We finished up our work with mathematics related to circles this week. We had a lot of fun on Thursday with the Internet Pizza Server. Students were asked to calculate the price per square inch of the four pizza sizes that are available, with and without the bizarre toppings that the site offers.

The idea of a unit price was new to many of our group. We talked about finding ways to compare prices in the grocery store when products come in different sizes made by different companies. The next time you and your child are shopping together, please take a few minutes to show them where the unit prices are posted and how they might influence a purchase decision. Consumer education is a very important part of math education in general.

On Friday, we used our Math in Focus texts to apply what students had learned about finding the diameter, radius, circumference, and area of circles. The end-of-unit problems were challenging to everyone because they required more reasoning than just running some numbers through the formulae.  Here’s one of the middle-level examples:

Students were asked to calculate the area of the black region. They were told that the square was 10 cm on each side. They had already solved problems related to finding the area of a quadrant (fourth) of a circle during the class. This one required realizing that there were two overlapping quadrants here — one was part of a circle whose center was at point A, and the other’s center was at point B.  The overlapped area was the black region. The radii of those circles were equal to the length of one side of the square, or 10 cm. To find the area of the black region, they needed to find the area of one of the quadrants and subtract it from the area of the square. That would leave the area of the other white region. When they know the area of the two identical white regions and the area of the square, they will be able to calculate the area of the black region.

So, at this point, they would have calculated the area of the square as 100 square cm and the area of each full circle as pi times 10 squared, or 314 square cm. (We were using 3.14 as an approximate value for pi.) So the area of each quadrant was 314 divided by 4, or 78.5 square cm. If we subtract one quadrant area (which is the black region plus one white section) from the area of the entire square, we get 100 – 78.5, or 21.5 square cm. The two white regions together must equal 21.5 x 2, or 43 square cm. If we subtract that from the area of the square, we are left with the area of the black region — 57 square cm.  This didn’t require any new formulae or algorithms, but it did require a lot of thinking and, for many students, the drawing of additional diagrams to “see” the full circles and understand the solution.  This is true mathematics. The part that a calculator plays is just about incidental. The math resides in the thinking that students are asked to do and which no amount of technology can replace.

We have begun our study of the foundations of Islam and its connection to West African kingdoms much later. In order to understand the core beliefs of Islam, we also need to know some basic things about Christianity and Judaism. We did several activities to build some of that understanding. One was asking students which of those three religions was the oldest. There was general disagreement within the group, with some students putting their vote behind each one of the three. As they are learning now, Judaism is the oldest and was followed by Christianity and then Islam. There are beliefs and traditions common to all three because of their shared history. To add to students’ general knowledge. we asked them to make a guess (educated or not) about the percent of the population in the USA and also in the world who are Christian, Jewish, Muslim, Hindu, Buddhist, non-religious, and “other.” They did this by coloring squares in a 10 by 10 grid to represent each percentage. We then looked up the information in several sources, found minor disagreement, but got fairly consistent answers. Students then colored new 10 by 10 grids to show that information and compared their first estimates to them.

As happens every time we do this, most of the group were amazed to find that Judaism (worldwide and within the US) is practiced by only about 1% of the population. This gave us a chance to talk about how much our sense of reality is influenced by narrow experience that we usually don’t recognize as narrow. We happen to live in an area and be part of a school community in which many more than 1% of the people are Jewish. But we are not representative of the world or even the nation in that respect. We also colored maps of the world to show where those belief systems are dominant. The next surprise was to see how much of the world is considered “animist” — which we defined as believing that there are spirits in all of nature. We talked a bit about how we might respond as animist believers  if someone came along and announced that those spirits didn’t exist and that there was just one all-powerful god. Most students agreed that it would be hard to accept, partly because of the evidence before us — natural disasters, fluctuation in success with crops and hunting, variations in the length of day and night (extreme in some places), disease, accidents, and more. It could seem very logical that those things were controlled by many different forces, and that they might be interpreted as deities and other spirit forms. So it would not be surprising if the idea of monotheism were met with a lot of resistance. We will go on doing some other activities that will enable students to learn more about the entire list of religions we mapped.

We also did some introductory vocabulary work relating to this and will continue with that. We talked briefly about terms such as heretic, infidel , theocracy, animism, clergy, adherent, secular, and more.

We started watching a long animated film called “The Life of Muhammad” and will finish it in the upcoming week. It gives us a fairly authentic perspective as it was made by Muslims and was approved by several authorities within that religion. One of the traditions is that the person of Muhammad is never drawn or shown. So the film-makers had to solve that challenge, and they did it by showing parts of the story from Muhammad’s viewpoint. That is, when he is in a scene, we often see it through his eyes. We’re having some good discussions about this kind of decision-making in films generally as well as taking time to talk about the story as it unrolls.

Junior Scholastic Magazine often contains articles that are not related to what we are doing in class but which are inherently interesting to our students as well as helping to build their general knowledge. They all enjoyed the cover article in our current issue which explained the science behind the attraction of snack foods such as Doritos. Another major article was about World War I. Since this is a period that none of our students have encountered in previous years as part of their exploration of history, we have decided to give it some time to create a broader understanding. Students have read the article but not done much with it.

This coming week, we will do some map work to learn more about the location of much of the fighting and also to see the extent of the Ottoman Empire, talk about the article’s explanation for how a small event led to a world war, read and discuss some related literature, and look at some images. The literature part will be twofold: the poem “In Flanders Fields” which was written by a soldier during the war — and the song “No Man’s Land” by Eric Bogle, written just after the end of the Vietnam war. The map work about the Ottoman empire will also connect with Islam, in that it was a Muslim theocracy whose influence is still present in much of the conflict in Europe and Asia today. Clearly this will not be a deep study as we are not giving it that much time, but we could probably spend the year on it and still leave a lot undone.

We hope you are hearing about some of this at home, or at least can get a conversation going if you ask.