1. Introduction
2. Pappas Problem of the Day
3. Blaugust: My Favorite “Rich Problem/Task”
4. Traditionalists and Rich Problems
5. Rich Problems in Geometry
6. Sixth Grade Rich Task: Tax Collector
7. Why No Rich Task in Seventh Grade?
8. Eighth Grade Rich Task: Mathematics of Love
9. Hawking, pages 51-80
10. Conclusion
Introduction
It's now August, and so I wondered whether there would be a Blaugust challenge this year. And so I visited the blog of Shelli -- the teacher who created the Blaugust challenge. Well, the first thing I see at her blog is -- a Blaugust post:
http://statteacher.blogspot.com/
And so yes, Blaugust is definitely back this year. But just like last year, I'm not worthy to add my name to the official Blaugust list. That list is only for real math teachers, not subs like me. If I'm hired at the last minute to become a teacher before the month is over, then I'll try to add my name to Shelli's list.
Nonetheless, it's worth it for me to think about and respond to Shelli's questions, even if I don't become an official Blaugust participant. And like last year, I'll add links to some of the real teachers who are blogging this month.
Pappas Problem of the Day
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
A square sheet of cardboard of area 64 sq. in. has 4 congruent squares removed from its corners. It is then formed into a box without a lid. Its surface area is 48 sq. in., and its volume is 32 cubic in. What is the dimension of the squares that are removed?
Well, since the original area is 64 in.^2 and the final area is 48 in.^2, we must have subtracted, or cut out, a total of 16 in.^2. This area consists of four congruent squares, so each square is 4 in.^2. And of course, a square of area 4 in.^2 must have a side length of 2 in. Therefore, the desired dimension is two inches -- and of course, today's date is the second.
This is actually a silly question. It's redundant to give both the surface area and volume of the box. In the past, Pappas has asked this sort of question before, but she gave only the volume and left out the surface area.
If we try solving this problem with only the volume, more algebra is needed. The original square sheet has an area of 64 in.^2, so each side must be 8 in. If we now subtracted a square of side length x from each corner, both the length and the width are decreased by 2x, while x itself is the height of the resulting box. So we obtain:
x(8 - 2x)^2 = 32
x(64 - 32x + 4x^2) = 32
4x^3 - 32x^2 + 64x - 32 = 0
4(x^3 - 8x^2 + 16x - 8) = 0
This is a cubic equation that can't be factored by grouping. We're forced to use trial and error. We might try dividing by (x - 1) first, which doesn't work. We could try (x + 1) next, but we know that x must be positive since it's a length, so we'll skip to (x - 2). Using synthetic division, we get:
2| 1 -8 16 -8
2 -12 8
1 -6 4 0
(x - 2)(x^2 - 6x + 4) = 0
So one solution is x = 2, and then we can use the Quadratic Formula to find the others:
x = 6 +/- sqrt(6^2 - 4(1)(4))
2(1)
x = 6 +/- sqrt(20)
2
x = 3 +/- sqrt(5)
Of these last two solutions, x = 3 + sqrt(5) makes no sense, because then 2x > 8 and so 8 - 2x, the length and width of the box, would be negative. But x = 3 - sqrt(5) is a valid solution. The volume of the box with length and width 8 - 2x and height x = 3 - sqrt(5) is indeed 32 in.^3.
Thus Pappas probably mentioned the surface area of 48 in.^2 in order to rule out x = 3 - sqrt(5) and thus force x = 2 as the only solution. But given the surface area of 48 in.^2, we can solve the problem much more easily without using algebra at all.
I like yesterday's problem much more than today's:
In Euclidean geometry how many lines parallel to m can be drawn through point P?
And of course, Playfair tells us the answer to this question. One line parallel to m can be drawn through P -- and of course, yesterday's date was the first. In hyperbolic geometry the answer would be "infinitely many," and in spherical geometry the answer would be "none."
By the way, speaking of spherical geometry, I've decided that we're doing with spherical geometry for the time being. I wish to shift my attention now to Blaugust.
Blaugust: My Favorite “Rich Problem/Task”
Let's get to today's Blaugust topic. I'll do it the same way I did it last year -- since today's date is the second, we'll just take the second topic from Shelli's list.
Here's the list:
http://statteacher.blogspot.com/2019/07/introducing-mtbosblaugust-2019.html
And here's the second item from that list:
We know that traditionalists like Barry Garelick have a low opinion of "rich problems." Indeed, this is what he wrote back in his July 5th post (which I quoted on my own blog two days later):
If you hang around long enough in the world of math education you’ll hear people refer to “rich problems”. What exactly are rich problems?
One definition is: “A problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”
My definition differs a bit: “One-off, open-ended, ill-posed problems that supposedly lead students to apply/transfer prior knowledge to new or novel problems that don’t generalize.”
For example: “What are the dimensions of a rectangle with a perimeter of 24 units?”
But then again, traditionalists like Garelick generally aren't Blaugust participants. I reckon if Shelli were to invite Garelick to Blaugust, he'd turn her down as soon as he saw the second topic on her list.
Anyway, let me answer Shelli's question. What is my favorite rich problem or task?
Well, so far in this post I posted a Pappas problem, since I always do so whenever she gives a Geometry problem on her calendar. I didn't intend it to be my response to Shelli's question, although it is arguably a "rich problem." Indeed, we might imagine giving this to students to see how many of them try to write and solve the cubic equation and how many see that they can get the answer from the areas without algebra at all.
But there's something that won't work if we tried giving this in a Geometry class -- solving the cubic equation isn't taught until Algebra II. Indeed, the only part of this problem that are Geometry are the formulas for area and volume -- and that's trivial. If we were to assign this problem and expected the students to solve it algebraically, we'd assign it to Algebra II students. (It's possible that there might be an Integrated Math II course where students are learning both the algebra and geometry needed for this problem, but I'm not sure.)
Even as an Algebra II problem, many students might have trouble coming up with the equation:
x(8 - 2x)^2 = 32
anyway unless they've seen this problem recently -- and if they have, they're likely just to write the equation without even considering the surface area shortcut. Thus I doubt there's any scenario where some students write the equation and others use the surface area shortcut.
And of course, it's also possible that the students solving the cubic see x = 2 and the two irrational solutions, and then simply assume the correct answer is x = 2 because they automatically think that any answer that's not an integer is wrong. In this case, one of the irrational solutions causes the length and width to be negative, and the other needs the surface area to rule out.
It's possible to rewrite this problem so that we don't need to specify the surface area for the sole purpose of ruling out the unintended solutions of the cubic:
A square sheet of cardboard of area 81 sq. in. has 4 congruent squares removed from its corners. It is then formed into a box without a lid. Its volume is 54 cubic in. What is the dimension of the squares that are removed?
This time, the side of original square must be 9 in.:
x(9 - 2x)^2 = 54
x(81 - 36x + 4x^2) = 54
4x^3 - 36x^2 + 81x - 54 = 0
It might be tricky to figure out what factor to try in the synthetic division. Here (x - 6) works:
6| 4 -36 81 -54
24 -72 54
4 -12 9 0
And 4x^2 - 12x + 9 = 0 is a perfect square:
4x^2 - 12x + 9 = 0
(2x - 3)^2 = 0
2x - 3 = 0
x = 3/2
So our solutions are x = 6 and x = 3/2. Since 6 is too big (as length 9 - 2x = -3), only 3/2 works. This is how we can create our problem without needing to give a surface area -- have it lead to a cubic with a double root, with the other root too big for the box.
If we don't like having a fraction as the correct answer, we can change the area of the original square to 36 in.^2 and the desired volume to 16 in.^3. Then the solutions are x = 1, 1, 4, with x = 1 as the only valid answer to the problem.
This should give students something to work with. In this case, I wonder whether they'd know why they need to rule out x = 4 (answer: because it would make the length be -2).
Traditionalists and Rich Problems
I've added the "traditionalists" label to this post, since it's difficult for me to think about rich problems without thinking about the traditionalists' disdain for them.
Recall that Garelick provides an example of a rich problem that he likes. He finds this problem in a Golden Age Dolciani text:
“If a+1 = b, then which is true? a>b, a<2b, or a<b?”
For some reason, Garelick likes this problem more than the infamous rectangle problem. Notice that the Equation to Inequality Property of Lesson 1-7 (of the U of Chicago text) tells us that b > a, from which we conclude that a > b is always false and a < b is always true. The last inequality a < 2b is sometimes true and sometimes false depending on the value of a.
Perhaps Garelick prefers this problem to the rectangle because at least two of the three possibilities have definite answers. Suppose we try to make the rectangle problem resemble this one:
If a and b are the dimensions of a rectangle with perimeter 24 units, then which is true? a > b, a < 2b, or a < b?
Then none of the three possibilities are always true so there is no answer, as opposed to the actual Dolciani question where one of the choices is always true.
By the way, speaking of traditionalists, SteveH added one more comment to Garelick's most recent post in addition to the one I described in my last post. Even though this comment isn't about rich problems, let's look at it anyway:
https://traditionalmath.wordpress.com/2019/07/27/advice-on-the-teaching-of-standard-algorithms-before-common-core-says-it-is-safe-to-do-so-dept/#comments
SteveH:
They increase the spread of student abilities and willingness in K-6 with Full Inclusion, but then do nothing about it? Many even resist the option of allowing a proper Algebra I in 8th grade. This goes against decades of common practice and success. However with Full Inclusion in K-6, they really need to offer different academic options for acceleration. They want to age-track all kids in one classroom, but that creates separate groups (tables?) for the advanced kids. The other kids know what’s going on. This is no surprise. Why are those tables filled with kids from affluent families – something that educators don’t want to see? It’s because those parents provide the missing facts and skills at home – things that many other kids could easily handle.
"The other kids know what's going on." I've seen a few other traditionalists use this phrase in connection with tracking. It refers to the reasons that tracking has gone out of fashion -- we want to avoid the situation where tracking segregates students by race or income. (Actually, let's focus on income since SteveH specifically mentions income in this comment.) In particular, we don't want low-income students to see themselves in a classroom surrounded by other low-income students and feel bad about being on the lowest track.
But SteveH's "The other kids know what's going on" implies that "full inclusion" has backfired. The classes are still divided among high and low groups, except by table, not tracking. And "the other kid know" whether they're in the high or low group, so that they feel just as bad at the low table as they would under full-blown tracking.
SteveH:
Why are high schools allowed to offer different levels of classes for same-age students? What’s different in K-6, especially with Full Inclusion increasing the academic range? Why do they resist the traditional split of students in math and foreign language in 7th and 8th grades?
Here's the problem -- suppose one middle school student is ready for Algebra I but not Spanish I, while another is ready for Spanish I but not Algebra I. A middle school can easily accommodate this because students have multiple teachers throughout the day. But at elementary schools, students have only one teacher the entire day. Thus a student in the high math class is forced to be in the high language arts class, and the same is true for the low math student.
There is another traditionalist -- momof4 -- who also asked this same question. She proposed that elementary school should place students in low and high classes "by subject." Thus she's essentially telling us that elementary students should have more than one teacher per day.
I once proposed a mild form of tracking as a compromise. Kindergartners should have only one teacher the entire day, but then we gradually add more teachers as they progress. (This is based somewhat on what my own elementary school did when I was a young student.) The plan I posted listed the following subjects to be taught by a teacher other than the homeroom teacher:
Grades 1-2: Math
Grades 3-4: Math, Elective
Grades 5-6: Math, Elective, Science
Grades 7-8: Math, Elective, Science, P.E.
But I once found a source which stated that even having fifth grade math taught by a teacher other than the homeroom teacher is bad (as in, it leads to lower state test scores). The article stressed that the relationship between a fifth grader and an all-day teacher outweighed any benefits to being taught in one's zone of proximal development. And of course, if having more than one teacher is bad for fifth graders, how much worse is it for first graders?
Traditionalists say that "ed-school pedagogues" are concerned too much with how students feel -- that is, their self-esteem. They also say that their opponents are too concerned with "engagement." In contrast, the traditionalists claimed that their concern is with how good an education the students are actually getting.
To me, "self-esteem" and "engagement" matter for only one reason -- students with low self-esteem or who aren't engaged tend to leave problems blank. As far as I'm concerned, we can replace "engagement" and "self-esteem" in every ed article with "not leaving problems blank."
And how do we get students not to leave problems blank? By giving them problems that they're more eager to answer -- and these are known as "rich problems."
Rich Problems in Geometry
The Exploration questions in the U of Chicago text can count as rich problems. Some chapters of the text lend themselves more to rich problems than others.
For example, Chapter 5 (polygons) contains many interesting questions. Here they are:
Lesson 5-1 (Isosceles Triangles)
20. Find three meanings of the word median other than the one in this lesson.
21. Use a ruler, compass, and protractor or an automatic drawing tool.
a. Draw an equilateral triangle. Draw all three medians. They should all intersect in a point called the centroid of the triangles. Measure the lengths of the medians and distances from the centroid to each of the vertices. What can you conclude?
b. Draw an isosceles triangle that is not equilateral. Repeat the steps in part a for your triangle.
c. Draw a scalene triangle. Repeat the steps in part a again.
d. Summarize your work by writing a sentence or two that begins: "In any triangle, the three medians...."
Lesson 5-2 (Types of Quadrilaterals)
28. Which of the types of quadrilaterals on the hierarchy can be nonconvex? Support your answer with drawings.
29. Biologists place living things in a hierarchy. Show a hierarchy containing the following terms: man, cat, animal, mammal, primate, chimpanzee, lion, feline, plant.
30. Think of another hierarchy outside of mathematics different from that in Question 29.
Lesson 5-3 (Conjectures)
26. A simple closed curve is a closed curve that does not intersect itself. Here are some examples (which of course I don't draw here).
An unsolved conjecture (as of 1989 -- and yes, as of 2019) is that on every simple closed curve there are four points which are the vertices of a square. In curve a are four points which might be the vertices of a square. Trace and try to find the square for curves b and c.
Lesson 5-4 (Properties of Kites)
23. Design and build a "flat" kite that flies (not a box kite). Does your shape agree with the mathematical definition of a kite?
Lesson 5-5 (Properties of Trapezoids)
23. Test this conjecture. If the midpoints of the sides of an isosceles trapezoid are connected in order, the resulting figure is a rhombus.
Lesson 5-6 (Alternate Interior Angles)
26. a. Other than Z, which printed capital letters usually contain alternate interior angles? Print them.
b. Which printed capital letters usually contain corresponding angles? Print them.
Lesson 5-7 (Sums of Angle Measures of Polygons)
24. Find a globe. Estimate the sum of the measures of the angles of the triangle determined by Los Angeles, London, and Rio de Janiero.
I've incorporated some of these problems (most notably those for Lessons 5-2 and 5-7) into some of the activity worksheets I've posted here. Lesson 5-7, of course, hints at spherical geometry.
Chapter 15, on circles, is also notable for its rich tasks. The last two lessons of the book are on the Isoperimetric Inequalities:
Lesson 15-8 (The Isoperimetric Inequality)
21. Give dimensions and draw a picture of a polygon whose perimeter is 100 feet and whose area is greater than 625 square feet.
Lesson 15-9 (The Isoperimetric Theorems in Space)
20. According to legend, the first person to use the Isoperimetric Inequality was Dido, the queen of Carthage.
a. Where is, or was, Carthage?
b. How did Dido use this inequality? (Hint: Look in an encyclopedia or dictionary under Dido.)
21. Develop an Isoperimetric Inequality for space. That is, find a relationship between the volume V and surface area S.A. of any space figure.
Indeed, the plan I mentioned in several posts over the past month suggested teaching spherical geometry at the end of the year (perhaps right after Lesson 15-9 -- a nice segue from the sphere as the figure with the most volume for its surface area). In this unit, we ask questions similar to the ones listed for Lesson 5-7 above. Thus this entire unit would consist of rich problems.
Shelli, the leader of the Blaugust challenge, mentions some rich tasks for Geometry herself:
http://statteacher.blogspot.com/2019/07/made4math-organization-for-win.html
For example, one of these problems fits quite nicely with Lesson 3-2 of the U of Chicago text:
COMPLEMENTARY AND SUPPLEMENTARY ANGLES
Directions: Using the digits 0-9 no more than once each, fill in the boxes to make the statements true:
_ _ and _ _ are complementary angles.
_ _ and _ _ _ are supplementary angles.
Hey, notice that it's possible to "Shelli-ize" Barry Garelick's rich problem:
PERIMETERS OF RECTANGLES
Directions: Using the digits 0-9 no more than once each, fill in the boxes to make the statements true:
_ and _ are the dimensions of a rectangle with perimeter 24.
_ and _ _ are the dimensions of a rectangle with perimeter 68.
_ _ and _ _ are the dimensions of a rectangle with perimeter 168.
We can even use the Isometric Inequality to "fix" Garelick's problem, if we wish:
- What are the dimensions of the largest rectangle with perimeter 24?
Sixth Grade Rich Task: Tax Collector
Let's go back to my classes at the old charter middle school. By the way, in case you missed it from my Pi Approximation Day post, I finally figured out that the old charter school is closed forever. I'll still refer to that school in my posts, but only in the past. There'll be no posts like "The old charter school is on its annual field trip to the fair this week. Here's what I would have taught during fair week if I were still a teacher there..." since the school no longer exists.
Anyway, many of the Illinois State projects that I taught during that year could be described as "rich problems," or at least "rich tasks." But I believe that my best project day was on Valentine's Day, when I gave two different rich tasks to my sixth and eighth graders. Let me cut and paste what I wrote about the sixth grade project on the blog -- which, as you'll soon see, doesn't come from the Illinois State text, but a different website:
The sixth grade activity comes from Denise Gaskins -- one of the other teachers I met during another challenge. I promised her that I'd do one of her activities in class and tell her about the experience, so here it is:
https://denisegaskins.com/2015/05/08/math-games-with-factors-multiples-and-prime-numbers/
The particular activity I choose for today is called "Tax Collector":
http://web.archive.org/web/20160222064756/http://mrlsmath.com/wp-content/uploads/2008/01/tax-collector-pdf.pdf
This is what I wrote in my comment to Gaskins:
Hello! I commented here during last month's MTBoS challenge. We agreed that if I used one of your games in my classroom, I tell you about it, so here goes!
Anyway, I played Tax Collector with my sixth graders today, and they seemed to enjoy it very much! I began with me as the tax collector and the class as the taxpayers. As it turns out, the first number chosen was 11 -- and the end, the class just narrowly lost to the tax collector, 110-100. If they had started with 19 (or even 17) and made all the same subsequent choices, they'd have won the game!
Afterwards, I divided the class into pairs, with one student as the tax payer and the other as the tax collector. No taxpayer wins, or comes quite as close as our initial game with the whole class, even though by now they knew that it was best to start with a large prime like 19. A few taxpayers believed they had won, but often it was because the tax collector didn't take all the factors correctly -- for example, a taxpayer started with 20, and the collector took only 10 instead of 1, 2, 4, 5 as well.
All in all, it was the most fun we had in class in a while. Thanks for the activity!
In fact, you can see what makes the game so difficult for the taxpayer. If the payer takes a prime, the collector can take only 1. But the payer can never take a prime on any move but the first, since no matter what number the payer takes on the first move, the collector takes 1. So the payer can only take at most one prime the whole game -- the collector is guaranteed all but one of the primes. This is why it behooves the payer to take the largest possible prime on the first move. I'm not sure that my students recognize 19 as a large prime number, but they do learn to start with a large odd number.
The tricky part of the game is when to take an abundant number like 12. If the taxpayer takes such a number too soon, the collector gets many of its factors and takes the lead. But if the payer waits too long to take the number, all of its factors are gone and the collector keeps the number itself. The girl who mistaken thinks that she beats the collector does learn that it's good to take 10 before 20 -- if done correctly, after the payer takes 20, the collector gets only 4.
As I wrote to Gaskins, this is a great activity to get the students thinking about factors -- which is great during the current lessons on fractions and percents.
Why No Rich Task in Seventh Grade?
Unfortunately, I didn't give a rich task in my seventh grade class that day. Let me cut and paste the reason I stated on the blog at the time:
As I wrote before, I have to do Student Journals in seventh grade because the class doesn't meet tomorrow. This class is learning about integer operations. This isn't exactly the topic I want to rush -- that this topic is already getting an extra day, but it really needs much more time. As you would expect, the students are confused when I had to jump from addition/subtraction to multiplication and explain why 7(-2) isn't 14 since the positive number is bigger -- or even worse, 5.
Last month, I wrote that for various reasons, seventh grade was the most important class that I taught that year, yet it's the class I had the least amount of time with. The day I gave the other two grades the rich task was a Tuesday. Mondays were for the coding teacher in all three grades, and Wednesdays were the day I didn't see the seventh graders. And that Thursday was the start of a big five-day weekend for President's Day. Thus if I didn't give the traditional integer ops lesson that day, it would have been pushed back a full week.
It's a shame that the schedule was the way it was. It would have been easy to create, say, a Shelli-style rich problem for the seventh graders.
Eighth Grade Rich Task: Mathematics of Love
This is what I wrote that day about the eighth grade rich task:
Now as it turns out, my eighth grade project is based on the Square One TV song "The Mathematics of Love." Part of the project has the students convert from Arabic to Roman numerals and vice versa. This is actually inspired by a project mentioned in the Illinois State text. Recall that two weeks ago was "Input, Process, Output," where the students are introduced to functions. There's a follow-up to this project -- but it's printed only in the teacher's edition, not the student texts. Anyway, the students learn that conversion to Roman numerals is also a function, with an input, process, and output.
The Illinois State teacher's text actually has the students focus on attempting to add, subtract, multiply, and divide Roman numerals. Instead, I mainly have the students convert the numerals and provide only one example for each operation. Naturally, the students find it easier to convert back to Arabic numerals before doing any calculations. The most common mistakes are just as you would expect -- confusing IV = 4 with VI = 6, and IX = 9 with XI = 11, and so on.
The text also asks whether the set of Roman numerals is closed under addition (or any of the other three operations). Notice that as soon as an additively closed set contains 1 (or Roman numeral I), it automatically contains infinitely many natural numbers, so we must ask whether one can write arbitrarily large natural numbers in Roman. At first glance it would appear not, since M is the largest number that has its own letter, and it only represents 1000. But sometimes we place a bar (which is called vinculum in Latin) over a Roman numeral to multiply it by 1000. So V-bar is 5000, M-bar is a million, and so on. So we can write Roman numerals as large as we please, if we use arbitrarily many vincula when writing them. Only then can the set of Roman numerals be closed under addition.
This is only part of today's activity -- the "mathematics" part. Since the activity is called "The Mathematics of Love," I need to represent the "love" part of the lesson. To accomplish this, I take a page from one of my usual fallback blogs -- Sarah Carter:
http://mathequalslove.blogspot.com/2013/09/relations-functions-and-dating-advice.html
Even back when I was the age my students are now, I often found myself thinking about the boyfriend and girlfriend functions. As we see from Carter's blog, boyfriend and girlfriend are supposed to be functions, but in practice, many people cheat and so they're no longer functions, as one input (boy) is paired with more than one output (girl, etc.).
This example fits perfectly today, since it fits both the holiday (V-Day) and the content (functions). I begin with some examples of functions (such as father) and non-functions (such as brother) before moving on to boyfriend. This is actually part one of my activity -- part two is Roman numerals.
Returning to the present, notice that this is "Sara(h)" season -- the time of the year when Sarah Carter and Sara VanDerWerf are the most active. During Blaugust, many teachers refer to the Sara(h) blogs to find first day of school activities, including VanDerWerf's famous "name tents."
I believe that the reason my rich tasks worked so well is that there was a worksheet there to help guide the students. As I wrote above, the eighth grade task came from the teacher's edition of the Illinois State text, and so I was forced to make it into a worksheet. For previous tasks, I just told the kids to follow the directions in the student texts -- and then they struggled. My regret, therefore, is that I didn't create worksheets for the rich tasks more often.
Hawking, pages 51-80
It's time for us to continue our summer side-along graphic novel, Jim Ottaviani's Hawking. We left off right in the middle of an explanation of what physicists prior to Hawking knew about the universe and its origins:
"The physicist and priest Lemaitre took this a step further," Hawking narrates. "He argued that general relativity and quantum theory implied that the universe must be expanding. He even proposed an actual origin of the universe, a real 'in the beginning' that started with what he called the 'primeval atom.'"
Lemaitre: I think that such a beginning of the world is far enough from the present order of nature to be not at all repugnant.
"Einstein didn't like this either. At all."
Einstein: Your calculations are correct, but your physics is atrocious.
"Of course, there was more than just arguing in front of blackboards and writing papers going on. Thanks to experimentalists, there's also a 'Meanwhile.' So...Meanwhile, astronomers were actually looking at the universe, and through bigger and bigger telescopes."
And this was right here in southern California -- the Mt. Wilson Observatory in Los Angeles. Here Edwin Hubble and Henrietta Leavitt:
"...showed two things," Hawking narrates. "First, we live on a small planet, circling an ordinary star, on the outskirts of an ordinary galaxy...one of who-knows-how-many such galaxies. Second, in the words of Douglas Adams, 'Space is big. Really big. You wouldn't believe how vastly hugely mind-boggingingly big it is.' At the time Hubble showed just how big it was, most astronomers couldn't believe it. Wouldn't believe it."
As the author explains, Hubble originally estimated the distance between galaxies as 900,000 light-years. Which is still plenty far, since a light-year is 9.5 trillion km.
At this point, Hawking discusses how he ended up leaving Oxford, his undergrad school, and moved on to Cambridge for grad school.
"I was back home in St. Albans after the Michaelmas term when two events transpired that -- eventually -- changed that attitude," Hawking narrates.
By the way, "Michaelmas" is a little-known Christian holiday that occurs in the fall. (On the blog, I mentioned "Michaelmas" as a possible precursor to Thanksgiving.) At the two schools that Hawking attended, Oxford and Cambridge, Michaelmas is the first of the three terms of the school year. The other two terms also have Christian names -- "Lent term" and "Easter term." The following scene takes place after Michaelmas term -- that is, during the Christmas holidays.
"The first event didn't seem that significant at the time."
Hawking: ...and I needed that First to go on to graduate study at Cambridge. It was a near thing, so I demanded a viva. I knew the idea of infecting Cambridge with the likes of me was all too tempting. So I told them that if they gave me a First, I'd leave, and if not, well...They could look forward to many more years of me right there at Oxford.
"And then I said..."
Hawking (flashback): I wager you're give me that First.
"I didn't even make it to the door before they had their decision."
Oxford guy: Mr. Hawking, wait.
Hawking (at the party): And off to Cambridge I went!
Friend: That is so totally, completely, and utterly...implausible. There's not an Oxford don in the history of Oxford dons that can think or decide on anything that fast. Nor are you anywhere near that bold.
Hawking: Nonono, I assure you! And I did have a backup plan. I was going to take the Civil Service exam. But then I forgot to go take the test...
Friend: Well, that part sounds plausible, anyway. Do you show up for classes now, at least? And how's this Sciama fellow?
Hawking: Well, our ideas about cosmology differ a great deal. But I don't know. I suppose he's fine.
Girl: Er. What's cosmology, then?
As Hawking finds out, the girl he meets is named Jane. As it turns out, the second event is his diagnosis with a sort of motor neurone disease.
Meanwhile, Hawking wants to go out with Jane, but she is having second thoughts after she learns about his disease:
Basil's sister: Are you all right?
Jane: Yes. No. I don't...Bit of a shock, that's all. We're -- to go on a date tomorrow. Italian dinner in Soho. Then Volpone at the Old Vic. And...I don't know. Should I beg off?
Basil's sister: Well, like I said, the whole family's mad. but my brother Basil adores him. And you like the theater. And him besides, yes? You should go.
"We had a fine evening," Hawking narrates, "but it proved a bit beyond my means. I was out of money before we made it to the bus going home."
Hawking: Er. This is terribly embarrassing, but...can you pay, um, this fare?
Jane: Of course I...No! No! I-I've lost my money purse!
They eventually find her purse inside the dark theater. Soon afterward, Hawking invites Jane to the May Ball -- which was held in June, in true Cambridge fashion. (OK, I know enough about the Cambridge calendar to explain Michaelmas term, but I can't say why the May Ball is in June.) But of course he continues to worry about his disease:
"But I couldn't imagine having much to smile about," Hawking narrates. "So I was determined to at least have a little fun in the time I had left. I headed home to pick up Jane when the term ended."
Jane: I was hoping for the family taxi this trip.
Hawking: Something a little nicer is in order, don't you think?
Jane: It appears to be...stuck. Perhaps if I...helped?
Mrs. Hawking: They're here! Well, don't you two look just lovely!
Edward (Hawking's adopted brother): Stephen has a new stick. Can I hold your stick please, Stephen?
Hawking: I thought Father was here.
Mrs. Hawking: He's in his study, doing research.
And here's the final scene for today's reading:
Mr. Hawking: The consensus is there will be no pain, at least. Just a loss of muscular control over...everything, eventually. But many victims suffer from depression, or panic. Did you notice...
Mrs. Hawking: Frank, He's off to the May Ball, and with a lovely girl. I shouldn't think so!
Jane (in the car): Perhaps...we...could...slow...down?
Hawking: We could, yes, but there's no time, and we don't want to be late.
They arrive at the ball.
Hawking: Champagne?
Jane: Yes, thank you.
Hawking: Sorry, I don't dance.
Jane: That's all right. It doesn't matter.
Hawking: The buffet's this way.
We'll pick up the story from here in my next post.
Conclusion
As I mentioned earlier, during Blaugust I like to link to a full participant of the challenge. Today I wish to link to "Algebra's Friend," Beth Ferguson:
http://algebrasfriend.blogspot.com/2019/08/im-not-good-at-math.html
Three years ago -- the same year that I taught at the old charter school -- Ferguson was the most prolific Blaugust poster (the "winner" of the challenge, you might say). But she hasn't written much since -- and here she explains why:
I love reading the blogs in August! Even though I retired a few years ago, I still love hearing about the start of school, looking at the classroom décor ideas, reading about the planning, the longing for students to excel.
In this post, she writes about a middle school girl she's assigned to tutor. The girl tells Ferguson that she isn't good at math. Ferguson writes the following three facts about the girl:
1) I don’t know my math facts so I’m not good at math.
2) I still use my fingers to count to figure out basic multiplication.
3) I was in an advanced math program in elementary school.
She also asks her mother more about the young student:
1) My daughter experiences a lot of anxiety.
2) She doesn’t know her math facts.
3) She struggles with math.
4) Her work since elementary school has been hit or miss because she has changed schools more than once.
After seeing the girl work, Ferguson reassures her that she is good at math after all:
I was hired in the summer to help the student with geometry. She had been told that in the upcoming year geometry would be a focus, and the mom and daughter realized she hadn’t had much instruction in the way of geometry. They gave me a workbook that had been suggested to them. After the first week of choosing a few workbook pages, I suggested we build a reference notebook together. I chose a sequence of introductory geometry topics, found foldables to use with them, and discovered that my new student loves foldables! She took great delight in working and building her notebook.
We notice that this girl enjoys foldables and interactive notebooks as she prepares for the start of her eighth grade year. (Yes, we know that geometry is indeed emphasized in Common Core Math 8.) I can only imagine that my own eighth graders might have succeeded with foldables and interactive notebooks at the old charter school -- but I never gave them that chance. Just imagine if I could have given them more rich tasks -- just like the Mathematics of Love -- on worksheets that they could then glue into their notebooks.
Her school this year started in July – on a modified year-round schedule. It is a tiny independent school that creates its own math curriculum. In the first week she had a packet on “work” problems, yes, two people painting a room or two pipes filling a tank. I was a little taken aback – it seemed the topic had been selected randomly.
I've written about the schools that start in July before. This is the opposite of the Oxbridge plan -- the Australian plan. Just as Australian schools start in January (a month after their summer solstice), the girl's school starts in July (a month after our summer solstice). Then there are four terms, with long breaks of at least a week between them. Both the Oxbridge (Michaelmas/Lent/Easter) and Australian schedules are based on the idea of having shorter summers and longer breaks during the year.
Let me end this post the same way that Ferguson does:
I’ll be searching YOUR blogs and following you on Twitter for all the good ideas!
END
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