In this post I'll focus on the sophomore class, since the aide is out of this class as well. The students are supposed to be finishing and submitting an essay on Chromebooks. For the most part, the students are quiet. The regular teacher specifically states that students are not to have free time -- those who finish are supposed to complete work for another class. But some students decide to play games on the computers anyway.
Then at the end of class, three football players claim that they're supposed to leave a few minutes early for a some meeting. If they had informed me at the start of class, I would have believed them, but instead they say it just as students are lining up to wait out the clock. Instead, I assume that they are lying just so that they can leave before the bell rings. Only one of them is wearing a uniform, so as far as I know, it's just one player and his two friends waiting a few extra seconds of lunch. (It's hard for me to stop them, since as usual, I'm near the Chromebook cart making sure that students are charging the laptops.) I place all three names on my bad list for the teacher.
The only other name I place on the bad list is in freshman English. The students are supposed to take a test. This time, they do have something definite to do afterward -- complete a word search. But this time, one guy starts drawing on another student with his word search marker. He also tries to get another girl in trouble -- she asks for a restroom pass (after lunch, so she must wait until the class is half over), and then he claims that she just left to talk to friends outside. Even though I catch him and write his name on my bad list, the girl is visibly upset and is unable to concentrate on the test.
Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Find this quadrilateral's perimeter to the nearest cm.
(Here is the given info from the diagram: the sides are 0.0622 m, 0.207 dm, 31.5 mm, and 756 um.)
This would have been a simple Lesson 8-1 perimeter question, except that Pappas expresses the side lengths using different metric units. (It's too bad that this question couldn't have appeared during National Metric Week last week!)
Since we want the final answer in centimeters, we might as well convert all of the given lengths to the desired units. The first three conversions are straightforward -- 0.0622 meters is 6.22 cm, 0.207 decimeters is 2.07 cm, and 31.5 millimeters, is 3.15 cm.
Let's look at the fourth side. The symbol "um" is ASCII -- it should be the Greek letter "mu" then the letter "m" for meters. The symbol stands for "micrometers" (or microns) -- we must use the letter "mu" because "m" is already taken for the "milli-" prefix.
Now here's the problem -- the prefix "micro-" means one millionth. So 756 um is 0.0756 cm. But this value clearly doesn't make sense as it doesn't make the perimeter equal to a whole number of cm (much less equal to the date). Instead, we obtain a perimeter of 12.5156 cm.
This is an error on the Pappas calendar. It appears that she intends "um" to mean a ten-thousandth of a meter rather than a millionth. But there is no prefix for 1/10000. If she wants to use microns, then she should have used 75600 um, not 756 um. Only then will the perimeter work out to be 19.00 cm --and of course, today's date is the nineteenth.
Speaking of powers of ten, the Mega Millions lottery jackpot has reached one billion dollars for the first time. Just as I did when the Powerball jackpot cracked a billion, the Square One TV Fat Boys song is in order:
By the way, in the classroom today, I see the following poster on the wall:
Multiple Intelligences: A Field Guide, by Marek Bennett
Linguistic: Read, Write, Talk, Listen
Logical-Mathematical: Quantify, Think Critically, Reason, Experiment
Visual-Spatial: See, Draw, Visualize, Color, Map
Bodily-Kinesthetic: Build, Act, Touch, Move, Dance
Musical: Sing, Rap, Drum, Play
Interpersonal: Share, Teach, Collaborate, Interact
Intrapersonal: Connect to self, Make authentic choices, Reflection
Naturalist: Experience, Connect to living things, Care for, Explore
Notice that this illustrated poster fits today's Teele chapter like a glove.
Chapter 3 of Sue Teele's Rainbows of Intelligence is called "The Theory of Multiple Intelligences." It begins as follows:
"Gardner argued that intelligence cannot be adequately assessed only by tests because paper-and-pencil measures are not always 'intelligence fair.' Gardner's theory contrasts markedly with the view that intelligence is based on a unitary or 'general' ability for problem solving."
Hmm, I might as well not even summarize this chapter since the Bennett poster says it all. Well, I'll do so anyway. Teele begins with linguistic intelligence:
"Linguistic students have highly developed auditory skills, enjoy reading and/or writing, like playing word games, have good memories for names, dates, and places, and prefer doing word processing on a computer."
She shows us the following chart for linguistic intelligence:
- Has highly developed auditory skills
- Enjoys reading and writing
- Has a good memory
- Spells words easily and accurately
- Uses language fluently
And here are some teaching methods for linguistically intelligent students:
- Lectures
- Word games
- Storytelling
- Debates
- Speech
- Reading aloud
- Reading, writing, spelling, & listening exercises
Then she warns us:
"Students who do not process linguistically should be taught in multidimensional ways with pictures, movement, and music."
But Teele tells us that linguistic intelligence is the most universal, since after all, everyone learns to speak a language at an early age. The author proceeds to the next intelligence:
"Logical-mathematical students like to explore patterns and relationships and are able to make connections."
She shows us the following chart for logical-mathematical intelligence:
- Explores patterns and relationships
- Likes to problem solve and reason logically
- Follows sequential, logical directions
- Enjoys mathematics
- Uses experiments to test things out
And here are some teaching methods for logically and mathematically intelligent students:
- Making charts, graphs, and lists
- Sequencing patterns and relationships
- Outlining
- Solving problems
- Calculating mathematically
- Predicting
- Questioning
- Categorizing
Teele writes:
"I often like to discuss Fibonacci numbers when I explain logical-mathematical intelligence to students. Fibonacci numbers were discovered in the 12th century by Leonardo of Pisa, who was designing a formula to explain the breeding of rabbits."
And of course I like to discuss Fibonacci numbers as well and have done so on the blog. I won't write out the next chart which shows how to generate this sequence, but instead I'll just list them:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...
Teele tells us that these numbers appear in nature:
"The inner core of an apple is divided into five segments, a lemon into eight, and a chile into three. A giant sunflower spirals out from the center with 89 rows of seeds going one way and 55 rows of seeds going the other."
She also describes the closely-related equiangular spiral:
"The horn of a sheep, the sea horse, the nautilus shell, a dying poinsettia leaf, a growing fern, a pinecone, the ocean waves, and even the galaxy are created by an equiangular spiral. From a discussion of Fibonacci numbers and equiangular spirals in nature, students can be introduced to the golden ratio and how equiangular spirals appear in mathematics and architecture."
And Teele tells us how to apply Fibonacci to music. The 12EDO chromatic scale, for example, can be said to have 13 notes if we count both the root and octave notes:
"Modern symphonic music and jazz sometimes use a chromatic scale. It is interesting that even in music we see the Fibonacci sequence at work."
I like to believe that I possess logical-mathematical intelligence, since I'm a math teacher. The goal, of course, is to teach math to students who possess one of the other six intelligences. The author proceeds to the next intelligence:
"Spatial students enjoy art activities, read maps, charts, and diagrams, and think in images and pictures. They are able to visualize clear images about things and can complete jigsaw puzzles easily."
She shows us the following chart for spatial intelligence:
- Enjoys art activities
- Reads maps, charts, and diagrams
- Thinks with images and pictures
- Does jigsaw puzzles
And here are some teaching methods for spatially intelligent students:
- Pictures
- Slides
- Diagrams
- Posters
- Graphics
- Movies
- Mind maps
- Colors to represent words or letters
I don't appeal to spatial intelligence often on the blog because pictures are difficult to post. I do post a few images and even videos from time to time. I do a lot better with spatial intelligence when I'm in an actual classroom. The author proceeds to the next intelligence:
Musical students are sensitive to the sounds in their environment, enjoy music, and may prefer listening to music when studying or reading. They appreciate pitch, rhythm, and timbre, and often sing songs to themselves or create their own melodies, rhythms, or rhymes."
She shows us the following chart for musical intelligence:
- Sensitive to sounds in their environment
- Enjoys music
- Listens to music when studying and/or reading
- Sings songs
- Taps or hums rhythms
And here are some teaching methods for musically intelligent students:
- Chants
- Clapping and snapping fingers
- Poetry
- Music that matches the curriculum
- Moving rhythmically
The "music break" that I had at the old charter school obviously appeals to musical students. Indeed, Teele explains:
"Music can engage students in the learning process by appealing simultaneously to both the rational and emotional parts of the brain."
I never thought about that when I first came up with "music break," but it makes sense. The author proceeds to the next intelligence:
"Students who are bodily-kinesthetic, process knowledge through bodily sensations and use their bodies in unique and skilled ways."
She shows us the following chart for bodily-kinesthetic intelligence:
- Processes information through body sensations
- Requires hand-on learning
- Moves and acts things out
- Uses body in unique and skilled ways and is often well coordinated
And here are some teaching methods for bodily and kinesthetically intelligent students:
- Manipulates
- Games
- Simulations
- Laboratory experiments
- Movement
- Hands-on activities
- Action-packed stories
Teele suggests that all teachers (even high school teachers) should:
"Establish a bodily-kinesthetic area in the classroom."
where students can go when they cannot sit still. She also warns:
"For long-term memory to occur with bodily-kinesthetic students, the brain has to be activated through movement."
Her example is in a math class where students can demonstrate the circumference and diameter of a large circle by using their bodies as a group. (I believe I subbed for such a lesson one day.) And of course, the old charter school used DIDAX manipulatives as part of Illinois State. The author proceeds to the next intelligence:
"Intrapersonal students prefer their own inner world, like to be alone, are aware, and evaluate their own strengths, weaknesses, and inner feelings. They often have a deep sense of self-confidence, independence, and a strong will, and motivate themselves to do well on independent study projects."
And here's the flip side:
"Interpersonal students enjoy being around people, often have many friends, and participate in social activities. They create and maintain long-term relationships. These students express empathy for the feelings of others, respond to their moods and temperaments, and can understand other perspectives."
Teele reminds us:
"Individuals vary in the way they process information."
At this point the author mentions another theorist, Daniel Goleman:
"If we examine the relationship between Goleman's emotional intelligence and Gardner's intrapersonal and interpersonal intelligences, we notice a close correlation between the two approaches. Emotional intelligence skills include self-awareness, empathy toward others, the ability to manage the emotions of both oneself and others, and an optimistic view of life."
Teele also informs us that Gardner came up with one final intelligence:
"Gardner introduces an eighth intelligence, which he called the naturalist intelligence."
She shows us the following chart for naturalist intelligence:
- Recognizes, categorizes, and classifies flora and fauna
- Sensitive to sounds in their environment
- Enjoys being outdoors
- Interacts with plants and animals
There are some basic principles of multiple intelligences. Teele highlights the following:
"Individuals utilize certain combinations of intelligences to perform different tasks."
That is, they don't always appear in isolation. She also writes:
"All of the intelligences can be developed more fully when positive educational and environmental circumstances exist."
In considering what qualifies as an intelligence, Gardner considered eight criteria:
- Its potential to be isolated in cases of brain damage
- Its existence in special populations such as prodigies, idiot savants, and other exceptional individuals
- The identification of core information processing mechanisms associated with the intelligence
- The presence of a characteristic developmental pathway and a set of expert end states
- An evolutionary history and evolutionary plausibility
- Support from experimental psychology that reveals relative autonomy for the ability
- Whether statistical analyses indicate a separate factor for the ability
- Whether the ability has been encoded in a symbol system
Teele tells us that some schools have embraced the multiple intelligences theory. Two of these is located right here in Southern California (in Victorville and Chino). At these elementary schools, it's realized that:
"When students are actually engaged in learning, they do not misbehave."
And moreover:
"Not all students can learn in the same way and on the same day."
Therefore, she suggests:
"Teachers who acknowledge all of the intelligences as legitimate ways to learn can encourage students to engage and use their strengths as springboards for translating from one intelligence to another."
As a final reminder:
"There is no one pedagogy or curriculum method that will work effectively in all classrooms and with all students."
Teele concludes the chapter as follows -- by embracing the theory of multiple intelligences:
"We will increase student achievement and provide opportunities for our students to process information at much higher levels of thinking."
Lesson 4-6 of the U of Chicago text is called "Reflecting Polygons." This lesson doesn't appear in the new Third Edition -- instead, its material is incorporated into Lesson 4-2.
This is what I wrote last year about today's lesson:
Section 4-6 of the U of Chicago text considers what happens when we reflect an entire polygon -- not just individual points or even a segment or angle.
Still, the section begins with a theorem on what happens when we reflect a single point twice. Suppose we have two points, F and G and a reflecting line m. Now suppose I told you that the mirror image of F is G. So where do you think the mirror image of G is? If we drew this out and showed it to a student, chances are the student will say that the mirror image of G is F. The book gives a proof of this fact -- by the definition of reflection, G as the mirror image of F means that m is the perpendicular bisector ofFG. But FG is the same segment as GF, so its perpendicular bisector is still m. And so, by the definition of reflection again, this would make F the mirror image of G. QED
The text calls this the Flip-Flop Theorem:
If F and F' are points or figures and r(F) = F', then r(F') = F.
Recall that the text often uses the function notation r(F) to denote the reflection image of F. So the theorem can be written as:
If F and F' are points or figures and the mirror image of F is F', then the mirror image of F' is F.
And one can use even more function notation than the text and write the theorem as:
If F is a point or figure, then r(r(F)) = F.
So here's a two-column proof of the Flip-Flop Theorem:
Given: r(F) = F'
Prove: r(F') = F
Proof:
Statements Reasons
1. r(F) = F' 1. Given
2. m is the perp. bis. of FF' 2. Definition of reflection (meaning)
3. FF' =F'F 3. Reflexive Property of Equality
4. m is the perp. bis. of F'F 4. Substitution Property of Equality
5. r(F') = F 5. Definition of reflection (sufficient condition)
Notice that this proof uses both the meaning and the sufficient condition parts of the definition of reflection -- this occurs in other proofs as well. For example, a proof of the theorem "all right angles are congruent" (Euclid's Fourth Postulate) uses both the meaning and the sufficient condition parts of the definition of right angle.
But the above proof is a little strange. We explained earlier the significance of Statement 3 in the above proof -- but the problem is that we need a reason as to whyFF' and F'F are the same segment. There is no actual definition, postulate, or theorem that states this directly. The reason I wrote "Reflexive Property" above is that this often occurs in other proofs -- especially triangle congruence proofs that are used to prove that certain quadrilaterals are parallelograms. For example, in Section 7-7, we wish to prove that quadrilaterals with opposite sides congruent are parallelograms. The proof at the beginning of that lesson divides quadrilateral ABCD into two triangles, ABD and CDB, which the text then proves are congruent by SSS. But Step 2 of that proof reads:
2.BD is congruent to DB 2. Reflexive Property of Congruence
And so I did the same in the above proof. Of course, it's awkward to follow a statement that uses the "Reflexive Property" (that some object equals itself) with one that uses the "Substitution Property." (So we're substituting an object for itself?)
Some people may point out that now we're being overly formalistic here. The Flip-Flop Theorem is obviously true -- the two-column proof only serves to confuse the students. Perhaps if even I, as a teacher, have trouble filling in all the steps in the "Reasons" column (like Step 3 above), it means that the proof is so simple that it's better written as a paragraph proof (as the U of Chicago text has done) and not as a two-column proof.
Here's one final way to state the Flip-Flop Theorem:
A reflection is an involution.
An involution is simply a function or translation such that performing it twice on a point or figure gives the original point or figure. Therefore composition of an involution with itself is the identity. In function notation, f(f(x)) = x.
Now the other concept introduced in this chapter is orientation. The important concept, added to the Reflection Postulate as part f, is that reflections switch orientation.
But what exactly is the "orientation" of a polygon? The text explains that, in naming the vertices of a polygon, we can move either clockwise or counterclockwise around the polygon. The important idea here is that if pentagon ABCDE is clockwise and we reflect it, then A'B'C'D'E' is counterclockwise.
Then the book proceeds to tell us that "orientation" is undefined -- just like point, line, and plane. As we mentioned earlier, we only discover what an undefined term is by using postulate. So we have the Point-Line-Plane Postulate to tell us what points, lines, and planes are, and we have part f of the Reflection Postulate to tell us what orientation is. We may not know what orientation actually is, but we do know that whatever it is, reflections switch it.
The idea that reflections switch orientation shows up later on. In particular, translations and rotations preserve orientation, because they are the compositions of two reflections -- so the first reflection switches it, and the second switches it back.
Also, a question that often comes up is, if translations and rotations are the compositions of two reflections, maybe reflections are the composition of two rotations, or two of something else. As it turns out, this is impossible. Reflections can't be the composition of two of the same type of transformation, because of orientation. Either the orientation is switched and switched back, or it isn't switched at all. (If you want a reflection to be some transformation composed with itself, you must do something complicated, such as cut the plane into strips, then translate some of the strips and reflect the others.)
Is it possible to define "orientation"? We think back to Chapter 1, where the term "point," although undefined, can be modeled with an ordered pair. If we know all of the x- and y-coordinates of the vertices of the polygon, then we can plug it into a complicated formula such that if the answer is positive, then the orientation must be counterclockwise, and if the answer is negative, then the orientation must be clockwise. (If it's zero, then the points are collinear, which means that they don't form a polygon at all.) What's cool about the formula is that the number -- not just the sign -- actually means something. In particular, if we divide the number by two, we get the area of the polygon! But I won't give the formula here.
There's also a simpler version of the formula, but it only works if the polygon is convex. Notice the picture of octagon FGHIJKLM in the text. The book points out that determining its orientation is more difficult because it's nonconvex.
A much more intuitive way of thinking about orientation is if the preimage and image aren't figures, but words. If we hold up words to a mirror, then unless we're lucky and choose a word like MOM, the image will be illegible, since reflections reverse orientation. But if we translated the words instead, then we can still read the words (unless by "translation" we mean translation into another language).
One final note about orientation: A well-known math teacher blogger named Kate Nowak -- she calls her blog "Function of Time" or f(t) in function notation -- recently gave an Opening Task to her geometry classes:
http://function-of-time.blogspot.com/2014/08/arguing-about-shapes.html
Now Nowak gave her classes pairs of figures, and the students had to identify whether the two figures are "the same" or "not the same." As it turned out, the students easily reached a consensus if the two figures have the same orientation, but they disagreed if the orientations were different:
One group: "We said set C is not the same because you have to flip it."
Me [Nowak -- dw]: "Great."
Other group: "Wait a minute, we said set C is the same because we thought flipping was okay."
Me: "Also great."
Yet another group: "So which is it? We said they are the same."
Me: "... ... ... because... ?"
Okay, let's return to 2018. Kate Nowak's website still exists, but she hasn't posted since the first day of school, and apparently she's creating a new middle school curriculum.
Today's an activity day. This is what I wrote about Euclid the Game:
A few weeks ago, I mentioned the math teacher Lisa Bejarano, who had posted something called "Euclid: The Game" in one of her recent posts. And when I saw that part of the game reminded me of ancient geometer's Proposition 1 from Lesson 4-4, I couldn't resist checking the game out.
First, here's a link to Euclid: The Game:
http://euclidthegame.com/
Apparently, this is a one-player game. The goal is, on each level, to construct the figure in the diagram at the top of each page. The possible moves are the same as those allowed in classical Greek construction -- drawing an arbitrary point, drawing a point at an intersection, drawing a segment given two endpoints, drawing a ray given the endpoint and another point, and drawing a circle given the center and a point on the circle.
Now Level 1 is indeed Euclid's first proposition -- to draw an equilateral triangle given a side. This one, despite being Level 1, may be tough for students seeing this for the first time -- but of course, our students who remember yesterday's Lesson 4-4 should have no trouble with this one. Notice that according to Kasper Peulen, the creator, this game is powered by Geogebra -- and we were just talking about John Golden and his Geogebra lessons this week. Yes, I'm definitely going to keep going back to Bejarano, Golden, and other teachers when looking for good geometry activities.
Level 2 requires students to construct midpoints. The usual way to perform this construction is to construct the perpendicular bisector -- it intersects the original segment at its midpoint. For our students, this will be a preview of next week's Lesson 4-5 on perpendicular bisectors.
Level 3 requires students to construct angle bisectors. As we've already seen here on the blog, angle bisectors appear on the Common Core tests, yet are given short shrift in the U of Chicago text. The construction is buried in a Question in Lesson 4-7. Here's how to bisect Angle AOB:
Step 1. Circle O containing A
Step 2. Circle O intersects Ray OB at C.
Step 3. Subroutine: Line PQ, the perpendicular bisector ofAC
BC, and then drew Ray AD. I passed the level with a minimum number of moves, three.
Level 4 requires students to find the perpendicular to a line through a point on the line. In the U of Chicago text, this is Example 2 of Lesson 3-6. This time, following the U of Chicago construction doesn't give me the minimum number of moves -- I needed four, but the minimum is three.
Level 5 requires students to find the perpendicular to a line through a point not on the line. It is the line given in this week's Uniqueness of Perpendiculars Theorem.
Of course, not every classroom has access to a computer -- then again, Euclid obviously didn't have a computer in ancient Greece either. So I decided to create worksheets for the first six levels of Euclid: the Game, and students will have to solve them the way that Euclid would have.
This is what I wrote last year about today's lesson:
Section 4-6 of the U of Chicago text considers what happens when we reflect an entire polygon -- not just individual points or even a segment or angle.
Still, the section begins with a theorem on what happens when we reflect a single point twice. Suppose we have two points, F and G and a reflecting line m. Now suppose I told you that the mirror image of F is G. So where do you think the mirror image of G is? If we drew this out and showed it to a student, chances are the student will say that the mirror image of G is F. The book gives a proof of this fact -- by the definition of reflection, G as the mirror image of F means that m is the perpendicular bisector of
The text calls this the Flip-Flop Theorem:
If F and F' are points or figures and r(F) = F', then r(F') = F.
Recall that the text often uses the function notation r(F) to denote the reflection image of F. So the theorem can be written as:
If F and F' are points or figures and the mirror image of F is F', then the mirror image of F' is F.
And one can use even more function notation than the text and write the theorem as:
If F is a point or figure, then r(r(F)) = F.
So here's a two-column proof of the Flip-Flop Theorem:
Given: r(F) = F'
Prove: r(F') = F
Proof:
Statements Reasons
1. r(F) = F' 1. Given
2. m is the perp. bis. of FF' 2. Definition of reflection (meaning)
3. FF' =
4. m is the perp. bis. of F'F 4. Substitution Property of Equality
5. r(F') = F 5. Definition of reflection (sufficient condition)
Notice that this proof uses both the meaning and the sufficient condition parts of the definition of reflection -- this occurs in other proofs as well. For example, a proof of the theorem "all right angles are congruent" (Euclid's Fourth Postulate) uses both the meaning and the sufficient condition parts of the definition of right angle.
But the above proof is a little strange. We explained earlier the significance of Statement 3 in the above proof -- but the problem is that we need a reason as to why
2.
And so I did the same in the above proof. Of course, it's awkward to follow a statement that uses the "Reflexive Property" (that some object equals itself) with one that uses the "Substitution Property." (So we're substituting an object for itself?)
Some people may point out that now we're being overly formalistic here. The Flip-Flop Theorem is obviously true -- the two-column proof only serves to confuse the students. Perhaps if even I, as a teacher, have trouble filling in all the steps in the "Reasons" column (like Step 3 above), it means that the proof is so simple that it's better written as a paragraph proof (as the U of Chicago text has done) and not as a two-column proof.
Here's one final way to state the Flip-Flop Theorem:
A reflection is an involution.
An involution is simply a function or translation such that performing it twice on a point or figure gives the original point or figure. Therefore composition of an involution with itself is the identity. In function notation, f(f(x)) = x.
Now the other concept introduced in this chapter is orientation. The important concept, added to the Reflection Postulate as part f, is that reflections switch orientation.
But what exactly is the "orientation" of a polygon? The text explains that, in naming the vertices of a polygon, we can move either clockwise or counterclockwise around the polygon. The important idea here is that if pentagon ABCDE is clockwise and we reflect it, then A'B'C'D'E' is counterclockwise.
Then the book proceeds to tell us that "orientation" is undefined -- just like point, line, and plane. As we mentioned earlier, we only discover what an undefined term is by using postulate. So we have the Point-Line-Plane Postulate to tell us what points, lines, and planes are, and we have part f of the Reflection Postulate to tell us what orientation is. We may not know what orientation actually is, but we do know that whatever it is, reflections switch it.
The idea that reflections switch orientation shows up later on. In particular, translations and rotations preserve orientation, because they are the compositions of two reflections -- so the first reflection switches it, and the second switches it back.
Also, a question that often comes up is, if translations and rotations are the compositions of two reflections, maybe reflections are the composition of two rotations, or two of something else. As it turns out, this is impossible. Reflections can't be the composition of two of the same type of transformation, because of orientation. Either the orientation is switched and switched back, or it isn't switched at all. (If you want a reflection to be some transformation composed with itself, you must do something complicated, such as cut the plane into strips, then translate some of the strips and reflect the others.)
Is it possible to define "orientation"? We think back to Chapter 1, where the term "point," although undefined, can be modeled with an ordered pair. If we know all of the x- and y-coordinates of the vertices of the polygon, then we can plug it into a complicated formula such that if the answer is positive, then the orientation must be counterclockwise, and if the answer is negative, then the orientation must be clockwise. (If it's zero, then the points are collinear, which means that they don't form a polygon at all.) What's cool about the formula is that the number -- not just the sign -- actually means something. In particular, if we divide the number by two, we get the area of the polygon! But I won't give the formula here.
There's also a simpler version of the formula, but it only works if the polygon is convex. Notice the picture of octagon FGHIJKLM in the text. The book points out that determining its orientation is more difficult because it's nonconvex.
A much more intuitive way of thinking about orientation is if the preimage and image aren't figures, but words. If we hold up words to a mirror, then unless we're lucky and choose a word like MOM, the image will be illegible, since reflections reverse orientation. But if we translated the words instead, then we can still read the words (unless by "translation" we mean translation into another language).
One final note about orientation: A well-known math teacher blogger named Kate Nowak -- she calls her blog "Function of Time" or f(t) in function notation -- recently gave an Opening Task to her geometry classes:
http://function-of-time.blogspot.com/2014/08/arguing-about-shapes.html
Now Nowak gave her classes pairs of figures, and the students had to identify whether the two figures are "the same" or "not the same." As it turned out, the students easily reached a consensus if the two figures have the same orientation, but they disagreed if the orientations were different:
One group: "We said set C is not the same because you have to flip it."
Me [Nowak -- dw]: "Great."
Other group: "Wait a minute, we said set C is the same because we thought flipping was okay."
Me: "Also great."
Yet another group: "So which is it? We said they are the same."
Me: "... ... ... because... ?"
Okay, let's return to 2018. Kate Nowak's website still exists, but she hasn't posted since the first day of school, and apparently she's creating a new middle school curriculum.
Today's an activity day. This is what I wrote about Euclid the Game:
A few weeks ago, I mentioned the math teacher Lisa Bejarano, who had posted something called "Euclid: The Game" in one of her recent posts. And when I saw that part of the game reminded me of ancient geometer's Proposition 1 from Lesson 4-4, I couldn't resist checking the game out.
First, here's a link to Euclid: The Game:
http://euclidthegame.com/
Apparently, this is a one-player game. The goal is, on each level, to construct the figure in the diagram at the top of each page. The possible moves are the same as those allowed in classical Greek construction -- drawing an arbitrary point, drawing a point at an intersection, drawing a segment given two endpoints, drawing a ray given the endpoint and another point, and drawing a circle given the center and a point on the circle.
Now Level 1 is indeed Euclid's first proposition -- to draw an equilateral triangle given a side. This one, despite being Level 1, may be tough for students seeing this for the first time -- but of course, our students who remember yesterday's Lesson 4-4 should have no trouble with this one. Notice that according to Kasper Peulen, the creator, this game is powered by Geogebra -- and we were just talking about John Golden and his Geogebra lessons this week. Yes, I'm definitely going to keep going back to Bejarano, Golden, and other teachers when looking for good geometry activities.
Level 2 requires students to construct midpoints. The usual way to perform this construction is to construct the perpendicular bisector -- it intersects the original segment at its midpoint. For our students, this will be a preview of next week's Lesson 4-5 on perpendicular bisectors.
Level 3 requires students to construct angle bisectors. As we've already seen here on the blog, angle bisectors appear on the Common Core tests, yet are given short shrift in the U of Chicago text. The construction is buried in a Question in Lesson 4-7. Here's how to bisect Angle AOB:
Step 1. Circle O containing A
Step 2. Circle O intersects Ray OB at C.
Step 3. Subroutine: Line PQ, the perpendicular bisector of
Level 4 requires students to find the perpendicular to a line through a point on the line. In the U of Chicago text, this is Example 2 of Lesson 3-6. This time, following the U of Chicago construction doesn't give me the minimum number of moves -- I needed four, but the minimum is three.
Level 5 requires students to find the perpendicular to a line through a point not on the line. It is the line given in this week's Uniqueness of Perpendiculars Theorem.
Of course, not every classroom has access to a computer -- then again, Euclid obviously didn't have a computer in ancient Greece either. So I decided to create worksheets for the first six levels of Euclid: the Game, and students will have to solve them the way that Euclid would have.
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