The joys of peas and toothpicks for all ages!

Translation of this post by Jane Kats.

A building set from peas and toothpicks is a simple and winning proposition. I can keep building from it myself again and again, inventing new structures every time.

You can make platonic solids, or you build a house or a rocket, a bird or an animal.

This is my favorite setup – where a free-style creative activity provides “development” on its own, without additional directions or instructions.

If you decide to build a cube, you will figure out the angles at which you have to put in the toothpick into the pea in the process of building. If you want to build an octahedron like the kid next to you, you will have to study it carefully, understand the shape of its faces and count its edges, figure out the angles at which the toothpicks connect.

For our math festivals, we soak 9-11 pounds of peas and buy more than 20 thousand toothpicks!

And like with any other building material or construction set, if there is a lot of it and many builders, then the buildings turn out more interesting than when there are only one or two builders.

When one person starts building a multistory building, her neighbors also get inspired to build something similar.


the buildings are growing


building houses is a good place to start!


some windmills appear


and rockets


and dinosaurs!


a tetrahedron and an octahedron


a crown


Parents are also excited to build with the kids.



The structures aren’t fitting on the dedicated tables and shelves anymore!


We have had this station at many of our math festivals and we aren’t tired of it! Some children stay with the toothpicks and chickpeas for over an hour. The parents get tired and try to move on, but the child continues his experiments with shapes and angles. Don’t rush them, they are learning math!


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Math enrichment – what is the value?

Another guest post by Yulia Shpilman on why we teach enrichment math and what it means to us.

When you look for additional math classes for your child, what do you look for?  We (and by we I mean Golden Key Russian School teachers) are often asked about what exactly we offer in our unique math classes and what skills we teach our students.  I think that we have not always been successful in articulating clearly and concisely what we actually do and why we do it.

Questions we get most often are: (1) Are we like Kumon? (we’re pretty much the opposite) (2) Do we teach “Russian math”? (not even sure what that is, though starting to realize people are probably referring to RSM) (3) Do we tutor kids who need help with their school math homework? (not really) (4) Do we train students for Kangaroo Math? (somewhat, in that we solve interesting problems, but that’s not the main goal of the program).

Well, what is it that we do, then? At GKRS, we invite the students to join us for enrichment math classes, and we take the word “enrichment” seriously in that we truly want the activities in the class to enrich their lives and be an enjoyable experience. That said, in our epoch of over-scheduled kids, any extracurricular activity must provide additional value beyond just pure entertainment (if you can even believe that math classes can be entertaining, which ours very much are 😊).

What we strive to teach our students, first and foremost, is the skill (and art) of problem solving.  

Why do we focus on problem solving?  Well, it’s a skill that is very broadly applicable to modern everyday life, regardless of whether you’re a mathematician, an engineer, a scientist or a writer.  All of us, on a daily basis, are faced with problems the answers to which are unknown and the approaches to solving which are undefined.  Both big societal problems like building the driver-less car, finding a cure for cancer or improving our education system, or smaller problems you might face in your job like figuring out a way to improve a production process or market something effectively on social media, don’t have a defined path to an answer.  It takes creativity, grit and a lot of trial and error to figure them out.  

If you think about it, traditional math classes in school teach the exact opposite lesson! There is always one (and only one!) correct answer, and you arrive at it by a set of predefined steps that you must memorize and then apply at just the right moment. This couldn’t be further from reality of most of today’s careers in just about any field.

But how do you become good at these skills? How do you develop creativity? How do you learn to persevere after trying several failed approaches? How do you know if your result makes sense? The answer is, of course, practice, and a lot of it, over long, continuous stretches of time.

And so, we will not guarantee that your child will win a math Olympiad or will manage to do mental math very quickly in her head (although over time, there will be improvements on these fronts also). That said, we can promise that she will learn to be excited, not scared, when she sees a new, unfamiliar problem, in math or other areas of her life. She will learn to analyze the information that she is given, think about what she might already know about it from previous experience (pattern recognition), try an approach and course-correct based on the results.  And we will argue that there’s tremendous value in that.

So back to my original question: what do you look for when you look for additional math classes for your child? I am most curious to hear from parents of elementary school children, but also from parents of older kids.

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3-digit numbers are tricky! Part II

This is a follow-up post to this one, about introducing 3-digit numbers to kindergartners.

I start this lesson by asking the students to come up to the board one by one and write down consecutive numbers starting with 100.  There are 4 students and they each have a chance to go up 4 times.  This is what we end up with:


Here are the numbers written by each student:

  • Student 1: 100, 1004, 1008, and 10012
  • Student 2: 101, 105, 109, and 1013
  • Student 3: 1002, 1006, 10010, 1014
  • Student 4: 103, 107, 1011, 1015

It was interesting that the students did not really pay attention to what others wrote, but each one had his/her own internal reasoning that she/he stuck to.

Next comes the most interesting part, in my opinion.  I ask whether anyone thinks that there are mistakes in the chart.  They all unanimously say that all the numbers are written correctly and there are no mistakes!

Okay, lets suppose that is true and make some comparisons.  What’s bigger, 1000 or 1002?  According to the chart, 1002 is one hundred and two, and they all right away recognized that 1000 is one thousand.  So without any hesitation, the students tell me that 1000 > 1002.

I then ask them to compare a few more pairs of numbers and we end up with the following:


Notice that everything is consistent with the original chart of numbers.  However, other than that, I had a hard time finding any logic to it.

Before fixing their mistakes, I decide to go back to making some 3-digit numbers using base-10 blocks.  Each student gets assigned their own number to make.



They then have to go up to the board and write down their number.  But this time, I tell them that they have to write down just three digits, one for the number of flats, one for the number of rods, and one for the number of unit cubes.

Then I ask each of them what number they wrote down, and surprisingly they have no trouble at all with this.  I point out how all of their numbers have exactly three digits.

We then go back to the original chart that they made with the numbers 100-115 in it.  I point to each of the numbers one by one, asking them whether it is correct or not.  For the ones that they tell me are incorrect, I ask them how to fix it.  They tell me to erase lots of extra zeroes, and at the end we are left with the following:


I confess that I did not expect them to fix all the mistakes just like that and was quite impressed.  At the same time, I have no illusions about them internalizing the concept.  I am sure that they will make their original mistakes again in the future and I am totally okay with that.  They will be solidifying their understanding of place value for many years to come, but I think this was a pretty good first step.

Do you remember your own first experiences with multi-digit numbers?  What about those of your students/children?


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Entertaining kindergartners with caterpillars, dots, and monsters

Below is a guest post by my sister, Yulia Shpilman.

We had a particularly wonderful class with my Kindergarten Math Circle last week and I wanted to share a few of our activities, in case someone else wants to try them out.

For context, this is a group of four kindergartners who come for a weekly math circle on Tuesday evenings after school.  Last week, our class was extended to 1.5 hours to make up for a snow day, so I had to get creative to hold the attention of five-year-old students for that long.  The key in this case was variety – of activities, of materials, of topics and of places in the classroom where we worked and played.  

Here is a quick description and some photos of three of our favorite activities from the class.

  1. Bunchem caterpillars.  We used a fun and very tactile toy, Bunchems, to build caterpillars based on specific instructions such as “Build a caterpillar that has two more green balls than red balls” or “Build a caterpillar that has seven balls and there are more orange balls than yellow ones”.  The cool thing about this activity (besides the super fun Bunchems) is the variety of correct answers!03281717250328171736a03281717270328171731
  2. Dice and Tiny Polka Dot.  Each student got a special die – 20-sided for those with advanced arithmetic skills and 12-sided for beginners.  Across the room, on the window sill and floor, I spread out the cards from the fantastic Tiny Polka Dot.  Each student rolled his or her die and had to find two cards that together had that number of dots on them. On one hand, it’s basic arithmetic practice.  But on the other hand, it’s visual and involves a lot of movement, so kids don’t get nearly as tired from it as they do from doing arithmetic problems at their desks. This activity also has the advantage of letting everyone work at their own pace without feeling inadequate or behind the others.0328171742a0328171743a0328171740a0328171742b0328171742c
  3. Number monsters.  We used dice again for this activity. Each student got a die and a piece of paper with eight ovals (which in retrospect was too many – six would have been better).  I announced that the ovals are the bodies of the monsters and we were going to add heads, arms and legs to make them look silly and crazy! It worked like this – you roll a die and add as many heads as the die showed (we used stickers for heads).  Then you roll again and add that many arms.  And then you roll one last time and add legs (using a different color pencil than arms so it’s easier to tell them apart for the game afterwards).  

After we were all done drawing the monsters, we used them to play a game

  • “If you have a monster that has four arms, stand up!”
  • “If you have a monster that has more heads than arms, stand up!”
  • “If you have a monster that has more heads than arms and legs put together, stand up!”

These commands were surprisingly challenging (partly because not everyone followed directions and could tell their monsters’ legs from arms 😊) but they were silly and active and we had a blast.



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3-digit numbers are tricky!

For the past several weeks, our kindergartners have been playing with 2-digit numbers: they made numbers out of base-10 blocks, wrote them on the board, translated between the two representations, counted by 10’s from different starting points, and more.  We also explored 2-digit addition with the base-10 blocks and the students had a great time with it.

Today I decided to spice things up a little and venture into making 3-digit numbers with the base-10 blocks.  The students right away predicted that 10 rods would make a flat and 10 flats would equal a cube.

They also had no trouble counting by 10’s to 100 and by 100’s to 1000 (I was actually surprised that no one except for me said 10 hundred.  They all knew it was one thousand!).  Writing one hundred and one thousand on the board using digits posed no problem as well.

Then I took 2 flats, 5 rods, and 7 little cubes and asked the students what number I just made.  With minimal help, they were able to figure out that it was the number two hundred fifty seven.  And then came the real challenge of writing that number on the board.

One student volunteered, and wrote:


I asked the students whether they agreed that this was two hundred fifty seven and they all said yes.

So then I wrote another number under it and asked the students what that one was:


Now half the students said that the number that I wrote was two hundred fifty seven and the other half said they didn’t know what my number was, but they thought that two hundred fifty seven was 2057.

I then decided to ask the students how to write the number two hundred.  Here they all agreed that it should be 200.  However, when I asked them how I should write two hundred one, opinions varied once again.  Half the students thought it should be 201 and they other half said 2001.

One student said that they thought that 201 is two hundred one and 2001 is two hundred ten.  So of course I had to write 210 on the board and ask them what that would be.  They said they didn’t know.


I decided to make one more 3-digit number with the blocks, have someone write the number on the board and see if there’s consistency to their thinking.  This time we had four flats, three rods and one little cube.

I asked one student to go up to the board to write down the number and they wrote 40031.  A second student raised their hand and asked, “I do not understand, why would you put two zeroes there?  It should be 4031!”  When I asked the student why one put the one zero there, the student replied, “That’s just the way you write it.”

So in the end we were left with the following three possibilities:


and no consensus as to which one is correct.  We left it at that.

My big question now is, how should I start tomorrow’s lesson?


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Games with tanks and mirror books

Translated from this post by Jane Kats.

In our childhood, there was a popular simple game: take a piece of paper, fold it in half and on one half of it draw a dozen small tanks (*translator’s note: drawing balloons instead of tanks works just as well).

On the second half of the page draw your shots with a soft pencil.

Then fold the paper again so that the tanks (or balloons) and shots are on the inside and flip it to the “shot” side.  The shots are visible through the paper and you can trace them again so that the shot imprints on the side of the tanks. Then you can open the paper and see whether you hit the tank or not.

We taught the kids to play this game in class – and they couldn’t stop playing!

Interestingly, not all the kids get the idea immediately and can’t always adjust their shots. Sometime a kid sees that he’s a little short, but draws the next shot even closer to the folding line.  (*translator’s note: graph paper works well for this game)


Folding the sheet of paper

We also brought mirror books and studied angles.
A right angle gives you 4 images.

And this way, you get 6 images.

Also 4 images, but a different way.

The mirrors are plastic mirrors from IKEA
(*translator’s note: paper mirror boards also work well and they are available on amazon – here’s an example).

It’s impossible to stop playing with these!

Everyone’s images are so beautiful!

What were your favorite games on a piece of paper?

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Fibonacci Trees

For two weeks in a row, in our joint 1st-5th grade math classes, a certain famous sequence made its appearance.  The activities were seemingly very different: in the first one we were climbing stairs one or two steps at a time and in the second one we were growing magical trees according to certain rules.

I was very happy when the sequence started showing up for the second time and the students immediately exclaimed, “It’s the same sequence as last week!”

This post will focus on the second of the two activities: drawing Fibonacci trees.  Here are the rules:

  1. Start with one branch.
  2. Branches never die and every year a branch grows by the same amount.
  3. The first year of its existence, a branch is green and has no offspring.
  4. After the first year, a branch becomes brown and sprouts one offspring branch per year.

We used lined paper (about an inch between consecutive lines) to keep track of the years.  The students also had to record the number of branches they had after each year.  The sequence that emerged was 1, 1, 2, 3, 5, 8, 13, 21, 34…  As I mentioned, this was not their first encounter with the Fibonacci sequence and the students recognized it after about the first five elements.

Here are some pictures of the students working.






Some of the younger students had a bit of trouble at first keeping track of which branches were supposed to be green and which ones were supposed to have offspring.  However, by the end, just about everyone had at least 7-8 correct levels/years.

Some students insisted on continuing while they could still fit the branches on the paper.  Keeping track of the branches beyond the year with 34 proved to be quite tricky.  However, several students made it as far as 89!

While some students focused on drawing as many levels of branches as possible, others tried to make theirs as aesthetically pleasing as they could (although the rule was that they had to at least get to the year with 21 branches before decorating).

Here are some of the beautiful mathematical works of art that the students created!

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