Tuesday, October 7, 2014

Earth as a Sphere and its Result

Last week we finished up the presentations in the section of of KotU geography album called "The Rotation of the Earth and its Consequence."  This week we moved on to the "Earth as a Sphere and its Result." I'll give you some of the highlights of our work.

We started with a rectangular prism made out of modeling dough.  The patio doors to the right represent "the sun."  The toothpicks represent rays of light.  We had the toothpick fly from the patio doors straight into the clay.  We discovered that the toothpick and clay form right angles and that if the Earth had been formed with flat sides all of the rays of light would strike the Earth perpendicularly.

Next, we did the same thing with more modeling clay this time in the shape of a sphere.  The boys measured the angles where the toothpicks meet the clay and determined that they form acute and obtuse angles in many places.  Most of the toothpicks meet our clay "Earth" obliquely.  Only the toothpicks at the equator reach the Earth perpendicularly. 

It helps to have done or to review the study of lines, and the study of angles presentations in geometry before you reach this presentation.

We pulled out our impressionistic charts to take another look at this idea.

The next part had us crammed in the windowless upstairs bathroom with construction paper, chalk and a flashlight.  The geography album seems to trap us in that bathroom pretty frequently.  This time I was able to get Me Too to hold the flashlight so I could get a picture.

I gave them each a chance to do this on their own piece of black construction paper.  We shined the flashlight onto the paper so that the light struck the paper perpendicularly and the boys traced the resulting circle of light.

Next we shined the flashlight onto the paper so that the light struck the paper obliquely and the boys traced the resulting ellipse.

Here is Me Too modeling his work.  We observed that the same number of "rays of light" covered different sized areas depending on whether they approached perpendicularly or obliquely.

The boys then asked me take a silly one.  Enjoy.

We looked at three impressionistic charts that show this same idea.  Yes, I have three different charts for this at home.  Because it's the type thing I find interesting, I'll show you the three charts and talk about some of the differences.

Above is the chart by ETC Montessori.  I like these charts a lot and they are certainly beautiful.  My complaint about them is always this:  The realistic images are neat but don't always clearly illustrate the idea at hand.  Sometimes something a little less realistic is better.  This chart was great for showing the boys how what we did in the bathroom with the flashlight relates to the actual Earth and light from the sun.  However, it doesn't really illustrate WHY this matters.  It matters that when the approach is perpendicular a large amount of light meets a relatively small area of the sphere.  It matters that when the approach is oblique that same amount of light covers a  larger area.  

There are two ways to illustrate that idea and apparently there is not a "standard" way to draw the chart for this as you'll see from the following two images.

This is one approach from the Mid America Geography album.  The have show a specific number (seven) of rays of light covering different sized areas as a result of their approach.  This is the chart I chose to use first and we actually took out a ruler and measured to see that the area reached was different.  I chose this chart first because I felt it actually matched the experiment we did in the bathroom:  A fixed amount of light coming from our flashlight hit different sized areas on our piece of paper.

This is the final chart we looked at for this concept.  In contrast to the other chart, on this chart the size of the area where the rays meet the Earth is the same and it is the amount of rays striking that specific sized area that is different.  This is helpful because what is important about this is that when the rays of the sun strike perpendicularly, as they do at the equator as we observed with the clay and toothpicks, that area is receiving a larger number of rays from the sun than the areas where the rays strike obliquely.  The KotU image does a good job of showing that.

There is another reason this is important.  The perpendicular rays have to pass through less of the Earth's atmosphere than the oblique rays.  Here is the ETC chart for that concept:

Again we pulled out a less realistic chart from one of the other albums so that the boys could actually measure.  You CAN do that on the ETC chart but it was easier on the other.

On our chart at home the atmosphere was about an inch thick for the perpendicular ray and about three inches for the oblique ray if I'm remembering correctly.  I explained that the ray loses heat as it travels through the atmosphere.  This is another reason it is hotter at the equator.  Less heat is lost en route.


  1. Many thanks for your continual posts. I really value them. Do you go through the geography manuals systematically in the order that they appear or do you tend to pick and choose what topics are of interest to your children. Not too sure which way to do it with my children as I would imagine some lessons would need previous lessons being covered so picking and choosing may not work, but I'm not sure as I'm starting out in elementary.

    1. Anon,

      Thank you for your kind words. My mentor, Jessica from KotU, emphasizes that you can pick and choose according to the child's interest. She will probably read this and correct me if I am misrepresenting. I absolutely do that, but not much. Based on the boys intereste, we did all of chapter 2 last year (Nature of the Elements) without doing the end of chapter 1. Just a couple days ago Kal-El asked me "Where does wind come from." I told him I have a whole bunch of lessons about that and he was really excited. I was going skip straight to it, but realized that a lot of the understanding of wind was going to be based on what we needed to learn about oblique and perpendicular rays. Here is a quote from one of the winds presentations: "Remember we’ve talked about how hot it gets around the equator where the
      perpendicular rays of the sun fall, more or less, all year long. As it gets hot there (indicate panel 1 on the chart), the air heats up and the hot air rises."
      So, while encouraged to follow the child's interest, I think that we are somewhat limited in that. I told Kal-El that we have to finish a few other presentations so that we understand the winds presentations and that we'll get to that soon. However, human geography and maps work are separate threads really so those parts of the albums can be done parallel.

  2. Great post! I totally agree about the ETC charts - they are attractive but don't often exemplify the concept.

  3. It is so very nice to have all of these visual pictures to support the presentations. I love seeming what is coming in our future as we work thru KOTU way behind you. :) Super inspiring.