math · Math and Book Arts

Fraction & more Fractions

Many Parts of a circle
Many Parts of a circle

I’ve been working with 9 different grade levels, nine different projects, this month, which is kind of wild, and even more wild because of all the snow days and other unexpected shifts in schedules. Most of the projects we’re doing are things I’ve written about enough on these pages, but I have managed to slide in a couple of new things with the fourth graders.

I had some extra time with some of the students some because they chose to stay after school for some extra time with me. Am still racing to finish prep for tomorrow, but want to quickly post about these two extra projects.

Dividing up a circle project
Dividing up a circle project

I brought in circles and sheets of regular shapes. Student cut up the shapes, and rotated them around a center point. The circles were marked with 12 evenly spaces dots around the circumference. We talked about other cyclic things that are divided up into 12 parts (clock, months) and talked about how 12 has so many divisors.

Rotating Shaper around a circle
Rotating Shape around a circle

I printed the shapes on heavy paper. I hadn’t done this with kids before so I didn’t know if they’d have trouble with this. It was no problem for them at all. They were excited, worked creatively, asked questions and were totally engaged.

Student rotations
Student rotations

Here’s the PDFs that I created for this project.

Circles with 12 dots

shapes to rotate in circles

I casually mentioned that ANY shape can be rotated. Well, they didn’t have to hear me say that twice before they were making new shapes.

Crazy Shape rotation
Crazy Shape rotation

The trick is to retain points that can still line up with the center and with a point on the edge of the circle.

Another Crazy Shape Rotation

During class time, I worked with students on a fractions/ bookmaking project that I’ve written about previously on my Books Are Fractions  post.

Fractions book
Fractions book

I knew some students would finish up early, so I showed them some images I had printed up some twitter posts. (If you want to see many more images like this, type in the words Fraction Museum in the twitter search bar and you will be well rewarded)

The kids were enthusiastic about creating fraction museum pieces, which I then photographed.

Fraction Museum hearts
Fraction Museum hearts

The idea is to collect items, see them as part of a whole, then write fractions that describe the collection.

Fraction Museum books
Fraction Museum books

There was some deeper thinking going on than I expected.

Mixed Fraction Museum
Mixed Fraction Museum

I’ve assembled all their images on to 2 large sheets of papers, and will present them to the kids tomorrow….but only if I stop this blogging and get back to work,

 

Addendum March 26 2018

During my fractions conversations with these kids (who, by the way, had a good grasp of fractions before I ever showed up) I talked about the confusion that can happen when trying to understand why, when the denominator is a bigger number, the unit fraction is smaller. I showed them a piece of paper folded into four sections, then said if I had to fold the same paper into eight sections (which we did) that the number of units had to be smaller to accommodate the larger number of sections. Then I asked “Imagine if we had to divide this paper into 100 sections, how small would those sections have to be? 

Hundreths
Hundreths

Well, that was it. They begged to see a page divided into 100 sections. Each time they saw me, they reminded me. Finally, today, I brought in TWO papers, and asked which one of them had fraction units that were each 1/100. Led by an particularly independent thinker, they figured it out. And figured out why, even though the divisions looked different, that they were all 1/100s.  It was a great conversation. Here’ the PDFs of you can ask kids this question yourself: hundreths

So much fun.

math · Math and Paper Folding

Piecing Together a Project, Over Land and Sea

Pieces of Paper
Pieces

This story begins in a teachers’ lunchroom, a couple of years ago, in Upstate NY. I was sitting with some teachers when another member of the staff started talking to a first grade teacher, Mrs. K, about a new math mandate. It was something about using manipulatives to create a variety of shapes. I’m a bit foggy on this part but it seems to me that they were required to use rhombuses (or rhombi, both are correct) for their shape building.

A Rhombus
A Rhombus

Upon being told that she would have to incorporate these manipulatives into her math unit Mrs. K asked if there was any money in the budget for manipulatives. The answer was no.

After school I sought out Mrs. K and showed her some paper-folding and shape transformations that referenced rhombuses. This teacher seemed delighted with what I was showing her. I volunteered to send her something that I thought she might find useful, then went home and created these images for her, which are equilateral triangles that become a rhombus.

screen shot rhombi

I never asked Mrs.K if she used what I sent her. I recognize that what I sent was, unfortunately, not a project. Instead, it was just the bones, the beginning of a project that needed to be developed.  Every so often I’ve revisited these images, wondering what I could do with them. Then a few nights ago Malke, from Indiana, asked me about projects for a family night.

It was late, and we decided to resume the conversation the next day. The next morning, before Malke and I reconnected, I saw this post from Simon Gregg, in France:

I had an Aha! moment. It suddenly came together. I sent off this note to Simon:

Malke, who I included in the conversation, responded with a reference to a beautiful manipulative that I wasn’t familiar with, but which also showed that she immediately recognized what I was getting at with my DIY paper version of manipulatives.

Since Malke seemed to know exactly what I was thinking about I got to work creating the pieces for this activity. I’m pretty happy with how this has developed. It requires triangle paper, and matching paper shapes that can be printed on colorful papers. My thought is that simple, bold shapes can be created in sort of a free form way…

Big Shapes
Big Shapes

…or more challenging shapes can be drawn on to the paper…

Drawn Shapes
Drawn Shapes

…and filled in, while trying to make as few cuts as possible and being mindful about cutting along the lines defined by the triangles.

Filling in drawn shapes
Filling in drawn shapes

So, where can you get these papers to do a do-it-yourself shape building set? Right here. I’ve created a couple of PDF’s to get you started:

Triangle paper and Rhombuses

Make beautiful shapes. Send photos. Thank you.

Addendum: Take a look at Malke’s post on hands-on math: she collected and organized many interesting perspectives. It’s a fabulous piece of writing.  http://mathinyourfeet.blogspot.com/2015/04/some-thoughts-on-hands-on-math-learning.html

Addendum #2 (April 2016)  Malke liked working with smaller rhombuses so I made her this  PDF rhombi with spaces So far she is planning on using them without the triangle grid paper. Here’s a link to some images she’s created as samples for an upcoming project https://www.facebook.com/MathInYourFeet/ rhombusphotos

Flip Book · math

Flip Books as a Bodacious Learning Tool

Flip Books and some Flip Book pages
Flip Books and random pages. The Birthrite flip-book by Ruth Hayes, published by The Real Comet Press, 1988, was a gift from Lewanne Jones and Jim Fleming when my son was born.

After all these years of teaching how to make books I have now become smitten with using books as a teaching tool. Yes, I know, using books for instruction is anything but a new idea. This said, flip books have captured my interest because they are fun and dynamic. Among other things, I’m pairing them up with mathematical equations to illustrate how the picture of an equation changes when there are changes in a variable. So far I’ve shown some of these flip books to a couple of my smart but not particularly mathy friends and what happened next is so worth writing about.

y equals mx plus zero by Paula Beardell Krieg
y equals mx plus zero ( this should play through 2 times. Click on it into play again)

The first thing that happens, of course, is that my friends flipped the pages. Everyone loves a show. The gif in the box above is a pretty good representation of what they saw in one of the books. A gif is fun, too, but it’s not as effective as a flip book in that it doesn’t allow the viewer to slow down and examine what’s going on. I saw this happening so clearly: my friends were drawn into the equation by the action of the flipping, then they slowed down, looked at the images more slowly and tried to understand what was going on. They had great questions.

50--50

For instance, Sarah thought I had made a mistake in labeling these pages. She hadn’t sorted out how the graph of y=50x could be so similar to y= negative 50, after all 50 is arguably a large number while negative 50 is indisputably a very small number. It was easy enough to explain how this works, and she absolutely understood it. What I understood was that, without the flip book, she would have never been interested in having had this conversation.

Comparing the slopes of a line
Comparing the slopes of a line

John also had some questions. He couldn’t fathom why I showed lines with slopes equaling 1 to 8 in sequence, then started skipping to 12, 20 and 50. When I pointed out how the lines were becoming increasingly indistinguishable from each other as the slope becomes larger, he better understood my choices of which slope values to use.

It’s taken me quite a bit of time to get to the point where I’m ready to show this first equation-of-a-line book to anyone. There will be two or three more books to go with this one: one that shows only “b” changing; one that shows the graph when the changes in “m” are between positive one and negative one; and a book that shows b and m changing at the same time.

I’m hoping to get some feedback on these from classroom teachers. Here’s my plan: after I have this y=mx Flip-Book finished I will make a few extra copies and send some out to teachers who are willing to point out flaws in my presentation. In my next post, when I’m ready (hopefully tomorrow) let me know if you want to be one of my collaborators. Even though I’ve already worked through at least a half-dozen different variations of this one book, there are still things that I am not sure about. More about that when I continue…

math

My Last Piece on Pi: Pi Post #4

Hen

I have one last thing that I want to cackle about before finishing with pi: exactly what number should we use for pi?

Pi is an irrational number, but it’s just not practical to use an infinite amount of decimal places when using pi in an equation, right? So, how many decimal places should we use? This depends, in part, on how accurate you want your answer to be.

Okay, so pi equals about 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510….. but it’s just silly to think about all these numbers in terms of measurement because most of us simply do not have the means  to measure anything this accurately. So, what number should we use? To answer this I used Microsoft Excel to help me show you a picture (!) of a comparison of approximations of pi. (I’m really excited about this.)

Pi- comparisons of approximations
Pi- comparisons of approximations

Ta-Dah! Now, take a look at this! It looks like using 3 as an approximation of pi  can really be off the mark. Going out one digit is better, and it’s close to the value that the Babylonians came up with (3 1/8)  two thousand years ago, but it’s still not so good. Now, 3.14 looks like it’s close to the rest of the representations of pi, which become increasing indistinguishable from each other, and it’s a value that I’m satisfied with. But for anyone who is not so easily satisfied we get to zoom in on this picture for a closer look.

Zooming in on PI
Zooming in on PI

I have to tell you that when I created these images I didn’t know how the various approximations of pi would look compared to each other.  You can see here that 3.14 looks like it’s less than 3.141, and 3.141 looks smaller than 3.1415 and maybe 3.1415 looks smaller than 3.14159 but after that, even though there are differences between the  subsequent values of pi, the differences are so small that, to the senses and  for the purposes of us mere mortals, these differences are indistinguishable from each other.

So there you have it. A real look at pi.  So, what’s so important about this number anyway? There are books on this subject, but, briefly, pi shows up predominantly in two places. One is standardized tests. Every student who takes standardized tests, which include the SATs and ACTs will absolutely 100% be asked a question that will test their understanding of pi. Is this a practical reason to understand it? I dunno, but that’s the way it is. Will you ever use it for any practical reason? The only time I have ever used it was when I needed to roll out a clay coil that was long enough to become the foot of a bowl I was making. The bottom of my bowl was about 6 inches across so I knew that my coil needed to be nearly 19 inches long plus about another inch for overlap.

 

I think that the next time pi shows up in a meaningful way is in Calculus class.  There’s a really cool thing that pi does with calculus formulas:

Egg and Egg Cup

If you ever wondered what calculus does, well one of the things it does is provide a formula using pi which can calculate the surface area of an egg-shell, which of course, is not a perfect circle. And other use of Pi in calculus is that it is used in yet another formula which can calculate the volume of an egg cup, which, again, is not a regular sort of shape. Now. just think, if you can figure out the surface area of an egg you can figure out how much material you will need if you ever want to build irregularly shaped buildings like Santiago Calatrava’s.

Curvilinear Architecture by Santiago Calatrava
Curvilinear Architecture by Santiago Calatrava

For more big eggs like this take a look at http://artmagonline.wordpress.com/2011/12/15/on-calatrava

And if you want to know how much liquid it takes to fill up your bathtub, your swimming pool or your oil tanker, calculus and pi (paired with the right calculus formula) comes in handy again.

Now, it’s a good thing that Chinese New Year is coming up, because I’m feeling inclined to abandon pi and start writing about accordion style bindings…

Addendum: I just came across a great post on pi in the Scientific American Blog. The articles reports that NASA uses 15 or 15 digits in their Pi calculations, in the comments reader suggested that the cube root of 31 is  an excellent pi approximation. Read for yourself http://blogs.scientificamerican.com/observations/2012/07/21/how-much-pi-do-you-need/