I’ve been reading on-line conversations lately about scaling and dilation (which are two terms for the same thing, I think). Repeatedly I’ve seeing questions from various people on scaling activities. This confused me at first, not the concept itself, but why did there seem to be such an interest in methods of teaching dilation? Turns out that this idea of proportional change is a key concept to grasp in math, and the nuances of it are tricky to teach. After thinking about it I thought about how artists and designers use scaling all the time, and I mean *all the time. *So, this week, when I was working with first grade students, I decided that I would present the decorative part of their project as an a math lesson, as an exercise in scaling.

Each student picked six strips of paper, which were 1.5″ x 3″ each. I told them that these strips are the shape of two squares on top of each other, and I knew that because they are twice as high as they are wide. I don’t think they understood that part, but that was fine. I just wanted them to hear the language, as I think that it conveys a sense of importance. I then showed them how if they lined up the papers so that they made an “L” they could see a square, which they could then cut, as in the photo above. From there we talked about how they could use the square for decoration. Since a plain square is,well, not too exciting I tilted the square for a more interesting look. Next I showed them what happened when the square was cut in half, corner to corner -one cut, two triangles! We then looked at what happened when the triangle was cut in half again, and then again.

I also showed the students that to make a scaled down square that they had to cut the original square twice, once in both directions. There is something about making “baby squares” that particularly captured their interest.

Baby triangles were a big hit too.

This is an activity that I’ve taught many times bu it seems to me that, judging from the results, this was the most successful day I’ve had of teaching this kind of embellishment.

Their project, by the way, was making a book for their animal reports.

Students have done their research, now they’ve made their books, next they will be adding writing to the books. This elephant will eventually be flanked by facts about his habitat and other interesting facts.

These students will be able to continue with these embellishments as they complete the rest of the books, I won’t be there for that part, but I think they’ve got the idea and these books well become even more wonderful.

## The Animated Equation Book

### May 3, 2015

My Do-It-Yourself *Equation of the Line* Flip Books are ready to share.

I’ve been writing about these, on and off for months, but my work on them has been steady. My goal has been to create PDF pages that I can distribute on-line which a class of students can assemble in about 10 to 15 minutes, and which show how the changing variables in the equation of the line, y = mx + b, changes the look of the graph. I am doing this because it seems to me that the understanding of this particular equation is either a gateway or an obstacle into the continuing study of mathematics. This is my way of contributing to the conversation of how to nurture a more math literate culture.

My thought is that if students can hold the equation in their hands, that it will give them the opportunity make sense of it for themselves.

I’ve made four sets of PDF’s. I find that heavier weight papers makes the best flip book. I use Hammermill Color Copy Digital Cover paper, 60lb, but that said, if all you have is standard weight copy paper, use it.

Here are the PDF’s!

The PDF for flip book of changes of b in y= mx +b

The PDF for flip book of changes of m between negative one and positive one in y = mx + b

The PDF for Flip Book to show the changes in m greater than positive one, less than negative on in y = mx + b

The PDF for Flip Book to show changes in x on the graph of y = mx + b

There, now that you’ve downloaded all the PDF’s you’ll notice that, oddly, they look like this:

What may or may not be obvious is that there are six pages on each panel, and half of these pages are upside-down. So now it’s time to give you some hints on how to make these pages into flip books, and tell you what’s up with the upside-down.

Each piece of 8 1/2″ x 11″ copy paper will contain six pages of the flip book. (To my **A4** friends, I will be making an A4 version of these, but they are just not ready yet.) The pages need be cut out on the solid gray line. Let students do this with scissors! The pages are numbered, so it should be easy enough to get them in the right order. The important part, when assembling these books, is that the FLIPPING EDGES are even!!! That’s why half the pages are upside-down on the page, so that the flipping edge always fall on the edge of the paper so it doesn’t have to be cut.

Now, how to bind these small books…

Bookbinders will figure out their own ways to bind these books. These simple binding solutions are meant for the classroom or home school venue.

Before you start, note that the front and back covers aren’t numbered. You can figure out which is which. Just be sure to *flip over the back page so the graphic shows *otherwise your back cover will look boringly blank.

The easiest way to put these paper together is to make sure that the flipping edge is even, then wrap a rubber band tightly around the spine. This simple solution works surprising well, though every so often you might have to remove the rubber band and realign the edges. The thinner the paper, then thinner your rubber band should be. Experiment. You’ll know what works and what doesn’t.

My favorite simple solution is to use strong clips, like on the upper right in the photo above. I just found out that these are called binders’ clips. Excellent name. Use the smallest size that works. The small ones, 3/4″ wide, work just fine.

If you have access to a drill, then doing a simple sewing is swell. Drill three holes evenly spaced, about 1/4″ from the spine and follow this sequence: go through the middle hole, leaving a tail of thread behind, sew through the top hole, travel down to and go through the bottom hole, then go back up through the middle hole and tie your ends together.

That’s it. Make lots of books. Let me know how it goes.

## The Autobiography of a Second Grader -back to school!

### September 3, 2014

I had planned that this post be about the Paul Johnson show, but I won’t be able to get in to photograph it just yet. Instead I’ve decided to seize the moment and write about this great back-to-school project that Gail DePace did with her 2nd graders a few years ago. Gail and I worked together many times, and I see my influence in these books, but Gail (now retired) was an inspired teacher in her own right, and just took off with any of the skills that she picked up from me. What she did here was make a template of a young person, which each student personalized in their own likeness. With some simple folding the students created a pocket, which was glued on to the front of the book and which held the little self.

For the body of the book I’m fairly sure that Gail started out with a somewhat large sheet of paper, probably 11 inch x 17 inch (A3). The folds are based on the Origami Pamphlet folds. That cut away window in the middle makes a place for a secret picture, which is only revealed with when the book is set up in a certain way. Like this…

Here the book is set up so you can peek into the lives of the author.

Here’s the bird’s eye view of the book. Have you noticed the bit of framing that is done around some of the little drawings? This is accomplished by providing each student with just one square post-it, which they mount, temporarily, in their book, then color around it, thus masking off what’s beneath.

As lovely as this structure is, it’s the content that makes them so fabulous. Student wrote about their family, about their favorite place to be (which was illustrated inside the window) and what they like and dislike.

Then they ended the book with hopes for the future.

Hmm. I love student work.