Highlights of 2017

2017 was a very exciting year for me. Here are some highlights in three categories.

Teaching

This has been my first full (calendar) year of teaching. The first words that come to mind when I think about teaching are “wonderful” and “amazing”, but “hectic” and “stressful” are close seconds. That being said, I feel extremely fortunate to have found something I feel so passionate about and to work with awesome kids on a daily basis.

Book

This year I became a published author. The excitement of holding my first published book in my hands compared only to that of holding my newborn baby. The main difference is that with a baby your initial concerns are all about protecting it from “the world” while with a book they are quite the opposite – how to get it out there (and in the spirit of getting it out there, here’s the link where you can get it or write a review 🙂 ).

Family

Zoe started Kindergarten this year and Katie entered 4th grade. Although I post about them much more rarely than I used to, we still have many exciting conversations on a variety of topics. They both like math, but they also have a number of other interests: gymnastics, singing, piano playing, and Harry Potter topping the list. And then there’s my husband, who supports us in all of our crazy endeavors 🙂

I am looking forward to continuing these projects in 2018 and perhaps adding to the collection 🙂

Happy New Year!

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Discovering the area of a trapezoid

Below is a guest post by Dmitry Bryazgin (originally written in Russian). Dmitry runs a small math circle near Princeton for students in grades 3-5. On the particular day described in the post, there were 4 students present.

The main task for the day was to find a general formula for the area of a trapezoid. The class had previously derived formulas for the areas of a rectangle, right triangle, acute triangle, and parallelogram.

For some additional motivation, I promised to give a prize to anyone who independently solves the problem.  I then drew all the previous area formulas that we had encountered and reminded the students of how we had derived them.  

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Each student received a piece of graph paper and a pencil, and the search began.

The children puffed, cutting corners off and drawing all sorts of add-ons to the trapezoid.  I allowed them to use the board if they thought that would help them.  Some of the kids rushed to the board, but this did not lead to any progress on the problem.

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Then I gave them the first hint: try to complete the trapezoid to a parallelogram because we already know how it find the area of the latter. This did not help.

Soon I started to notice that the students were getting tired, so I decided to give them a second hint: try to add a second, upside down, trapezoid to the original one.

The students once again took to their drawings and soon started to yell that they solved the problem.  But a quick check of their work revealed that the solution was not yet in sight.  Nonetheless, I was pleased to see that the children were not giving up and were showing a real interest in the problem’s solution.

At one point I realized that one of the students was close to the first step of the problem – the construction of the parallelogram – but the drawing on his paper was very small.

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So I drew a grid on the board with a trapezoid on top of it and asked the student to show his construction. He came up and accurately transferred his drawing. This was the first successful step towards solving the problem. And then came the most interesting part.

I said that now we need to make that next step and find that “general formula”. The formula for the area of a parallelogram was written slightly higher on the board. Applying it required a substantial leap inside a child’s mind: realizing that the formula could be used for a DIFFERENT picture and understanding where in this new picture are the sides “a” and “b”. This, by the way, is already an abstract reasoning skill.

I pointed to the lower base of the trapezoid and asked the students for its length – all the answers were incorrect. I gave the students some time to think and then asked the question again. And suddenly, one girl exclaimed, “it’s a+b since we have the upside down trapezoid!”

If I could have have leaped to the heavens, I probably would have done so. I was really waiting for this answer, but had started to lose hope. After this, two other girls almost simultaneously exclaimed that we need to multiply by this by “h” since we have a parallelogram. And the student who had made the first step now added the finishing touch by pointing out that we need to divide by 2, since the parallelogram contains two identical trapezoids.

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Until the very last moment, I did not know whether we would be able to solve this problem with the students.  This unpredictability of the lesson is what made it so interesting and what I wanted to share that with you.

By the way, no one ended up getting the prize, but it seemed that this was no longer important.

Afterthoughts

Looking back at the lesson, I thought of two mistakes/improvements that would have simplified the task for the students.

Mistake #1: I should have had the students draw a specific trapezoid on their papers with given dimensions. Without this, everyone ended up drawing whatever they wanted, the trapezoids were crooked and this got in the way.

Mistake #2: I initially did not draw the parallelogram on the board, whereas this was the key to solving the problem. I should have drawn it on a grid, along with the trapezoid, and had the students transfer both to their papers. Only after this should the search for the area have begun.

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“Math you can play” – books with games

Games are an important part of the elementary school math curriculum that I develop and teach.  Through different games students learn and reinforce a variety of skills, ranging from arithmetic to spacial reasoning to logic.

In my classroom, I have many store-bought favorites: SET, SWISH, Tiny Polka Dot, Q-bits, and many games by The Brainy Band (HurriCount, Numberloor, TraffiCARS, Multibloom, Splitissimo, …), to name a few.

However, a number of games that we play require nothing more than a printed game board, a deck of cards or some counters.  I have found a wonderful collection of such games in the “Math you can play” series by Denise Gaskins.

There are three books in the series: Counting & Number Bonds, Addition & Subtraction, and Multiplication & Fractions.  Many of the games in the first two books  can be played with kindergartners but also have sophisticated logic and strategy elements that can be enjoyed by much older students.  I use games from the third book primarily with grades 3 and up.

Here are 2 pictures of my 3rd grade class playing a game called Thirty-one that I learned about from the Addition & Subtraction book:

This is a 2 player game but we first played it in teams.  For setup, you just lay out the Ace through 6 of each suit in rows.  Then the players (teams) take turns flipping cards upside down and adding the values to the running total of all flipped cards (Ace has a value of 1).  The goal is to get exactly 31 or force the other team to go over 31.

After the first game, the students realized that making 24 is good, and after the second game they noticed that the same thing is true of 17.  I then split them up into pairs to play individually and they all made more discoveries for themselves.

We had a great class discussion about the game but were still far from figuring out the “full” optimal strategy.  I hope to come back to the game a bit later in the year.

Some of my other favorites from the books are Number Train, Snugglenumber, Tiguous, The Product Game, and Ultimate Multiple Tic-Tac-Toe.  Most of the game descriptions have the history of the game as well as a number of variations.  The books have contributed and inspired many wonderful additions to my curriculum.

 

 

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Isometric graph paper and 3D pictures

Translated from this post by Jane Kats.

At our math festivals and during Mousematics lessons we actively use isometric graph paper, aka “triangular grid paper”.  Sheets with a triangular grid can be downloaded on our site here in the free downloads section, next to symmetric butterflies and a big color-in Christmas tree.

We use this paper in a variety of ways.  For example, to draw diagrams of structures we build using pattern blocks – this is a great fit because these consist of hexagons, equilateral triangles, rhombuses, and trapezoids with 60 degree angles.

Triangular grids are also very useful for teaching children to draw their constructions in 3D.  With younger kids, we draw the pictures ourselves and have them build the structure based on the diagram.  Starting in grades 2-3, students are perfectly ready to draw their own constructions.

It is easy to build a structure out of 2 pieces, and we start with these as a warmup.

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We then increase the number of pieces to 4.

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Building based on three projections is much harder.

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We also discovered that it is easiest to learn to draw your constructions using these rectangular blocks that have convenient dimensions, 1x3x7 cm.  It is much easier than drawing pieces that are constantly blocking each other.

You can also play domino run with these blocks if you have sufficiently many pieces. 

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We have also used isometric graph paper to draw “continue the pattern” designs, and for many children this assignment is much more challenging than the equivalent on a regular, square grid.

How do you use standard and isometric graph paper in your classes?

 

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Fun with pentacubes

Today we started a unit on geometry with the second graders.  The first activity had them explore how many different structures can be made out of 4 snap cubes.  They very quickly came up with the five tetrominoes:

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After a bit more tinkering, they declared that those were all the possibilities, but I told them to keep trying.  It was at least a few minutes before the leap into the third dimension was made.  But once they found one, they quickly found the remaining 2 as well, for a total of 8:

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They even immediately saw the non-equivalence of the top right and the bottom second from the left (I was kind of hoping that there would be some disagreement and discussion about it 🙂 ).

Since this planned part of my lesson went by so quickly, I decided to take a leap into uncharted territory and see what the students could do with 5 cubes.  I had never tried enumerating all the pentacubes myself and didn’t know how many there are (I knew that there are 12 pentominoes).   However, I think that this fact made the exploration process even more exciting for the students.

By far the most interesting part for me and for them was comparing the different 3D structures that they were building and determining whether they are equivalent.

This structure

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was built multiple times and each time we were comparing two of them, one of the students would say that they’re different and the another would rotate them to be in the same orientation, thus convincing the first one that they are the same (and the two roles kept switching!).

By the end of class, the students were able to come up with 26 different structures that we all collectively agreed on being different (including all 12 pentominoes).

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After class, I decided to look up how many different pentacubes that are altogether (I admit, I was too lazy / didn’t have enough time to do a thorough check myself).  It turns out, there are 29, so the students found almost all of them!  I was impressed.  But even more so, I was thrilled by how excited they were by each new one that they found.

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And now the question is how to keep that excitement going tomorrow.  Should I tell them that there are 29 and have them try find the remaining 3?  Should I just show them the remaining 3?  Should I switch to a new topic entirely?

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Funville Adventures launched!

Funville Adventures, our math-inspired fantasy adventure with Allison Bishop, is officially published! It is available on amazon and directly from Natural Math, where you can read more about the book.

Here are some early reviews of the book:

“You too will want to visit Funville, a delightful land where magical and strangely mathematical powers run rampant!” – Jordan Ellenberg, author of How Not to be Wrong

“Mathematical words can sound scary, but the concepts they describe are not: Funville Adventures proves this so!” – James Tanton, MAA Mathematician at large

“Enjoy the story. Stay for the math. Emmy and Leo’s magical adventure will encourage families to play with ideas together.” – Denise Gaskins, author of Let’s Play Math series

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Counting crocodile legs

Katie, at nine years old, rarely asks me to give her math problems.  She always has a lot on her mind and gets plenty of problems from school and other math-related activities (interesting problems do sometimes come up spontaneously in “everyday life” and she is often willing to think about those).

Zoe, on the other hand, at almost five, often asks me to give her problems to work on.  It usually sounds like this, “Mom, can you give me a math problem, but without numbers and without pluses and minuses?”  I have discovered that what she means by this is that she doesn’t want a straightforward arithmetic problem like 6+7, but rather wants a “story” problem.  And if she has to add/subtract numbers in the process of solving, then it’s okay.

A few days ago, Zoe used her usual formulation to ask me for a problem.  I decided to try out on her some problems similar to ones I had been recently doing with my first graders in class.

First, I asked Zoe for a 4-legged animal.  Inspired by a recent picture that her grandparents sent while vacationing in Florida, Zoe named the crocodile.  The problem I gave her was as follows:

Some crocodiles are swimming in a lake.  An underwater photographer takes their picture from below.  His picture comes out with only the feet visible, and there are 20 of them.  How many crocodiles are in the picture?

In class, the students would solve this problem by drawing a picture.  However, we were in the car so Zoe did not have that option.  She came up with many wrong answers before arriving (with some help) at the correct one.  However, we had a great conversation and many laughs discussing and interpreting her proposed answers.

Zoe: I know, the answer is 10.
Me: How did you come up with that?
Zoe: Because 10 plus 10 is 20.
Me: So your crocodiles only have 2 feet each?
Zoe: Oh yeah. Well then it must be 20.
Me: 20 crocodiles and 20 feet, so how many does each one have?
Zoe: One foot! (Some laughter and silliness). Ok, then 40.
Me: Ah, now some of your crocodiles don’t have any feet! (This caused even more amusement).

We did get to the correct answer eventually, by starting with the legs of one crocodile and working our way up.

Whereas Zoe was seriously proposing her answers, she was very quick at realizing what was wrong with each one. I think she learned a lot more from our conversation than if she had just gotten the correct answer right away, and the process was definitely a lot more enjoyable for both of us.

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