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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Counting beads

Translated from this post by Jane Kats.

Recently, at our math lessons with 5-6 year olds, we were … threading beads.
Actually, we were making very simple abacuses.
We were putting 5 light beads and 5 dark beads on a string.
Then we played a game where the kids would hide some of the beads in their hands and I had to guess how many they hid of each color.
I was guessing correctly, and the five year olds were very surprised.
They can all count to 12, and many can count further, but they still don’t fully understand how I can know how many beads they are hiding in their hand…

Putting on the beads

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Who can hide as many as I did?

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Later, we played with these beads with the 4-year olds.

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And at the math festival we also played with these beads on a string.
You can see some beautiful pictures from it here.

Now we are playing “a game of 10”.
The goal is to find a pair of cards that have a total of 10 black squares on them.

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Each card has 10 squares, 2 rows of 5. Some of them are colored and the total number of colored ones is written in a corner of the card.

Hiding all but 6 beads on the string and seeing how many need to be added to make a total of 10.

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It sounds easy, but many 6 year olds search for a while, pointing with their finger and going through the possibilities.

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It is not obvious to them that if here we have 2 white ones and there 2 black ones then together there are precisely 10.

Eventually they find pairs that “work”.

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I think that this is precisely one of those cases where one needs to realize and understand before memorizing.

Pairs that are found are given to me and then I give the students additional cards…

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We also play games where to answer one needs to use fingers – say,
add fingers so that together we have 10.

With younger kids we first play add until we have 5, and that
too is not always easy.

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Playing Broken Telephone

What follows was translated from this post by Jane Kats.  The game described was played by children ages 5-6, but I feel that a variation of it could be fun for older kids, or even adults.  Ideas?

When I was little, we played a game called broken telephone.
During our lesson today, we invented a new variation of this game.

Children are running around, around, “snowflakes were twirling and joined in pairs”. Now one child in each pair has to use a finger to draw some dots on the back of another. The second child has to guess how many dots were drawn.

Next, the snowflakes were twirling and joined in threes.
The triples make a train. The third person draws dots on the back on the second and the second reproduces those dots on the back of the first. The first person now has to say how many dots reached them.

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With two people it’s a lot easier.

And if the train consists of three or four people, then the children are not always able to
– draw the dots without counting out loud
– correctly determine the number of dots
– correctly draw the same amount on someone else’s back.

This turned out to be a great game.

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Drawing heads, tails, knees, and toes – useful?

Drawing a useful picture or diagram is an important problem-solving skill.  From teaching different groups of elementary school children, I have noticed that some students are very reluctant to draw anything whereas others will draw elaborate pictures with lots of extraneous details.

It can then be tricky to push both types towards the happy medium of drawing just enough to help one solve the problem.

My first graders so far have mostly worked on problems where the pictures are drawn for them and they just have to interpret them.

Last week, I decided that they were ready for a little less hand-holding.

For my first experiment, I chose the following problem:

How many legs and tails do 5 cats and 4 birds have altogether?

I chose the numbers to be big enough so that they wouldn’t be tempted to do it in their heads, but not so big that the drawing would get overwhelming.

The students did not disappoint.  Their solutions included a full range of drawings – from none to very detailed.  Everyone got the correct answer, although it took some several attempts.

First, there were the students who drew very elaborate animals and spent much more effort on the pictures than on counting the legs and tails.

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Then there were those that still clearly drew out all the animals but in a much more minimalist way.

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Then there was the student that just drew lines for all the items that needed to be counted.  What impressed me the most here is that not only were the correct number of lines drawn, but they were also counted correctly from the first try.  I’m not sure that I’d be able to do that!  Oh, and then the student went on to draw an elaborate farm scene, but only after solving the problem.

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Finally, there was the student that didn’t draw anything at all but computed everything in their head.  I did try to convince them several times that drawing would help with keeping track, but the student either didn’t believe me or wanted to prove that they could figure it out without those silly pictures.

The student obtained the correct answer from the third attempt but then was able to explain it by saying that 5 fives is 25 and 4 threes is 12 which together make 37.

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We have since solved several similar problems and whereas the tendencies of the students have remained the same, I did notice ever so slight movements towards that happy medium.

We had nice discussions with the students about whether pictures help and whether it is important to draw that fancy mane on the lion to convey that it has 4 legs.  They laugh and agree that it is not necessary to draw that mane.

But they draw the mane anyway.  Because it’s just too much fun.  And I let them.  At least for now.

 

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A math-inspired love story in verse

For post #100 on this blog I wrote a poem.  This is post #200 and I decided to start a tradition.

This is a poem written jointly with Allison Bishop.  It is an epic love story.  Math is involved.  The title is still in the works.

The triangle was born in a triangle land
Where lived all triangles, hand in hand
They lived in three-sided houses and went to three-sided schools
They ate off three-sided plates and sat on three-legged stools

But one day the triangle met a square
And was stopped in her tracks by his four-sided stare
And the square himself realized in surprise
He’d never seen such beauty with his four-sided eyes

So their lives together they decided to share
But with this resolved, the question was where?

For the triangle world the square proved unfit,
‘Cause through triangle doors he just could not fit.
After breaking a few when he could not make way,
The other triangles chased him away.

So the triangle followed him to four-sided land
But quickly found that wherever she’d stand
Her pointy end would poke and tear
So the quadrangle world they could not share.

The square had an idea, he would save the day.
He could simply give one of his angles away!
Since he had four angles, and she had three
He could cut one off, and well-matched they’d be

So he took some shears and chopped off a corner
Then looked in the mirror and found to his horror
Where four sides had been, there were now five
Oh what a cruel day to be alive!

But the triangle thought fast, she knew what to do!
If the square could gain one side, then she could gain two!
She picked up the shears, she was not afraid,
With a snip and a snap, two cuts were made.

He looked at her with sadness no more,
So filled with joy that he could soar.
Now they both had five sides, and went off together
To live happily in five-sided land forever.

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A figure with pointy things and a line, a line, and a line

Look at the title.  Did you guess that this was a description of an acute triangle?

This description was from a student in my third grade class.  Let me assure you, this student knows how to recognize an acute triangle and how to draw one.  But verbal descriptions are much trickier for them.

I believe that verbal descriptions are just as important for communicating math as they are in literature, history, science, etc.  That is why we play games and do activities in my math class that practice precisely this skill.

The description in the title came about from one such game that we played.  We had just finished a unit on shapes (mostly focused on polygons) where the students encountered a lot of new terminology about angles, triangles, quadrilaterals, and other polygons.

I gathered all of the terms and wrote each one on its own slip of paper.  The papers were then all placed into a “hat” (this can be a box, a bag, an actual hat, or anything that can store the papers and the students can reach into without seeing the words on the slips).

Play proceeds as follows: two students are called to the front, one to be the “explainer” and one the “guesser” (I made up a schedule so that all the students had an equal number of opportunities to play both roles and so that everyone had a chance to explain to everyone else).  One minute is put on the clock (we tried 30 seconds at first, but that didn’t seem to be sufficient).

When the moderator says “go”, the explainer takes out a piece of paper from the hat at random and attempts to explain the word to the guesser.  The rule is that you can’t use any word or part of word that is written on the paper (so if the term is “right angle” or “triangle” you can’t use the word “angle” in your explanation).

When a word is guessed, the explainer takes out another word and goes on to explain that one.  The explainer continues to take out new slips and explain the words until the time runs out.

If the explainer gets stuck on a word (either because they don’t know what it means, they don’t know how to explain it, or the guesser is just not getting it), they can put it aside and take another one.  The rule, though, is that you can do that with only one word per turn (later I thought that perhaps allowing to put aside 2 or 3 would have worked better).

The first few rounds that we played, no words were guessed.  This was mostly due to descriptions being like the one in the title and the students still getting a feel for the game.

But soon the students started feeling a little more comfortable with what they were doing and putting a bit more thought into their explanations.  Here is what some of the early ones to be guessed sounded like:

“They can be of different types.  A rectangle can be one, a square can be one, and there are many others of these.  Well, you know, with 4 sides.”

Up until the explainer mentioned the 4 sides, the guesser looked very confused, but as soon as the 4 sides were mentioned, the word quadrilateral was said immediately.

Often, concepts were described by what they are not.  For example, “acute angle” was described as “it is not obtuse or right.”  Or “rectangle” might be described as “it is not a square but a…”

But overall, the descriptions got more precise as the game went on.  Some of my favorites were, “It is the shape of a stop sign” (octagon), “It is both a rhombus and a rectangle” (square), “It is a stupid shape with three sides” (obtuse triangle).

What sort of games and activities do you do with your students to practice communicating mathematics?

 

 

 

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Protecting against magic

Yesterday, Zoe and I were playing a game of Bingo.  In this version, there are cards with the pictures and names of the items (all related to the ocean) and you flip them over one by one to figure out which spots to cover on your board.

The cards were all in a pile facing down and Zoe would pick them up one at a time and without showing me have me guess what was on it.  I had to find the correct item on one of our boards (or say that it wasn’t on either of them).

The first time that I pointed to the correct item, Zoe looked quite surprised and said, “yes, how did you guess?”  Well, one time can be a coincidence, right?  But then she did it several more times and for every one of them I guessed correctly what was on the card.  Now Zoe was very intrigued.

What Zoe didn’t realize, is that the names of the items (along with their descriptions) were written on the backs of the cards.  So even though I couldn’t see the picture, I could still identify the item based on the name.

I told Zoe that I could see through the card and tell what was on the other side of it.  Zoe seemed a bit skeptical, but finding no other explanation she bought into this one.

To correct for this phenomenon, Zoe began picking up the cards in such a way as to cover the back of the card right away and keep it covered until I made a guess.  But this didn’t seem to change much – I still guessed all of them correctly.

How? The back of the card would become visible as soon as she’d pick up the previous one, so I had plenty of time to see it before she picked up the card.

But Zoe was not ready to give up.  She put the whole pile of cards behind her and started pulling the cards out from the middle of the deck.  She would then look at the card and put it face down on the floor behind her back.

I thought that this was quite clever of her (especially the part of taking the card out of the middle), but the problem was in the mechanics.  The way she’d leave the card lying behind her, I could still see what was written on the back.

Finally, Zoe resorted to desperate measures – she made me close my eyes while she was selecting the card and would hold it in such a way that I couldn’t see any part of it when I opened them back up.

I was beaten and my magic powers of guessing the item disappeared.  Later she took pity on me and would give me clues about the item, which turned it into a totally different game entirely.

Zoe never figured out how I was magically guessing what was on her cards but I was impressed with her persistence to figure out a way to prevent me from doing it.

I think that one day, perhaps several years from now, Zoe will be playing or looking through the game and have a great revelation.  But I don’t mind if she thinks that I have magic powers for just a little while longer.

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