Stewart Chapter 12: Great Sky River

Table of Contents

1. Introduction & Return of Blaugust???
2. Calculating the Cosmos Chapter 12: Great Sky River
3. Lemay Chapter 13 Part 1: Creating User Interfaces with the awt
4. More on Blaugust, Java, and Card Layouts
5. Other Blaugust Participants
6. Shapelore Learning 14-6: Laws of Arrows
7. Music: "Count the Ways," "Linear or Not," "Patterns," "Same Sign, Add and Keep"
8. A Rapoport Math Problem
9. Rapoport Geometry Problems from This Week
10. Conclusion

Introduction & Return of Blaugust???

Today is August 6th. Hey, it's August -- and you know what that means! This is the time when we visit math teacher Shelli's blog to find out whether there's a Blaugust challenge this year.

Well, Shelli certainly doesn't disappoint:

http://statteacher.blogspot.com/2020/08/mtbosblaugust-living-in-limbo.html

As you can see, Blaugust is back, but there are a few differences. There is no sign-up list, nor is there a list of suggested Blaugust topics. Just as Shelli writes in her first Blaugust post, we're all "living in limbo" during the coronavirus pandemic as we wait to see whether our schools will reopen. And so this year's challenge reflects that -- no sign-up list, no topic list. Just get out there and post about whatever's on your mind.

Then again, it wasn't as if I would have signed the official Blaugust list anyway. To me, that honor is only for real teachers, not subs like me. And that's even more so this year -- as my districts remain online for the foreseeable future, I won't be in a classroom at all in August, or even September. So I'm the last blogger whose name belong on a sign-up list. But of course, I'll still be posting and even responding to some of the prompts, just not as an official participant.

Also, I won't be posting as often for this Blaugust either. In past years, I would post once or twice a week until the first day of school, when I'd post every school day. But since my schools aren't reopening, I'm sticking to my summer blogging schedule, and that means fewer Blaugust posts.

Even though there is no topic list this year, I can still use Shelli's list from last year. And since today is August 6th, I choose the sixth topic from her list. I've never posted on the sixth before, so this topic is a new one for me:

Share a lesson that uses tech (Desmos AB, Geogebra, applets, etc)

Hey, that's the perfect topic for this year -- every lesson is going to use tech this year, as long as my counties remain on the watch list and the schools remain closed.

I think back to my days at the old charter middle school, where we definitely used technology. The main software we used back then was IXL. For each lesson, each student received a score, 0-100, and this technology was adaptive. If students answered a question correctly, their score would increase and they would get a harder question. But if they answered a question wrong, their score would decrease and they would get an easier question. To reach 100, or mastery, required a minimum of 28 questions (if they were all correct), including nine in the "challenge zone."

We also used Study Island software from time to time. And finally, as the SBAC approached, we visited the practice SBAC website. That's the closest we ever came to using Desmos, one of the apps mentioned in Shelli's prompt above -- the SBAC online calculator was powered by Desmos.

In my current pair of districts, both will have live synchronous instruction on Zoom on Mondays at least, and then asynchronous instruction the rest of the week. There's certainly room for apps like IXL during the asynchronous part of the week. Indeed, teachers can tell whether the students have logged in, as well as what their 0-100 score is.

Of course, we should be wary of naively taking the IXL score and mapping it to grades -- that is, an IXL score from 90-100 is an A, from 80-89 is a B, and so on. After all, if many students struggle to understand the material -- and this is likely given that the Monday instruction -- then lots of students will get IXL scores below 60. I'd say that if most of the IXL scores are high, then we can count these as grades, but if they're mostly low, then we should try reteaching the material first. IXL will remember the scores, so they can start from where they left off to improve the score.

The last thing Shelli mentions in her prompt is "applets" -- does this mean Java applets? Well, I've never used Java applets in any class that I've taught -- but learning how to code in Java is, after all, one of my summer projects.

So instead of linking and describing my favorite Java applet, why don't I write my own applet -- one that can be used in the classroom -- and post it here to satisfy this Blaugust prompt?

And indeed, that's exactly what I'm going to do -- later in this post, when we reach the Java section.

Calculating the Cosmos Chapter 12: Great Sky River


Chapter 12 of Ian Stewart's Calculating the Cosmos is called "Great Sky River." As usual, it begins with a quote:

See yonder, lo, the Galaxy,
Which men call the Milky Way,
For it is white.

-- Geoffrey Chaucer, The House of Fame

And the proper chapter begins:

"In ancient times there was no street lighting beyond the occasional torch or fire, and it was virtually impossible not to notice one amazing feature of the heavens."

And this feature is our galaxy, the Milky Way -- the topic of this chapter. It isn't the only galaxy in our universe, of course:

"A few are visible to a keen eye: the tenth-century Persian astronomer Abd al-Rahman al-Sufi described the Andromeda Galaxy as a small cloud, and in 964 he included the Large Magellanic Cloud in his Book of Fixed Stars."

Here Stewart includes his first pictures of the chapter. The first is the Milky Way over Summit Lake, West Virginia. On the left is a galaxy seen edge-on, with central bulge. On the right is an artist's impression of our own Milky Way Galaxy.

One of the first to study the Milky Way was American astronomer Harlow Shapley. About 100 years ago, he studied the nebula M31, which is outside the Milky Way:

"He argued that if M31 is like the Milky Way, it must be about 100 million light years away, a distance considered too big to be credible."

As it turned out, M31 is about 2.5 million light years away. Another astronomer who studied the shape of galaxies was Edwin Hubble (for whom the famous telescope was named):

"He distinguished four main types: elliptical galaxies, spiral galaxies, barred spirals, and irregular galaxies."

The author shows us these shapes in a picture -- Hubble's tuning fork classification for galaxies. It begins with the elliptical nebulae (with codes starting with E) and branches out to the normal spirals (starting with S) and barred spirals (starting with SB). Irregular galaxies are omitted.

This raises the question -- why are the galaxies in this shapes? We might use a computer model:

"Since the constituent stars of a galaxy are fairly thinly spread, and don't move at near light speed, Newtonian gravity ought to be sufficiently accurate. Many theories of that type have been studied."

And one of these spiral theories was suggested by Swedish astronomer Bertil Lindblad:

"He proposed that stars are repeatedly recycled through the arms. In 1964 Chia-Chiao Lin and Frank Shu suggested that the arms are density waves, in which stars pile up temporarily."

Stewart's next picture is a graph. It shows how stars orbiting in ellipses can create a spiral density wave -- here, a barred spiral.

A "barred spiral" is a spiral with a bar across the middle:

"A typical example is NGC 1365. One way to approach galaxy dynamics is to set up an n-body simulation with large values of n, modelling how each star moves in response to the gravitational pull of all the others."

To put it simply, the "bar" rotates, and gravity cases the other stars to move in a spiral. But the calculations to prove this are quite complex:

"Mathematically, this is a 150,000-body simulation in a fixed but rotating gravitational landscape. Three of the Lagrange points, L3, L4, and L5 are stable."

These points appear in the author's picture. On the left are Lagrange points for the rotating bar. On the right are stable (S) and unstable (U) manifolds near L1.

This point, L1, is called a saddle point (as opposed to a stable valley or unstable hill). But the true picture is more complicated, because we must consider extra dimensions again:

"Here the two dimensions of position, shown directly in the picture, must be complemented by two more dimensions of velocity."

This explains the spiral galaxies, but not necessarily the elliptical galaxies:

"However, real n-body dynamics doesn't yield elliptical orbits, because all the bodies perturb each other, so the proposed pattern doesn't really make sense, unless it's a reasonable approximation to something that does."

Stewart includes many pictures of galaxy shapes here. First, on the left is a projection of the n-body problem on the galactic plane. On the right is a spiral pattern with the unstable invariant manifolds emanating from L1 and L2. Next are many graphs of instantaneous positions of particles belonging in different populations of the regular and chaotic orbits (black dots) superimposed on the backbone of the galaxy on the plane of rotation (grey background).

Finally, there is a set of images showing ring and spiral-arm morphologies. In the top row are four galaxies, namely NGC 1365, NCG 2665, ESO 325-28, and ESO 507-16. In the second row are schematic plots of these galaxies, bringing out the spiral and ring structures. In the third row are examples from stable/unstable manifold calculations with similar morphology, projected in roughly the same way as the observed galaxy or schematic plot. In the fourth row is a face-on view of these manifolds with the bar along the x-axis.

Once again, chaos theory explains some galaxy shapes:

"Force patterns artificially, and use risk getting nonsense. There's further confirmation that sticky chaos plays a role in the formation of spirals in barred spiral galaxies."

We move on to the spectroscope, which is used to determine how fast galaxies are moving:

"Hydrogen is the commonest element in the universe, so the hydrogen-alpha line is often prominent. It's even possible -- for galaxies not too far away -- to measure the rotation speed at different distances from the center of the galaxy."

This appears in the author's final graphs of this chapter. On the left is the simple rotation curve predicted by Newtonian laws. On the right are the observed rotation curves for six galaxies, which are more complex:

"It's hard to measure the mass distribution directly, but one prediction is independent of such considerations: how the rotation curve behaves for large enough radii."

And Stewart ends the chapter by describing the problem with these graphs:

"For comparison, the right-hand picture shows observed rotation curves for six galaxies, one of them ours. Instead of decaying, the rotation speed grows, and then stays roughly constant. Oops."

Lemay Chapter 13 Part 1: Creating User Interfaces with the awt

Here's the link to today's lesson:

http://101.lv/learn/Java/ch13.htm

Lesson 13 of Laura Lemay's Teach Yourself Java in 21 Days! is called "Creating User Interfaces with the awt." Here's how this chapter begins:

For the past five days you've concentrated on creating applets that do very simple things: display text, play an animation or a sound, or interact with the user. When you get past that point, however, you may want to start creating more complex applets that behave like real applications embedded in a Web page-applets that start to look like real GUI applications with buttons, menus, text fields, and other elements.

It's this sort of real work in Java applets and applications for which Java's Abstract Windowing Toolkit, or awt, was designed. You've actually been using the awt all along, as you might have guessed from the classes you've been importing. The Applet class and most of the classes you've been using this week are all integral parts of the awt.

Lemay does give us a warning here:

This is by far the most complex lesson so far, and it's a long chapter as well. There's a lot to cover and a lot of code to go through today, so if it starts becoming overwhelming, you might want to take two days (or more) for this one.

OK, Lemay, that's exactly what I'll do. It's been a while since I've split one of her chapters into two of my posts, but I'll follow her suggestion and split this lesson. This appears to be an important lesson, and if my goal is to learn Java well enough to get hired in a non-teaching job during the coronavirus era, then I need to take the time to master these chapters well.

The author explains:

The basic idea behind the awt is that a graphical Java program is a set of nested components, starting from the outermost window all the way down to the smallest UI component. Components can include things you can actually see on the screen, such as windows, menu bars, buttons, and text fields, and they can also include containers, which in turn can contain other components. Figure 13.1 shows how a sample page in a Java browser might include several different components, all of which are managed through the awt.

As usual, I don't post any figures here. You'll have to click the link above to see Figure 13.1.

The first example of a UI component is a label. Here is a code snippet that produces a label:

import java.awt.*;

public class LabelTest extends java.applet.Applet {

  public void init() {
    setFont(new Font ("Helvetica", Font.BOLD, 14));
    setLayout(new GridLayout(3,1));
    add(new Label("aligned left", Label.LEFT));
    add(new Label("aligned center", Label.CENTER));
    add(new Label("aligned right", Label.RIGHT));
  }
}

When I tried entering this into my compiler, there was a message that the class LabelTest already exists somehow. It turns out that Lemay called one of her classes from earlier LabelTest. So I just replaced the old one with this one, since it's actually about Labels.

Ah, so now I see how these labels work. When I change the size of the applet window, the labels are automatically aligned to the left, center, or right.

The next example is a button -- something that the user can click on. Oops -- I notice that Lemay forgets to import java.awt.* here, so I'll do so myself.

import java.awt.*;

public class ButtonTest extends java.applet.Applet {

  public void init() {
    add(new Button("Rewind"));
    add(new Button("Play"));
    add(new Button("Fast Forward"));
    add(new Button("Stop"));
  }
}

The next example is a check box:

import java.awt.*;

public class CheckboxTest extends java.applet.Applet {

  public void init() {
    setLayout(new FlowLayout(FlowLayout.LEFT));
    add(new Checkbox("Shoes"));
    add(new Checkbox("Socks"));
    add(new Checkbox("Pants"));
    add(new Checkbox("Underwear", null, true));
    add(new Checkbox("Shirt"));
  }

}

The next example is a special kind of check box often called a radio button. Here only one box in the group (here called cbg) can be checked at a time:

import java.awt.*;

public class CheckboxGroupTest extends java.applet.Applet {

  public void init() {
    setLayout(new FlowLayout(FlowLayout.LEFT));
    CheckboxGroup cbg = new CheckboxGroup();

    add(new Checkbox("Red", cbg, false));
    add(new Checkbox("Blue", cbg, false));
    add(new Checkbox("Yellow", cbg, false));
    add(new Checkbox("Green", cbg, true));
    add(new Checkbox("Orange", cbg, false));
    add(new Checkbox("Purple", cbg, false));
  }

}  

The next example is a choice menu:

import java.awt.*;

public class ChoiceTest extends java.applet.Applet {

  public void init() {
    Choice c = new Choice();

    c.addItem("Apples");
    c.addItem("Oranges");
    c.addItem("Strawberries");
    c.addItem("Blueberries");
    c.addItem("Bananas");

    add(c);
  }
}

The last example of a UI component is a text field. Lemay gives several examples here but doesn't include them as a full class, so let me do so here. The last one is a password field -- it's hidden so that no one looking over our shoulder can see our password:

import java.awt.*;

public class TextFieldTest extends java.applet.Applet {

  public void init() {
    add(new Label("Enter your Name"));
    add(new TextField("your name here", 45));
    add(new Label("Enter your phone number"));
    add(new TextField(12));
    add(new Label("Enter your password"));
    TextField t = new TextField(20);
    t.setEchoCharacter('*');
    add(t);
  }
}

This works, although my compiler gives the warning setEchoCharacter is deprecated. I wonder why obscuring my password is considered deprecated.

At this point the author moves on to layouts, which determine how the components are displayed on our screen. The first example of a layout is the flow layout:

import java.awt.*;

public class FlowLayoutTest extends java.applet.Applet {

  public void init() {
    setLayout(new FlowLayout());
    add(new Button("One"));
    add(new Button("Two"));
    add(new Button("Three"));
    add(new Button("Four"));
    add(new Button("Five"));
    add(new Button("Six"));
  }
}

Again, I can see how the buttons "flow" across the screen as I change the window size -- at minimum size only two buttons fit per row, and as I enlarge the window, eventually all buttons fit on one line.

The next example is a grid layout. Unfortunately, Lemay omits a parenthesis near the end of the line setLayout, so I fix it.

import java.awt.*;

public class GridLayoutTest extends java.applet.Applet {

  public void init() {
    setLayout(new GridLayout(3,2));
    add(new Button("One"));
    add(new Button("Two"));
    add(new Button("Three"));
    add(new Button("Four"));
    add(new Button("Five"));
    add(new Button("Six"));
  }
}

Now there are always two buttons per row -- and the buttons get bigger when I enlarge the window.

The next example is a border layout:

import java.awt.*;

public class BorderLayoutTest extends java.applet.Applet {

  public void init() {
    setLayout(new BorderLayout());
    add("North", new Button("One"));
    add("East", new Button("Two"));
    add("South", new Button("Three"));
    add("West", new Button("Four"));
    add("Center", new Button("Five"));
    add(new Button("Six"));
  }
}

I wonder whether this is supposed to happen -- button #6 is added to the center, and so button #5 is no longer visible. Lemay doesn't explain this fully.

The next example is a card layout. Again, the author doesn't write a complete class for this one, so I try to write the class myself.

But then I get several errors. Some of them are because Lemay omits some semicolons at the ends of the lines. But unfortunately, the following lines produce fatal compile-time errors:

    // move around
    show(this, "second"); //go to the card named "second"
    show(this, "third");   //go to the card named "third"
    previous(this);       //go back to the second card
    first(this);          // got to the first card

According to my compiler, the method show is supposed to take just one argument -- and that argument should be a boolean, not a CardLayoutTest (the name of my class, which is what the word this refers to) or a string. And the methods previous and first aren't defined at all.

The last example is a grid bag layout -- which, according to Lemay, are "extremely complicated." I'm sure whether I really want to tackle these after my card layout failed.

How about this -- I'll just skip to the first listing, which contains the entire layout. Then I'll wait to see whether my compiler produces any errors:

Listing 13.1. The panel with the final grid bag layout.

 1:import java.awt.*;
 2:
 3:public class GridBagTestFinal extends java.applet.Applet {
 4:
 5:  void buildConstraints(GridBagConstraints gbc, int gx, int gy, 
 6:      int gw, int gh,
 7:      int wx, int wy) {
 8:      gbc.gridx = gx;
 9:      gbc.gridy = gy;
10:      gbc.gridwidth = gw;
11:      gbc.gridheight = gh;
12:      gbc.weightx = wx;
13:      gbc.weighty = wy;
14:  }
15:
16:  public void init() {
17:      GridBagLayout gridbag = new GridBagLayout();
18:      GridBagConstraints constraints = new GridBagConstraints();
19:      setLayout(gridbag);
20:      
21:      // Name label
22:      buildConstraints(constraints, 0, 0, 1, 1, 10, 40);
23:      constraints.fill = GridBagConstraints.NONE;
24:      constraints.anchor = GridBagConstraints.EAST;
25:      Label label1 = new Label("Name:", Label.LEFT);
26:      gridbag.setConstraints(label1, constraints);
27:      add(label1);
28:
29:      // Name text field
30:      buildConstraints(constraints, 1, 0, 1, 1, 90, 0);
31:      constraints.fill = GridBagConstraints.HORIZONTAL;
32:      TextField tfname = new TextField();
33:      gridbag.setConstraints(tfname, constraints);
34:      add(tfname);
35:
36:      // password label
37:      buildConstraints(constraints, 0, 1, 1, 1, 0, 40);
38:      constraints.fill = GridBagConstraints.NONE;
39:      constraints.anchor = GridBagConstraints.EAST;
40:      Label label2 = new Label("Password:", Label.LEFT);
41:      gridbag.setConstraints(label2, constraints);
42:      add(label2);
43:
44:      // password text field
45:      buildConstraints(constraints, 1, 1, 1, 1, 0, 0);
46:      constraints.fill = GridBagConstraints.HORIZONTAL;
47:      TextField tfpass = new TextField();
48:      tfpass.setEchoCharacter('*');
49:      gridbag.setConstraints(tfpass, constraints);
50:      add(tfpass);
51:
52:      // OK Button
53:      buildConstraints(constraints, 0, 2, 2, 1, 0, 20);
54:      constraints.fill = GridBagConstraints.NONE;
55:      constraints.anchor = GridBagConstraints.CENTER;
56:      Button okb = new Button("OK");
57:      gridbag.setConstraints(okb, constraints);
58:      add(okb);
59:  }
60:}

Well, except for the same warning message about the password field from earlier, it works. I'll cut and paste in some steps about creating this grid bag layout:

Step One: Design the Grid

Step Two: Create the Grid in Java

Step Three: Determine the Proportions

Step Four: Add and Arrange the Components

Step Five: Futz with It

I'll have to practice these layout more in order to get the hang of it. For now, we read:

If you stopped reading today's lesson right now, you could go out and create an applet that had lots of little UI components, nicely laid out on the screen with the proper layout manager, gap, and insets. If you did stop right here, however, your applet would be really dull, because none of your UI components would actually do anything when they were pressed, typed into, or selected.

For your UI components to do something when they are activated, you need to hook up the UI's action with an operation. Actions are a form of event, and testing for an action by a UI component involves event management. Everything you learned yesterday about events will come in handy here.

And of course, "yesterday" doesn't really mean yesterday. We learn that all of the UI components that we've looked at so far can generate actions:

• Buttons create actions when they are pressed and released with the mouse, and a button's extra argument is the label string of that button.
• Check boxes, both exclusive and nonexclusive, generate actions when a box is checked. The extra argument is always true.
• Choice menus generate an action when a menu item is selected, and the extra argument is the label string of that item.
• Text fields create actions when the user presses Return or Enter inside that text field. Note that if the user tabs to a different text field or uses the mouse to change the input focus, an action is not generated. Pressing Return or Enter is the only thing that triggers the action.

The second listing for today tests for the button actions:

Listing 13.2. The ButtonActionsTest applet.

 1:import java.awt.*;
 2:
 3:public class ButtonActionsTest extends java.applet.Applet {
 4:
 5:  public void init() {
 6:    setBackground(Color.white);
 7:
 8:    add(new Button("Red"));
 9:    add(new Button("Blue"));
10:    add(new Button("Green"));
11:    add(new Button("White"));
12:    add(new Button("Black"));
13:  }
14:
15:  public boolean action(Event evt, Object arg) {
16:    if (evt.target instanceof Button) {
17:      changeColor((String)arg);
18:      return true;
19:    } else return false;
20:  }
21:
22:  void changeColor(String bname) {
23:    if (bname.equals("Red")) setBackground(Color.red);
24:    else if (bname.equals("Blue")) setBackground(Color.blue);
25:    else if (bname.equals("Green")) setBackground(Color.green);
26:    else if (bname.equals("White")) setBackground(Color.white);
27:    else setBackground(Color.black);
28:
29:    repaint();
30:  }
31:}

This chapter contains four official listings and we've done two of them, so we're halfway done. Thus this is a great dividing point for the lesson. We'll continue this chapter in our next post.

More on Blaugust, Java, and Card Layouts

It's a shame that I couldn't get the card layout to work on my compiler. Recall from above that for today's Blaugust/Java project, I'd like to create a math lesson that uses applets. And one layout that works well for education is a card layout.

When I was a young student, I often watched a local PBS channel that aired educational shows like Guitar with Frederick Noad, Homework Hotline, and Solve It (all of which I've described in previous blog posts). Well, another show I used to watch on that channel was -- actually, it wasn't a show as much as an online college class for teachers, "Introduction to Computers in Education." Even though I had no idea at the time that I wanted to become a teacher, I watched the show anyway because I enjoyed learning about computers.

The presenter was Professor Peter Desberg. And one of his favorite lessons was on a card layout, perhaps on (a 1990's version of) PowerPoint. The lesson would consist of four cards:

• The first card teaches a simple lesson, say on multiplication.
• The second card asks a question on this lesson, such as "What is 2 times 3?" Students are given two choices, 5 or 6.
• If the students choose 5, they reach the third card, "Sorry, you added instead of multiplied." If they click again, then they return to the second card.
• If the students choose 6, then they reach the fourth card, with a congratulatory message.

I'd love to be able to implement Desberg's card layout here on Java. But unfortunately, Laura Lemay's card layout methods don't work on my compiler. Then again, all this means is that it will be a coding challenge for me to get this to work -- and what is Blaugust, after all, if not a blogging challenge? I'll learn more about Java if I can figure out card layouts for myself.

Let's try writing a card layout applet for the addition of vectors. We'll follow the Desberg format -- one card teaches, the second card asks a question, one card has the wrong answer, and the last card has the right answer.

This is what I ultimately came up with:

import java.awt.*;

public class VectorAddingCards extends java.applet.Applet {

public void init() {
setLayout(new CardLayout());
Panel one = new Panel();
add(one,"first");
    setFont(new Font ("Helvetica", Font.BOLD, 14));
    one.setLayout(new FlowLayout());
    one.add(new Label("Vector Addition Theorem:"));
    one.add(new Label("The sum of the vectors (a,b) and (c,d) is the vector"));
    one.add(new Label("(a+c,b+d)."));
    one.add(new Button("OK"));
Panel two = new Panel();
add(two,"second");
two.add(new Label("Add the vectors (300,15) + (100,212)."));
CheckboxGroup cbg = new CheckboxGroup();
two.add(new Checkbox("(512,115)",cbg,false));
two.add(new Checkbox("(400,227)",cbg,false));
Panel three = new Panel();
add(three,"third");
three.add(new Label("Make sure you add x- and y-coordinates separately."));
three.add(new Button("OK"));
Panel four = new Panel();
add(four,"fourth");
four.add(new Label("Congratulations! You answered correctly."));
}

public boolean action(Event evt, Object arg) {
CardLayout cl = (CardLayout)getLayout();
if (evt.target instanceof Button) {
      cl.show(this,"second");
      return true;
    }
    if (evt.target instanceof Checkbox) {
    if (((Checkbox)(evt.target)).getLabel() == "(512,115)") {
    cl.show(this, "third");
    return true;
    }
    else {
    cl.show(this, "fourth");
    return true;
    }
    }
    return false;
  }
}

As you can see, I figured out how to get show to work. Lemay never explains that show is a method of the CardLayout class, so we need a way to get an object in order to use it. Check the above version of action to see how I was able to get a CardLayout object.

Other Blaugust Participants

Shelli herself has made five posts so far in Blaugust. Here's a link to yesterday, her most recent post:

http://statteacher.blogspot.com/2020/08/mtbosblaugust-junior-calendar.html

It's time to get serious about this planning thing, y'all!

Our district decided on Monday night to start back in distance learning until the numbers decline in our county.  I know it was a really tough decision and honestly, it was a no-win situation as there wasn't a decision would make everyone happy.  However, that means I really need to get myself into gear because teachers report back next week and I've gotten nothing physically done for school! Eeeekkkk!

So apparently, Shelli's county (wherever that is) is just like the Southern California counties -- there are still so many cases that online instruction must continue.

In past years, I would refer to this time of year as "Sara(h) season." That's because of the two big math teacher bloggers, Sarah Carter and Sara van der Werf. Both of them are famous for their first day of school activities, and so many other Blaugust participants write about how they were able to implement the Sara(h) activities in their classes.

Of course, everything's different this year. The Sara(h) activities work best for face-to-face instruction and aren't necessarily suitable for online instruction.

Well, Sarah Carter wrote that she'll be a full Blaugust participant this year -- even though she's often quoted during the month, she hasn't actually participated in Blaugust since the year I taught at the old charter school, four years ago. She even set up a calendar for herself with a goal of blogging for all 31 days in August. No, she won't meet that lofty goal -- she hasn't posted since Monday the 3rd. So here's a link to her August 3rd post:

I started writing this as a blog post about how I've been working on cultivating healthy and productive habits in my life since school shut down in March. It turned into a rambling reflection on my morning walk and what I can take away from it and apply to the upcoming school year. While it's not exactly what I was going for, I guess it still means I have a blog post idea for tomorrow which will be Day 4 of the MTBoS Blaugust Blogging Challenge.


And of course, she hasn't made that Day 4 Blaugust post yet.

Sara van der Werf, meanwhile, hasn't posted in August at all, but she does link to an online seminar that is ongoing:

https://www.saravanderwerf.com/resources-from-my-august-3rd-building-math-minds-virtual-math-summit-session/

NOTE 8/2/20: Many of you have reached out to me asking for a similar conference if you are NOT Elementary. Though there are great Secondary Conferences – in my opinion the ‘Building Math Minds Virtual Math Conference’ is for all K-12 Math teachers, not just Elementary. Most of us secondary teachers were never taught how to build numeracy in our students. I became a much, much better (secondary) math teacher when I started getting smart about Elementary Math content and pedagogy 15 years ago. I still have a lot to learn. Secondary teachers – you need this almost more than any other math conference out there (except for info for the digital era) . Join me! Christina recruits great people to learn from.

The last Blaugust participant we'll look at today is Beth Ferguson, aka "Algebra's Friend":

http://algebrasfriend.blogspot.com/2020/08/parent-function-ideas.html

Wow -- Ferguson was a regular Blaugust participant last year, but somehow this year I missed most of her posts and never quoted any of them on the blog. (Once again, I linked to her blog four years ago when I was at the old charter school.) Well, I'm correcting that error this year and highlighting Ferguson's posts for Blaugust:

I'm feeling inspired by my teacher friends on Twitter ... they are the BEST of the BEST!

Anyway ... someone on Twitter was asking about Parent Functions.  My activities are "old-school" ... but maybe some of these ideas will inspire creativity in lesson planning.

One of my goals in studying parent functions is to engage students in identifying key characteristics, and then comparing/contrasting the various functions to build fluency with them.

So first we build a parent function "notebook."
Then we identify key characteristics.

Next we compare/contrast.

And last we explore some questions that hopefully get students to thinking!

This appears to be for an Algebra II class. The concept of a "parent function" is a relatively new idea that wasn't taught back when I was a young Algebra II student -- but it does follow directly from Common Core Geometry and transformations.

For example, f (x) = x^2 is considered to be a parent function to all parabolas. That's because given any parabola, there exists a sequence of translations, rotations, reflections, and dilations such that the image of f (x) = x^2 under the composite of these transformations is the given parabola. The image under the mirror y = x is a special case -- this isn't a function unless we restrict it to one branch. The resulting square root function is considered to be a parent function in its own right.

For some functions such as the absolute value function  f (x) = |x|, it's helpful to consider "vertical stretching" or "horizontal stretching" as a transformation -- for example, vertical stretching by a factor of 2 produces the function g(x) = 2|x|. This isn't a Common Core Geometry transformation, but it's important in Algebra II classes.

On Ferguson's worksheet she lists f (x) = mx + b as a parent function, but this is a bit too general to be considered a parent function by most teachers. The fact that she writes "cuts Quadrants I and III at a 45-degree angle" gives away what the true parent function is -- the identity f (x) = x. The composite a vertical shift by m followed by a vertical translation by b maps f (x) = x to the general linear function.

Even though she doesn't list them, the trig functions like f (x) = sin x are often considered to be parent functions as well.

Ferguson's assignment includes a "notebook" and "concept cards" -- these are definitely intended for face-to-face instruction, though I suppose that they can be printed at home by students in those districts with full distance learning.

Shapelore Learning 14-6: Laws of Arrows

Lesson 14-6 of the U of Chicago text is called "Properties of Vectors." I know that we just saw this in the Java language, but now we're redoing the lesson in the Anglish language -- words derived from Old English, rather than French, Latin, or Greek (which students often struggle with). In my last post, I wrote that a good Anglish word for "vector" is arrow. Meanwhile, the use of law to mean "property" has already been established with the "Laws of Exponents."

Let's do the main lesson in Plain English -- as usual, many mathematical words will be written in Anglish, while non-mathematical and simpler mathematical words will remain in English:

Any arrow can be placed so that its starting point is the zero. (I haven't decided yet what to do about the word "origin." The word "zero" is Arabic, so it shouldn't be in Anglish. But more students know what "zero" is, so we can use it in our Plain English here. Perhaps we can it the "zero point" -- yes, we're combining the Arabic "zero" with the French "point," but both words are simple.)

If its ending point is (a, b), the arrow named is (a, b). The breadthwise deal of (a, b) is a; the upright deal is b. (OK, so breadthwise and upright are the suggested words listed on the Anglish website for "horizontal" and "vertical." Well, I don't like breadthwise here, since we're already using breadth to mean the measure of an angle. Upright is better but still troublesome -- it's not the up that's the problem here, but the right. Meanwhile, deal is the suggested word for "component." Here I would prefer something like "right part" and "up part," even though "part" is French. Using right and up also reminds us that "left" and "down" are negative.)

This is the tidy twosome arecking of an arrow. (On the Anglish website, the word for "ordered" is tidy and the word for "pair" is twosome. The word "pair" is simple enough to be kept, though I'm not sure whether to keep "ordered" or not -- indeed, we need to come up with some definite terms for the coordinate plane soon, though I admit tidy twosome is fun to say. Arecking is the suggested word for "description," but this is best thought of as a non-mathematical word that we don't change. So at best we should say "tidy pair description of an arrow.)

The arrow (a, b) can be interpreted as the adding of a right force a and an up force b. (The text uses the word "resultant" here, which is rare even for this text. Instead, we use either "adding" or "sum," even though neither word is Anglish.

For instance, if a bullet is fired rightward at a starting speed of 200 meters per second, gravity will pull it down about 5 meters the first second, and its location after one second could be described by the arrow (200, -5). (OK, now I see how awkward it is to use the word up to mean "vertical" -- the first example of a vertical force is gravity, which is clearly a down force. How about this -- let's use updown to mean "vertical" and rightleft to mean "horizontal.")

When arrows are described as tidy pairs, they can be easily added.

Arrow Adding Provedsaying ("Theorem," even though "prove" is French):
The adding of the arrows (a, b) and (c, d) is the arrow (a + b, c + d).

Proof:
The idea is to use the akinside ("parallelogram," but fourakinside might be better if we prefer to save akinside for "trapezoid") law. We let O = (0, 0), M = (a, b), N = (c, d), and P = (a + c, b + d). To show that OP is the adding of OM and ON, it needs to be shown that OMPN is an akinside. For this, all that is needed is to show that MP and ON are worthsame ("equal") arrows. They are worthsame if they have the same way (slope) and greatness (length). This is left to you in Question 14.

The Arrow Adding Provedsaying can be exemplified using forces. To combine two forces, add their rightleft parts and their updown parts. If the bullet mentioned above is fired from a plane going 300 meters per second (about 675 mph), in the direction of the plane, its location after one second is (200, -10) + (300, 0) which is (500, -10).

Arrow adding is so named because it has many of the laws of ordinary real number adding. (The Anglish website suggests using true scores to mean "real numbers." I've already decided to keep "number" instead of score, and "real" is also simple enough to keep.)

These real number laws, in fact, help to prove the matching laws for arrows.

Laws of Arrow Adding Provedsaying:
(1) Arrow adding is swapping ("commutative," from Anglish website).
(2) Arrow adding is linking ("associative," from the Anglish website).
(3) (0, 0) is a sameness ("identity," from the Anglish website) for arrow adding.
(4) Every arrow has a withering ("opposite," from the Anglish website) (-a, -b)

(On the Anglish website, these are called the swapping, linking, etc., laws of "eking," which is the Anglish word for "adding." But "adding" is simple enough to keep -- we don't need "eking" here.)

Proof:
All the parts use the strategy of adding appropriate arrows and using the matching real number law:
(1) It must be shown that (a, b) + (c, d) = (c, d) + (a, b). That is left to you.

Actually, all the parts of this proof are left to you, since this will simply become an exercise in typing lots of variables and parentheses, rather than much Anglish or Simple English.

Parts (1) and (2) of the Laws of Arrow Adding Provedsaying are important because they imply that any number of vectors can be combined in any order. Part (2) of the Laws of Arrow Adding is obvious when adding tidy pairs, but not so obvious with wayed liths ("directed segments").

The arrow (0, 0), called the zero arrow, represents a force whose greatness ("magnitude") is zero. A helicopter is able to hover because the adding of all the forces that act on it is the zero arrow. The arrows (a, b) and (-a, -b), whose adding is the zero arrow, are called withering arrows. If two teams engage in a tug-of-war, and neither is moving the other, the forces they have applied are withering arrows.

There are operations with arrows other than arrow adding. Perhaps the simplest is timesing of an arrow by a real number, called number timesing. (The word "scalar," as in "scalar multiplication," is an interesting one. What is a scalar? A scalar appears to be just a real number -- but that's because we're working with vector spaces over the reals. It's possible to have vector spaces over fields other than the real numbers, such as the rational or complex numbers. In a vector space over the rationals, a "scalar" is a rational number. In a vector space over the complexes, a "scalar" is a complex number -- therefore "scalar" doesn't simply mean "real number." To avoid confusing our Geometry students here, I decided just to use "number" to mean "scalar" here.)

The idea of number timesing is simple. If 3 people each push with the same force v, then the total force is v + v + v. This is the force 3v.

It is easiest to give the general definition of number timesing in terms of tidy pairs:

Meaning:
Let k be a real number and (a, b) be an arrow. Then k(a, b), the number timesing of k and (a, b), is the arrow (ka, kb).

Recall that the point (ka, kb) is the image of (a, b) under a size change of greatness k. (We already have flip for "reflection" and "turn" for "rotation," but what about "dilation"? It might be best just to keep the French words "size change" here. Also, I'm not sure whether to use greatness for the magnitude of a dilation, where other texts use "scale factor." Then again, we might choose to use both "scale (factor)" and "scalar," as these are clearly related concepts as explained right here.) Thus, when k is forward ("positive"), the arrow (ka, kb) has the same way as (a, b) and k times the greatness of (a, b). When k is backward ("negative"), the arrows (a, b) and (ka, kb) have withering ways.

Number timesing has many laws, some of which are touched upon in the Questions.

We'll only look at a few of the questions here.

1. Name the rightleft and updown parts of the arrow (2, 3). (Answer: right 2, up 3)

7. What arrow is the adding sameness? (Answer: (0, 0))

10. Write 2(-9, 5) as a single arrow. (Answer: (-18, 10))

18. a. Determine whether this statement is true for all arrows (a, b) and (c, d) and number timesing k:
k[(a, b) + (c, d)] = k(a, b) + k(c, d). (Answer: yes)
b. What law have you proved or unproved in part a? (Answer: The Dealing Law of Number Timesing over Arrow Adding.)

19. An arrow has greatness 2 and direction 30 degrees north of east. What are its parts? (Answer: east sqrt(3), north 1)

Music: "Count the Ways," "Linear or Not," "Patterns," "Same Sign, Add and Keep"

There are four songs that I sang at the old charter school during Weeks 21-22 of the school year.

The first song is the Square One TV song "Count the Ways," performed by the country duo the Judds. This song prominently mentions the normal human heart rate, 70 beats per minute (or 100,000 beats per day). It is not posted on YouTube except as part of a full Square One TV episode (song starts about 5 1/2 minutes in):

Here is a transcription of the lyrics -- which can't be found on Barry Carter's site. My version changes it up a bit so that I, the singer, am the "mystery man" referred to in the song:

COUNT THE WAYS

My love is three-dimensional, it has width, depth, and length,
I'll run that by you one more time.
Like an equilateral triangle, my love has special strength.
Yes, I have mathematics on my mind.

Refrain:
Let me count the ways that I love you,
I'll calculate the rhythm of my heart.
Let me count the ways that I love you,
And count each fraction of a second we're apart.
My heart beats for you 70 times a minute.
My heart beats for you 4200 times an hour.
My heart beats for you 100,000 times a day.
Three million times a month, 36 million times a year.

Yes, I want you to know that I'm not your average guy.
I'm the missing factor in your equation.
I'll multiply your happiness 'til your love equals mine,
Hey, you don't need any more persuasion,

(Repeat Refrain)

The second song is called "Linear or Not":

LINEAR OR NOT

Exponent on the variable?
It's not linear!
Multiply two variables?
It's not linear!
Variable on the bottom?
It's not linear!
Everything else?
It's linear!

This is a short, simple song whose tune is lost. Thus it's a prime candidate for a new tune based on the new scales that I mentioned in my last post.

I'll put a temporary Mocha BASIC computer tune here, based on the 18EDL scale, since that's the scale I'd like to tune/fret my guitar to. I'll probably change the tune based on what sounds best later on. I notice that the bars that I randomly selected avoid both the minor third on Degree 15 and the supermajor third on Degree 14 -- while Degree 13, a wide fourth, is emphasized -- and so this will sound neither major nor minor.

https://www.haplessgenius.com/mocha/

10 N=6
20 FOR V=1 TO 2
30 FOR X=1 TO 40
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
70 RESTORE
80 NEXT V
90 END
100 DATA 13,3,17,1,18,4,13,2,18,1,17,1,9,3,10,1
110 DATA 9,6,12,2,10,4
120 DATA 13,3,17,1,18,4,13,2,18,1,17,1,9,3,10,1

130 DATA 9,6,12,2,10,4
140 DATA 13,3,17,1,18,4,13,2,18,1,17,1,9,3,10,1

150 DATA 9,6,12,2,10,4
160 DATA 13,3,17,1,18,4,12,8

170 DATA 10,6,12,2,9,4


As usual, click on Sound before you RUN the program.

The third song is Weird Al's "Patterns," another Square One TV song.

Barry Carter, who has transcribed the lyrics to many Square One TV songs, doesn't have the lyrics to this particular song. But Weird Al is famous enough that his lyrics appear at other websites:

http://www.azlyrics.com/lyrics/weirdalyankovic/polkapatterns.html

POLKA PATTERNS

Everywhere, I see them there
I stop and stare at patterns
I don't care, I must declare
I've got a flair for patterns

On my hair, the clothes I wear
My savoir faire is patterns
All I see is patterns
The patterns that repeat

Let's go into the bathroom
I know we're in a room where you would not expect much math
Usually you're in here for a shower or a bath
But if you gaze upon the floor, and if you're kinda smart
You'll see the repetition is like geometric art

Wow, haha
Look

Everywhere, I see them there
I stop and stare at patterns
I don't care, I must declare
I've got a flair for patterns

On my hair, the clothes I wear
My savoir faire is patterns
All I see is patterns
The patterns that repeat

Hey!

A polkameister like myself never has to be bored
I just grab my ax and play some patterns on my keyboard
Now's the time for earplugs if you care about your health
So stand back, everybody, I'm gonna express myself

Look at this, patterns
I've got blisters on my fingers
Woo, hey, aw, get down
Yeah, help me, somebody, woo

Still there? Okay

Next time you find yourself at an exciting polka party
You can make some patterns with your feet and with your body
If you don't know the steps yet, here's the gang with all the answers
Ladies and gentlemen, introducing, the "Weird Al" Polka Dancers

Here they are

Everywhere, we see them there
We stop and stare at patterns
We don't care, we must declare
We've got a flair for patterns

On our hair, the clothes we wear
Our savoir faire is patterns
All we see is patterns
The patterns that repeat

Wallpaper, skyscrapers, funny papers, patterns
Evergreens, nouvelle cuisine, human beings, patterns
Garden rakes, wedding cakes, rattlesnakes, patterns
Golden wheat, little feet, my heartbeat

I gotta stop

Patterns, patterns, patterns, patterns

The fourth and final song is "Same Sign, Add, and Keep," based on "Row, Row, Row Your Boat." I'd already sung the "Measures of Center" version for sixth grade earlier in the year, but this week was when seventh grade learned how to add integers:

ADDING INTEGERS SONG

Same sign, add and keep.
Different signs, subtract,
Keep the sign of the bigger number,
Then you'll be exact!

A Rapoport Math Problem

Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:

| 2   3   5 |

|19 23  7 |

|17 13 11|

|1 -9    9  |

|2 -21 58 |

|2 -18 55 |

This definitely isn't Geometry -- these are matrix determinants. So let's do the work to calculate both of these determinants and then divide:

2*23*11 + 3*7*17 + 5*19*13 - 5*23*17 - 2*7*13 - 3*19*11

-1*21*55 - 9*58*2 - 9*2*18 + 9*21*2 + 1*58*18 + 9*2*55

= 506 + 357 + 1235 - 1955 - 182 - 627

   -1155 - 1044 - 324 + 378 + 1044 + 990

= 2098 - 2764

   -1479 + 1368

= -666

   -111

= 6

So the desired answer is six -- and of course, today's date is the sixth.

Rapoport Geometry Problems from This Week

There is one Geometry problem on the Rapoport calendar this week:

The number of scalene triangles with integral side lengths and perimeter at most 12.

This is mainly a Triangle Inequality problem. For example, we learn that no side of our triangle can have length 1. Here's an indirect proof of this -- suppose one side has length 1. Since this is a scalene integer triangle, the other two sides must have difference at least 1. Then the shorter of these two sides, plus the shortest side of length 1, add up to at most the length of the longest side, which contradicts the Triangle Inequality. Therefore no side of our triangle can have length 1.

So the simplest possible scalene integer triangle is 2-3-4, of perimeter 9. This is our first triangle.

There is no scalene integer triangle of perimeter 10. The best we can do is 2-3-5, but once again, this violates the Triangle Inequality. The longest side of an integer triangle of perimeter 10 must be 4, but then the only possible such triangles are isosceles.

There is one possible triangle of perimeter 11 -- a 2-4-5 triangle. This is our second triangle. The longest side can't be raised to 6 without violating Triangle Inequality, and the only other triangles with longest side 5 are isosceles.

There is one possible triangle of perimeter 12 -- a 3-4-5 triangle. This is our third triangle (and yes, it's our familiar Pythagorean right triangle). The longest side can't be raised to 6 without violating Triangle Inequality, and the only other triangle with longest side 5 is isosceles.

And this is where we stop, at perimeter 12. Therefore there are only three possible scalene integer triangles of perimeter at most 12 -- and of course, the date of this problem was Monday the third.

There is one more problem this week that's somewhat related to Geometry -- the Four-Color Theorem (which I count as Geometry since it's in the U of Chicago text). The question is:

What is the minimum number of colors needed to color this map?

Here's how I describe the map -- begin with a regular pentagon. Inside this is a smaller regular pentagon, formed by joining the midpoints of the original pentagon. Inside the smaller pentagon is a triangle formed by joining three vertices of the pentagon, not all consecutive. Such a triangle is often called a "golden triangle."

Begin by coloring the golden triangle black. Then we can color four more triangles black -- these are the smaller triangles bounded by the two pentagons and are often called "golden gnomons." There are five gnomons bounded by the two pentagons, and we color them all black, except for the one that borders the golden triangle. This fifth gnomon, as well as the remaining two triangles that border the golden triangle (which are themselves larger gnomons), are all white.

This colors the entire map, and we only needed two colors, black and white. Therefore the minimum number of colors needed is two -- and of course, the date of this problem was Sunday the second.

Conclusion

Today I took a visit to the Huntington Library and Botanical Gardens:

https://www.huntington.org/

I first learned about the Huntington Library as a young high school student, when our school went there on a field trip. Even though I didn't attend the field trip, I try to visit there once a year -- often during the spring, usually on a Free Day (the first Thursday of the month).

Of course, the coronavirus upended my plans to go there this year. This is the first time that I went there as late at August (but I do think I went as late as July one year). Because I went there so late, this is the first time I saw fully mature sunflowers in the gardens, as well as young grapes.

The Huntington Library is the first place that required me to have my temperature taken upon entering -- a new normal here in the coronavirus era. I fully expect that -- if the schools ever reopen -- I'll be subject to temperature checks and other health precautions upon stepping onto any campus. So this is something I'd better get used to.