Lesson 2-6: Unions and Intersections of Figures (Day 26)

Before we begin today's lesson, let's continue to take a peek at that seventh grade class I subbed for last week -- the one whose Google Classroom notifications I'm still getting in my email.

Well, today another IXL lesson was assigned. The due date isn't until next week, so it appears that this teacher is already looking ahead to next week. I notice that while one class only has to reach a maximum score of 80, the other class has to reach a score of 90 -- the honors class. It takes a minimum of nineteen questions to reach a score of 90 if all of them are correct. As we already know about IXL scores, getting from 80 to 90 is harder than going from 70 to 80, yet easier than getting from 90 to 100.

A new lesson for next week has also been posted -- one-step equations, multiplication and division. As we may recall, in the year at the old charter school, I didn't reach one-step equations until February. I sang the song "Solving Equations," and the important verse for one-step equations is, "Whatever we do to the left side, the same done to the right side. A letter alone on the left side, a number alone on the right side. That's all you have to do to solve it!"

Meanwhile, let's finally get to that first episode of Neil DeGrasse's science show -- the one I missed six months ago on National Geographic but didn't watch until last night on FOX.

Episode 1 of Cosmos: Possible Worlds is called "Ladder to the Stars." Here is a summary:

• Humans were originally hunters and gatherers, until we became voyagers to new worlds.
• Our "ship of the imagination" can take us anywhere through space and time.
• NASA's Voyager 1, the most distant object in space, is currently traveling in interstellar space.
• Our first destination is a pair of black holes that will cause a huge collision that will slow time.
• We can watch Cosmos through our devices because we can manipulate electromagnetic waves.
• If we could manipulate gravitational waves, we'd understand the start and future of the universe.
• On the Cosmic Calendar, life began (on September 15th) four billion years ago.
• DNA -- the ladder of the double-helix -- evolved from materials formed in the middle of stars.
• On the Cosmic Calendar, mammals (on December 26th) developed a neonatal cortex in the brain.
• About 9,000 years (20 seconds) ago, the first cities were built, yet society was still egalitarian.
• The first maps were devised, and humans were able to locate "home" on their maps.
• In the 17th century, Enlightenment philosophers began to study the world around them.
• Dutch-Jewish philosopher Baruch Spinosa wrote The Book of Nature -- to him, God is nature.
• There are two kingdoms in the world -- those who build the world and those who destroy it.
• Plants and insects developed a symbiotic relationship that lasts for millions of years.
• Every third bite of food we eat ultimately goes back to insect-pollinated plants.
• It's up to us to prevent the sixth great extinction event, the Anthropocene.
• A few decades from now, a fleet of starships will depart from the driest Atacama desert.
• Their destination is Alpha Centauri -- our first exploration of a world outside our solar system.

As the first episode of the season, this is in some ways a summary of the season to come. We've already watched episodes about the Cosmic Calendar, and the Anthropocene extinction event last spring, and traveling to Alpha Centauri is the topic of the second episode -- which, unfortunately, was not the second episode that aired on FOX last night.

Instead, for some reason FOX aired Episode 11 last night, "Shadows of Forgotten Ancestors." I already described this episode on the night it aired on National Geographic, in my April 13th post.

This is one reason why I like to watch episodes when they first air, rather than be on the mercy of networks who don't rerun their shows properly. I didn't watch the first two episodes in March because the schools were still open that week (and Cosmos was something I decided to watch during the lonely weeks of the coronavirus shutdown), and now there's still an elusive Episode 2 that I haven't seen.

And who knows when Episode 2 will finally air on FOX? It certainly won't be this upcoming Tuesday, because that's when the first of the Presidential Debates will air (and there was a commercial for those debates during Episode 1) -- oh well!

Lesson 2-6 of the U of Chicago text is called "Unions and Intersections of Figures." (It appears as Lesson 2-5 in the modern edition of the text.)

This is what I wrote last year about today's lesson:

Lesson 2-6 of the U of Chicago text focuses on unions and intersections. This is, of course, the domain of set theory.

In many ways, set theory is the basis of modern mathematics, and so many textbooks -- including higher math such as Precalculus and beyond -- mention set theory early on. Of course, the focus in this text is on unions and intersections of geometric figures. In particular, unions are used to define both polygon and angle, while intersections are used to define parallel lines.

The first three examples in the text, where the underlying sets contain natural numbers, real numbers, and points -- are OK. But I didn't like the fourth example, on airlines. I've decided to throw this one out -- if we want a non-mathematical example, why not just use sets of letters, such as {a, e, i, o, u}, the set of vowels?

One of the most important sets in mathematics is the null set, or empty set. According to the text, this set can be written as either { } or an O with a line through it (often called O-slash by students). Once again, since I can't represent that symbol on Blogger, let's use the strikethrough instead: O.

Now the text mentions that the intersection of two sets might be the empty set. But it doesn't mention what happens when one finds the union, or intersection, of the empty set and another set. As it turns out, the union of the empty set and another set is that other set -- so the empty set acts as the identity element for union, just as 0 is the identity for addition and 1 is the multiplicative identity. But the intersection of the empty set and another set is the empty set -- so the empty set acts as the absorbing element for intersection, just as 0 is the absorbing element for multiplication.

One question students often ask is, if { } is the empty set and O is the empty set, what's {O}? When I was young, I once heard a teacher point out that this is not the empty set because it's no longer empty -- it contains an element.