TUTORIAL: Constructing Parabolic Curves (for little hands)

One evening recently, I found myself in need of a little time to unwind. While the kids played, I grabbed a few sheets of graph paper and doodled parabolic curves using straight lines. An hour later, I posted this photo to my social media, with a caption about "fun math."

I received some comments asking how to do this, so I drew up some instructions and posted those as well. This got me thinking about the math behind parabolic curves, and how I might introduce that math to my young children. I posted some of my pictures and ideas to one of my homeschooling support groups, and the idea for this blog was born!

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I used graph paper and a ruler to construct the above images. However, my children are all young, and I don't believe they yet have the fine motor skills necessary to draw parabolic curves using graph paper and a straight edge (though I am going to try it with my 5-year-old, soon...he might be able to do it, with assistance). I decided to create a large version on the wall, using yarn.

What you need:

(*Not pictured: tape)

-Large sheet of paper (I used 18x24 drawing paper, medium-weight, and cut it square, but you could use anything. I just wanted it BIG.)

-Ruler

-Scissors

-Markers

-Yarn in various colors to match the markers

The set-up:

(*Hey, look. There's the tape. 😉)

For a basic parabolic curve, draw two perpendicular lines of equal length, intersecting in the bottom-left corner (Quadrant I on the XY-plane).

Make an equal number of marks on each axis, equally spaced. Each of my lines is 16 inches long, with four marks spaced every 4 inches. 

Color-code your marks as shown. 

     -Choose your color order.

     -Starting at the top of your vertical axis and using your markers in order, draw a dot over each mark.

     -Do the same with your horizontal axis, working from left to right with your colors in the same order.

     -Note that the intersection of your two lines will *not* have a colored dot.

Cut lengths of yarn in colors to match the marks on your paper.

To make it easier for my kids, I wrapped each string around its coordinating marker.

Tape your paper to the wall, then call your kids!

The experience (part 1):

Ask your kids if they know it is possible to make a curving line using only a bunch of straight lines. (They'll probably say no! They might not believe you when you tell them it's possible, and that you'll show them how.)

I let my kids take turns placing the strings on the graph. We started at the top of the vertical line and worked our way down.

Here's my 5-year-old, stretching the red yarn between the two red dots. I helped him tape it in place.

First string is placed!

My daughter (almost 3) placed the green string, as shown:

String #3 was purple:

The blue string finished our curve:

We traced the curve formed by the lines, then pointed out the corners where the strings crossed. The lines basically function as tangent lines to the curve (I'll talk more about tangent lines later.)

I pointed out that the corners were pretty obvious because we only used a few strings and they were spaced pretty far apart. I asked them what they thought would happen if we used more strings with less space between them. 

The experience (part 2):

We decided to try it out with more strings. I created another graph using the method outlined above. This time, I drew lines that were 15 inches long and placed 12 dots on each line, spaced 1 1/4 inches apart. You could use different lengths and numbers of dots, but ensure that you use the same number of dots on each line and that they are spaced equidistant from one another.

Have your kids place the strings in the same way that they did on the first graph.

Your curve will be much more smooth this time, and the corners much less obvious. Have your kids look closely and see if they can find them! The contrasting colors of your yarn will help with this.

My son points to one of the corners:

The math:

What is a parabola?

A parabola is a conic section. In other words, it's a shape you get by cutting out a paper-thin slice of a cone. Other conic sections include ellipses and hyperbolas. 

• A parabola occurs when you take a slice of a cone that cuts through both the base and the side of a cone. 
• An ellipse occurs when your slice goes only through the sides of the cone, without intersecting the base. A circle is a special kind of ellipse. All circles are ellipses, but not all ellipses are circles (in the same way that all squares are rectangles, but not all rectangles are squares). 
• A hyperbola occurs when you set two cones tip-to-tip vertically (like an hourglass) and then take a vertical slice that passes through both cones.

To learn more about conic sections, including parabolas, click here. 

What is a tangent line?

A tangent line is a straight line that intersects a curve at exactly one point. The slope (steepness) of a tangent line is equal to the slope of the curve at the intersection point.

Any given curve has an infinite number of tangent lines, as there are an infinite number of points on that curve. When we construct a parabolic curve using line segments, we essentially approximating that curve by using a finite number of tangent lines.

To learn more about tangent lines, click here.

Next steps:

Now that you have learned how to make parabolic curves using straight lines, take another look at the first picture in this post. Can you figure out how to modify the method to create the figures in that picture? Can you create any designs of your own? Have fun with this concept! 

When I described these figures as "fun math," one friend commented that she wasn't so sure about the math part, and that it looked more like structured art. My response was that structured art = math. Can you see why?

I will be sharing more about parabolic curves (and why a circle might be more accurately described as a figure with infinite corners, rather than as a figure with no corners at all) in a future post. Stay tuned!