This week the class is learning about the Distributive Property. It makes sense to cover the Distributive Property after integers, since students must learn how to distribute negative signs. On the other hand, back at the old charter school I tried following the "STEM text" order (which, in hindsight, wasn't a good idea) and actually taught Distributive before negatives. It was one of my seventh graders' most successful lessons -- probably because my lesson was much easier, because they didn't have to distribute negatives!
Meanwhile, tonight is the long awaited rebroadcast of Neil DeGrasse Tyson's Cosmos on FOX. Recall that I already watched most episodes on National Geographic in the spring and discussed them here on the blog, but I missed the first two episodes of the season. Since the FOX broadcast is scheduled for two hours tonight, I assume that both Episodes 1-2 are airing now.
Earlier, when I blogged on the night that two episodes aired, I would discuss the first episode that night and the second episode in my next post. This time, I'll write about both episodes in tomorrow's post. As soon as I complete tomorrow's post, we'll be done with Cosmos (unless Tyson makes more episodes of the show someday).
Lesson 2-5 of the U of Chicago text is called "Good Definitions." (It appears as Lesson 2-4 in the modern edition of the text.)
This is what I wrote last year about today's lesson:
Lesson 2-5 of the U of Chicago text deals with definitions -- the backbone of mathematical logic. Many problems in geometry -- both proof and otherwise -- are simplified when students know the definition.
Consider the following non-mathematical example:
Given: My friend is Canadian.
Prove: My friend comes from Canada.
The proof, of course, is obvious. The friend comes from Canada because that's the definition of Canadian -- that's what Canadian means. But many English speakers don't think about this -- if I were to say to someone, "My friend is Canadian. Prove that my friend comes from Canada," the thought process would be, "Didn't you just tell me that?" Most people would think that "My friend is Canadian" and "My friend is from Canada" as being two identical statements -- rather than two nonidentical statements that are related in that the second follows from the first from the definition of Canadian. Yet this is precisely how a mathematician thinks -- and how a student must think if he or she wants to be successful in mathematics.
And so, let's take the first definition given in this section -- that of midpoint -- and consider:
Given: M is the midpoint ofAB
Prove: AM = MB
The proof is once again trivial -- AM = MB comes directly from the definition of midpoint.
The text proceeds with the definition of a few other terms -- equidistant, circle, and a few terms closely related to circles. Then the text emphasizes biconditional statements -- that is, statements containing the phrase "if and only if." Some mathematicians abbreviate this phrase as "iff" -- but very few textbooks actually use this abbreviation.
Notice that Dr. Franklin Mason does give the "iff" abbreviation in his text. Last year, I noticed that his Lesson 2.4 on biconditionals had an (H) symbol -- which stands for honors. It's interesting how Dr. M once considered this to be an honors-only topic. Now, he wants to emphasize the importance of definitions and biconditionals to all his students, not just his honors students.
Every definition, according to the U of Chicago text, is a biconditional statement, with one direction being called the "meaning" and the other the "sufficient" condition. Mathematicians often use the terms "necessary" and "sufficient." Many texts use the word "if" in definitions when "if and only if" would be proper -- but our U of Chicago text is careful to use "if and only if" always with definitions.
This is what I wrote last year about today's lesson:
Lesson 2-5 of the U of Chicago text deals with definitions -- the backbone of mathematical logic. Many problems in geometry -- both proof and otherwise -- are simplified when students know the definition.
Consider the following non-mathematical example:
Given: My friend is Canadian.
Prove: My friend comes from Canada.
The proof, of course, is obvious. The friend comes from Canada because that's the definition of Canadian -- that's what Canadian means. But many English speakers don't think about this -- if I were to say to someone, "My friend is Canadian. Prove that my friend comes from Canada," the thought process would be, "Didn't you just tell me that?" Most people would think that "My friend is Canadian" and "My friend is from Canada" as being two identical statements -- rather than two nonidentical statements that are related in that the second follows from the first from the definition of Canadian. Yet this is precisely how a mathematician thinks -- and how a student must think if he or she wants to be successful in mathematics.
And so, let's take the first definition given in this section -- that of midpoint -- and consider:
Given: M is the midpoint of
Prove: AM = MB
The proof is once again trivial -- AM = MB comes directly from the definition of midpoint.
The text proceeds with the definition of a few other terms -- equidistant, circle, and a few terms closely related to circles. Then the text emphasizes biconditional statements -- that is, statements containing the phrase "if and only if." Some mathematicians abbreviate this phrase as "iff" -- but very few textbooks actually use this abbreviation.
Notice that Dr. Franklin Mason does give the "iff" abbreviation in his text. Last year, I noticed that his Lesson 2.4 on biconditionals had an (H) symbol -- which stands for honors. It's interesting how Dr. M once considered this to be an honors-only topic. Now, he wants to emphasize the importance of definitions and biconditionals to all his students, not just his honors students.
Every definition, according to the U of Chicago text, is a biconditional statement, with one direction being called the "meaning" and the other the "sufficient" condition. Mathematicians often use the terms "necessary" and "sufficient." Many texts use the word "if" in definitions when "if and only if" would be proper -- but our U of Chicago text is careful to use "if and only if" always with definitions.
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