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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Who’s the Oldest: Conversation with Kindergartners

Yesterday, I overheard a wonderful conversation between our Kindergarten teacher and the Kindergartners.  The kids needed to line up to exit the classroom and the teacher told them to line up by age, oldest to youngest.  Immediately, one of the kids (K1 from now on) had a question.  “But how can we do it?  I’m five, K2 is five, and K3 is 5, so that means we’re all the same age!”

Teacher: Are you all the exact same age?
K1: Yes.
Teacher: So you were all born on the exact same day?
K1: Noooo. (giggling from the other kids)
Teacher: Ah, so some of you were born before others.  When are your birthdays?
K1: July.
K2: May.
K3: May.
Teacher: When in May?
K2: May 5.
K3: May 17.
Teacher: So who is older, who was born first?
K1: K2 is older.
Teacher: Why?
K1: Because she is taller!
Teacher: So taller people are always older than shorter ones?
All kids: Noooo.
Teacher: So in order to figure out who is older we need to determine what comes first, May 5 or May 17?
Teacher: Well when you count, do you say 5 or 17 first?
K1: 17.
Teacher: So we count 1, 2, 3, 4, 17, and then five comes at some point later?
K1 (after much giggling): Noooo, it’s 1,2,3,4,5.
Teacher: So who’s older?
All kids: K2!

After that conversation it still took them a moment to get into the correct order, but they did it, and off they went! I love hearing kids of this age group reason because they are, for the most part, still not afraid of being wrong and they will say whatever comes to mind. This allows you to analyze how they think and is just plain lots of fun.

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Story Math in Kindergarten: Two of Everything

Friday is story day in our Kindergarten math class.  For our first book we read Two of Everything, a Chinese folktale.  We then had a wonderful discussion and the kids asked some very insightful questions.

Here is a brief synopsis of the story: A poor elderly couple find an old brass pot in their garden and it turns out to be magic.  Whenever you put something into the pot, two of that thing come out!  The couple started doubling everything and soon became very rich.  One day, the husband accidentally pushed his wife into the pot and then fell into it himself.  After some initial arguing, the two couples realized that they could become the best of friends and use the pot to create two of everything, one for each couple.

At the end of the story, one of the kids asked, “But would there also be two pots?”  What a great question!  I said that I thought there would be only one pot, but some kids disagreed.  They spent several minutes debating whether it was possible to put the pot inside of itself to create a second one.

The discussion then moved on to how one would make lots of something.  The kids suggested that you could just keep putting the same object into the pot over and over again, creating one more each time.  I then asked them what would happen if we put two of the same object into the pot at the same time.  They all immediately yelled out that you would get three of that object.  My next question was whether only one of the objects would be doubled or both.  This led one of the kids (and then the rest) to realize that in fact, four of that object would come out.

I wanted to ask them about putting three or more objects into the pot, but it was time to move on.  Perhaps that was for the best because they already had a lot to take in.  I hope to come back to this topic and can’t wait to read more stories with them.  I feel that stories engage this age group like nothing else does.  And I absolutely love the questions and thoughts that the kids come up with!





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Playing Math Detectives: First Week of Second Grade Math

The first week in our second grade class we did lots of time traveling.  We played the role of math detectives and helped people from different time periods solve problems.  We also learned about some ancient number systems.

On the first day we went back to several million years ago.  (The idea for our scenario was taken from the wonderful book by Julia Brodsky, Bright, Brave, Open Minds: Engaging Young Children in Math Inquiry.)  During this time period, there lived ferocious saber-toothed tigers with sharp teeth, crocodiles with awful jaws, and the first cave people, who had no strong jaws, long teeth, or sharp claws.  How could we help those early people survive in their unfriendly world of dangers?  The kids came up with making weapons out of sticks and stones, building fires, hiding in caves, running away, and climbing trees.  I think they would have had a good chance of survival! 

On the second day, we went back just 10-20-30 thousand years, to a time before numbers were invented but people had a need for keeping track.  The kids’ task was to help a farmer determine whether his shepherd was bringing back all of his sheep at the end of day or whether he was stealing or losing some along the way.

The kids were split up into groups and each group got a bag of coins (which stood for sheep).  They were told that when they were ready, I would take the “sheep” on a walk and bring them back.  They would have to determine whether any were missing.  The main rule was that they were not allowed to count in any way!

Here is their solution:

They made holes/homes for each of the coins/sheep, and when I brought back two fewer coins than they gave me, they were easily able to detect that because they had two empty holes.  I was very impressed with their inventiveness.  We then discussed and looked at pictures of how people actually did use dots, tally marks, stones, and knots to keep track of animals, money, and anything else they needed to.

Finally, on the third day we went back only several thousand years, to several locations around the globe.  We visited the Babylonians, the Mayans, and the Romans, and learned how they wrote the numerals 1 through 10 in their number systems.  The detective work consisted of helping them decide how they should write 11.

The kids examined the patterns closely, made suggestions, and discussed the merits of each one.  In the end, they came up with versions that I think the ancient people would have been happy with.

Here is a picture of what it looked like (the Babylonian version did later get modified to be a horizontal wedge followed by a vertical one).


Next week we will begin our in-depth exploration of the Hindu-Arabic number system.  More specifically, we’ll focus on the usefulness and meaning of place value and the importance of zero!

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