Math with Art Supplies · pop-up

Inversion Two: Pop-Ups

I had wasn’t sure I’d write about seeing pop-ups through the lens of inversion, but then the title of this post got into my head, and I so much like that it kind of reads as “aversion to pop-ups”  that I had to to use it.

The generalized thinking here is  about bridging disciplines of design, bookmaking and math by using similar language to describe the various manifestations of the same exact concept. In this case I’m looking at the action of going back and forth, as in inverse operations.

This is the shortest post ever because I’m going to let this three minute video say it all.

The video isn’t really a tutorial about making pop-ups Instead, it’s about a way of thinking about the action of making a pop-up.

Even so, the final result is pretty nice.

Math with Art Supplies

Perks of Backwards Thinking, or designing with Inverses

Let’s say you just made this “No Strings Attached” pamphlet, designed by John C. Woods in the 1970’s and demonstrated here by Annie Perkins this past week. You might notice that you can’t readily tell the front from the back so you start thinking of what to do to decorate the front cover. It may occur to you that this binding method is created by making an inverse cutting pattern so that the pages can line up and be locked together. This might suggest making a design using inverses.

The understanding and using the concept of inverse relationships are all over design and math. This is something I’m going to be focusing on for a few days, starting with this post.

This cut paper method for decorations is perfect for classroom bookmaking because it uses little in the way of materials, can be done in a short period of time, and creates surprising results.

The size of the book  above is what you get when folding a piece of regular copy paper. That blue center piece started out as a two-inch wide strip, which, at first glance, doesn’t seem possible.

Here’s what’s happening.

I start at a lower corner, then cut a winding path to the upper right corner. Once the papers are cut the two strips can be realigned in a variety of ways.

Same cut, different arrangements

The four grouping of paper above are the exact same cut, just arranged differently. Considering the negative/positive, exchanging sides, or flipping horizontally, these are variations of thinking about inverting the original arrangement. To me, these different looks coming from the same exact cutting pattern, is exciting to me every time. Well, at least until I’ve had my fill of using just one color.

If I cut two different colors at the same time, other possibilities open up.

Two colors can give me these offset possibilities to explore. There’s so many ways to go with this.

The biggest problem with this way of working is that it’s hard to stop.

Here’s my demonstration video below. At the end I flip through a number of designs that aren’t in this post. Happy cutting!

Math with Art Supplies

Symmetry and Multiplying Negative Numbers

Where I left off in my last post was young children making lovely symmetrically reflective arrangements.

Where I am taking you from there is how to understand, through the lens of symmetry, why multiplying two negative numbers together makes a positive.

Not being able to visualize this has been a chronic thorn in my side.

Then I saw a post by David Wees

He didn’t mention symmetry by name but he did talk about reflection, so, yeah, symmetry.

Reflected shapes
Reflected Shapes

Above you see reflective symmetry. created by a couple of young children.  (Please think of the shades of greens as just green) They reflect across an imaginary line.

Now think of a number line, one which includes negative numbers. It’s the same idea as visualized above. except now it’s the numbers that are reflected. The negative just indicates which side of zero the numbers are situated.

Think of the numbers as defining the distance they are away from zero. For instance, the 4 is four spaces away from zero. The negative four is also four spaces from zero, but the negative sign indicates its position is a reflection of the placement of the positive four.

(Just to let you know, this is not something I am making up on the fly. Mathematicians often look at numbers as their distance away from zero, sometimes referring to this as the number’s magnitude, or its absolute value. This is not something you need to sort out, I just want you to know I’m not making this stuff up.)

In the photograph above -Reflected Shapes- you can see that both deep purple shapes are four units away from either side of the center, right? We could say that the deep purple shape on the right is at the positive four spot and the deep purple shape on left is at the negative four spot.

So what does this have to do with multiplication?

When multiplying, what the negative sign does is that it REFLECTS the quantity across the zero.



Here’s  three times two. There’s no negatives. Three times two means two groups of three, so, six.

Now, what happens, visually, when just one of the numbers in our problem is wearing a negative sign?

In this equation, negative three times two, I’m first thinking as three times two because that’s the distance from zero that this equation is expressing. So, again, I get six BUT NOW, because there is a negative sign, that’s my signal to reflect the product (6) over across the zero line.  Multiplying by a negative does this: it reflects across the zero line.

Now, what about if there are two negative signs in the multiplication problem?

The same thinking applies. First, multiply the numbers three times two. Because the three is negative, reflect the product over the zero line. Now, because the two is also negative, reflect across the zero line again, which lands you back into positive territory.

To see me reflecting these quantities via youtube, click below:

I shouldn’t gush over my own posts, but this particular one, is one I am so excited by because it’s something I’ve failed to visualize for such a long time.

Many thanks to David Wees.

This is the section from his post that turned the light on for me:
Art and Math · Arts in Education · Math with Art Supplies

Thinking Hands


Oh look, that’s me. For once, I’m not the one taking the picture!

Third and last post in this group of postings that are meant to help me clarify and remember where I am at in my thinking about the work that I do with students in schools.

After years doing bookmaking projects to make with children, I realized that many of the art and design skills I use every day align with some of the skills that mathematicians aim to develop. Part of the reason this alignment caught my attention is that I have a great affection for the mathematical thinking that I want to encourage.

Rotating Shapes after school with third graders
Rotating Shapes after school with third graders

In the relatively short time I am in schools with students, I hope to have a positive influence. My experiences and interests have led me to an unusual place where I can use colorful, artful materials to help kids create projects that enrich mathematical thinking. My place isn’t to teach art or teach math, but rather to plant seeds of engagement and excitement.

It seems to me that children intuitively understand concepts that are recognizably both artful and mathful. More and more, my thinking is centered around how to engage and encourage that which is already inside of students.

Spontaneous Scaling of Origami Pocket by First Grader. I see this happen frequently.
Spontaneous Scaling of Origami Pocket by First Grader. I see this happen frequently.

For instance, kids absolutely understand the idea of scale. They realize that their hands are the same, but smaller versions, of dad’s hand. Same with their shoes, their shirts, everything in their world.

Scaling shapes to make decorative border
Scaling shapes to make decorative border

There is no room in the school day for formal study of scale until the intuitive connection to it seems to have long disappeared. Turns out that scale doesn’t only have to do with making large models smaller, but it also is intrinsically connected to relationship thinking, predictive thinking and to the recognition of trends.

Symmetrical frame for a drawing of outer space by a first grade student

Discovering that children are naturally inclined to embrace symmetry has been another exciting area for me to explore with kids. When making books or other structures with students, there is nearly always symmetrical folding going on. I have choices when I teach folding: I can introduce what I do as step-by-step directions, or I can nudge the  students to see the symmetry of what’s going on so that they can predict for themselves what the next fold will be. The latter way gets them to see the project in a more global way, draws them in because they have understanding which includes them, rather than being like a little robot that is being programmed to do this then do that.

Doing some spontaneous symmetry during a few extra minutes at the end of a kindergarten session
Doing some spontaneous symmetry during a few extra minutes at the end of a kindergarten session

Symmetry is deeply embedded in math thinking, so I have been talking to children about connecting symmetry to what they are learning right now in math. Specifically, I talk to them about how when they are looking at an equality, such as 5+3 = 8, that this expression is balanced on both sides. It can also be understood as 5+3= 4+4. If I add 6 to one side of the equation, then I have to add 6 to the other side so that the symmetry of the equation remains true. Talking to students about equations as balanced forms just might help them, later on, when they will have to maintain balance in an equation to solve for x.

Symmetry, Scaling and Fractional Thinking all on display
Symmetry, Scaling and Fractional Thinking all on display

As far as I can tell, the only time symmetry is formally taught in elementary school it’s part of the examination of lines of symmetry in regular geometric objects. I like to be able to at least offer hints that symmetry has richer applications.

Making shapes from other shapes, parts and whole
Making shapes from other shapes, parts and whole

Children seem to have an innate sense of parts that make up the whole, which seems antithetical to the reality that teaching fractions is unfathomably difficult. Is it possible, though, to focus on having students work fractionally from a very early age, way before we introduce the numbers that describe the fractions?

Hexagon, Cube and Thing made from equilateral triangles
Hexagon, Cube and Thing made from equilateral triangles

Playing with blocks was one of my favorite activities as a kid. I certainly noticed halves, fourths, and wholes, but I didn’t make this connection between the blocks and fractions until I was much older. This makes me value not only exposing kids to artful mathematical thinking, but also, sooner rather than later, to help students connect their hands-on activities to the numbers.

Four Fold symmetry Pre-K
Four Fold symmetry Pre-K

There’s more I have to say about all this. but I reminding myself that I have to get to work getting ready for classes.

Am going to end with my list of ideas that I want to keep in mind, not all of which are explained here. Maybe I will get to writing about all these here and there through my teaching season. If not, at least I will have them here to keep me on track.

Art/Math concepts to explore with kids:

  • symmetry
  • Pattern recognition
  • Pattern Building
  • Scale
  • Inverses
  • Continuous magnitudes
  • Trends
  • Relationship thinking
  • Problem solving
  • Parts of a whole
A drawing of mine, that's artful/mathful
A drawing of mine, that’s artful/mathful

The first two posts in this set are here and here.