Sometimes it can be useful to be able to quickly determine is divisible (meaning it can be divided evenly) by a certain number. Whether you are trying to evenly divide a set of objects among a group of people or simplifying a fraction, knowing these divisibility rules can be a big help.
You can download a B&W pdf version here.
My previous post about platonic solids included a link to a site with printable templates for making your own solids. However, there are other easy ways to make paper platonic solids. Here are directions for making a tetrahedron using an envelope.
Step 1. Seal the envelope and then fold length-wise to mark the center line. I have marked the center line with a dotted line. (Marking the line is optional, I was just trying to make it more visible.)
Step 2. Fold the top corner down to the center line. The fold should extend down to the bottom corner.
Step 3. Fold opposite side over until it touches the corner that was folded down. This is to find the line perpendicular to the center line that touches point where the corner touches. I’ve marked this with another dotted line.
Step 4. Cut along the vertical line and unfold the corner.
Step 5. Fold the two corners on the open end of the envelope from the center line to the opposite top corner. These folds are marked with the highlighted lines. Fold forward and back to make a good crease.
Step 6. Open up envelope at the cut end and pop out the 4 folds you made (the highlighted folds plus the corresponding ones on the opposite side). Tape the open end together and you have your completed tetrahedron.
Conic sections are the curves created when a cone is intersected by a plane. There are 4 types of conic sections: circle, ellipse, parabola, and hyperbola. Sometimes, you may see it stated that there are 3 types of conic sections, because the circle can be considered a special kind of ellipse (we’ll discuss that more another time). The picture below shows how the curves are formed from the intersecting planes. As you can see, cutting the cone so that you miss the upper cone, but slice through the bottom gives a parabola. Cutting the cone so that you do not intersect the top/bottom of either cone gives a ellipse (a circle if the cut is parallel to the top/bottom). And, cutting the cone so that you slice through both the top and bottom cones gives a hyperbola.
As with many concepts, doing a hands-on activity can help demonstrate conic sections, so why not make your own? You can easily make your own cone with modeling clay and slice some conic sections. To make your cone:
- Cut a half-circle from sturdy paper or card stock and a second half-circle from parchment paper or wax paper.
- Curl cardstock half-circle into a cone and secure with tape and then place parchment paper inside (do not tape parchment paper).
- Pack cone to the top with air-dry modeling clay. Use small pieces and press down to the tip of the cone. You can pull the parchment paper out a little to peek as you go, to make sure there aren’t any big gaps or air bubbles.
- Flip over onto a paper plate or sheet of parchment paper. Remove cardstock cone, then peal away the parchment paper.
Once you have your cone you can use dental floss to make your sections. Cut straight across near the top to make a circle.
Below your circle, make a slanting cut to make your ellipse.
Below your ellipse, make a cut that slants enough to pass through the bottom of the cone to make your parabola.
You could do another cut that goes straight down, but without the top cone, it is hard to see it as a hyperbola. So with my students, I chose to just show them a picture of the hyperbola. Once the pieces dry, you can color your curves. Below, is the cone that my 6-year old daughter made. She chose to color her circle blue (she also colored the bottom blue, since that is also a circle). She colored her ellipse yellow and her parabola orange.
I recently wrote about using the Sierpinski triangle to teach students about fractals. Another relatively easy fractal for young students to make is the dragon curve fractal. The dragon curve is created by folding a strip of paper in half over and over (in the same direction) and then placing all folds at 90o. The pictures below show what the curve looks like after 1, 2, 3, and 4 folds. (This is a great time to introduce the term ‘iteration’ to students. Iteration means the process of repeating. The curve with 4 folds is the 4th iteration.)
If you start with a really long piece of paper, you may be able to fold it 5 or 6 times, but that is about the limit. If you could fold it 14 times, it would look like the picture below. If you kept folding, the curve would have the same shape, it would just be folded in on itself even more. Notice how each “section” of the curve looks like a smaller version of the whole curve.
It can be difficult to make all the folds to lie at 90o angles, even with a small number folds, so I like to have students tape their curve down onto a piece of paper. If you size things correctly, you can put your folded paper dragon curve onto a picture of a high-iteration dragon curve, as shown below – a great example of how a fractal looks similar at different scales. You can print my directions and handout here.
You can also use colored paper or cardstock to make some dragon fractal art. Take a look at these lovely dragon curves at CutOutFoldUp.com. If you are interested in the math behind the dragon curve, this page at The Fractal Umbrella explains how to mathematically describing the fold sequences.
Numberphile has a great video about the dragon curve (below) that covers many fascinating facts about the curve, including why it is also known as the Jurassic Park fractal.
A great activity for teaching kids about fractals is making Sierpinski triangles – one of the easiest fractals to draw.
The Sierpinski triangle is formed using this process:
- Take a point-side-up triangle
- Connect midpoints of the three sides to create a point-side-down triangle.
- Repeat process with the resulting point-side-up triangles.
- Repeat as many times as you want.
This example shows taking the process through 4 iterations.
You can print out the starting triangle from:
Fractal Foundation: Sierpinski Triangle (includes worksheet and instructions)
Spitz: Fractal Pack 1: Educators’ Guide (includes Sierpinski triangle template and other information for teaching about fractals)
If you have a classroom of kids, you can take their Sierpinski triangles and put them together to make a larger triangle – you just need a power of 3 to make a complete Sierpinski triangle (3, 9, 27, etc.).
Also, you can blow their minds by showing this Sierpinski Zoom video to show that the triangles can go on forever!
A couple of weeks ago, I did a post about origami. In origami, figures are made with only folding. But, there is a variation of origami, called kirigami, that involves folding and cutting and then opening up the folded paper. Familiar examples are the snowflake and paper dolls, that you may have made as a kid.
This Kirigami for Kids
website has directions for making several kirigami figures, including 5
and 6-pointed stars, snowflakes, and paper doll chains and rings.
In the field of mathematics, Erik Demaine of MIT has been working on what is called the fold-and-cut problem. The fold-and-cut process involves folding the paper, making one straight cut, and then unfolding the paper. Any figure formed from straight lines can be produced this way. The problem is figuring out how to fold the paper – that’s where the computational geometry comes in. Demaine has several patterns on his website, and the video below shows the process for creating a swan.