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

FunvilleCover2

 

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Secret 3-digit numbers

For the past two weeks, in my first grade math class, we had been playing with 3-digit numbers. Many of the students in this class are the same ones that participated in the lessons described in the posts 3-digit numbers are tricky!, Part I and Part II, from last year.

But now, 3-digit numbers are their old friends and they are ready to do much more sophisticated things with them. I have found a number of interesting activities in the Super Source – Base Ten Blocks book for grades K-2 for us to try out.

One of the activities was a particular hit with the students. Here is how it worked.

One student would go to the special table and build a 3-digit number inside a box using base-10 blocks. That student would then give out clues about the number until another student built the same number.

Here are lists of clues that three different students came up with. Can you guess the numbers? For reference, a unit is 1 little cube, a rod consists of 10 such cubes and a flat consists of 100 cubes.

Student 1:
Clue 1: The number of units is 1
Clue 2: My number is bigger than 100 and smaller than 500
Clue 3: The number of tens in the flats is 40.
Clue 4: There is only 1 rod.

Student 2:
Clue 1: There are 9 flats.
Clue 2: There are 14 blocks total.
Clue 3: The number is bigger than 909.
Clue 4: The number of rods is smaller than 6.
Clue 5: There are 5 units.

Student 3:
Clue 1: The number of units is smaller than 9.
Clue 2: There are 20 blocks altogether.
Clue 3: The number of flats is smaller than 7.
Clue 4: The number of rods is smaller than 7.

Notice that both Student 1 and Student 2 have redundant clues, which was very typical. But it was the clues of Student 3 that I found particularly interesting.
It was not immediately obvious, even to me, that they were sufficient to determine the number.

The “guessing students” took some time after Clue 4 to come up with a number that fit all 4 clues, but to my surprise they all came up with the same number, which turned out to be the correct one. Only then did I realize that the clues were in fact sufficient.

I tried to discuss this fact with the students, but I’m not sure that they followed much of what I said. They were excited to have guessed the number and the next student wanted their turn at the special table.

What are your favorite 3-digit number activities to do with this age group?

 

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Witnessing the woes of math homework

Every week, I spend several hours in the waiting area of a large gym where my kids take gymnastics lessons.  The room has several large tables where parents and siblings can do their work while waiting for their gymnasts.

Many kids use this time to do their homework and by far the most common subject that I see them working on is math.  About half the time, the parents are helping them.  The kids rarely look excited but they are generally resigned.

Last week, I witnessed a scene that left me very disturbed. A mom was helping her daughter, who looked to be about 9 or 10, with math homework. The girl would alternate between crying, yelling, or even occasionally hitting the mom. She also kept repeating the following phrases, “I don’t understand this,” “You are wasting my time,” and “I hate you/this.”

The mom was generally much calmer than I would expect someone to be in this situation. Her most common response was, “Sweetie, you just have to memorize that this is the way you do it.” When she was explaining something, all I would hear is, “You have to put this here,” and “You have to write this there.”

At some point, the mom said, “It’s ridiculous that they’re having you do this. I didn’t have to do this sort of thing until college.” I don’t know what this specifically referred to, but later, I did overhear the mom reading a different problem to the daughter and it was definitely not anything beyond standard third grade level.

This all went on for over an hour. There were moments when I was very tempted to come over and offer my help, but I couldn’t bring myself to do it and wasn’t sure that it would be appropriate.

I know that math anxiety exists – I’ve seen it. I’ve also witnessed bad parent/child interactions. But this prolonged scene really staggered my imagination.

And now a few questions for the audience. How common is this? Are there many kids out there crying over math homework for hours? Is this hurting their relationship with parents who are trying really hard but for one reason or another are not able to help them?

And on a more specific note, should I have tried to help somehow in this particular case? Would you have?

 

 

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The mathematical boundary between a joke and a lie

Like all siblings, my children occasionally quarrel. I generally try not to get involved unless it gets loud or one of them complains to me. Sometimes, however, I hear something that amuses me and I tune in or even join the conversation.

Today, it began with Zoe (5 yo) accusing Katie (9 yo) of lying to her. Katie, in her turn, was claiming that it was just a joke. This is how the conversation continued:

Katie: You don’t even know the difference between a joke and a lie.
Zoe: Yes I do.
K: Oh yeah, what is it?
Z: For example, if you say “I ate 100 crepes”, then that’s obviously a joke. But if you say “I ate all the remaining crepes”, then that’s either the truth or a lie.
K: That depends on how many crepes were left.

This is where I joined the conversation. I was very intrigued to find out exactly at which point the joke turns into a lie. So I started asking some questions of my own.

The girls agreed that at 50 crepes it was still a joke and at 5 crepes it was a lie (assuming you didn’t actually eat them). When I asked about 20 crepes, there was a brief pause, and then Katie said, “You have to know the person.”

So the mathematical boundary between a joke and a lie, like so many other things, depends on the person.

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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.  

trapezoid1

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.

trapezoid3
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.

trapezoid2

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.

trapezoid4
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.

Jane_isometric1

We then increase the number of pieces to 4.

Jane_isometric2

Building based on three projections is much harder.

Jane_isometric3

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. 

Jane_isometric4

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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