: all of them, but for different reasons.  Doesn’t sound very interesting?  That’s because the real answers are the reasons and observations that the kids come up with, and these are truly wonderful and creative.

Here are some highlights from the discussions around two of the pages from both age groups.

Page I:


On this page, the Kindergartners right away noticed the big square and pointed it out as different, whereas with the second graders this was the last shape that they found a reason for. In both classes someone pointed to the rectangle and said that it has two short sides and two long sides.  In the second grade group, one kid also pointed out that the rectangle is the only non-square.  

It was interesting that it took both groups some time to notice that one shape was a different color (perhaps because they weren’t thinking about color as a “geometric” reason?).  In K, someone even pointed to it and said “the blue one is different because…” and then they couldn’t articulate a reason.  After a pause I couldn’t help myself and suggested, “maybe because it’s blue?”  The kid got very excited and agreed.  When one of the 2nd graders pointed out that the shape was blue another one said, “Color doesn’t count.”  I had to gently point out that every reason counts and all observations are welcome.  When the kid kept insisting that he won’t count that, I told him that he was welcome to come up with a different reason as well.

The bottom left figure didn’t belong because it was a diamond (same formulation in both age groups).  I asked the kids what makes it a diamond and the reply was, “it’s standing on it’s corner.”  (Here the kid that didn’t want to count color as a difference said that this shape was just another square that was turned around, so he wasn’t going to count that either.  I told him that he was making interesting observations but that he has to be respectful of other people’s reasons.)

Page II:



This one was a little more complicated for the Kindergartners, but they were truly fascinated by the shapes.  I also loved one kid’s observation that the top two are essentially the same shape but the heart is more curvy.  The other reasons they came up with were: the bottom left is different because it has lots of pointy ends and the bottom right is different because it has circles.  

The second graders had a ton of fun with this one.  Some of their reasons:

  1. The bottom left is the only one without something pointing in (what better way to introduce convexity?).  
  2.  It also looks like a cupcake.
  3.  The bottom right is the only one with 4 identical sides.  
  4.  It is also the only one with circles for its corners.  
  5. The top right is the only one with all straight lines.  
  6. The top two are the only ones that have very pointy corners.  
  7. The bottom left is the only one not with 4 sides.

There was a fun discussion about how many sides the heart and the cupcake have. Someone argued that they both have 4 (one kid said, “I know it’s not a straight line, but it’s like a side”).  Others said that the heart had none.  Still others thought that you couldn’t tell.  

When we work on problems with the second graders, the most common question I hear from them is “Is this right?”  One of the many things that I loved about this activity was that I did not hear this question a single time.

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Little Kids and Big Numbers: Reading How Many Jelly Beans?

I have noticed on a number of occasions that little kids are fascinated by big numbers.   They like to hear them said, they like to see them written, they like to have them visualized.  So it was not surprising to me that our kindergartners loved the book How many Jelly Beans? by Andrea Menotti, which we read last week in class.  Their reactions, on the other hand, were wonderful and unpredictable as always.

Before starting the book, I told the kids that this book had some pretty large numbers in it and asked each of them to name the biggest number they know.  One kid said one hundred, another said a million, a third one said 700, and the fourth one said a gazillion.  I told the last kid that I wasn’t sure quite how big a gazillion was, but everyone seemed to agree that it was very big.

We then started reading the book.  At first the kids insisted on counting all the jelly beans that were drawn, but I warned them that pretty soon that would become a very difficult task.  We first counted a set of 10 jelly beans, then 20 (discussing how it is 10 and another 10 in the process), followed by 25, and then 50.  The next frame consisted of 75 beans and I told them that if we continue counting we may never get to the really big numbers later in the book.  The kids felt a bit torn between the two desires but ultimately decided to just looked at the pictures from then on.


The kids were very excited with each new picture that contained more jelly beans than the previous one.  As there became more and more of them, the jelly beans became smaller.  “That’s because they are far away” one kid said.  “Oh, like the moon and sun?” I asked.  Yes, exactly like that.  In fact, the kid repeated this several times throughout the story.

When we got to 1 million jelly beans the excitement went through the roof.  I asked them if they thought it was a lot of jelly beans and they looked at me like “really?  You have to ask that question?”  and then said in unison “Yes!!!!”

I then decided to ask them if they thought that there were more or fewer than 1 million people in the world.  Some thought that it was fewer and others were unsure.  So I told them that there are more than 7 billion people on earth and that a billion is 1000 million.  I don’t think that they had a real sense of how many that is, but looking at the million tiny jelly beans they sensed that it was a lot.


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Drawing stars and triangles

Last week we did several really fun activities from the book A Collection of Math Lessons, A: Grades 1-3  by Marilyn Burns and Bonnie Tank.  Here is an account of one of them.

The activity was “How many stars can you draw in a minute?”  It turned out that some of the kids in the class knew how to draw 5-pointed stars and some didn’t.  I told the ones who didn’t know how to draw a star that they could draw triangles instead and we’d work on their stars later.  

After practicing drawing the stars and triangles, the kids had to predict how many of them they could draw in one minute (everyone had to draw either only stars or only triangles).  Then I actually timed one minute and they tried to draw as many of their figure as they could.  Finally, they had to count how many they drew and determine how close this was to their estimate.

The kids really enjoyed the drawing of the stars and the triangles and I was impressed at how close many of their predictions were.  However, a number of them made mistakes when counting.  I told them that they could try grouping their shapes by 10s (something we’ve been doing a lot of in class lately), which would make the counting easier.  This was a bit hard for them because usually there weren’t 10 figures on the same line.  However, when they succeeded at grouping the 10s correctly, they could easily see how many there were.  The kids got to practice estimation, drawing, time sense, subtraction, and place value skills all in one activity! 

Observing the activity, there were two things that I found particularly interesting.  In the first round, the kids who made higher predictions drew more stars/triangles.  Was this a coincidence or a self-fulfilling prophesy?  Second, one of the kids who drew triangles drew way more of them than anyone else (triangles or stars).  Seeing this, the kids who drew stars in the first round chose to draw triangles for the second time around and made much higher predictions than before.  However, the number of triangles they drew was very close to the number of stars they were previously able to draw.  Not sure what conclusions to draw from this, if any.


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Spontaneous Estimation in Kindergarten

One of the games we played in class with the kindergartners consists of two decks of cards.  The game is called Turbo-count and I will write about the it separately, as it is a great one for this age group, but this post is about an unplanned math conversation that happened around the two decks.  My favorite conversations with the kids are usually the ones that are not initiated by me.

I took out one of the decks and immediately one of the kids said, “Wow, that’s like 1000 cards!”  I asked him if he really thought it was 1000.  He looked at it again and said “Nah, more like 100.”

“Hey, lets count them,” I suggested.  “But first, everyone tell me how many you think there are.”


Kid 1 stuck to his 100, kid 2 estimated 40, kids 3 and 4 both guessed 25.  Then we counted together.  There turned out to be 48 cards.  I then asked who was the closest and all the kids chimed together that it was kid 2.  This made for one happy and proud kid.

Kid 1 then pointed to the second set of cards in the game and said with a smile, “I meant 100 including those.”

This second set was clearly somewhat smaller, but not by a whole lot.  


I had the kids try guess how many they thought was in that one.  The kids’ estimates were 29, 25, 26 and “something in the 20s”.  It was hard for me to tell how much the first few estimates affected the later ones, but I was impressed that all the kids understood that the amount should be less than 48 but at the same time not too small.  We then counted and got 36.  The kids seemed satisfied with that.  It was time to start the actual game!

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Kids are like marbles

In addition to teaching Kindergarten, 2nd grade, and partially 3rd-4th grades at the Main Line Classical Academy, I am also continuing to teach enrichment math at the Golden Key Russian School.  There, I have two groups of Kindergartners, one group of 3rd-4th graders, and one group of 5th-6th graders.


Last week with the Kindergartners we focused on different ways of making 10 out of two smaller numbers.  We first figured out the ways using our fingers and then wrote them out using digits on the board.

The next activity was a bit more tricky.  We took out ten marbles and counted them together carefully.  Then, all the kids had to close their eyes and I put some of the marbles into the cup.  When they opened their eyes, the kids had to guess how many marbles were hiding in my cup.

At first this was very challenging for the kids; they would make random guesses and ask questions like “how can I know how many you hid?”  But then, one by one, they started to catch onto the trick.  The cup and marbles then went around the room and each kid had a turn at hiding some of the marbles while the other closed their eyes.

How many did I hide?


Now lets take them out and see whether you guessed correctly.


Now the next person gets to hide the marbles.


And a few more.



At the end of the class we played a similar game but only with kids instead of marbles!  One person would close their eyes and some of the remaining kids would hide under a blanket.  Then the person who was “it” would open their eyes and had to determine how many kids were under the blanket.  There were seven kids total.

Here is what it looked like:




The classic mistake was for kids to forget to count themselves.  Then I would ask them, “How many kids are not hiding under the blanket?”  When they would say the number of kids they saw, I’d follow up with, “So you’re hiding under the blanket?”  And then they’d laugh.

I love activities that use kids as manipulatives!

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Exploring Rectangles

Friday is a special day in our math classes at the Main Line Classical Academy.  We read and discuss mathematical stories and we do exploration projects.  Here is the project that we did with the 2nd-4th grades last Friday.

It began with one of my favorite questions to discuss with kids: What is a rectangle?  Some of the kids in each class had participated in previous discussions with me on this topic, but this was close to 2 years ago and so probably had very little effect on the outcome.

Here is what the boards looked like after the 2nd grade and the 3rd/4th grade discussions respectively:



The kids used a lot of hand motions in their initial descriptions, but I told them to pretend that we were talking on the phone and I couldn’t see them.  They would also sometimes come up with very long and convoluted explanations, which I also refused to write on the board.  After each initial set of properties, I’d try to draw a shape on the board that fit them all but was not a rectangle or did not fit some of them and was a rectangle (some of the shapes unfortunately did not make it into the pictures).  The kids had a lot of laughs when I would draw a silly shape and ask them “is this a rectangle?”  In the end though, I believe that we settled on a set of properties that succinctly characterized rectangles.

The second part of the class consisted of making all possible rectangles out of a given number of squares.  The kids had to make them out of snap cubes and then draw them on graph paper.  The second graders all got 12 snap cubes while the 3rd/4th graders initially got 12 and then each their own different number between 18 and 32.

I was very surprised that no one tried to draw the same rectangle in different orientations.  Some kids did, however, try to make and draw rectangles with holes in them.  A few of the second graders initially had trouble because the squares on the graph paper were smaller than the snap cubes, so tracing the structure did not work.  However, after a brief discussion, they were all able to make the one-to-one correspondence between the cubes and the squares.

Here are some pictures of the process:


In the end, we discussed with both groups how to make sure that we have made all the possible rectangles.  One of the older kids pointed out the connection with factors/divisors of a number.  None of the kids had formally studied area or multiplication (although most know what those are to various degrees), but those will both be big topics in the 3rd/4th grade class this year.  I think that this served as a good indirect introduction to them.

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