1. Introduction
2. Calculating the Cosmos Chapter 2: Collapse of the Solar Nebula
3. Shapelore Learning 13-3: Leading Out Maybehoods
4. Shapelore Learning 13-4: Unstraight Showing
5. Some Rapoport Math Problems
6. Lemay Chapter 6 Part 2: Creating Classes and Applications in Java
7. More on Goldbach in Java
8. Mocha Music: "The Need for Speed," "Ratios," "Another Ratio Song"
9. Reblogging: Preparing for the Old Charter School
10. Conclusion: Traditionalists
Introduction
Today would have been Day 179 in my new district (as my old district ended last week). Graduations would have been this week -- but of course, they aren't, since mass gatherings are still banned. Both of my districts are hoping to hold graduations for the Class of 2020 at the end of July -- but once again, this will be only if local and state officials allow mass gatherings by then. Otherwise 2020 won't have a traditional ceremony.
This week also marks the end of school for my students at the old charter school. I have absolutely no idea where any of them are now. Perhaps some of them signed up for charter high schools, while others returned to the LAUSD. At any rate, they should be just finishing up Grades 9-11 now. As much as I feel sorry for the Class of 2020 in losing its graduation, the classes that I'm more interested in are 2021-2023, since these are the cohorts I taught at the old charter school.
Calculating the Cosmos Chapter 2: Collapse of the Solar Nebula
Chapter 2 of Ian Stewart's Calculating the Cosmos is called "Collapse of the Solar Nebula." As usual, it begins with a quote:
Two thousand million years or so ago two galaxies were colliding or rather, were passing through each other...At about the same time -- within the same plus-or-minus ten percent margin of error, it is believed -- practically all of the suns of both of those galaxies became possessed of planets.
-- Edward E. Smith PhD, Triplanetary
And the proper chapter begins:
"Triplanetary is the first of Edward E. Smith's celebrated 'Lensman' series of science fiction novels, and its opening paragraph reflects a theory about the origin of planetary systems that was in vogue when the book appeared in 1948."
This chapter is all about the formation of our solar system. As Stewart explains, Smith's idea isn't the current theory of how our solar system formed. Instead, he tells us that our galaxy began as a cloud of gas that tore into pieces:
"One such piece, the solar nebula, formed the Sun, together with its solar system of eight planets, five (so far) dwarf planets, and thousands of asteroids and comets."
So there were five dwarf planets at the time the author wrote this line. As of now, there are nine objects that are candidates for being dwarfs -- Ceres (in the asteroid belt), Orcus, Pluto, Haumea, Quaour, Makemake, Gonggong, Eris, and Sedna (much more distant that the others).
Why do all planets go around the sun in the same direction? We learn that it has something to do with momentum -- we read:
"It governs the tendency of a body to travel at a fixed speed in a straight line when no forces are acting, as Newton's first law of motion states."
To be more precise, it's angular momentum, which refers to objects that spin like a top:
"It spins around the line that runs through the middle of the top, so every particle of matter in the top rotates around this axis."
Therefore the theory is that the original gas cloud spun with a lot of angular momentum, so when it collapsed, all of its parts continued to spin, since angular momentum is conserved:
"That's why early astronomers guessed that the Sun and planets all condensed from a cloud of gas, after it had collapsed to create a protoplanetary disc."
But that still doesn't explain why the planets have more momentum than the massive sun. One theory was that lumps of material passed by the Sun and became captured in orbit. But this is problematic:
"Capture is tricky because what comes down must go up again (unless it hits the Sun and gets engulfed)."
Computer simulations have been made to recreate the early solar system, showing with how tiny "pebbles" can grow into the cores of the gas giants:
"In theory this process can build up a core ten times the mass of the Earth in a few thousand years. Previous simulations had thrown up a different problem with this idea: it generates hundreds of planets the size of the Earth."
But not only were scientists able to fix the computer simulation to get it to produce the correct number of planets, but they also discovered another distant solar system where a similar process is currently taking place:
"It would be difficult to find a more dramatic confirmation of a theory. It's easy to believe that gravity can cause things to clump, but how can it also pull them apart?"
At this point Stewart gives the only picture of the chapter -- the Atacama Large Millimetre Array image of HL Tauri, showing concentric rings of dust and gaps between them.
He moves on to discuss the laws of thermodynamics -- namely entropy. Entropy causes the gas molecules in a room to be spread out rather than clump together in one spot:
"However, the thermodynamic model of gas in a room is not appropriate for working out how the solar nebula, or the entire universe, should behave."
That's because it leaves out a major force that pulls on objects as massive as planets -- gravity:
"The short-range repulsion between molecules that collide might tell us something about the state of a small region in a planet's atmosphere, but it won't tell us about the planet."
If there is one large object, then gravity pulls everything towards it. But something else can happen if there are two or more large objects in a system:
"If two stars are close enough together, or approach each other for chance reasons, they can end up orbiting their common center of mass."
Again, we can use computers to simulate the early solar system, but they must conserve energy and momentum, or otherwise it might show the planets colliding with the sun:
"Methods that avoid this danger are of recent vintage. The most accurate ones are known as symplectic integrators, after a technical way to reformulate all relevant the equations of mechanics, and they conserve all relevant physics of quantities exactly."
Even so, the planets weren't always in the order from the sun that they are today. Originally Saturn was the farthest planet from the sun, until, as the author explains, it moved into a 2:1 resonance, where one Saturnine year became exactly equal to two Jovian years:
"Resonances have a strong effect on celestial dynamics, because bodies in resonance orbits repeatedly align in exactly the same way, and I'll be saying a lot more about them later."
And the planets may continue to fall out of their current orbits. Stewart concludes the chapter with a warning here:
"The Earth is then likely to collide with Mercury, Venus, or Mars. And again there's a slight chance that Mars will be ejected from the solar system altogether."
Shapelore Learning 13-3: Leading Out Maybehoods
Lesson 13-3 of the U of Chicago text is called "Ruling Out Possibilities." We must convert the words "Ruling" and "Possibilities" into Anglish. The recommended word for "ruling" at the Anglish website is leading although using leading out for "ruling out" sounds a bit weird. The word "ruling" is short enough that we might just grandfather it in as a simple word in our Plain English.
As for "possibility," the suggested word is likelihood. But I prefer to save that word for "probability," another math-related term. Just as "possibly" isn't the same as "probably," I won't want "possibility" to be the same as "probability." Instead, I'll use one of the other words listed under "possibility" -- and that's maybehood:
Law of Ruling Out Maybehoods:
When p or q is true and q is not true, then p is true.
The original name for this rule of inference is "Modus Tollens," which is clearly Latin. Another word for "ruling out" is "eliminate," which is also derived from Latin. Compared to "Modus Tollens" and "eliminating," I've decided that yes, I will keep "rule" grandfathered in even though it is French. (On this blog, I've decided to keep all titles in pure Anglish, and so I wrote "leading out" above. In practice though, we'll still call it "ruling out" in Plain English.)
Many of the logic lessons from Chapter 13 are non-Geometrical, even non-mathematical. For this post, we'll focus only on the mathematical examples as we convert them into Plain English:
You know every angle in a threeside is either inward, right, or outward. So if an angle in a threeside is not inward or right, then it is outward. There is no other maybehood.
Once again, "angle" has been grandfathered in, so we only needed to introduce the words threeside, inward ("acute"), outward ("obtuse"), and maybehood in this paragraph above.
And that's it. The rest of this lesson consists of logic problems, not Geometry. (Some of the Exercises do mention Geometry, but we're only converting the lesson itself, not the Exercises.) And so we're already done with converting this lesson.
Shapelore Learning 13-4: Unstraight Showing
Lesson 13-4 of the U of Chicago text is called "Indirect Proof." A good word for "direct" is straight, and so "indirect" can become unstraight. We might also grandfather in "direct" if we feel that it's already a familiar word.
I've already decided that "proof" is simple enough to be grandfathered in. The suggested word for proof on the Anglish website is witness. Actually, I notice that in Geometry, we often ask students to "show" something when we really mean to "prove" it. And so show is a goodAnglish word for "proof" in this setting, and thus showing can replace "proof."
Let's keep "proof" and "reason(ing)" in our Plain English but use straight and unstraight instead of "direct" and "indirect." Then here's what our new lesson looks like:
In straight reasoning, a person begins with given knowledge known to be true. (We're only changing mathematical words here, so "person" doesn't change. But information might be considered math here, so I must changed it to knowledge.) The Laws of Following and Threefollowing are used to reason from that knowledge to a finding. The proofs you have written so far in this book have been straight proofs.
In unstraight proofs, a person tries to rule out all the maybehoods but the one thought to be true.
Let's move on to the first mathematical example:
You can rule out a maybehood if you know it is untrue. (While "true" is English, "false" is French, and so we can change it to "untrue" -- grandfathering it in works, too.) But how can you tell that a saying is untrue? One way to tell is if you know its notsaying is true. For instance, suppose you know y = 5 is true. Then y != 5 is untrue. (Here != is ASCII/Java for "is not equal to." Meanwhile, notice that "equals" can be replaced in Anglish with is, but we'll probably grandfather "equals" in.) Suppose you know that threeside ABC is sameside ("isosceles"). Then it is false to say it is unsameside ("scalene").
We move on to the word "contradict," which appears on the Anglish website as gainsay:
Two sayings p and q are gainsayful if and only if they cannot both be true at the same time.
We'll do the first two examples here, which are indeed Geometry:
1. Let p be the saying: Angle V is inward. Let q be: Angle V is right. Are p and q gainsayful?
Yes. An inward angle has breadth less than 90. (The Anglish website uses awkward words like bowlengthworth for the measure of an angle. But I know of a simpler English word, breadth, that even sounds like length and width. So I prefer it for angle measure.) A number cannot be both less than 90 and 90 at the same time, so an angle cannot be both inward and right at the same time. (In Anglish, "number" should be score, but "number" is so familiar that we must grandfather it in.)
2. Let p: ABCD is a foursameside. Let q: ABCD is a fourrightside. (I'm still not sold on rout for "rhombus" or righthook for "rectangle.") Are p and q gainsayful?
No. p and q can be true at the same time. Square (grandfathered) ABCD at the left is both a foursameside and a fourrightside.
A gainsaying is a situation in which two gainsayful sayings p and q are both asserted as true. (The words "situation" and "asserted" aren't mathematical.)
Law of Unstraight Reasoning:
If valid reasoning from a saying p leads to an untrue finding, then p is untrue.
Let's do the final example of the lesson, which contains an indirect proof:
5. Use an unstraight proof to show that no threeside has two outward angles.
First rewrite in if-then form: if a shape is a threeside, then it does not have two outward angles. Now draw a representative threeside. ("Representative" isn't mathematical.)
Either threeside ABC has two outward angles, say A and B, or it does not.
1. Guess both Angles A and B are outward. ("Assume" is math enough for us to say as guess instead.)
2. Then, by meaning of obtuse, Angle A > 90 and Angle B > 90. By the Adding Law of Unsameness, Angle A + B > 180. (I haven't quite decided what to do with words like "addition" and "inequality" -- we might grandfather "add" and "equal," yet still prefer adding to the word "addition.") Then, since Angle C > 0 for any angle in a threeside, Angle A + B + C > 180. But the Threeside-Adding Provedsaying says that Angle A + B + C = 180.
3. The last two sayings in step 2 are gainsayful. An untrue finding has been reached. By the Law of Unstraight Reasoning, the guess of step 1 is untrue. Angles A and B cannot both be outward.
In Byword 5 (Example 5 -- but we might keep "example" as non-math), the following saying has been proved: (1) if a shape is a threeside, then it does not have two outward angles. Its othernotsaying is: (2) If a shape has two outward angles, then it is not a threeside. By the Law of the Othernotsaying, saying (2) is also true.
Notice that I also used the word adding to mean "sum." The word "sum" is Latin -- it's in fact related to the Latin phrase "summa cum laude." Even though "sum" has only three letters, it (and other similar words like "difference," "product") are easily confused by students, so we'd definitely like to replace those words in Plain English. Even though both "add" and "plus" are also Latin, students seem to understand what those words mean more than "sum," and so we should replace "sum" with something to do with either "add" or "plus."
I believe that British pupils are more likely to hear phrases like "doing sums" than Americans, so "sum" might be less problematic across the pond. (By the way, how American/British dialects relate to Anglish is interesting. Americans use a Latin word "elevator," while the British use the pure Anglish word lift. On the other hand, the British use a Latin word "autumn" while Americans use the pure Anglish word fall.)
Then again, Threeside-Sum (or Threeside-Adding, whatever) sounds a bit awkward because it seems as if we're adding the three sides, not the angles. So Threenook-Sum sounds better, but I'm not sure whether I want to use nook just for that reason alone. Then again, there's nothing stopping us from just saying Threeside-Angle Sum to emphasize that we're adding the angles.
Some Rapoport Math Problems
Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:
The smallest number with two representations as the sum of two primes.
The fact that 2 is the only even prime helps us greatly -- it means that any odd number can have at most one prime partition -- an odd prime plus 2. So we're left looking for an even number -- one which can be written two different ways as a sum of two odd primes.
After 3 + 3 and 3 + 5, we might notice that 3 + 7 can also be written as 5 + 5. Therefore the desired sum is 10 -- and of course, today's date is the tenth.
There is one Geometry problem this week:
What is the area of the annulus as a multiple of pi?
[Here is the given info from the diagram: the annulus is bounded by two circles of radii 5 and 6.]
This one's straightforward using A = pi r^2 -- the outer circle has an area of 36pi and the inner circle has area 25pi. Therefore the area of the annulus is 36pi - 25pi = 11pi. Since we're already provided the factor of pi, all we need is the 11 -- and of course, this problem will be for tomorrow, the eleventh.
An "annulus," by the way, is simply a ring. In Anglish/Plain English, we'd just call it a ring.
Lemay Chapter 6 Part 2: Creating Classes and Applications in Java
Here is the link to today's lesson:
http://101.lv/learn/Java/ch6.htm
Lesson 6 of Laura Lemay's Teach Yourself Java in 21 Days! is called "Creating Classes and Applications in Java." Here's where we left off:
Passing Arguments to Methods
When you call a method with object parameters, the variables you pass into the body of the method are passed by reference, which means that whatever you do to those objects inside the method affects the original objects as well. This includes arrays and all the objects that arrays contain; when you pass an array into a method and modify its contents, the original array is affected. (Note that primitive types are passed by value.)This differs from C++, where everything is passed by value unless we specifically say so.
This takes us straight to our first listing for today:
Listing 6.4. The PassByReference class.
1: class PassByReference {
2: int onetoZero(int arg[]) {
3: int count = 0;
4:
5: for (int i = 0; i < arg.length; i++) {
6: if (arg[i] == 1) {
7: count++;
8: arg[i] = 0;
9: }
10: }
11: return count;
12: }
13: public static void main (String arg[]) {
14 int arr[] = { 1, 3, 4, 5, 1, 1, 7 };
15: PassByReference test = new PassByReference();
16: int numOnes;
17:
18: System.out.print("Values of the array: [ ");
19: for (int i = 0; i < arr.length; i++) {
20: System.out.print(arr[i] + " ");
21: }
22: System.out.println("]");
23:
24 numOnes = test.onetoZero(arr);
25: System.out.println("Number of Ones = " + numOnes);
26: System.out.print("New values of the array: [ ");
27: for (int i = 0; i < arr.length; i++) {
28: System.out.print(arr[i] + " ");
29: }
30: System.out.println("]");
31: }
32:}
This code takes the array and changes all of its 1's to 0's.At this point we learn about class methods, which are like instance methods, except that they are considered members of the entire class and not a particular object of that class. Lemay's example here is about the Math class -- including its sqrt method that I've already discovered on my own when I was writing my own Distance Formula method:
float root = Math.sqrt(453.0);
System.out.print("The larger of x and y is " + Math.max(x, y));
She explains that sqrt and max are class methods, which is why we call them using Math as opposed to a new Math object. If we were to write a class method ourselves, we use static:
static int max(int arg1, int arg2) { ... }
We also learn that while there is a primitive int type, there's also an Integer class which has its own methods. I still find this confusing, because this isn't anything like C++:
int count = Integer.parseInt("42", 10) // returns 42
Now Lemay finally explains all those extra words we see at the start of every main function:
public static void main(String args[]) {...}
Here's a run-down of the parts of the main() method:- public means that this method is available to other classes and objects. The main() method must be declared public. You'll learn more about public and private methods in Week 3.
- static means that this is a class method.
- void means that the main() method doesn't return anything.
- main() takes one parameter: an array of strings. This argument is used for command-line arguments, which you'll learn about in the next section.
But we are indeed doing the "next section" right now -- command-line arguments. This exists in C++ as well. Lemay writes:
How you pass arguments to a Java application varies based on the platform you're running Java on. On Windows and UNIX, you can pass arguments to the Java program via the command line; in the Macintosh, the Java Runner gives you a special window to type those arguments in.
Hmm, I wonder how I'm supposed to enter command-line arguments in the JRE that I downloaded last month (Eclipse). Well, I guess I'll find out:
Here, args is the name of the array of strings that contains the list of arguments. You can actually call it anything you want.
And indeed, Lemay accidentally calls it arg[] instead of args[] in the last lesson -- which is a bit awkward, since the parameter to the onetoZero method is also called arg[]. The usual name in C++ for this array is argv[].
This takes us to our next listing:
Listing 6.5. The EchoArgs class.
1: class EchoArgs {
2: public static void main(String args[]) {
3: for (int i = 0; i < args.length; i++) {
4: System.out.println("Argument " + i + ": " + args[i]);
5: }
6: }
7: }
And I compile and execute this program using the menus at the top of the screen. Thus there is no command line, and so this code produces no output. So args.length is 0, and so the for loop executes zero times. Not even "Argument " displays on the screen since the loop doesn't run.I'll try to figure out how to do command-line arguments later. But I will give this warning from the author about command-line arguments in Java vs. C/C++:
The array of arguments in Java is not analogous to argv in C and UNIX. In particular, arg[0], the first element in the array of arguments, is the first command-line argument after the name of the class-not the name of the program as it would be in C. Be careful of this as you write your Java programs.
Because of this, I can't really run Listing 6.6, since it averages the numbers on the command-line. I'll just skip to where the Lemay tells us how to convert the Strings in the command-line into Integers so that we can take the average:
sum += Integer.parseInt(args[i]);
Let's conclude with one item from Lemay's FAQ list, since it compares Java to C++ again:
| static and final are not exactly the most descriptive words for creating class variables, class methods, and constants. Why not use class and const? | |
| static comes from Java's C++ heritage; C++ uses the static keyword to retain memory for class variables and methods (and, in fact, they aren't called class methods and variables in C++: static member functions and variables are more common terms).final, however, is new. final is used in a more general way for classes and methods to indicate that those things cannot be subclassed or overridden. Using the final keyword for variables is consistent with that behavior. final variables are not quite the same as constant variables in C++, which is why the const keyword is not used. |
More on Goldbach in Java
OK, so I can't use command-line arguments on my computer yet. But it is time for me to come up with something I can actually code on my computer.
First of all, I've actually thinking about today's Rapoport question again:
The smallest number with two representations as the sum of two primes.
A representation of a number as the sum of two primes is called a Goldbach partition. So this question is asking for the smallest number with two Goldbach partitions. And this is an excellent problem for us to code in Java, considering that we already have a Goldbach method:
boolean isprime(int n) {
if (n<2)
return false;
for (int i=2; i<n; i++)
if (n%i==0)
return false;
return true;
}
boolean isgoldbach(int n) {
for(int i=2; i<n; i++)
if (isprime(i) && isprime(minus(n,i)))
return true;
return false;
}
The isgoldbach method stops as soon as it finds one Goldbach partition, so it won't help us find a second one. So we'll create a new method, is2goldbach, that determines whether a number has two Goldbach partitions.
We might think is2goldbach could call isgoldbach, just as isgoldbach calls isprime -- but unfortunately, that won't work. This is because isgoldbach simply returns a Boolean value, true or false -- the actual value of i (the prime in the found partition) is a local variable that disappears at the end of the method.
So instead, we'll write a new method for is2goldbach that repeats some of isgoldbach -- the difference here is that we'll have local Boolean value, first, that starts as false and switches to true when the first partition is found. Then the loop continues until the second is found:
boolean is2goldbach(int n) {
boolean first = false;
for(int i=2; i<=n/2; i++)
if (isprime(i) && isprime(n-i))
if (first)
return true;
else
first = true;
return false;
}
By the way, when I first tried this method, I accidentally got is2goldbach(8) to be true, because the compiler counted 3 + 5 and 5 + 3 to be distinct partitions. To fix this, I let the for loop go up only to i<=n/2 instead of n. I also got rid of that silly minus method and used n-i -- it's weird to have a minus method (which was there for the benefit of BlooP, not Java) and then have division in the for line.
Let's now write a main method that will solve the Rapoport problem -- it keeps going until it finds the first number with two Goldbach partitions. (Recall that our class here is Hofstadter.)
public static void main(String args[]) {
Hofstadter h = new Hofstadter();
int i;
for(i=2; !(h.is2goldbach(i)); i+=2)
;
System.out.println(i);
}
Now this program correctly prints out 10 -- and of course, today's date is the tenth.
Here is what our Hofstadter class looks like now:
public class Hofstadter {
double twotothethreetothe(int n) {
int cell0 = 1;
for (int i=1; i<=n; i++)
cell0 *= 3;
double cell1 = 1.0;
for (int i=1; i<=cell0; i++)
cell1 *= 2.0;
return cell1;
}
int minus(int m, int n) {
if (m<n)
return 0;
for (int i=0; i<=m; i++)
if (i+n==m)
return i;
return 0;
}
boolean isprime(int n) {
if (n<2)
return false;
for (int i=2; i<n; i++)
if (n%i==0)
return false;
return true;
}
boolean isgoldbach(int n) {
for(int i=2; i<n; i++)
if (isprime(i) && isprime(minus(n,i)))
return true;
return false;
}
boolean is2goldbach(int n) {
boolean first = false;
for(int i=2; i<=n/2; i++)
if (isprime(i) && isprime(n-i))
if (first)
return true;
else
first = true;
return false;
}
public static void main(String args[]) {
Hofstadter h = new Hofstadter();
int i;
for(i=2; !(h.is2goldbach(i)); i+=2)
;
System.out.println(i);
}
}
There are other ways to refine this program further. It's inefficient to have both isgoldbach and is2goldbach methods as such. It might be better to have a helper method that takes two ints -- n and a start value, and returns not a Boolean value, but an int representing the first found prime (or some dummy value like 0 or -1 if no prime partition is found). Then isgoldbach can call this helper method with 2 as the start value, and isgoldbach2 can call it twice -- first with 2 as the start value, and then with the first return value plus 1 as the second start value. But I won't write that right now.
Mocha Music: "The Need for Speed," "Ratios," "Another Ratio Song"
Here's a link to the Mocha website:
https://www.haplessgenius.com/mocha/
In today's post, I'll cover the songs I performed during Weeks 3-4 at the old charter school. Because Labor Day weekend occurred during this stretch, I sang only three songs during that time.
Our first song is "The Need for Speed":
The Need for Speed:
Life is full of patterns.
They show us the way.
School starts at the same
Time everyday.
The need for speed
To tell us every time
We can go how far
When we build a better mousetrap...
Mousetrap car!
Life is full of patterns.
Circumference follows patterns.
We can use math
To learn about patterns.
The need for speed
To tell us just how fast
We can be a star
When we build a better mousetrap...
Mousetrap car!
10 N=1
20 FOR V=1 TO 2
30 FOR X=1 TO 46
40 READ A,T
50 SOUND A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 13,2,13,2,12,2,12,2,13,4,13,2,13,2
110 DATA 11,4,13,2,13,2,15,8
120 DATA 15,2,15,2,13,2,13,2,11,4,13,4
130 DATA 11,4,11,4,11,6,11,2
140 DATA 12,4,11,4,15,6,15,2
150 DATA 15,2,15,2,12,2,12,2,18,4,18,2,18,2
160 DATA 18,4,15,4,12,4,15,2,15,2
170 DATA 13,2,13,2,13,2,13,2,15,4,15,4
180 DATA 13,4,16,4,18,24
I originally wrote this song in a minor key, and so I use 18EDL to represent minor. The root, third, and fifth are just, but the fourth (Degree 13) and sixth (Degree 11) are both off, although 13/11 itself sounds as an acceptable minor third. As for the seventh, I often left it out when singing, but there are a few changes we can make above where the minor seventh at Degree 10 is acceptable. The rhythm of this song -- 1-and-2-and-3-4-(and) based on eighth notes -- developed organically as I performed the song.
Here are the other two songs that I performed at that point in the school year:
RATIOS:
Ratios are everywhere.
Ratios surround you,
Probably here and there.
For every "for every"
There's a ratio.
Divide at the colon,
And away we go.
That's all there is
To a ratio!
Bridge:
Ratios are everywhere,
But not everything's rational.
Square root of two and pi,
Are proved to be irrational.
ANOTHER RATIO SONG
6th Grade:
What can we do with fractions?
What can we do with fractions?
As everyone knows
We write ratios.
You can do no worse
If you write the 1st one 1st.
In between write dots
Then the 2nd -- that's a lot
We can do with fractions!
7th Grade:
What can we do with fractions?
What can we do with fractions?
We see over there
With ratios we can compare.
The fractions to divide
Flip the 2nd & multiply.
Remember to simplify
And now you know why
We can use fractions!
8th Grade:
What can we do with fractions?
What can we do with fractions?
We can make them decimal
And that is not all.
Another major feat
Is that decimals repeat.
They go on forever.
Know that whenever
We can use fractions!
I consider the songs I sang at the start of the year -- from "The Dren Song" up to "The Need for Speed" -- to be one of my stronger stretches, But these two songs about ratios are weaker efforts on my part. I never sang them again after that week at the old charter school, and they were never requested when I started singing again as a sub (and for good reason). I wrote the tunes of these in my songbook, and so they, along with the rest of that book, are lost forever.
So let's randomize new tunes on Mocha for these two. This will be something that I'll do a lot in these posts -- go back and fill in missing tunes. I might also convert some of these old songs into raps, where only the rhythm is significant and not the melody.
I'll try writing the first song in 12EDL and the second song in 18EDL. I'll write the Mocha code for those tunes in a future post, when I've finished them.
Of course, I wrote so many ratio songs because I knew how important this topic is -- and in fact, I wrote so exactly four years ago today.
Reblogging: Preparing for the Old Charter School
The last time I blogged on June 10th was in 2016. That year, June 10th was Day 180, the last day of school in a certain district. (Again, the calendar in that district was similar to that in my new district, where today is Day 179.)
And so in that post I looked ahead to the following year, when I'd start at the old charter school. (It saddens me to think about how optimistic I was at the time that my year of teaching would go well, now that we know that it didn't go so well at all.)
Here is the reblogged post:
Of course, the biggest thing on my mind now is my new charter middle school and the classes that I will be teaching in the fall. So here are some things I know so far about my first teaching assignment:
First, I know that I will be teaching all three middle school grades -- sixth, seventh, and eighth. Let's look at the Common Core standards for these three grades:
Grade 6-7 (same strands for each):
Ratios and Proportional Relationships
The Number System
Expressions & Equations
Geometry
Statistics & Probability
Grade 8:
The Number System
Expressions & Equations
Functions
Geometry
Statistics & Probability
On this blog, I will write only about the eighth grade class. So let's look at some of those eighth grade standards in more detail:
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.
CCSS.MATH.CONTENT.8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.
CCSS.MATH.CONTENT.8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
CCSS.MATH.CONTENT.8.EE.C.7
Solve linear equations in one variable.
CCSS.MATH.CONTENT.8.F.A.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
CCSS.MATH.CONTENT.8.F.B.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
CCSS.MATH.CONTENT.8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations.
CCSS.MATH.CONTENT.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
CCSS.MATH.CONTENT.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
CCSS.MATH.CONTENT.8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
The most important concept learned throughout middle school is fractions -- or to be more precise, it's ratios. Students learn arithmetic with fractions in sixth grade, the field of rational numbers in the seventh grade, and the application of ratios to similarity in the eighth grade. Indeed, we've seen one Common Core curriculum refer to middle school math as "A Story of Ratios."
This blog is named "Geometry, Common Core Style." Here Geometry refers to the high school class of that name, usually taken between Algebra I and II. When I begin writing about my eighth grade class, this will become a misnomer, since my focus will be on ratios, not geometry. Still, we see that eighth grade is the year when reflections, rotations, translations, and dilations are introduced, so there will still be some geometry. Besides, we found out yesterday that there's a lack of middle school blogs in the MTBoS, so I'm happy that I'll be able to contribute to the MTBoS from that perspective.
In the fall, I will be posting only three days a week. This fits my new school's block schedule. Indeed, this is another reason that I chose to participate in MTBoS30 this year despite writing only 21 posts in May due to skipping weekends -- next year, I'll write only about a dozen posts in May and be even further away from 30 (that is, if there even is an MTBoS30 challenge next May).
Now juggling three preps will certainly be a challenge for any teacher, especially a new teacher. I am already looking ahead to next year's school calendar to figure out how I will avoid having to grade tests for all three preps at the same time.
Last year, the first day of school at my new school was the third Tuesday in August, and so I'm anticipating that Tuesday, August 16th, will be the first day of school. Today is the last day of school at both my old and new schools, and so we can pencil in Friday, June 9th, 2017, as the last day of school next year. Because my new charter school is associated with the LAUSD, the school calendar is very similar to the district calendar.
When considering how to teach my new eighth grade class, I especially want to look at the Statistics & Probability strand. I mentioned that last week was the final math class I subbed for this year, and that day the students had to take a quiz -- eighth graders on statistics, seventh graders on geometry. I wrote about how much trouble the eighth graders having with their stats quiz:
"I observe how much these students struggle on this quiz -- especially the question where they must identify and write an equation for a trend line. (Considering how much the eighth graders struggled with stats, maybe it's a good thing that the seventh graders didn't make it to their own stats chapter!)"
This is a good time to segue into our traditionalists-labeled topic of the week, because some traditionalists have written about teaching stats in middle and high school.
Conclusion: Traditionalists
Yes -- returning to 2020, let's see what the traditionalists have to say right now. Specifically, we're looking at our two main traditionalists:
https://traditionalmath.wordpress.com/2020/06/07/nctm-dept/
A recent article proclaims with great fanfare two new publications of the National Council of Teachers of Mathematics (NCTM). One is for elementary teachers and the other for middle school teachers. In the words of the article, the publications address “how to help all students view themselves as “capable learners and doers” of math.”
Something tells me that the report isn’t strong on students learning their addition/subtraction and times tables by heart. And it probably is not big on practice, or worked examples, and scaffolded problems, but I’m just guessing here. I could be completely wrong.
Of course, Barry Garelick knows that he's probably right. The NCTM is the common denominator between the U of Chicago text and the Common Core, which explains why some Common Core Geometry standards appear in our text despite predating the Core by two decades. And once again, I'm all for memorizing addition and multiplication tables, just not in the way that the traditionalists insist on teaching.The main commenter is -- well, you already know:
SteveH:
“All stakeholders must examine beliefs about who is capable of doing and understanding mathematics, disrupt existing inequitable practices and catalyze change toward creating a just, equitable and inclusive system in early childhood and elementary mathematics.””
As stakeholder parents who have been through this process, we must catalyze change toward a just, equitable and inclusive system that gives all students what we parents had to do at home and with tutors to create all of their STEM-ready traditional math high school students they hide their incompetence behind.
And you already know what I have to say about tutors, traditional math, and blank papers.Since I mentioned traditionalists in this post, I might as well bring up George Floyd again. One teacher who has tweeted a lot about the Floyd situation is Eugenia Cheng:
https://twitter.com/DrEugeniaCheng?ref_src=twsrc%5Egoogle%7Ctwcamp%5Eserp%7Ctwgr%5Eauthor
I don't like to repost tweets that much, so I'll only repost a little here. Her first tweet on June 1st is:
Eugenia Cheng:
Black lives matter. If you reply "All lives matter" you are missing the point. Nobody is saying all lives don't matter. The point is that some of us believe we currently have a system in which black lives are not valued as much as other lives, and that this needs to change.
Indeed, she writes about "black lives matter" and "all lives matter" in her third book on logic. And her last post that day was:
Eugenia Cheng:
Besides which, that's just what I believe in as a basic principle of humanity. It doesn't mean you can't acknowledge your own struggles and sufferings. We all suffer things in the context of our own limited lives.
Oh, and since I was at Cheng's Twitter anyway, I notice that she's announced that her fourth book will be coming out soon. It's called x + y -- and just as she writes much about race in her third book, this fourth book will have much to do with gender.
Her fourth book -- which I'll probably discuss as our next side-along reading book -- will come out just after the first day of school -- assuming, of course, that schools reopen as scheduled.
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