But let's start with substitute teaching. Today I subbed in my old district -- believe it or not. Thus it really is Day 44 in this district. (In my new district, today is Day 36.)
It's a special ed class -- in fact, it's the same teacher whose class I covered the last time I subbed in this district -- which was exactly one month ago, September 17th. Thus you can refer back to my September 17th post in order to recall some information about this class. Again, I won't bother to do a "Day in the Life" for this special ed class (since I didn't do one for September 17th either). And yes, it seems as if the only time I sub in this district these days is for this class.
This district has an block schedule, with all classes on Fridays (which are also late days) so that the schedule aligns with days of the week. Then Wednesdays become odd period days. Thus first, third, and fifth periods meet today. (The fact that there was no school in this district on Monday throws this week's schedule off a bit, but it had nothing to do with today.)
First period is observing (co-teaching) Algebra I -- which marks only the third time I've been in a math class in this district (with the first two being observing this same class)!
And as it turns out, the resident teacher for this class is also out with a sub. But unlike September, today's sub is actually a long-term sub who knows math. Therefore she runs the class, and I can only help a few students out when needed.
The students are preparing for the Chapter 3 Test, which will be given on Monday. As I mentioned back in September, this isn't exactly the same as the Glencoe text. In particular, Chapter 3 is on graphing linear equations.
The teacher uses the ever-popular Interactive Notebooks, and so she tells the students to prepare their notebooks as they'll be due on test day. Here I reproduce her notebook plan, which includes the notebook page numbers, dates, and lessons:
Page Date Lesson
32-33 10/3 3.1: Functions
34-35 10/3 3.2: Function Notation and Plotting Points
36-37 10/5 3.3: Standard Form, Intercepts, Vertical/Horizontal Lines
38-39 10/8 3.4: Slope of a Line
40-41 10/8 3.5: Graphing Equations of Lines in Slope-Intercept/Standard Form
42-43 10/12 3.6: Graphing Absolute Value Equations
44 10/17 Chapter 3 Review
45 10/17 My Goals/Reflection
I warn the students that the unit on linear equations is often difficult for many learners. I've student taught Algebra I, and student grades dropped after this unit. And as I mentioned a few months ago, even the summer school students (in the summer class that I never got to teach) struggled during the graphing linear equations unit. And neither of those Algebra I classes had to graph absolute value equations -- the subject of Lesson 3.6 here.
When reviewing functions with the students, of course I had to mention DIXI-ROYD. I know -- Sarah Carter no longer teaches Algebra I (and she doesn't post as much any more now that she's an Algebra II teacher). Yet I still use her DIXI-ROYD mnenonic every time I teach her favorite Algebra I unit on functions.
The other two classes are Academic Enrichment. First, there is a short assignment where the students are supposed to write down a hypothetical question they would ask each of their teachers. (For example, they might ask their Algebra I teacher how to graph lines in slope-intercept form.)
One junior is enrolled in Algebra II -- yes, it's a rarity when a special ed student makes it that far. He is currently working on imaginary numbers. A few other students are working in Algebra I, including this same Chapter 3 on graphing linear equations.
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Find BD.
(Here is the given info from the diagram: in Quadrilateral ABCD, the midpoints of
This is a midpoint quadrilateral problem. The important thing to note is that the sides of a midpoint quadrilateral are parallel to, and half the length of, the diagonals. The side parallel to
Midpoint quadrilaterals don't appear in the U of Chicago text except as an exercise (and we're a few weeks away from that exercise). They don't appear in the Moise text that I purchased over the weekend either (though they're also hinted at in an exercise).
And speaking of the library book sale, I want to discuss two other books that I purchased at that time, since I usually select a side-along reading book at that time. Over the summer, I've notice that when other math teachers discuss their reading books, it's usually not the recreational math books that I tend to read during our side-along reading. Instead, it's books about teaching. Ruth Parker's book on number talks is typical (though once again, number talks are something you do rather than merely read about). Eugenia Cheng's newest book sort of takes us away from recreational math, but it's still not what I'd call a teaching book.
So I decided that I'd rather choose a teaching book for our side-along reading book. At the library book sale, I found two possible books, and bought both of them (as each was 50 cents). One of them is Horace's School: Redesigning the American School, written in 1992 by Theodore R. Sizer. This book sounds interesting, but it's apparently the sequel of an earlier book, Horace's Compromise. I don't wish to confuse the readers of this blog by reading a sequel when we've never read the first book together.
Thus instead, we'll read the other book, Rainbows of Intelligence: Exploring How Students Learn, written in the year 2000 by Sue Teele. It refers to the concept of multiple intelligences, with which many teacher should be familiar. Thus this will be our side-along reading book, starting today. Yes, I know that we just finished side-along reading Eugenia Cheng's book. But I've always timed our reading books to the library books sales. So let's dive in. In addition to the "milestone" label, I'm adding a new "Sue Teele" label to this post.
(And yes, I started Eugenia Cheng's book the last time I subbed in the old district, and now that I've returned to that district, I'm starting another book.)
Chapter 1 of Sue Teele's Rainbows of Intelligence is called "A Focus on How Students Learn." Here's how it begins:
"It has been stated that all students can learn and succeed, but not in the same way and not on the same day."
This is, of course, the basic idea behind the concept of multiple intelligences:
"Intellectual performances may vary on different days and in different ways when measured by a variety of criteria. Individuals appear to process their own rainbows of intelligence; no two individuals process the same way."
As readers quickly find out, the rainbow is Teele's dominant metaphor throughout this book. She tells us about driving between two Northern California towns in a rainstorm -- she gets lost, until rainbows suddenly appear in the sky. She uses them as a guide to make it back to safety. She continues:
"When I witnessed the rainbows, a new idea started to evolve, focusing on the colors of the rainbow and the primary, secondary, and complementary colors of the color wheel."
At this point, Teele summarizes how she organizes her book:
"In the first six chapters of this book, I present several views of the nature of intelligence and discuss how the educational community must achieve a more balanced approach to teaching and learning. Chapter 1 explores different approaches to understanding intelligence."
The second part of her book is more practical:
"Chapter 7 presents units I developed that revolve around the idea of colors. These units can be adapted for any grade level and content area."
Returning to the current Chapter 1, in this chapter Teele describes four approaches to intelligence:
"The psychometric approach acknowledges a single, unitary, quantitative concept of intelligence."
She highlights the following statement:
"The psychometric approach focuses only on two ways to learn: linguistic and logical-mathematical."
And indeed, the PSAT that students took last week, as well as the SAT to which the PSAT leads, has three sections that correspond directly to those two ways to learn (reading and writing sections for linguistic, and of course the math section). This approach is the only one accepted by traditionalists and other defenders of the PSAT and SAT. But Teele warns:
"Students' minds are multifaceted instruments that cannot be completely defined through any one type of assessment."
The author proceeds:
"A second approach to intelligence is developmental progressions."
This approach is strongly associated with Jean Piaget. Teele tests out Piaget's theory of development on a six-year old boy, "Shawn":
"He was shown two rows of six pennies each and asked to identify if the rows were the same or different. For the first two rows, he stated they were the same. He was then shown a second two rows."
In the first row, the pennies were organized 2, 1, 1, 2, and in the second row, it's 1, 2, 1, 2:
"He experienced no difficulty seeing the similarities and differences between the coins."
Even though he was a six-year-old, his understanding represented a higher stage, more representative of an older student:
"The third [stage], from 11 to 12 years, is when the child is able to understand the process of random mixture and see an operative system of permutation."
But even though Shawn showed the third stage of understanding how many pennies there were, he was still in the first stage when Teele changed them to marbles. She emphasizes the following:
"Each individual is unique in each of his or her processing skills."
Notice that Parker's number talks remind us of this counting exercise. Meanwhile, the author also writes about Vygotsky here:
"Children's developmental progressions of ability differ individually and depend on the unique combinations of the ways they process information and the different tasks they are given."
And she writes:
"Teachers must consider the students' zones of proximal development and move from teacher to facilitator."
Traditionalists occasionally mention Vygotsky's "zones of proximal development" as well. But of course, they don't believe that a teacher should be demoted to facilitator. Instead, Vygotsky's zones are used to justify tracking.
Like Piaget, Vygotsky divides childhood into various stages:
0-2 years: Emotional contact
2 years: Manipulation of objects
3-7 years: Role play and symbolic activity
7-11 years: Emphasis on formal study in school
12-18 years: Blending of interpersonal relations and exploration of careers
Teele observes and mentions her own grandchildren as an example. For example, she writes about her five-year-old grandson Davey. (OK, I admit it -- I write about Davey since my name is "David."):
"He creates symbolic places with his [toy] cars and trucks. He has begun to show interest in learning about shapes, colors, letters of the alphabet, and counting."
Let's skip to her older grandchildren -- the ones closer to the ages that I actually teach:
"Brian, who is 11, is very musically talented and quite bodily-kinesthetic. He is an excellent soccer player and loves working with spatial games on the computer. Kristin, who is 14, is a strong academic student who is very linguistic and interpersonal. She is exploring careers and is considering broadcasting."
Teele reminds us:
"Different teaching methods are required to encourage students to learn from their dominant ways of processing and to build on previous successful experiences."
Reuven Feuerstein is the last theorist Teele mentions in this section:
"His theory of structural cognitive modifiability states that learning can be transformed to enable one to learn better."
The author proceeds with the third approach to intelligence, psychobiological:
"Several researchers have studied the brain from a biological perspective to explore new ideas about what intelligence is and how to measure it."
And of this psychobiological perspective, she writes:
"It incorporates awareness of gender differences in some cognitive activities. The brain remains flexible and evolves throughout an individual's life. It is altered based on life experiences, situations, and circumstances."
Finally, the author writes about one last approach to intelligence:
"Substantial research supports the fact that individuals process in multiple, interactive, and complex ways. This leads directly to the fourth approach, multiple forms of intelligence."
The idea of multiple intelligences is most strongly associated with Howard Gardner, but Teele also cites Robert Sternberg of Yale. Here is his Triarchic Theory:
- Analytical
- Creative
- Practical
In case you don't already know, Gardner's theory consists of seven types of intelligence -- linguistic, logical-mathematical, spatial, musical, bodily-kinesthetic, intrapersonal, and interpersonal:
"These two theories focus on individual differences among students."
Teele concludes the chapter with a preview:
"There continue to be remarkable discoveries in brain research and in how individuals learn. Chapter 2 highlights some of these discoveries. We must examine ways to redesign our educational system to allow all students to develop their intellectual abilities to full potential."
Lesson 4-4 of the U of Chicago text is called "The First Theorem in Euclid's Elements." This lesson from the Second Edition, like yesterday's 4-3, has no exact counterpart in the Third Edition. The closest corresponding chapter would be part of Lesson 5-4.
This is what I wrote last year about today's lesson:
Lesson 4-4 of the U of Chicago text introduces formal, two-column proofs. As a first example of a proof, the text gives Proposition 1 from Euclid. Since Euclid was among the first to write formal proofs and this was his first theorem, the students' first proof will be one of the oldest proofs written in the whole world.
I already mentioned David Joyce's website and his Euclid pages. Here's a link to Proposition 1:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html
And here's what Joyce writes about why Euclid chose this one to be first:
"This proposition is a very pleasant choice for the first proposition in the Elements. The construction of the triangle is clear, and the proof that it is an equilateral triangle is evident. Of course, there are two choices for the point C, but either one will do.
"Euclid could have chosen proposition I.4 [SAS Congruence -- dw] to come first, since it doesn’t logically depend on the previous three, but there are some good reasons for putting I.1 first. For one thing, the Elements ends with constructions of the five regular solids in Book XIII, so it is a nice aesthetic touch to begin with the construction of a regular triangle. More important, though, is I.1 is needed in I.2, and that in I.3. Propositions I.2 and I.3 give constructions for moving lines, and I.4, although not logically dependent on I.2 or I.3, does use the concept of superposition which involves, in some sense, moving points and lines."
The U of Chicago text gives a two-column proof of Euclid's Proposition 1. There are a few differences between a two-column proof in this text and those found in most other texts. First, the U of Chicago labels the two columns "Conclusions" and "Justifications" -- whereas most other texts label them "Statements" and "Reasons." Also, most books start their two-column proofs with the "Given" information, but the U of Chicago skips the "Given" lines. On this blog, I plan on doing proofs the way most traditional texts do them, with "Statements," "Reasons," and "Given."
Notice that this theorem is truly a construction -- if available, teachers can have the students use a compass to draw the circles and a straightedge to draw the segments. This sounds like something mentioned in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
The question is, must the equilateral triangle and the square be inscribed in the circle, or must only the regular hexagon be so inscribed? After all, Euclid's equilateral triangle is not inscribed in the circle -- for a triangle to be inscribed, all three vertices must lie on the same circle, but the vertices of Euclid's triangle lie on different circles. Unfortunately, the standards are vague here.
Now Example 1 in the text gives another example of a two-column proof. Many books refer to this as the Alternate Exterior (not Interior) Angle Theorem. But we're saving proofs based on parallel lines until a little later. The midpoint proof in Example 2, of course, can be given.
I decided that this is a good time to include two of the proofs that I gave last year -- the Uniqueness of Perpendiculars Theorem and the Line Perpendicular to Mirror Theorem. The first theorem is actually Euclid's Proposition 12:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI12.html
I chose these two theorems because they fit here the best. After all, Joyce points out that the same double-equilateral-triangle construction is used here as in Proposition 1. Also, Example 2 in Lesson 4-1 hints at this same proof. The difference between our proof and Euclid's is that we explicitly mention reflections in our proof, since Common Core demands that we use reflections in proof. The Line Perpendicular to Mirror Theorem follows directly from the Uniqueness of Perpendiculars Theorem, and tells us that a line perpendicular to the reflecting line (the mirror) is invariant -- that is, its image is identical to the preimage. Finally, these two theorems on perpendiculars provide a great segue into Lesson 4-5, the Perpendicular Bisector Theorem.
Now let's do the reflections. The key is that the graph of x = h is a line perpendicular to the x-axis for any value of h, and likewise y = k is perpendicular to the y-axis for any value of k. We then reflect the point (h, k) by finding the images of the lines x = h and y =k. For example, if the mirror is the y-axis, then any line of the form y = k must be mapped to itself -- this is because y = k is perpendicular to the y-axis, and so by the Line Perpendicular to Mirror Theorem, y = k is an invariant line. That the image of x = h must be x = -h follows from the Reflection Postulate and Ruler Postulate applied to the x-axis.
I'm not sure whether I want to post these proofs for the students yet -- not because the proofs are hard, but because they have enough to worry about as they first learn about proofs. The cases where the mirror is either of the axes can be proved using today's theorems. The mirrors y = x and y = -x can wait until the first part of Chapter 5, where we prove properties of isosceles triangles and kites, but still before we give a parallel postulate. The mirrors x = h and y = k require a parallel postulate, but we only need to consider the properties of rectangles, not similar triangles. No other mirrors appear on the Common Core test -- thank goodness for that!
Regarding the worksheets, as I mentioned earlier, I like the idea of having the students experiment with each theorem by folding the paper (for reflections) before actually proving the theorem. In some ways, this is the Michael Serra approach, except that the proof is given immediately after -- not near the end of the book. Because of this, I'm including several worksheets today. (Don't say I didn't warn you!) I have the statement of each theorem on one side of the page -- large enough to encourage folding -- and a two-column proof on the other, with a few Reasons left blank for the students to write in.
And as for the Exercises, notice that Question 12 contains a flow proof. Many other geometry texts emphasize flow proof as a third type of proof, after paragraph and two-column proofs. But in the U of Chicago texts, a flow proof appears in this question and then never again in the text. Since it's a rewriting of the Alternate Exterior Angle Theorem from Example 1, I'm throwing it out. Let's not confuse the students with flow proofs just yet -- and I'll probably leave flow proof out altogether.
Question 24 is the Exploration/Bonus Question. It directs students to discover that the altitudes of a triangle are congruent -- that is, that they meet at a point (called the orthocenter). Notice that many texts cover the four concurrency proofs for triangles (centroid, circumcenter, incenter, and orthocenter), but these aren't covered fully in the U of Chicago text. The orthocenter never appears in the text again after this Exploration Question. Actually, one concurrency question appears in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.CO.C.10
Prove theorems about triangles. Theorems include ... the medians of a triangle meet at a point.
The proof that the medians meet at the centroid requires similar triangles, and so the proof must wait until the second semester. But the proof that the perpendicular bisectors meet at the circumcenter is the easiest of the concurrency proofs, and it appears in the very next section, 4-5. Incidentally, the proof that the altitudes meet at a point entails constructing a larger triangle and showing that the altitudes of the smaller triangle extend to the perpendicular bisectors of the larger triangle (so the orthocenter of the smaller triangle is the circumcenter of the larger triangle)! But this proof requires parallel lines and Playfair, so it must wait. Still, the students can still explore this in the Exercises.
This is what I wrote last year about today's lesson:
Lesson 4-4 of the U of Chicago text introduces formal, two-column proofs. As a first example of a proof, the text gives Proposition 1 from Euclid. Since Euclid was among the first to write formal proofs and this was his first theorem, the students' first proof will be one of the oldest proofs written in the whole world.
I already mentioned David Joyce's website and his Euclid pages. Here's a link to Proposition 1:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html
And here's what Joyce writes about why Euclid chose this one to be first:
"This proposition is a very pleasant choice for the first proposition in the Elements. The construction of the triangle is clear, and the proof that it is an equilateral triangle is evident. Of course, there are two choices for the point C, but either one will do.
"Euclid could have chosen proposition I.4 [SAS Congruence -- dw] to come first, since it doesn’t logically depend on the previous three, but there are some good reasons for putting I.1 first. For one thing, the Elements ends with constructions of the five regular solids in Book XIII, so it is a nice aesthetic touch to begin with the construction of a regular triangle. More important, though, is I.1 is needed in I.2, and that in I.3. Propositions I.2 and I.3 give constructions for moving lines, and I.4, although not logically dependent on I.2 or I.3, does use the concept of superposition which involves, in some sense, moving points and lines."
The U of Chicago text gives a two-column proof of Euclid's Proposition 1. There are a few differences between a two-column proof in this text and those found in most other texts. First, the U of Chicago labels the two columns "Conclusions" and "Justifications" -- whereas most other texts label them "Statements" and "Reasons." Also, most books start their two-column proofs with the "Given" information, but the U of Chicago skips the "Given" lines. On this blog, I plan on doing proofs the way most traditional texts do them, with "Statements," "Reasons," and "Given."
Notice that this theorem is truly a construction -- if available, teachers can have the students use a compass to draw the circles and a straightedge to draw the segments. This sounds like something mentioned in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
The question is, must the equilateral triangle and the square be inscribed in the circle, or must only the regular hexagon be so inscribed? After all, Euclid's equilateral triangle is not inscribed in the circle -- for a triangle to be inscribed, all three vertices must lie on the same circle, but the vertices of Euclid's triangle lie on different circles. Unfortunately, the standards are vague here.
Now Example 1 in the text gives another example of a two-column proof. Many books refer to this as the Alternate Exterior (not Interior) Angle Theorem. But we're saving proofs based on parallel lines until a little later. The midpoint proof in Example 2, of course, can be given.
I decided that this is a good time to include two of the proofs that I gave last year -- the Uniqueness of Perpendiculars Theorem and the Line Perpendicular to Mirror Theorem. The first theorem is actually Euclid's Proposition 12:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI12.html
I chose these two theorems because they fit here the best. After all, Joyce points out that the same double-equilateral-triangle construction is used here as in Proposition 1. Also, Example 2 in Lesson 4-1 hints at this same proof. The difference between our proof and Euclid's is that we explicitly mention reflections in our proof, since Common Core demands that we use reflections in proof. The Line Perpendicular to Mirror Theorem follows directly from the Uniqueness of Perpendiculars Theorem, and tells us that a line perpendicular to the reflecting line (the mirror) is invariant -- that is, its image is identical to the preimage. Finally, these two theorems on perpendiculars provide a great segue into Lesson 4-5, the Perpendicular Bisector Theorem.
Now let's do the reflections. The key is that the graph of x = h is a line perpendicular to the x-axis for any value of h, and likewise y = k is perpendicular to the y-axis for any value of k. We then reflect the point (h, k) by finding the images of the lines x = h and y =k. For example, if the mirror is the y-axis, then any line of the form y = k must be mapped to itself -- this is because y = k is perpendicular to the y-axis, and so by the Line Perpendicular to Mirror Theorem, y = k is an invariant line. That the image of x = h must be x = -h follows from the Reflection Postulate and Ruler Postulate applied to the x-axis.
I'm not sure whether I want to post these proofs for the students yet -- not because the proofs are hard, but because they have enough to worry about as they first learn about proofs. The cases where the mirror is either of the axes can be proved using today's theorems. The mirrors y = x and y = -x can wait until the first part of Chapter 5, where we prove properties of isosceles triangles and kites, but still before we give a parallel postulate. The mirrors x = h and y = k require a parallel postulate, but we only need to consider the properties of rectangles, not similar triangles. No other mirrors appear on the Common Core test -- thank goodness for that!
Regarding the worksheets, as I mentioned earlier, I like the idea of having the students experiment with each theorem by folding the paper (for reflections) before actually proving the theorem. In some ways, this is the Michael Serra approach, except that the proof is given immediately after -- not near the end of the book. Because of this, I'm including several worksheets today. (Don't say I didn't warn you!) I have the statement of each theorem on one side of the page -- large enough to encourage folding -- and a two-column proof on the other, with a few Reasons left blank for the students to write in.
And as for the Exercises, notice that Question 12 contains a flow proof. Many other geometry texts emphasize flow proof as a third type of proof, after paragraph and two-column proofs. But in the U of Chicago texts, a flow proof appears in this question and then never again in the text. Since it's a rewriting of the Alternate Exterior Angle Theorem from Example 1, I'm throwing it out. Let's not confuse the students with flow proofs just yet -- and I'll probably leave flow proof out altogether.
Question 24 is the Exploration/Bonus Question. It directs students to discover that the altitudes of a triangle are congruent -- that is, that they meet at a point (called the orthocenter). Notice that many texts cover the four concurrency proofs for triangles (centroid, circumcenter, incenter, and orthocenter), but these aren't covered fully in the U of Chicago text. The orthocenter never appears in the text again after this Exploration Question. Actually, one concurrency question appears in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.CO.C.10
Prove theorems about triangles. Theorems include ... the medians of a triangle meet at a point.
The proof that the medians meet at the centroid requires similar triangles, and so the proof must wait until the second semester. But the proof that the perpendicular bisectors meet at the circumcenter is the easiest of the concurrency proofs, and it appears in the very next section, 4-5. Incidentally, the proof that the altitudes meet at a point entails constructing a larger triangle and showing that the altitudes of the smaller triangle extend to the perpendicular bisectors of the larger triangle (so the orthocenter of the smaller triangle is the circumcenter of the larger triangle)! But this proof requires parallel lines and Playfair, so it must wait. Still, the students can still explore this in the Exercises.
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