Dots in a Square from Math Without Words

Recently, my third grade class and I have fallen in love with Math Without Words by James Tanton. This is a delightful collection of math puzzles, ranging in difficulty from fairly easy to quite hard, but all mathematically beautiful. Occasionally, I find one that is somehow related to the topic that we’re studying, but usually I pick out ones that I think the students would enjoy and that seem just barely within their reach. I also often have them work in pairs or groups of 3.

True to the book’s title, the puzzles come with no word descriptions; one has to figure out what is being asked based on a diagram or picture. Whenever I give out the puzzles, the reaction is predictable: “Huh? What is this? What are we supposed to do here?” But fairly quickly, the students start getting some ideas about what the pictures might mean, and soon there are heated discussions about how to best solve the problems.

One of my favorite discussions ever happened when I gave the students a puzzle as we were just starting our perfect squares unit. We had at this point spent several weeks on multiplication. Here was the picture “description”



And then came the problems. The first one was 1 + 2 + 3 + 2 + 1. Well who needs a diagram to explain to them how to do that? The next few were slightly larger, but still the students could do the additions fairly easily, which they proceeded to do. But then came… 1 + 2 + 3 + …+ 99 + 100 + 99 + … + 3 + 2 + 1. And that’s when they started analyzing the diagram more closely. A few fearless students started on the additions, but gave up pretty quickly.

Here is a conversation that I witnessed among a group of three students. This isn’t an exact transcript (partially because I don’t remember who said what exactly), but it is very close to it.

S1: Hey, 25 is the number of dots in the square!
S2: (after counting them carefully) Oh yeah, you’re right.
S1: And 5 is the number of dots in that longest line in the middle.
S3: So we just need to draw a square with 100 dots in the middle and count the total number of dots.
S2: Somehow I don’t think that’s going to be any easier than adding up all those numbers.
S3: You have a better idea?
S2: Wait, 5 is also the number of dots in each row of the square.
S1: And there are 5 rows.
S3: Well this problem has a 5 in it and the one we’re trying to solve has 100.
S1: I know, I know, it’s 100 times 100!
S2: Oh yeah, you’re right! 100 rows of 100.

At this point, the problem was solved as far as I was concerned and I was about to turn my attention to another group, when I heard:

S3: Wait, what is 100 times 100?
S1: I think it’s a thousand.
S2: No, I know it’s not a thousand because I remember my dad telling me that 100 times 100 is not a thousand. I think it’s a thousand ten.
S3: That doesn’t make sense.
S1: Well if it’s not a thousand, then it’s a million.

This part of the conversation continued for a bit longer, but they finally figured it out, with a tiny bit of my guidance. When I recently reminded these students that not so long ago they had an argument over the value of 100 times 100, they had a good laugh.

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Conservation of fingers and toes

Zoe likes to play with my hands. She will examine the fingers, moving them around and pulling on them. And I, being who I am, find myself thinking, “Fingers are a great math manipulative. I should seize the moment and make some math talk.” Luckily, so far she hardly ever objected. Here is a conversation that the two of us recently had.

Me: How many fingers do I have on each hand?
Z: Five and five.
Me: And how many fingers do I have on both hands?
Z: Ten.
Me: Now what if I give one finger from the left hand to the right hand, how many fingers will the left hand have then?
Z: (Hiding one finger on my left hand) Four.
Me: And the right hand?
Z: (Closing the first four) Six.
Me: And together there will still be…
Z: Ten!

We then moved some more fingers from one hand to the other to make different combinations that make 10. And then Zoe exclaimed, “And if we move one toe from a foot and make it a finger, you’ll still have 20 fingers and toes!”

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Lots of fun with Tiny Polka Dot

For a week in our kindergarten class we have been playing with Tiny Polka Dot!  I had been excited about these cards arriving for a while, and I was not disappointed.  They are colorful, sturdy and creative, and my students fell in love with them right away.  But the most wonderful thing about them is that there are a myriad of ways that one can use them, including a ton of games (some of them come as suggestions with the cards, but I’ve been coming up with a bunch of my own).


The cards come in six colors and there are cards with representations of the numbers 0-10 for each color.  The representations are: ten-frame, “standard dice”, in a circle, 2-color, randomly arranged different size dots, and numeral.

Here is a sample of the activities that we did and games that we played.

The first time I showed my students the Tiny Polka Dot cards, I gave them some time to just examine them by laying some out and passing some around.  Then, as a warm-up, I gave each student five cards at random and they had to arrange them from smallest to largest.



For the next activity, the students took turns rolling a 10-sided die (too bad there’s no 11-sided die!) and then finding a card on the board with that many dots on it.  I noticed that for numbers up to 5, the kids had no trouble finding a card without any counting, but larger numbers often presented more of a challenge.  Most students recognized 6 as 3 and 3 or 8 as 4 and 4, and some know that 3 rows of 3 is 9, but beyond that the kids had to count the dots.  By far the greatest challenge was counting the dots in a circular arrangement on the orange cards.



The following day, we played a game to practice their number bonds.  I separated out those cards with up to 6 dots on them and then laid three colors out on the floor and shuffled the remaining three into a draw pile.  The students took turns taking a card from the draw pile and then finding a card on the board to make a pair with a total of 6 dots.  Then, to up the challenge a bit, we played the same game but with a target total of 8.




Later in the week, we played different versions of memory with the cards.  In the simplest version, we used two suits of cards and the students had to find pairs with equal numbers of dots.  In other versions, students had to find pairs with a target total number of dots (anywhere between 5 and 10).





Both the students and I had a super fun week playing with the Tiny Polka Dot cards and I plan on taking them out many more times throughout the year!


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Avoid Hard Work! – A book for problem-solvers of all levels (toddler to mathematician)

It is the beginning of winter break and the family is off to a vacation.  The car ride ahead is many hours.  Ten minutes into the drive, Katie (8 years old) realizes that she forgot her book.  In desperation, she asks me, “Mom, can I look at your book?”  I check my bag for what I have.  There are two books in it: one is definitely not kid friendly (many pages, small font, no pictures) and the other one is “AVOID HARD WORK!…And Other Encouraging Mathematical Problem-Solving Tips For the Young.  The Very Young.  And the Young at Heart,” which had just arrived in the mail and I was eager to read myself.

I take out the book, with its colorful and intriguing cover, and Katie upon seeing it immediately exclaims, “Yes, that one.  I want that one!”


I give the book to her, expecting her to flip through it and give it right back to me.  She opens the book to the first picture/problem (one of interlocking gears) and asks me what one is supposed to do.


I ask her a few questions about what she sees in the picture and what she thinks the arrows mean, and soon she is pointing to the gears one by one, saying which direction they’ll turn in.  She then moves on to a much bigger problem of the same type on the next page in the book and proceeds to do the same thing.


Meanwhile, Zoe (4 years old) has noticed that Katie is repeating the same two strange words (clockwise and counter-clockwise) over and over, while looking at a fun picture with lots of question marks.  She asks Katie to explain what she’s doing, and instead of telling her younger sister to not bother her (like you can imagine sometimes happens), Katie goes on to carefully explain that the spiky circles in the picture are gears (which Zoe is quite familiar with as she’s often played with a construction set that had many of them), that the arrows show in which direction some of them are spinning and that you have to figure out the spinning directions of the rest.  She then proceeds to help Zoe work through the problem that she had just done.

I was in awe!  Here was a problem that is interesting to me and that my eight year old and my four year old can think about together.  And having now read through the whole book myself, I can say that this is very much in the spirit of the book.  Many of the problems in it, if not immediately approachable for the little ones, are easily adaptable for them, and a large part of the book describes how to do just that.

But in fact the book contains much more than that.  Not only is it filled with wonderful, thought-provoking problems, it also contains discussions of how one may approach the problems, how to deal with children’s (and adults’) anxiety and frustration, as well as the big mathematical ideas behind the problems.  Katie and I have since worked on a number of other problems in the book and there hasn’t been a single one that we did not enjoy.  I have also successfully used a few in my second and third grade classes.  There is something in the book for everyone, from age 4 to 104!

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Functions in Kindergarten – A favorite

A few weeks ago I did a series of lessons with the kindergartners about one of my favorite topics to do with this age group: functions!  I recall being quite surprised when I introduced the concept to my daughter when she was five and she really understood it.  She wanted me to come up with more and more rules for her to figure out, and these guys were no different!

Here are some of the functions that they had to guess.  Usually, I would draw the first few input/output pairs and the students would try to guess the rule.  Then I would draw a few more inputs and the students would tell me what to write/draw for the output.  Sometimes the students couldn’t explain the rule in words but I could tell that they understood it because they could predict outputs for various inputs.




As you can see, I got more creative with the drawings of the actual machines.  The students were very excited to tell me what type of hair or nose to draw.  And then they got to create some functions machines of their own!


In this machine, a wolf and a chicken went in, two wolves and two chickens came out.


This machine changed functions half way through.  First it was the doubling machine: One mouse went in, two came out; two candies went in, four came out.  But then the student got bored of that function and turned it into a related one: small turtle goes in, large turtle comes out.

The following lesson, we talked about pairs of functions that undo each others work (posts about the discussions with Katie on this topic here and here).  We started with a very simple pair: one turns squares into circles and the other one turns circles into squares (color and size is kept the same).

Sometimes they quarrel: One says “don’t turn my circles into squares” while the other one replies, “don’t you dare turn my squares into circles.”  But other times they are friends because they can undo each other’s mistakes.

Here is one where I gave the students the first function and they had to come up with and tell me what to draw for the inverse:


And finally, a pair of inverse functions that the students came up with completely on their own:


I loved it!  We talked a bit about what one would need to do to turn bad candy into healthy candy (remove all the chemicals, extra sugar, and artificial colors) and vice versa.  I asked the students why we’d want to have a machine that turns healthy candy into bad candy and they said that bad candy is more yummy  :-(.

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Our Second Fun Math Festival

We had another wonderful math festival, hosted jointly by Golden Key Russian School and Main Line Classical Academy.  Here is an account of it, written by Yulia Shpilman.

We did it, we did it, we did it again! Last Sunday, we hosted our second annual GKRS/MLCA Fun Math Mosaic Festival (a mouthful of a name ☺).  With mostly new stations, many new faces among volunteers and participants and smooth sailing on the organization front, I will solemnly and humbly declare it a success.  And now for a little bit of detail…

Stats: 10 stations (with multiple sub-stations in each), ~25 volunteers, ~75 kids ages 3-12 with many parents in tow, 2 hours

Stations: we had many new stations this year (mostly because we wanted to try new things and there is only so much volunteer capacity to fit everything in).

Folding paper polyhedra: our wonderful Tatiana brought hundreds of paper plates and bobby pins to create visually stunning icosahedrons out of paper plates.



Jean the Function Machine: Dasha created Jean out of a cardboard box and Allison and Joe made it come to life during the festival by “operating” it and teaching kids about functions (kids wrote down numbers and put them in the machine and the machine spit out the results. The goal was to figure out the rule).  The beautiful design and the little bell inside were key to the success, and the station had many “repeat customers”.


Symmetry: this was a popular, multi-faceted station, where younger kids cut out paper masks, colored symmetrical pages and played with magical mirror books. Older kids explored the symmetry of numbers, letters, words and arithmetic problems and solved a multitude of problems on what happens when you fold a square sheet of paper and punch holes through it in many different configurations. 




And here are two sample problems for those curious.


Hand-tying and t-shirt flipping: a very fun and physical station where the kids had to figure our how to untangle the rope and flip a t-shirt with tied hands.


Projections: we created literally tons of projection problems of various levels for the Nikitin blocks and geoblocks (top view, side view, front view is given – build the structure).  But the blocks are so beautiful and fun that many kids (from youngest to oldest) preferred just to build, mostly structures on the Equilibrio cards using geoblocks.




BLOKL: Lhianna, with baby in tow, showed kids how to play with Soma cubes.


Building extravaganza: It’s always fun to build something huge at an event like this. Last time, we built structures from newspaper rolls, this time we had straws and Kapla blocks.  Geoboards and Connectaballs sat humbly on the sidelines – kids played with them a bit, but I’d like to think of something more structured for next time with these awesome materials.




Games: our fearless teenagers Dmitriy and Katherine played games such Ghostblitz, Set and Swish tirelessly for two hours straight. The Elf Castle game was mostly used for building castles – I guess that works too.


Problem quest:

One of the biggest (surprising) hits of the festival were deciphering problems – each age group had a mini-quest to find and solve four of them. I was worried that with all the hands-on building and creating, the problems would hang ignored on the walls, but I was wrong! I am going to subscribe this enthusiasm to the atmosphere of the festival and the quality of the problems and not to the little chocolate penguins and Santas that kids got as prizes for solving all four problems.  


Here is a sampling of them, if you want to try them for yourselves.

Age 5-6


Age 7+


Age 9+


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Measuring everything in sight!

Last week, our second grade math class was filled with fun measurement activities, more specifically, measuring length.  Here are some highlights.  A number of the activities were adapted from the Georgia Mathematics Standards of Excellence.

For the first activity, I cut up many colorful inch long strips.  The students had to use them to measure different objects of their choice.  Lining them up without gaps or overlaps was quite tricky, but this did not stop the students from picking the largest objects they could see: teacher’s desk, mirror, blackboard.  Measuring them was tedious work, but the students persevered!




For the second activity, the students used the strips to make rulers.  Now measuring the desk or the mirror was a much easier task.  Students came up with 2 ways of doing it: 1) using one ruler and moving it along while keeping track of where the end was and 2) using multiple rulers and lining them up.  However, measuring the whole room with 10-12 inch rulers was still very tedious.  The students suggested using measuring tape or longer rulers such as yardsticks.  We then discussed the need for different units of measurement.  Would you measure the distance from the school to your house in inches?  How about the length of a pencil in yards?

Perhaps my favorite activity involved the students estimating various lengths given the length of one unit.  They were given an inch long line segment and a centimeter long line segment, and they had to make estimates for lengths such as 3 or 7 inches and 5, 14, or 20 centimeters.  Most students measured the single unit with their fingers and tried to use that to estimate the multiple units.  At the end they got to use actual rulers to see how far off their estimates were.  It was interesting to hear how the opinions of how well they did varied.  Within the same minute I heard both “I was so close, only off by an inch!” and “I was a whole inch off!”  Finally, the students had to find objects around the room that they thought were approximately of the given lengths.




The students also measured lots of objects in both centimeters and inches.  Everyone noticed that you always get more centimeters than inches and was able to explain that this was because an inch is larger.   Some even noticed that the number of centimeters is always between 2 and 3 times more than the number of inches.

What fun measurement activities have you done or would like to do with your students or kids?



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