This is a major milestone, and so it's time for a new background. I hope you enjoy it.
I have many things to discuss in this milestone post. First of all, I taught in a middle school special ed class today. Three of the classes are actually math, and so I wish to do "A Day in the Life" despite having aides and co-teachers today.
8:15 -- This is one of the schools where all classes rotate. But today the rotation just happens to start with first period -- and at this school, homeroom is the same as first period.
First period isn't a math class, though. Instead, it's co-teaching seventh grade World History. The class is learning about the Aztec Empire. As often happens, the co-teacher is out as well. (Since both teachers must often attend the same meetings, both teachers need subs.)
I might as well link to the video that the class watched -- Tenochtitlan, the Venice of Mesoamerica:
I walk around the classroom and notice a worksheet that these students did earlier in this unit -- it's all about the Mayan numeral system. I'm not sure whether this unit is timed to line up with Columbus Day or not, but it's fascinating to learn about the ancient cultures that the Spanish explorers (in this case, Cortes) would have first encountered in Mexico.
9:20 -- Second period is the teacher's conference period, which leads directly into snack.
10:30 -- Third period arrives. This is the first of the seventh grade math classes.
The students begin with Bellwork (that is, a Warm-Up). They must convert fractions to decimals:
- -3 4/7
- 17/21
Afterwards, they take a quiz. It is based on Chapter 2 of the seventh grade Big Ideas text, which is on rational numbers.
Since I'm a math teacher, of course I'm going to analyze how these students perform -- and since it's middle school, I'll also compare it to the math I taught at the old charter school.
First of all, the special ed students are allowed to use a multiplication table (25 * 25) as well as notes, which appear to be from an interactive notebook. A few students have accommodations that allow them to use calculators.
Of the dozen students in this class, a majority -- seven -- fail even to complete the quiz. The aide grades the quizzes, and of the five finishers, here are their scores out of ten: 9, 7, 6, 6, 4. That's an A, C, two D's, and an F. The aide informs me that the students struggle with the following questions:
Convert decimals to fractions:
6. 0.98
7. -1.25
8. 3.26
The students are able to figure out that 0.98 = 98/100, but they have trouble reducing. The teacher has told them to look for factors of the numerator and denominator on their multiplication chart -- but since it only goes up to 25 * 25, the value 98 = 49 * 2 doesn't appear. With -1.25, some students divide the numerator and denominator by five instead of 25.
11:25 -- Third period leaves and fourth period arrives. This is another seventh grade math class.
Recall that one of my resolutions is to spend less time on Warm-Ups. And so the aide and I work harder to get through the Bellwork and start the quiz. This time, only a third of the 15 students in this class run out of time. The other students' scores are as follows: 10*, 9, 8, 8*, 7, 7, 5, 5, 2, 0. That's two A's, two B's, two C's, and four F's. (The starred scores are special ed students with a calculator as an accommodation.)
One thing I find interesting is that in this class, some students in fourth period try to convert mixed numbers to improper fractions before reducing, while no one in third period does so. For example, they might write -1.25 = -125/100. But all this does is confuse the students -- even though the product 125 = 25 * 5 barely makes their table, no one correctly reduces -125/100 to -5/4.
The aide is now concerned with some problems in the fraction-to-decimal direction. Some decimals like 4/5 = 0.8 obviously terminate, but others (like those in the Bellwork) don't. The teacher expects them to write the first three decimal places. But some students either write only two decimal places, or make a mistake in finding the third one -- and they make this mistake for both 5 7/12 (which is truncated after the thousandths place) and 5 1/8 (which terminates there). And so the aide suggests requiring only two decimal places in their answers. (I also wonder whether the students are finding the first two digits and then getting tired of long division, so they just make up the third digit.)
12:15 -- Fourth period leaves for lunch. In the hallway where the teachers are eating lunch, I see the regular teacher, who also has a lunch break during his meeting. I inform him that the aide will only require the next class to write two decimal digits. His response is, "She's the boss!"
1:00 -- Fifth period arrives. This is the last of the three seventh grade math classes.
Ordinarily, this school has scheduled silent reading right after lunch. But the aide, after seeing so many students in earlier classes who are unable to finish the quiz in time, decides to skip silent reading and just use the extra time for the quiz.
Classroom management is a huge issue in this class, as three or four guys continually whisper, make sound effects, and laugh during the quiz. The aide says that she'll inform the regular teacher, and consequences will be given.
And of course, even with this extra time, some students (especially the talkers) don't finish. Since this is the last math class and the aide leaves afterward, she doesn't grade the quizzes for this class.
2:05 -- Fifth period leaves and sixth period arrives. This is a study skills class.
Like the math classes, this class begins with Bellwork. The prompt is: "Did you meet the goals you set for this week? Why or why not?"
After the Bellwork, the students are allowed to play games, either cards or Chromebooks. The lesson plan states that he ordinarily does this on Fridays during the last class of the day -- presumably because this class is usually study hall and at the end of the week they have nothing to study for. But recall that the pure period rotation has nothing to do with the day of the week at this school. Thus it's rare when sixth period is actually the last class on a Friday.
2:10 -- Just after the students turn in their Bellwork for the week and take out the games, the regular teacher returns from his meeting. Thus my day essentially ends here.
OK, so now let's compare these math classes to math at my old charter school. Notice that only once did I give my students a quiz on fraction-decimal conversion -- and it was to my eighth grade class, not my seventh graders! Let's look at the relevant Common Core Standards:
CCSS.MATH.CONTENT.7.NS.A.2.D
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
CCSS.MATH.CONTENT.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Recall that the Illinois State traditional text was organized by standards, not chapters -- and these standards naively follow the naive order they appear in the Common Core document. For eighth grade, both Number System (NS) standards appear first, but in seventh grade, all Ratio and Proportion (RP) standards appear before any NS standards. Thus decimal conversion, an NS standard, appeared earlier in eighth grade than in seventh grade.
But then before I ever made it to 7.NS, I switched to follow the order of the Illinois State STEM text, where standards appeared in a haphazard order. This was a huge mistake on my part -- Illinois State actually wants us to follow the order of the traditional text, not the STEM text. In the end, the only standard for 7.NS that I fully covered was 7 NS1a, which merely introduces negative numbers.
Yet as I know now, the NS standards are probably the most important in seventh grade. Familiarity and comfort with negative (and ultimately rational) numbers lead to success in Algebra I. Because I never reached the meat of 7.NS, I can't be sure whether my students were fully prepared for high school math.
It would have been much easier for me if the Illinois State text were organized more like the Big Ideas text. Such texts have a dozen chapters (more or less), and we know in advance that we should follow the text in order and spend about three weeks on each chapter. There's no mystery as to what order the teacher should follow. If there's to be a STEM text, such as text should make it crystal clear which projects correspond to Chapter 1, 2, etc., of the traditional text.
I notice that the teacher has his own interactive notebook from last year. In it, I see that the teacher uses many mnemonics (such as "cowboy/horse" and "In 'n' Out," -- named for a restaurant that's prominent on the West Coast) to help the students with fractions and long division. And peeking ahead in his notebook to the next chapter, I see that he even has a song for solving equations -- unlike my "Solving Equations" song, this one contains a rhyme to remind students when to divide and when to multiply to solve them. ("If the variable's on the side, divide. If the variable's up high, multiply.")
(Oh, and of course I don't sing any songs today. I rarely do so on quiz days -- when there's not enough time for the students to finish the quiz, why should I waste even more time on singing?)
But all of these mnemonics and songs don't lead to better grades on the tests. As I wrote above, there are still lots of D's and F's in the classes. The teacher says that he's given up trying to teach this chapter to the special ed students.
But the students at the old charter school might have fared better if I had used this teacher's method, if only because most of them were not special ed. Once again, interactive notebooks most likely would have been beneficial.
I still have a record of how my eighth graders fared on a review worksheet leading up to the test on decimal conversion, if only because it's the only time I gave out such a worksheet. (Subsequent reviews were done on whiteboards, which have long since been erased!) Of the seven students whose papers I still have, only one (maybe two) of them struggled. Then again, there was more emphasis on problems like converting 0.333... to fraction (the eighth grade standard) as opposed to 0.98. My students had no problem with 0.2, but as far as I know, 0.98 likely would have tripped them up.
Arguably, this lesson was one of my most successful eighth grade lessons that year. I had students convert both terminating and repeating decimals, even though the eighth grade standard mentions only repeating decimals (because terminating decimals are in seventh grade). This is how most texts (such as the Big Ideas text) present material in their chapters. (Later on, I would teach them volume formulas for the three solids mentioned in the eighth grade standards, cylinders, cones, and spheres -- thanks Archimedes -- even though most texts would review volume formulas from previous years, and my eighth graders suffered for it.)
For seventh (and for that matter, sixth) grade, I should have emphasized NS more strongly. If I had followed the intended Illinois State order and pacing plan, I would completed all RP and NS standards by the midpoint of the year, and the students would have had a stronger foundation with fractions, decimals, and negative numbers. At least my gen ed kids should have fared better. I also could have come up with the same accommodations for special ed -- they get to use calculators, and they must take the test in another classroom (perhaps that of the history teacher, since he had the strongest classroom management and thus would provide a quiet room for the quiz/test).
As for multiplication tables, this is a tough one. My philosophy is that every student should memorize the products of single-digit numbers or be labeled as a "dren." But these times tables go up to 25 * 25, well past the dren range. Today, I see the problem with these tables -- the students are taught to use the table to reduce fractions, but then they can't simplify 98/100 because 98 = 49 * 2 doesn't appear on the table. (In fact one student tries to use 98 = 14 * 7 to simplify 98/100 even though 100 isn't divisible by seven, only because 98 = 14 * 7 is in the table.)
I see from the teacher's notebook that he did try to emphasize that if both numerator and denominator are even, then we can simplify by a factor of two. (This appeared in our proof that sqrt(2) is irrational if you recall.) Perhaps we can drive this home for the students by giving them a worksheet where the only common factors to cancel are 2, 5, and 10 -- but they may not use the table. Then we give them the table with a different worksheet to simplify factors other than 2, 5, and 10.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
Determine the measure in degrees of the central angle of a regular icosagon.
The word "icosagon" doesn't appear in Lesson 2-7 of the U of Chicago text (not even in the Exploration question). But "icosahedron" does appear in the Exploration question of Lesson 9-7 (as well as in yesterday's post). We infer that "icosa-" means 20, so an icosagon is a 20-gon.
The measure of the central angle of a regular n-gon is 360/n. Thus the central angle of a regular icosagon must be 360/20 = 18. Therefore the desired angle is 18 degrees -- and of course, today's date is the eighteenth.
Chapter 3 of Ian Stewart's The Story of Mathematics is called "Notation and Numbers: Where Our Number Symbols Come From." It begins as follows:
"We are so accustomed to today's number system, with its use of the ten decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (in Western countries), that it can come as a shock to realize that there are entirely different ways to write numbers."
This chapter is all about the development of our numeration system as well as others. Stewart begins with Roman numerals, in which 2007 (the year he wrote the book) is written MMVII. I don't need to review Roman numerals -- just watch Square One TV's "The Mathematics of Love." He continues:
"Most of us are also aware, at least if reminded, that we employ two distinct methods for writing numbers that are not whole numbers -- fractions like 3/4, and decimals such as 0.75."
Where have I seen fractions and decimals like 3/4 and 0.75 before? Oh, that's right -- the seventh graders had to convert between the two forms in math class. (Yes, today's chapter is perfect timing!)
The author moves on:
"Later, base-20 numbers were used in Central America by the Mayan civilization. Only recently did humanity settle on the current methods for writing numbers, and their use became established through a mixture of traditional and convenience."
Where have I seen Mayan numerals before? Oh, that's right -- the seventh graders had to convert between the two forms in history class. (Yes, today's chapter is perfect timing!)
Stewart now describes ancient Greek numerals:
"Similarly, 10 was written (delta), the first letter of deka; 100 was written H (eta), the first letter of hekaton; 1000 was written (chi), the first letter of chilioi, and 10,000 was written M, the first letter of myrioi."
He later mentions another system where every letter of the Greek alphabet corresponds to a digit:
"Some Greek astronomers, notably Ptolemy, employed the Babylonian sexagesimal system for precision, but using Greek symbols for the component digits."
Now Stewart moves on to the forerunners of our current system -- number symbols from India. This is where our symbols for our digits 1-9 were created. But:
"In positional notation, where the meaning of a symbol depends on its location, it is important to specify that location without ambiguity."
This lead to the development of a new concept -- zero. Meanwhile, Stewart goes on to describe three key South Asian mathematicians -- Brahmagupta, Mahavira, and Bhaskara:
"[Mahavira's book] included fractions, permutations and combinations, the solution of quadratic equations, Pythagoream triangles and an attempt to find the area and perimeter of an ellipse."
The author also tells a sad story about Bhaskara's daughter, who tried to use a water clock with cups to determine the best time to get married:
"The cup did not sink, which meant that Lilavati could never get married. To cheer her up, Bhaskara wrote a mathematics textbook for her."
I've heard this story before -- it's mentioned in the user manual for the TI-34 calculator. (This manual is also the first place where I've heard of the four 4's activity.) Anyway, the author adds:
"So great was Bhaskara's reputation that his works were still being copied around 1800. The Hindu system started to spread into the Arabic world, before it was fully developed in its country of origin."
This is why we refer to our number system as Hindu-Arabic. This was during the Dark Ages in Europe, where advances in knowledge were particularly slow. The exception was Italy:
"Venice, Genoa, and Pisa were significant, and merchants sailed from these ports to North Africa and the eastern end of the Mediterranean."
One important mathematician during that time was Leonardo Fibonacci. In addition to his famous sequence, his book was the first to introduce modern fractional notation:
"For typographical convenience, fractions are often written on a single line in the form 3/4, or sometimes in the compromise form" (that I can't easily display in ASCII).
Speaking of Fibonacci, I notice that yesterday, elementary math coach Jenna Laib (whose blog I've linked to before, a few months ago), just so happened to write about the medieval mathematician:
https://jennalaib.wordpress.com/2019/10/17/the-story-of-fibonacci-and-the-math-ethnic-studies-framework/
In the late 12th century, a young Leonardo Bonacci — better known as Fibonacci — traveled with his merchant father from their home in Italy to Algeria (North Africa). They landed in the gorgeous port city of Bugia, now known as Bejaïa, where Fibonacci set about learning mathematical ideas from the local vendors. It was in the marketplace that he observed the use of Hindu-Arabic numerals, 0-9. They were so much more efficient than the Roman numerals that dominated Europe.
To Laib, the fact that Fibonacci learned about the numerals in Algeria especially hits home because her husband is Algerian. Indeed, Stewart writes more about this story in a sidebar:
"Leonardo, Italian born, grew up in North Africa, where his father Guilielmo was working as a diplomat on behalf of merchants trading at Bugia (in modern Algeria). In his Liber Abbaci of 1202 he writes: 'When my father, who had been appointed by his country as public notary in the customs at Bugia for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting.'"
As for decimals, these were introduced by Dutchman Simon Stevin in the 16th century. This is the mathematician for whom the Stevin algorithms -- the standard algorithms -- are named:
"His notation did not include the familiar decimal point, but it lead quickly to today's decimal notation."
We now move on to the development of negative numbers. (That's right -- all the numbers the seventh graders are learning now are mentioned in today's chapter.) Stewart credits the Chinese with the invention of negative numbers. They used rods for numbers -- red for positive, black for negative:
"So to solve equations that we would write as 3x - 2y = 4, x + 5y = 7, they would set out the two equations as two columns of a table: one with the numbers 3 (red), 2 (black), 4 (red), and the other 1 (red), 5 (red), 7 (red)."
This eventually became our modern system for signed numbers, though their acceptance was slow:
"But interpretation aside, their arithmetic worked perfectly well, and they were so useful as a computational aide that it would have silly not to use them. Our number system is so familiar that we tend to assume that it is the only possible one, or at least the only sensible one."
The author reminds us -- and our young students -- why it's important to learn arithmetic nowadays:
"And if your only arithmetical ability is reading what's on a calculator, you probably won't notice if the supermarket gets your bill wrong. You might not worry about that, but modern civilization would quickly break down if we stopped teaching arithmetic, because you can't spot the future engineers and scientists at the age of five."
On this note, Stewart concludes the chapter as follows:
"But, just as you won't learn to walk by always using a crutch, you won't learn to think sensibly about numbers by relying solely on a calculator."
And this explains today's regular teacher and his calculator policy -- those who truly need them (the special ed students) can use them, while everyone else learns how to avoid the calculator "crutch."
There are also several sidebars in this chapter. I already mentioned the one about Fibonacci, and there is another sidebar that discusses Mayan numerals in more detail.
Of the dozen students in this class, a majority -- seven -- fail even to complete the quiz. The aide grades the quizzes, and of the five finishers, here are their scores out of ten: 9, 7, 6, 6, 4. That's an A, C, two D's, and an F. The aide informs me that the students struggle with the following questions:
Convert decimals to fractions:
6. 0.98
7. -1.25
8. 3.26
The students are able to figure out that 0.98 = 98/100, but they have trouble reducing. The teacher has told them to look for factors of the numerator and denominator on their multiplication chart -- but since it only goes up to 25 * 25, the value 98 = 49 * 2 doesn't appear. With -1.25, some students divide the numerator and denominator by five instead of 25.
11:25 -- Third period leaves and fourth period arrives. This is another seventh grade math class.
Recall that one of my resolutions is to spend less time on Warm-Ups. And so the aide and I work harder to get through the Bellwork and start the quiz. This time, only a third of the 15 students in this class run out of time. The other students' scores are as follows: 10*, 9, 8, 8*, 7, 7, 5, 5, 2, 0. That's two A's, two B's, two C's, and four F's. (The starred scores are special ed students with a calculator as an accommodation.)
One thing I find interesting is that in this class, some students in fourth period try to convert mixed numbers to improper fractions before reducing, while no one in third period does so. For example, they might write -1.25 = -125/100. But all this does is confuse the students -- even though the product 125 = 25 * 5 barely makes their table, no one correctly reduces -125/100 to -5/4.
The aide is now concerned with some problems in the fraction-to-decimal direction. Some decimals like 4/5 = 0.8 obviously terminate, but others (like those in the Bellwork) don't. The teacher expects them to write the first three decimal places. But some students either write only two decimal places, or make a mistake in finding the third one -- and they make this mistake for both 5 7/12 (which is truncated after the thousandths place) and 5 1/8 (which terminates there). And so the aide suggests requiring only two decimal places in their answers. (I also wonder whether the students are finding the first two digits and then getting tired of long division, so they just make up the third digit.)
12:15 -- Fourth period leaves for lunch. In the hallway where the teachers are eating lunch, I see the regular teacher, who also has a lunch break during his meeting. I inform him that the aide will only require the next class to write two decimal digits. His response is, "She's the boss!"
1:00 -- Fifth period arrives. This is the last of the three seventh grade math classes.
Ordinarily, this school has scheduled silent reading right after lunch. But the aide, after seeing so many students in earlier classes who are unable to finish the quiz in time, decides to skip silent reading and just use the extra time for the quiz.
Classroom management is a huge issue in this class, as three or four guys continually whisper, make sound effects, and laugh during the quiz. The aide says that she'll inform the regular teacher, and consequences will be given.
And of course, even with this extra time, some students (especially the talkers) don't finish. Since this is the last math class and the aide leaves afterward, she doesn't grade the quizzes for this class.
2:05 -- Fifth period leaves and sixth period arrives. This is a study skills class.
Like the math classes, this class begins with Bellwork. The prompt is: "Did you meet the goals you set for this week? Why or why not?"
After the Bellwork, the students are allowed to play games, either cards or Chromebooks. The lesson plan states that he ordinarily does this on Fridays during the last class of the day -- presumably because this class is usually study hall and at the end of the week they have nothing to study for. But recall that the pure period rotation has nothing to do with the day of the week at this school. Thus it's rare when sixth period is actually the last class on a Friday.
2:10 -- Just after the students turn in their Bellwork for the week and take out the games, the regular teacher returns from his meeting. Thus my day essentially ends here.
OK, so now let's compare these math classes to math at my old charter school. Notice that only once did I give my students a quiz on fraction-decimal conversion -- and it was to my eighth grade class, not my seventh graders! Let's look at the relevant Common Core Standards:
CCSS.MATH.CONTENT.7.NS.A.2.D
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
CCSS.MATH.CONTENT.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Recall that the Illinois State traditional text was organized by standards, not chapters -- and these standards naively follow the naive order they appear in the Common Core document. For eighth grade, both Number System (NS) standards appear first, but in seventh grade, all Ratio and Proportion (RP) standards appear before any NS standards. Thus decimal conversion, an NS standard, appeared earlier in eighth grade than in seventh grade.
But then before I ever made it to 7.NS, I switched to follow the order of the Illinois State STEM text, where standards appeared in a haphazard order. This was a huge mistake on my part -- Illinois State actually wants us to follow the order of the traditional text, not the STEM text. In the end, the only standard for 7.NS that I fully covered was 7 NS1a, which merely introduces negative numbers.
Yet as I know now, the NS standards are probably the most important in seventh grade. Familiarity and comfort with negative (and ultimately rational) numbers lead to success in Algebra I. Because I never reached the meat of 7.NS, I can't be sure whether my students were fully prepared for high school math.
It would have been much easier for me if the Illinois State text were organized more like the Big Ideas text. Such texts have a dozen chapters (more or less), and we know in advance that we should follow the text in order and spend about three weeks on each chapter. There's no mystery as to what order the teacher should follow. If there's to be a STEM text, such as text should make it crystal clear which projects correspond to Chapter 1, 2, etc., of the traditional text.
I notice that the teacher has his own interactive notebook from last year. In it, I see that the teacher uses many mnemonics (such as "cowboy/horse" and "In 'n' Out," -- named for a restaurant that's prominent on the West Coast) to help the students with fractions and long division. And peeking ahead in his notebook to the next chapter, I see that he even has a song for solving equations -- unlike my "Solving Equations" song, this one contains a rhyme to remind students when to divide and when to multiply to solve them. ("If the variable's on the side, divide. If the variable's up high, multiply.")
(Oh, and of course I don't sing any songs today. I rarely do so on quiz days -- when there's not enough time for the students to finish the quiz, why should I waste even more time on singing?)
But all of these mnemonics and songs don't lead to better grades on the tests. As I wrote above, there are still lots of D's and F's in the classes. The teacher says that he's given up trying to teach this chapter to the special ed students.
But the students at the old charter school might have fared better if I had used this teacher's method, if only because most of them were not special ed. Once again, interactive notebooks most likely would have been beneficial.
I still have a record of how my eighth graders fared on a review worksheet leading up to the test on decimal conversion, if only because it's the only time I gave out such a worksheet. (Subsequent reviews were done on whiteboards, which have long since been erased!) Of the seven students whose papers I still have, only one (maybe two) of them struggled. Then again, there was more emphasis on problems like converting 0.333... to fraction (the eighth grade standard) as opposed to 0.98. My students had no problem with 0.2, but as far as I know, 0.98 likely would have tripped them up.
Arguably, this lesson was one of my most successful eighth grade lessons that year. I had students convert both terminating and repeating decimals, even though the eighth grade standard mentions only repeating decimals (because terminating decimals are in seventh grade). This is how most texts (such as the Big Ideas text) present material in their chapters. (Later on, I would teach them volume formulas for the three solids mentioned in the eighth grade standards, cylinders, cones, and spheres -- thanks Archimedes -- even though most texts would review volume formulas from previous years, and my eighth graders suffered for it.)
For seventh (and for that matter, sixth) grade, I should have emphasized NS more strongly. If I had followed the intended Illinois State order and pacing plan, I would completed all RP and NS standards by the midpoint of the year, and the students would have had a stronger foundation with fractions, decimals, and negative numbers. At least my gen ed kids should have fared better. I also could have come up with the same accommodations for special ed -- they get to use calculators, and they must take the test in another classroom (perhaps that of the history teacher, since he had the strongest classroom management and thus would provide a quiet room for the quiz/test).
As for multiplication tables, this is a tough one. My philosophy is that every student should memorize the products of single-digit numbers or be labeled as a "dren." But these times tables go up to 25 * 25, well past the dren range. Today, I see the problem with these tables -- the students are taught to use the table to reduce fractions, but then they can't simplify 98/100 because 98 = 49 * 2 doesn't appear on the table. (In fact one student tries to use 98 = 14 * 7 to simplify 98/100 even though 100 isn't divisible by seven, only because 98 = 14 * 7 is in the table.)
I see from the teacher's notebook that he did try to emphasize that if both numerator and denominator are even, then we can simplify by a factor of two. (This appeared in our proof that sqrt(2) is irrational if you recall.) Perhaps we can drive this home for the students by giving them a worksheet where the only common factors to cancel are 2, 5, and 10 -- but they may not use the table. Then we give them the table with a different worksheet to simplify factors other than 2, 5, and 10.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
Determine the measure in degrees of the central angle of a regular icosagon.
The word "icosagon" doesn't appear in Lesson 2-7 of the U of Chicago text (not even in the Exploration question). But "icosahedron" does appear in the Exploration question of Lesson 9-7 (as well as in yesterday's post). We infer that "icosa-" means 20, so an icosagon is a 20-gon.
The measure of the central angle of a regular n-gon is 360/n. Thus the central angle of a regular icosagon must be 360/20 = 18. Therefore the desired angle is 18 degrees -- and of course, today's date is the eighteenth.
Chapter 3 of Ian Stewart's The Story of Mathematics is called "Notation and Numbers: Where Our Number Symbols Come From." It begins as follows:
"We are so accustomed to today's number system, with its use of the ten decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (in Western countries), that it can come as a shock to realize that there are entirely different ways to write numbers."
This chapter is all about the development of our numeration system as well as others. Stewart begins with Roman numerals, in which 2007 (the year he wrote the book) is written MMVII. I don't need to review Roman numerals -- just watch Square One TV's "The Mathematics of Love." He continues:
"Most of us are also aware, at least if reminded, that we employ two distinct methods for writing numbers that are not whole numbers -- fractions like 3/4, and decimals such as 0.75."
Where have I seen fractions and decimals like 3/4 and 0.75 before? Oh, that's right -- the seventh graders had to convert between the two forms in math class. (Yes, today's chapter is perfect timing!)
The author moves on:
"Later, base-20 numbers were used in Central America by the Mayan civilization. Only recently did humanity settle on the current methods for writing numbers, and their use became established through a mixture of traditional and convenience."
Where have I seen Mayan numerals before? Oh, that's right -- the seventh graders had to convert between the two forms in history class. (Yes, today's chapter is perfect timing!)
Stewart now describes ancient Greek numerals:
"Similarly, 10 was written (delta), the first letter of deka; 100 was written H (eta), the first letter of hekaton; 1000 was written (chi), the first letter of chilioi, and 10,000 was written M, the first letter of myrioi."
He later mentions another system where every letter of the Greek alphabet corresponds to a digit:
"Some Greek astronomers, notably Ptolemy, employed the Babylonian sexagesimal system for precision, but using Greek symbols for the component digits."
Now Stewart moves on to the forerunners of our current system -- number symbols from India. This is where our symbols for our digits 1-9 were created. But:
"In positional notation, where the meaning of a symbol depends on its location, it is important to specify that location without ambiguity."
This lead to the development of a new concept -- zero. Meanwhile, Stewart goes on to describe three key South Asian mathematicians -- Brahmagupta, Mahavira, and Bhaskara:
"[Mahavira's book] included fractions, permutations and combinations, the solution of quadratic equations, Pythagoream triangles and an attempt to find the area and perimeter of an ellipse."
The author also tells a sad story about Bhaskara's daughter, who tried to use a water clock with cups to determine the best time to get married:
"The cup did not sink, which meant that Lilavati could never get married. To cheer her up, Bhaskara wrote a mathematics textbook for her."
I've heard this story before -- it's mentioned in the user manual for the TI-34 calculator. (This manual is also the first place where I've heard of the four 4's activity.) Anyway, the author adds:
"So great was Bhaskara's reputation that his works were still being copied around 1800. The Hindu system started to spread into the Arabic world, before it was fully developed in its country of origin."
This is why we refer to our number system as Hindu-Arabic. This was during the Dark Ages in Europe, where advances in knowledge were particularly slow. The exception was Italy:
"Venice, Genoa, and Pisa were significant, and merchants sailed from these ports to North Africa and the eastern end of the Mediterranean."
One important mathematician during that time was Leonardo Fibonacci. In addition to his famous sequence, his book was the first to introduce modern fractional notation:
"For typographical convenience, fractions are often written on a single line in the form 3/4, or sometimes in the compromise form" (that I can't easily display in ASCII).
Speaking of Fibonacci, I notice that yesterday, elementary math coach Jenna Laib (whose blog I've linked to before, a few months ago), just so happened to write about the medieval mathematician:
https://jennalaib.wordpress.com/2019/10/17/the-story-of-fibonacci-and-the-math-ethnic-studies-framework/
In the late 12th century, a young Leonardo Bonacci — better known as Fibonacci — traveled with his merchant father from their home in Italy to Algeria (North Africa). They landed in the gorgeous port city of Bugia, now known as Bejaïa, where Fibonacci set about learning mathematical ideas from the local vendors. It was in the marketplace that he observed the use of Hindu-Arabic numerals, 0-9. They were so much more efficient than the Roman numerals that dominated Europe.
To Laib, the fact that Fibonacci learned about the numerals in Algeria especially hits home because her husband is Algerian. Indeed, Stewart writes more about this story in a sidebar:
"Leonardo, Italian born, grew up in North Africa, where his father Guilielmo was working as a diplomat on behalf of merchants trading at Bugia (in modern Algeria). In his Liber Abbaci of 1202 he writes: 'When my father, who had been appointed by his country as public notary in the customs at Bugia for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting.'"
As for decimals, these were introduced by Dutchman Simon Stevin in the 16th century. This is the mathematician for whom the Stevin algorithms -- the standard algorithms -- are named:
"His notation did not include the familiar decimal point, but it lead quickly to today's decimal notation."
We now move on to the development of negative numbers. (That's right -- all the numbers the seventh graders are learning now are mentioned in today's chapter.) Stewart credits the Chinese with the invention of negative numbers. They used rods for numbers -- red for positive, black for negative:
"So to solve equations that we would write as 3x - 2y = 4, x + 5y = 7, they would set out the two equations as two columns of a table: one with the numbers 3 (red), 2 (black), 4 (red), and the other 1 (red), 5 (red), 7 (red)."
This eventually became our modern system for signed numbers, though their acceptance was slow:
"But interpretation aside, their arithmetic worked perfectly well, and they were so useful as a computational aide that it would have silly not to use them. Our number system is so familiar that we tend to assume that it is the only possible one, or at least the only sensible one."
The author reminds us -- and our young students -- why it's important to learn arithmetic nowadays:
"And if your only arithmetical ability is reading what's on a calculator, you probably won't notice if the supermarket gets your bill wrong. You might not worry about that, but modern civilization would quickly break down if we stopped teaching arithmetic, because you can't spot the future engineers and scientists at the age of five."
On this note, Stewart concludes the chapter as follows:
"But, just as you won't learn to walk by always using a crutch, you won't learn to think sensibly about numbers by relying solely on a calculator."
And this explains today's regular teacher and his calculator policy -- those who truly need them (the special ed students) can use them, while everyone else learns how to avoid the calculator "crutch."
There are also several sidebars in this chapter. I already mentioned the one about Fibonacci, and there is another sidebar that discusses Mayan numerals in more detail.
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 2019. Kate Nowak's website still exists, but at a new location:
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 2019. Kate Nowak's website still exists, but at a new location:
But the most recent post there is dated April of last year, so even with the new location, she still hasn't posted this year:
http://fxnoftime.com/2018/04/time-to-noodle/
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.
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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