"Definitions are new terms we describe/define using undefined terms or previously defined terms. Theorems are ideas which must be proven by using existing axioms, definitions, or theorems."
This is the second and final page of the section on forming mathematical worlds. Pappas gives us the definition of, um, "definition," and then defines theorem.
Pappas now creates her new mathematical world -- a sort of mini geometry -- with definitions, axioms (or postulates), and theorems:
Undefined terms: Points and lines.
Definition 1: A set of points is collinear if a line contains the set.
Definition 2: A set of points is noncollinear if a line cannot contain the set.
Axiom 1: Our mini world contains only 3 distinct points, which do not lie on a line.
Axiom 2: Any two distinct points make a line.
Theorem 1: Only three distinct lines can exist in this world.
Proof:
Axiom 1 states that there are 3 distinct points in this world. Using Axiom 2 we know that every pair of these points determines a line. Hence three lines are formed by the three points of this world. QED
Clearly this is a much simpler geometry than Euclidean geometry, where at least four points exist. I do take issue with Pappas and her two axioms (or postulates). First she defines "collinear" and "noncollinear," then fails to use these terms in her axioms. She should have written:
Axiom 1: Our mini world contains only 3 distinct points, which are noncollinear.
Axiom 2: Any two distinct points are collinear.
Pappas concludes the page:
"The following section introduce you to some mathematical worlds and their inhabitants."
Recall that in yesterday's Putnam A1 question, we created our own mathematical world. After all, we could have called the rules from yesterday "axioms":
Undefined term: Sexy.
Axiom a) 2 is sexy
Axiom b) If n is sexy, then sqrt(n) is sexy.
Axiom c) If n is sexy, then (n + 5)^2 is sexy.
Axiom d) No other positive integers are sexy unless a)-c) force them to be sexy.
Theorem 1: 7 is sexy.
Proof:
Statements Reasons
1. 2 is sexy. 1. Axiom a)
2. 49 is sexy. 2. Axiom c) [1]
3. 7 is sexy. 3. Axiom b) [2]
Axiom a) 2 is sexy
Axiom b) If n is sexy, then sqrt(n) is sexy.
Axiom c) If n is sexy, then (n + 5)^2 is sexy.
Axiom d) No other positive integers are sexy unless a)-c) force them to be sexy.
Theorem 1: 7 is sexy.
Proof:
Statements Reasons
1. 2 is sexy. 1. Axiom a)
2. 49 is sexy. 2. Axiom c) [1]
3. 7 is sexy. 3. Axiom b) [2]
By the way, in Problem B1 we didn't create any new worlds, but worked in our Euclidean world. But I've been thinking more about the proof we gave yesterday.
If we were to present this proof to high school students, then I think it's better to use the same proof for both the intersecting and parallel cases, to make it easier. For intersecting lines, we showed that a dilation with center P and magnitude k maps the first line to a new line intersecting the second. So we should do the same in the parallel case -- forget about similar triangles, and just look at a dilation with center P and magnitude k. It turns out that the the "correct k" maps the first line directly to the second, while the "wrong k" maps the first line to a new line parallel to the second. Therefore only one k is the "correct k," contradicting the assumption that all values of k work.
Of course, the problem with even with mentioning dilations is that if we're following the U of Chicago text, the students won't see dilations until Chapter 12 -- and most other texts don't reach dilations and similarity by the first week in December. Ironically, the Third Edition of the U of Chicago text briefly mentions dilations in Lesson 3-7. But these are only the "S_k" dilations whose center is the origin (0, 0). Putnam B1 has no coordinates, and it's not obvious that point P, which could be anywhere in the plane, should be labeled (0, 0).
Lesson 7-3 of the U of Chicago text is called "Triangle Congruence Proofs." Let's forget about dilations and get back to the proofs we've been working on.
This is what I wrote two years ago about today's lesson:
Lesson 7-3 of the U of Chicago text discusses triangle congruence proofs. Finally, this is what most Geometry students and teachers think of when they hear about "proofs."
There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.
Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.
Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.
Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.
Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.
Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.
Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.
There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.
Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.
Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.
Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.
Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.
Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.
Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.
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