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.

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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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Why not count on our fingers?

Translated from this post by Jane Kats.

I would love to know the source of the very prevalent idea that counting on your fingers is bad.

Our school is currently teaching counting and basic arithmetic using a ruler, i.e., a number line, which doesn’t seem like a great or logical solution to me.

The students are told that 6+2 is a grasshopper that hops 2 steps to the right of the number 6.  Look, see, we stopped at the number 8, so 6+2 is 8.  And when it’s 6-2, the grasshopper hops to the left.

From my experience, this method doesn’t add any visual value and it’s not terribly convenient for kids who confuse left and right (and there’s quite a large percentage of young kids who do!)

If we instead took craft sticks (or any other counting manipulatives) and took 6 red sticks and added 2 blue ones,  the result is much more visual and much clearer than a ruler.

You can do this same problem on your fingers and there’s nothing bad or harmful about it. Show me 6 fingers (by the way, can you show me 6 fingers not as 5+1? How many ways can you think of?)  Now lets do 6 of your fingers and 2 of mine – how many does that make in total?

The important thing is not to memorize, but to understand that I can figure out the correct answer by myself (even if it’s through the help of my fingers)!

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Show the same number of fingers as I am showing.  How many fingers are we showing altogether?

You three showed the same number and got 23 fingers in total?  Count again please!

Can you figure out another way to show 16 fingers between the two of you? How do you show 16 fingers if there are three of you?

With five year olds, we use not only counting on their fingers but also many other manipulatives, such as these bright stickers.

 

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Glue as many petals to each flower as the number in the center tells you.  Then roll a die and add leaves to the stem based on the number that you rolled.

Now lets use base 10 blocks. We have small cubes that are ones, sticks that are 1 x 10 blocks that are tens and big squares 10 x 10 that are hundreds. The kids like to count and check that there are indeed 100 little ones blocks in the big square.

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I am going to give you a stick and 3 blocks and you have to draw that in your notebook and write what number that is.  How about 3 sticks and 2 blocks? Can I have one hundred square please? And it’s completely ok if some kids need to count every block in each stick to be sure of their answer.

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Others work with hundreds with no problems. But for me, it’s not important to figure out who can count better or worse. I want to give them a chance to get a feel for the numbers.  What does 10 look like? And what about 100?

There is also a big 10x10x10 block in this set of blocks.  One six year old girl was recently examining it during class. She counted all the faces – 100 on each side, 100 on the top and on the bottom, and then exclaimed, “I know, this block is 600!”

Sometimes we also take rubber bands and tie wooden matches into batches of 10.  It’s tricky for some kids to count out exactly 10 matches and then tie them up.  And now the manipulatives look different, so some kids need to count every matchstick again to get exactly 23 matches.

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The following is a great story from my colleague.  A five year old in one of her classes asked what “five squared” means.  She explained and, at the end of class, he wrote out a problem on the board for her as he interpreted it.

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What materials do you use with your kids and students at school and at home?

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Time, symmetry, and unexpected turns

This is a story of how a kindergarten lesson on telling time turned into a lesson on symmetry.  It all started with a fun, colorful clock, with numbers 1 through 12 made out of shapes (some related to the number, others arbitrary) that could be taken out and put back in.

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We were playing a game where one person would roll two 12-sided dice, another one would set the hands of the clock to point to those numbers, and a third would tell the time.  They were doing a decent job at the beginning, but very quickly the colorful shapes with numbers became too distracting and they started playing with them, taking them out and putting them back in.

Soon, it became quite hard to keep them on track, but then one student put a piece in upside down and that gave me the idea of completely shifting gears and changing topics.

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I asked the students which numbers could be put in incorrectly and in how many ways.  We went through the numbers one by one, and the students tried putting them in incorrectly in all possible ways.

The number 2, for example, could be put in upside down:

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But it could also be put in sideways in two different ways.  The number three, on the other hand, could not be put in fully upside down, but there were two ways of putting it in incorrectly.  After going through all the numbers, the students discovered that the 1 and the 11 were the only numbers that could not be put in incorrectly (unless you flipped them over so that you couldn’t see the number at all).  Seven, nine, and ten were the numbers that could be put in incorrectly in just one way, and the rest of the numbers had multiple ways of being disoriented.

Here is a picture of the clock with all the numbers put in incorrectly (except for the two for which this was not possible).

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The students were very engaged with this new activity and it easily lasted us until the end of the lesson.  Even though this was not what I had planned originally, I was very happy with the way the class turned out.  I have recently discovered that I really enjoy when lessons take an unexpected turn, as long as it leads us to other rich and beautiful mathematics to explore.

And of course we went back to playing with telling time in future lessons, but only to the point that they were interested in and had fun with.  I believe that at this age it is much more important to instill the joy of doing mathematics rather than have the kids acquire any particular skill.

 

 

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The Piaget Phenomenon

I am very excited to let you know that I am now collaborating with the amazing Jane Kats and that translations of her insightful and inspirational posts will now regularly appear on my blog under the tag “Jane Kats”.  Among her many other activities, Jane writes a prolific and popular blog (in Russian), with many posts dedicated to early childhood math education.  For those who don’t know Jane, here is a link to the interview that I did with her last summer.  And, not to keep you waiting, here is the first translated post, on the topic of the Piaget phenomenon.

There is a lovely story that talks about the fact that you shouldn’t get too far ahead with kids, that there is a perfect time for everything, etc.
The point is that oftentimes, children don’t immediately understand the concept of numbers, of quantity, and decide whether there is “more” or “less” of something based more on how much space something takes up and not based on the number of objects.
At 3-4 years old, many kids confidently assert that there are “more of” 3 large blocks than of 5 small blocks.   Piaget wasn’t the only one who described these experiments – and I periodically conduct them myself – and have a blast with it.
Usually, the trick works best with those kids who have learned the mechanics of pointing to objects and reciting “one-two-three-four-five” but who haven’t yet embraced the concept of a number as a universal measuring tool.
For example, many 4-5 year olds are completely thrown off if you ask them…to walk 6 steps and then stop.

Zvonkin conducted a classic experiment (and his book Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers is very inspiring and generates a ton of interesting thoughts – it’s too bad he didn’t record the experience of the math circle in its first year).

He laid out 8 coins in a row on the table.  Next to each coin, he put a stick, and then he and the kids counted that there are indeed exactly 8 sticks.

Then, in front of the kids, he moved the sticks closer together, so that there was less space between them and moved the coins further apart, so that there was more space between two coins.  And then he asked – are there more of sticks or coins?
And many kids said “Now, there are more coins!” even though they saw that we didn’t get rid of any sticks.
Moreover, you can actually get rid of a few coins (so the rows are now equal in length) – and many kids will argue that there are equal amounts of each objects now.

At this stage, there is no sense trying to explain anything – it’s just a neat game, and they won’t understand anything more even if you try to explain. They’ll get to understanding these concepts themselves later, through experimentation!

This is what it looked like in our class:

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I took cards with animals on them and asked the kids to place the cards next to appropriate numbers.  So next to the number 6, they had to place cards with 6 animals, etc.

The kids got the idea and were able to place all the cards.

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Then they counted the cards with 1, 2 and 3 animals and made sure that there is an equal number of each type of card.

And then I decided to test my hypothesis, and moved the cards in front of the kids.  Now one of the rows looked longer.

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And what happened?  These same kids, who just counted every row, confidently said that now there are more of “2s” than any other kind of card, and there are very few “1s”.

I could never catch my own kids with this trick, and I am really amused every time I see this phenomenon.

And interestingly, in about a year, this goes away on its own, it settles in their heads somehow – and kids start counting and using numbers confidently.

Have you seen such phenomena in your own kids? At what age?

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Four colors or more?

When you find your students working during break time on a problem that was the topic of a lesson three weeks prior, you know that something has gone right and your heart is filled with joy and excitement.

What they are trying to do is to create a map that cannot be properly colored using just 4 colors.  What does this mean?  Roughly, The Four Color Theorem states that given a map in the plane (think map of USA or map of Africa for example), one can always color the regions (think countries/states) using just 4 colors so that regions that share a border get different colors.  These students cannot believe that such a thing could possibly be true and are therefore trying hard to find a counterexample.

This theorem (which only became a theorem 40 years ago, and before that was just a problem) has fascinated and stumped mathematicians young and old for several centuries.  How can a statement so easy to state that it can be explained to a first grader be so difficult to prove?   Even today, the only known proofs require the aid of a computer.

But the proof is not what we dealt with at our lesson with second and third graders. Instead, the students had fun coloring maps with 2, 3 and 4 colors.

Coloring the following map with 2 colors was an easy task:

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Finding a 3-coloring of this one was slightly harder:

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And coloring this map with 4 colors was a very big challenge:4colors

As I already mentioned, some students refuse to believe that 4 colors will suffice for any map and are convinced that they will find a counterexample soon.  One student claims that she will grow up and find a flaw in the proof!  She has my full support and encouragement for doing so!

And here are some more pictures of the students coloring away:

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Double perfect squares

We have recently done a unit on perfect squares in school. The following conversation with Katie (8yo) happened completely spontaneously on a peaceful, evening ride home.

K: 10,000 is a double perfect square.
Me: What do you mean?
K: Well, 100 is already a perfect square, and 10,000 is 100 times 100.
Me: I see. Can you think of another double perfect square.
K: Yeah, 81.
Me: Indeed. Now, double perfect squares are also called perfect fourth powers.
K: What does that mean?
Me: It means that you can get them by multiplying a number by itself 4 times.
K: I don’t really understand.
Me: Well, 100 is 10 times 10 so 100 times 100 is 10 times 10 times 10 times 10, that is, 10 times itself 4 times. So what is 81 the fourth power of?
K: 3, because it is 3 times 3 times 3 times 3.
Me: Yep. Now if a number can be obtained by multiplying some number by itself 3 times, then it is called a perfect cube.
K: How come?
Me: Well it has something to do with cubes. Why don’t you think about it.

I am totally going to use the term “double perfect square” now!

To be continued…

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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”

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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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