Thursday, May 31, 2018

SBAC Practice Test Questions 25-26 (Day 176)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

It's 3 pm. At 3:?? pm the [minute] hand will have traveled 186 degrees.

(Pappas has a typo here -- she writes "second" where "minute" is intended.) In 60 minutes the minute hand travels 360 degrees, so we can write a proportion:

x/186 = 60/360
x/186 = 1/6
6x = 186
x = 31

So the answer is 31 minutes -- and of course, today's date is the 31st. Technically this is a Geometry problem since it involves angles and degrees, but it's also an Algebra problem.

Today I subbed in a seventh grade special ed class. Math is indeed one of the subjects I cover today, and so I am doing "Day in the Life" today. But the two main math classes aren't until the end of the day, and so bear with me while I discuss what happens the rest of the day. The focus resolution for today will be the fourth resolution, on warm-ups, since they're a part of the math classes today.

8:15 -- This is the middle school that starts with homeroom and first period everyday. The teacher I subbed for had first period conference -- except that I must cover for another teacher who calls for a sub late. The other sub doesn't arrive until first period is almost over, and so I cover what turns out to be a digital video class. The students already have projects to work on, so all I do is make sure that they don't start playing games or YouTube on the computers.

9:20 -- Just before first period ends, the other sub arrives. At this time, there's an announcement that the school is now on a "rainy day schedule." So it's time for me to explain California weather once again, since today is a rainy day (viz. a so-called "California 'Snow' Day").

Last year, I explained that snow is almost nonexistent at sea level in Southern California. As for rain, it usually occurs only between late October and early May. The exception is the so-called "May Gray" and "June Gloom." This refers to a marine layer consisting of low clouds, fog, and drizzle. I usually don't consider "May Gray" or "June Gloom" to be a true rainstorm since it usually falls in the morning and is gone by noon (though earlier this month the rain stayed an entire day).

My second/fifth grade teacher was born in Washington State. She once told the class that in her hometown, the schools let students have recess in the rain since there's so much precipitation. It's only here in California where if there's a little rain, outdoor recess is cancelled. Yet even I was surprised that the middle school declared a rainy day schedule after a mere morning drizzle.

In elementary schools, a rainy day schedule means no outdoor recess. Here at this middle school, rainy days mean no nutrition break, and instead of a single lunch there are two lunches, A and B. In theory, this means that there's more room to eat indoors or in a covered area, since only half the students eat during either lunch. All periods on a rainy day schedule are a few minutes off of the regular schedule except for first, which is why the schedule must be declared so early.

9:25 -- All periods except for first rotate at this school. On Thursdays, fifth period is after first. All special ed teachers co-teach at least one period -- and in fact, this regular teacher co-teaches two classes, both fifth and sixth period.

Fifth period is a science class. Whenever I sub for science -- especially middle school science -- I lament my failure to teach science properly last year. The students are working in the final unit of the year, which is the engineering design process. This is a brand new unit that's part of the Next Generation Science Standards.

These students take a test on engineering. Then they watch The Bee Movie, since there was a recent project where they design a bee hotel. (And yes, I know that tonight is a very different type of bee -- the National Spelling Bee.) The resident teacher is also out -- and instead another sub is there. He explains that he used to work in industry as an engineer, and so he's able to tie the lesson to his former job experience. (He was also recently long-term sub for a math class, and so some of the students recognize him.)

10:20 -- Sixth period I co-teach a math class. (Notice that I still haven't actually taught my own math class yet -- that's just the luck of the period rotation.) These seventh graders are learning about volume and surface area. The resident teacher uses nets of prisms to introduce surface area. The U of Chicago Geometry text is a bit similar -- nets appear in Lesson 9-7, and then Lesson 10-1 uses nets to explain surface area.

11:20 -- Some students head for lunch -- but I don't, as I'm scheduled for second lunch. Along with losing both conference period and nutrition, it means that I have four straight classes without a break.

The second period class, my first special ed class, is science. These students are also working on engineering, but this time they are to use Chromebooks to look up the definitions of words related to the engineering design process (including engineering, design, and process).

12:15 -- It is finally time for lunch.

1:00 -- Third period is the first of two special ed math classes. Unlike science (where the special ed students are more or less at the same point as the gen ed kids), this math class is well behind. The students are just now learning about circumference. Recall that I spotted a gen ed seventh grade class at this same school learning this lesson around Pi Day.

Here's how the special aide and I run this class. Let's watch for the fourth resolution:

1:13 -- We complete the Warm-Up -- the aide stamps their homework while the students answer six perimeter questions (all regular polygons). One student at a time goes up to the document camera to answer one of the Warm-Up questions -- the worksheet is placed in a dry-erase packet and then placed under the camera.

1:19 -- We finish going over the homework. The main lesson begins about a third of the way into the period.

1:45 -- We finish the main lesson (actually I teach most of it), with the students completing the notes in their notebook. The students are given the last 10-11 minutes to do the homework.

This is something I must keep in mind as I prepare for my summer school Algebra I class. Then again, the fact that my class is computer-based would affect the pacing of my class.

2:00 -- Fourth period is the last math class. Checking the clock, I notice that the pacing for this period isn't quite as good as the previous class -- perhaps because this class is slightly more talkative. Thus time management is affected into classroom management.

2:55 -- Fourth period ends and I go home.

Let's get back to music and the 18EDL scale:

The 18EDL scale:
Degree     Ratio     Note
18            1/1         white D
17            18/17     umber D#
16            9/8         white E
15            6/5         green F
14            9/7         red F#
13            18/13     ocher G
12            3/2         white A
11            18/11     amber B
10            9/5         green C
9              2/1         white D

How can we make music in this scale? Well, we notice that this scale contains a minor triad on the root, just as 12EDL does. The main difference is that 12EDL has a perfect fourth, while 18EDL lacks this white fourth. In exchange, 18EDL has a neutral sixth (which might go better in a minor scale than the supermajor sixth) and a minor seventh (as opposed to no seventh at all in 12EDL).

But 18EDL also has a supermajor third and hence a supermajor triad, 18:14:12 (or 9:7:6). Since 18EDL is the first EDL to have any sort of major triad, we can think of this as the supermajor EDL.

One use of the prime 17 is to make a fully diminished seventh chord, 10:12:14:17 (otonal). In Mocha, we might try playing the corresponding utonal diminished chord, 17:14:12:10. This chord can join the root supermajor triad as the start of a riff (D supermajor, D#dim7). But the next chords that are part of that riff (Em, A) aren't playable in Mocha. We've been using our utonal chords for descending rather than ascending, and so maybe D#dim7 could lead into D supermajor.

On the guitar, we might play D supermajor as D ordinary major. There's a way to finger D#dim7 on the guitar (take a D7 chord and add a "lute index finger" for D#), but this is difficult to play.

Possible 18EDL root notes in Mocha:
Degree     Note
18            white D
36            white D
54            white G
72            white D
90            green Bb
108          white G
126          red E
144          white D
162          white C
180          green Bb
198          amber A
216          white G
234          ocher F
252          red E

Question 25 of the SBAC Practice Exam is on inequalities:

A student earns $7.50 an hour at her part-time job. She wants to earn at least $200.

Enter an inequality that represents all of the hours (h) the student could work to meet her goal. Enter your response in the first response box.

Enter the least whole number of hours the student needs to work to earn at least $200. Enter your response in the second response box.

This is a first-semester inequality problem. Inequalities appear in Chapter 5 of Glencoe, and so it's the last topic we'll cover this summer:

7.50h > 200
h > 13 1/3

So the inequality is 7.50h > 200 ("at least" = "greater than or equal to") and the smallest whole number value that satisfies it is 14 hours.

Question 26 of the SBAC Practice Exam is on comparing statistics:

Michael took 12 tests in his math class. His lowest test score was 78. His highest test score was 98. On the 13th test, he earned a 64. Select whether the value of each statistic for test scores increased, decreased, or could not be determined when the last test score was added.

(The possible stats are standard deviation, median, and mean.)

Stats appears in the last chapter of Glencoe Algebra I, and so this is a second-semester question. The other summer teachers don't like how Edgenuity requires students to find standard deviation by hand!

Anyway, the mean must decrease because Michael's last test score is lower than that of any previous test that he has taken. The standard deviation must increase when that low value is added. But we can't be sure about the median. The median of 12 values is the mean of the 6th and 7th value, but the median of 13 values is the 7th value (the old 6th value before the 64 happened). So median could decrease if the original 6th test is less than the 7th -- but the median could stay the same if the original 6th and 7th scores were equal. (It's impossible for the median to increase here!)

SBAC Practice Exam Question 25
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

SBAC Practice Exam Question 26
Common Core Standard:
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Commentary: The inequality I wrote for h can be solved as early as Lesson 4-6 of the U of Chicago Algebra I text. Meanwhile, stats isn't covered in the text at all, except for Lesson 1-2 where mean and median appear, but not standard deviation.

Note that I'm posting two additional Geometry pages from today's subbing. One is a Kuta Software worksheet about Nets of Solids from the gen ed class. (I've posted Kuta Software on this blog before, since anyone can access the free trial pages.) The other is a half-sheet with a circumference, diameter, and radius from the special ed class.

Wednesday, May 30, 2018

SBAC Practice Test Questions 23-24 (Day 175)

Today is the second day of training for my summer school Algebra I course. I met up with the other algebra teachers, and we're starting to reach a consensus on the grading.

First of all, the administrators agreed with the second semester cohort of teachers. The assignments don't count at all. Instead, the quizzes is 40%, the tests 50%, and the district final 10%. There are also minimum thresholds required to advance in Edgenuity. A score of 68% is required to advance beyond a quiz and 65% to advance beyond a test, and 60% is needed on the final.

But as I wrote last week, I still don't like how some of the opening lessons are taught in Edgenuity. In the first unit on solving equations (Chapter 2 of Glencoe), so many of the simplest examples contain decimals and fractions. And the students are taught to solve equations by graphing f (x) = LHS and g(x) = RHS and finding their point of intersection. As I wrote last week, the reason for Edgenuity teaching it this way is a pair of Common Core Standards:

Represent and solve equations and inequalities graphically.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

But Glencoe teaches these standards in Lesson 3-2, after solving linear equations is covered. On Edgenuity, students are expected to graph linear equations in the plane first, and then they are taught to solve linear equations in one variable. This means that, for example, students aren't taught to convert a linear equation from standard form to slope-intercept form in order to graph it -- since at that point they don't know how to solve equations for y.

Of course, keep in mind that many of Edgenuity's equations have decimal or fractional solutions -- which aren't exactly easy to graph, much less locate as the intersection of two graphs.

In fact, there's only two situations when I'd teach this as the primary method of solving equations. If the equation is nonlinear, then an algebraic method is impossible (such as 2^x = x + 3.) The other situation is when a graphing calculator is available, so it's easy to graph y_1 = LHS and y_2 = RHS on the calculator. Otherwise, I prefer to do it like Glencoe's Lesson 3-2 -- after solving equations, so that students can see the connection between equations and graphs in preparation for solving equations by graphing in the two aforementioned cases (nonlinear equation or calculator available).

The Algebra II teachers (and probably the Algebra 1B teachers as well) watch the first few Edgenuity videos as well, and they don't necessarily like how these lessons are set up either. One of them remark that the teaching method in the video is how she'd set up her Honors Algebra II class -- not a summer class where two-thirds of the students received F's during the school year.

We fear that by the end of the first week of classes, the students will be completely lost unless we resort to supplementing Edgenuity with packets. That's right -- I might have to bring back that old "Packet Song" from last year in music break. I could make the packets out of worksheets published by Glencoe -- or I might create my own worksheets based on the U of Chicago Algebra I text!

Let's get back to Fawn Nguyen's Memorial Sunday post. She's describing how similar house cleaning is to lesson planning. Since we're still in the middle of lesson planning for summer school today, I'll post the rest of her list today.

  1. Swiffer products should be regarded as essentials like toothpaste and TP.
  2. The person who did not put the TV remote control away in a designated spot shall be banished from the home (or get punched in the face).
  3. Make your bed every morning.
  4. Never go to bed unless the kitchen is clean. (If you dread this, then don’t cook.)
  5. If you find the above 9 steps difficult to implement, then try step 1 again.
And here's how Nguyen compares these house cleaning steps to lesson planning. For #6, she points out that the essential products of the classroom are equity and access. She admits that she isn't perfect when it comes to equity and access in her class. We summer school teachers are also worried about the accessibility of the Edgenuity lessons. One Algebra II teacher warns us that there will probably be a few special ed students in each class -- will Edgenuity be understandable to them? And one Algebra I teacher -- the one who's currently student teaching -- is worried about her current students who will be taking her class again in the summer.

For Nguyen, step #7 is all about respect. In lesson planning, this means that teachers should respect other teachers by giving proper credit to the source of each lesson, and sharing good lessons. This is exactly what she and I are doing on our respective blogs.

Steps #8-9 are all about fresh starts or do-overs. Nguyen reminds us that just as we expect out students to persevere after a setback, we teachers need to pick ourselves up and move forward after lessons that don't work as well as they should have. And I've devoted numerous blog posts to unsuccessful lessons and how I'd improve upon them.

For the last step, Nguyen tells us that lesson planning can be a pain, especially on weekends. This is why we summer school teachers have been given these Wednesdays to do our lesson planning. I admit that Edgenuity has made the lesson planning easier, even though we don't necessarily like how the software presents some of the material.

I want to use some of these posts to revisit my most popular posts from the past twelve months -- and that includes my Lee Canter classroom management post. Last summer, I looked at his Succeeding with Difficult Students and wrote about his worksheets #4-5. I considered three hypothetical, but common, classroom situations and how I'd handle them.

I'd like to continue in this book especially in preparation for my summer class. But unfortunately, worksheets #4-5 are the only exercises I can do easily. Many of the worksheets require watching either a video that I can't access (the book was written a quarter-century ago, before YouTube) or students in our actual classrooms. I could go back to whom I remember of my class from last year (such as the "special scholar" from my January 6th post). I wish I'd thought about this book during the multi-day subbing assignments (such as the middle school Digital Video class from back in March, or the senior Economics class from about a month ago).

There's still some time to try out these exercises in these last few days of subbing before summer. I want to make sure that my management skills are much improved from last year.

Question 23 of the SBAC Practice Exam is on comparing rates:

Nina has some money saved for a vacation she has planned.

  • The vacation will cost a total of $1600.
  • She will put $150 every week into her account to help pay for the vacation.
  • She will have enough money for the vacation in 8 weeks.
If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation? Enter the result in the response box.

OK, I'd count this as a first semester Algebra problem that our students might be required to solve this summer. First of all, let's ignore the given value $1600 completely, since it has nothing to do with the solution of this problem. What we want to know is, how long will it take Nina to save the same amount of money at $200/wk. as she does at $150/wk. for eight weeks? Thus this is the equation:

200w = 150(8)
200w = 1200
w = 6

So it takes her six weeks to save the money as opposed to eight weeks. Therefore it will take her two fewer weeks (that is, 8 - 6) to save the money -- students should enter the number 2.

Sometimes students might forget that even though w = 6, they must enter the number 2, since the question is not how many weeks will she save, but how many fewer weeks. Sometimes I avoid this problem by letting x = 8 - w, so that as soon as I find x, I have the number I need to enter:

200(8 - x) = 150(8)
1600 - 200x = 1200
-200x = -400
x = 2

I often solve Pappas problems this way -- using the variable x only for the final value that I need to find rather than any intermediate values. Also, if there's a system of equation and Pappas asks us to solve for x, I might eliminate y even if x is easier to eliminate, so that I find the actual asked-for value more directly.

Question 24 of the SBAC Practice Exam is on quadrilaterals:

Consider parallelogram ABCD with point X at the intersection of diagonal segments AC and BD.

Evelyn claims that ABCD is a square. Select all the statements that must be true for Evelyn's claim to be true.

  • AB = BD
  • AD = AB
  • AC = BX
  • Angle ABC isn't 90
  • Angle AXB = 90
Here's some Geometry -- yes, it's been some time since I posted Geometry on this Geometry blog. I'll look at the three length equations first. AD = AB is obviously true, since the sides of a square must indeed be congruent. On the other hand, AB = BD is definitely false -- the diagonal of a square is sqrt(2) times the length of a side, not the same length as a side. And AC = BX is false as well -- the diagonals of a square (or any rectangle) are congruent, and so AC can't be half of BD (which is what BX is, as the diagonals of a parallelogram bisect each other).

Now let's look at the angle statements. The statement that ABC isn't 90 is obviously false -- the angles of a square must be exactly 90. The other statement that AXB = 90 is true, since the diagonals of a square are also perpendicular. So students must select two correct answers, AB = AD and AXB = 90.

Notice that if all Evelyn knew about her parallelogram is AB = AD and AXB = 90, that would not be sufficient for her to conclude that it's a square. A rhombus, after all, also has congruent sides and perpendicular diagonals. (In fact, given that ABCD is a parallelogram, just one of these two is sufficient for her to conclude that ABCD is a rhombus.) On the other hand, adding the falsity of the claim that ABC isn't 90 (that is, the truth that ABC = 90) to either of the two selected statements is enough for Evelyn to conclude that ABCD is a square.

SBAC Practice Exam Question 23
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

SBAC Practice Exam Question 24
Common Core Standard:
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Commentary: The first equation I wrote for w can be solved as early as Lesson 4-4 of the U of Chicago Algebra I text, although the distributive property in the x equation must wait for Lesson 6-8 a little later on. Meanwhile, the properties of rectangles, rhombuses, and squares are covered in Lesson 5-4 of the Geometry text.

Tuesday, May 29, 2018

SBAC Practice Test Questions 21-22 (Day 174)

Today I subbed in a high school science class. There were three sections of Biology and two sections of Anatomy/Physiology. In the bio classes, students are supposed to create a comic strip where the characters are T-cells and other members of the immune system. I won't do a "Day in the Life" for today, but I do point out that several students in all three classes failed to complete three panels (and of these, half failed even to begin one panel) despite my repeating over and over again that I'd leave for the regular teacher the name of any student who didn't do three panels.

The Anatomy classes are mostly seniors, but one class has three juniors. This is significant because today is the infamous California Science Test -- the test that caused me many headaches when trying to set up my science classes last year. I haven't though much as much about the high school test as the middle school tests. I know that one grade (which could be any grade from 9-12) administers the test each year -- it makes sense that it would be juniors (since they test English and math as well).

Apparently, the California Science Test was supposed to be last week, but for some reason (computer failure?) it was delayed to this week. The three Anatomy juniors are sent to another classroom, presumably a science class with more juniors than Anatomy.

Meanwhile, today there is a science Google Doodle -- S.P.L. Sorensen, the Danish chemist who developed the pH scale. It's too bad that I didn't sub in a Chemistry class today (instead of Biology) in order to fit the doodle. The pH scale is often mentioned in Algebra II classes as an application of the common logarithm. (During tutorial, I am able to help a few students in Algebra II. One student has a logarithm problem for review, but it's not a pH problem.)

Meanwhile, the Queen of the MTBoS has made her latest post. That's Fawn Nguyen, in case you don't recall who our queen is:

This is what she writes on Memorial Sunday:

There is something else that I do way better than teaching mathematics, even though teaching has been a 25-year plus career. That something is house cleaning.

Nguyen proceeds by listing ten steps required for house cleaning. I'll post only the first five steps here, since you can just click the link if you really want the whole list:

  1. Throw everything out.
  2. When done with step 1, repeat step 1 again bc we both know you really didn’t throw everything out.
  3. With remaining [ideally just 3] items, ask, “Is it really really pretty?” If so, it should be displayed in your home in a pretty spot. Ask, “Is it useful, like a wine-bottle-opener type of necessity?” If so, keep it in a drawer.
  4. Unless it’s a piece of furniture, a houseplant, or a 4-legged friend, forbid it from touching your floor.
  5. Counter space is only for items that do not fit inside a drawer/cupboard and are used almost daily — e.g., toaster, Nutribullet, knife block.
Then Nguyen compares these steps for house cleaning to those of lesson planning. I, of course, am currently thinking about lesson planning for my summer Algebra I course, so let's follow this closely.

The first two steps are to throw everything out. Well, last week's lesson planning was all about throwing things out -- the course as posted on Edgenuity is too long for a three-week course. But Nguyen goes on to explain that she often throws out questions in order to make some of her problems more open-ended. This I won't be able to do -- Edgenuity will ask the students questions, and it's difficult for a computer to evaluate open-ended answers.

Nguyen rejects the third step in her own classroom -- her students don't have time to make their work look "pretty." It does remind me of Foldable notes, where students take time to decorate their Foldables before the unit begins. The idea of Foldables actually comes from the Glencoe text, and I was considering having my students make them to take notes from the text. But Nguyen's right -- there won't be time in a summer course for Foldables, and besides, the main text for the course is Edgenuity, not Glencoe. In a computer-based class, there's no time for Foldable notes.

Steps 4-5 are connected to decorating the room itself. I don't know what my room will look like -- indeed, I probably won't have access to it until the Friday before summer school begins, since it's currently occupied by a regular school year class. So we'll worry about these two steps later.

It's time for our next musical scale. This week we'll look at 18EDL:

The 18EDL scale:
Degree     Ratio     Note
18            1/1         tonic
17            18/17     septendecimal chromatic semitone (Arabic lute index finger)
16            9/8         major tone
15            6/5         minor third
14            9/7         supermajor third
13            18/13     tridecimal semiaugmented fourth
12            3/2         perfect fifth
11            18/11     undecimal neutral sixth
10            9/5         minor seventh
9              2/1         octave

20 FOR D=9 TO 18
30 SOUND 261-N*D,4

This is a descending scale. To make the scale ascend, use:

20 FOR D=18 TO 9 STEP -1

We notice that this scale contains a new prime, 17. Indeed, 18EDL is the only 17-limit EDL, and it contains the entire 17-limit tonality diamond.

The name "septendecimal" for the 17-limit is a bit awkward. It reminds us of the name "hexadecimal" for base 16 (going back to my last post, on number bases). But notice that linguistic purity is violated, as "hexa-" is Greek but "septen-" is Latin.

Of course, this ordinarily isn't a problem. We know that number base enthusiasts don't even like odd bases, much less prime bases. But musical intervals are named for their prime limit. So it's unlikely we'll ever have to worry about confusing number bases with interval names. As a prime base, 17 will be ignored, so it doesn't matter whether it's "septendecimal" or "heptadecimal." And intervals such as 16/13 won't be called "hexadecimal" or "sexadecimal" neutral third, but "tridecimal" neutral third, after the prime 13.

(And besides, number bases often take Latin names anyway. There's been much discussion lately of base 6, called senary, not "hexary." And base 11, when it appears, is "undecimal," not "hendecimal.")

Let's check out the notes of 18EDL using Kite's color notation:

The 18EDL scale:
Degree     Ratio     Note
18            1/1         white D
17            18/17     umber D#
16            9/8         white E
15            6/5         green F
14            9/7         red F#
13            18/13     ocher G
12            3/2         white A
11            18/11     amber B
10            9/5         green C
9              2/1         white D

The new prime 17 requires two new colors -- "tan" for otonal and "umber" for utonal. Since EDL scales are utonal, the color "umber" appears in the scale, with "umber D#" the first note of that color.

The color "umber" is a odd choice, since it sounds so much like "amber." In fact, Mocha has a note at Degree 187 (Sound 74) called "umber-amber A#." But Kite, when choosing colors for his notation, was more concerned with the "u," not the "-mber" part. He wanted each color to start with a different letter, and he needed a color starting with "u," so "umber" it is. (His notation uses every letter of the alphabet except "z.")

Finally, you might ask why our first umber note is called D# rather than Eb. Well, there are two reasons for preferring sharps over flats. In utonal music, fourths are preferred over fifths (since the largest prime of 4/3 is in its denominator, unlike 3/2). If Degree 17 is umber D#, then Degree 51 (that is, 17 * 3) is umber G# and Degree 153 is umber C#. Then Degree 459 would be umber F#, and Degree 1377 would be umber B (even though these last two are out of Mocha range). So at least in theory, we can reach notes without a classical accidental by proceeding fourthward from D#. On the other hand, proceeding fourthward from Eb takes us into more flats before reaching double flats.

And here's another reason that Kite prefers sharps here. The interval 18/17 is 99 cents, while the interval 17/16 is 105 cents. Thus Degree 17, in cents, is closer to Degree 18 than to 16. And so we'd prefer 17 to have the same letter name as 18, which again means that D# is used over Eb. Both of these reasons generalize to higher primes, so if we have a choice between a flat name and a sharp name for a new prime, the latter is preferred.

What good are umber notes, anyway? The use of the other limits is obvious -- the 3-limit gives us fifths while the 5-limit gives us thirds. The 7-limit is used in jazz (including barbershop). Even the 11- and 13-limits can be used for neutral intervals, plus we might want harmonic 11th and 13th chords (following the pattern for harmonic 7th chords). But 17 seems to be just a useless prime.

Well, in Helmholtz-Ellis notation, the 17-limit can be indicated using a symbol for 256/255. This comma is so small that H-E calls it a "schisma" rather than a "comma." (At 6.8 cents, it's slightly smaller than the 225/224 comma that Kite calls the "sub.")

The 256/255 schisma is one of the smallest intervals playable in Mocha. Degree 256 (Sound 5) is called "white E," while Degree 255 (Sound 6) is called "umber-green E" (since 255 has the factor 5 as well as 17). In other words, the difference between "white" and "umber-green" is the schisma. As the color that cancels "green" is yellow, it follows that "yellow" differs from "umber" by the schisma.

In Mocha, otonal colors like "yellow" are unplayable, but utonal colors like "umber" are playable. So we can play major chords in Mocha using "umber" instead of "yellow." For example, an A umber major triad would be Degrees 192:153:128 (Sounds 69-108-133). Unfortunately, there still aren't enough umber notes to make a complete major scale (so for now, green Bb major remains the only playable major scale).

There's one more thing I'd like to say about the prime 17. Let's look at the 18EDL scale once more:

The 18EDL scale:
Degree     Ratio     Note
18            1/1         tonic
17            18/17     septendecimal chromatic semitone (Arabic lute index finger)
16            9/8         major tone
15            6/5         minor third
14            9/7         supermajor third
13            18/13     tridecimal semiaugmented fourth
12            3/2         perfect fifth
11            18/11     undecimal neutral sixth
10            9/5         minor seventh
9              2/1         octave

Hmm, that's a strange name for the 18/17 interval, "Arabic lute index finger." I'm very curious about the reason for that name -- and I wonder if there's an "Arabic lute index finger," are there also an "Arabic lute middle finger," "ring finger," and "little finger"?

First of all, a "lute" is a string instrument, similar to the modern guitar. On a modern guitar, we can place an index finger on the first fret to raise the pitch -- for example, fretting the D string at the first fret produces D#. So perhaps we can place an index finger on an Arabic lute to raise its pitch by the semitone-like interval 18/17.

Well, a traditional "Arabic lute" -- also known as an oud -- has no frets. Still, I can imagine raising the pitch of an oud string by using the index finger, thereby justifying the name.

The Xenharmonic website uses the name "Arabic lute" to refer to two intervals. One of these is 18/17, but the other is the "Arabic lute acute fourth," 49/36:

But 49/36 is a septimal interval -- it has nothing to do with 17. And you'd think that as easy as it is to use Google to research ouds, no website seems to explain why 18/17 is "Arabic lute index finger."

Well, here's my conjecture -- 18/17 is actually part of a simple 18EDL system. The middle, ring, and little fingers can be placed on the strings to sound out 18/16, 18/15, and 18/14 -- that is, the fingers divide the string into 18 equal parts. But 18/16 isn't called "Arabic lute middle finger" because it reduces to 9/8, which already has a name ("major tone"). Likewise, 18/15 = 6/5 (minor third) and 18/14 = 9/7 (supermajor third), so these don't have Arabic names.

The next interval that could have an Arabic name would be 18/13 -- but by this point, we've run out of fingers! And besides, 18/13 is a sort of superfourth, while the interval between the strings are perfect fourths (just as they mostly are on a modern guitar). Thus once we reached the fourth, an oud player would just jump to the next string.

From the perfect fourth, it's easy to use the middle finger to add 9/8 to this to obtain a perfect fifth, and the octave is likewise reached on the next string. And within each string, we can play a sort of semitone (18/17), whole tone (9/8), minor third (6/5), and major third (9/7). In other words, from 18EDL we have the beginning of a twelve-tone scale, centuries before 12EDO was created.

Before we leave music, I like to point out that for one of the higher primes, Kite actually has a color called "fawn" -- which reminds us of Fawn Nguyen, the Queen of the MTBoS. We'll find out what prime the color "fawn" corresponds to in a later post.

 We'll continue our discussion of 18EDL on Thursday.

Question 21 of the SBAC Practice Exam is on simplifying exponents:

Write an expression equivalent to b^11/b^4 in the form b^m.

Exponents are definitely a second-semester Algebra I topic. We know the Laws of Exponents, and the rule that to divide powers, we subtract exponents. Therefore the answer is b^(11 - 4) = b^7. The hardest part of students (provided they know the Laws of Exponents) is entering b^7 properly.

Question 21 of the SBAC Practice Exam is on the average rate of change:

The depth of a river changes after a heavy rainstorm. Its depth, in feet, is modeled as a function of time, in hours. Consider this graph of the function.

[The graph passes through many points, including (9, 18) and (18, 21) -- which, as you'll soon see, are the only two points that matter.]

Enter the average rate of change for the depth of the river, measured as feet per hour, between hour 9 and hour 18. Round your answer to the nearest tenth.

This is considered to be a first-semester Algebra I problem, but it's worded strangely. The phrase "average rate of change" confuses many students and teachers alike.

The first time I, as a young student, ever heard the phrase "average rate of change" was in an AP Calculus class. Our teacher fave an average rate of change problem, and polled the students whether they needed to find an integral or a derivative to find the solution. I forgot which answer I chose, but I remember that the correct answer is neither. Here's the reason why, in a nutshell -- the word "average" implies an integral (as in "the average value of a function"), while "rate of change" obviously implies a derivative. Thus in "average rate of change," the integration and differentiation cancel each other out, and so neither is needed (which had better be the case, otherwise this question has no business being on the SBAC).

In fact, the average rate of change of a function between is just the slope of the line passing through the two points. Here's somewhat of a proof, from Calculus:

"Average" means integral:
1/(b - a) times the integral from a to b of something dx

"Rate of change" means that "something" is derivative:
1/(b - a) times the integral from a to b of f '(x) dx

By the Fundamental Theorem of Calculus, the integral of f '(x) is f (x)
1/(b - a) times f (x), evaluated from a to b
1/(b - a) times (f (b) - f (a))
((b) - (a))/(b - a)

which is indeed the slope of the line through (a, f (a)) and (b, f (b)). QED

Of course, Algebra I students don't deal with the proof. Instead, they're taught that the average rate of change through two points is simply the slope of the line passing through them. It's mentioned as a real-world example of slope and an instance of the Common Core Standards on modeling.

Oh yeah, let's solve the problem. The average rate of change, or slope, is:

(21 - 18)/(18 - 9) = 1/3

The directions ask students to round this to the nearest tenth, so the correct answer is 0.3 ft./hr.

SBAC Practice Exam Question 21
Common Core Standard:
Rewrite expressions involving radicals and rational exponents using the properties of exponents.

SBAC Practice Exam Question 22
Common Core Standard:
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Commentary: The standard listed for Question 21 is the closest standard in the high school section -- the true standard is an eighth grade standard. Quotients of Powers appear in Lesson 9-7 of the U of Chicago Algebra I text. The phrase "average rate of change" doesn't appear in the text, but "rate of change" appears in Lesson 8-1, with "average" implied. Constant rates of change appear in the next lesson. Notice that the first eight chapters of the U of Chicago text correspond to the first five chapters of Glencoe and the first semester in Edgenuity.

Friday, May 25, 2018

SBAC Practice Test Questions 19-20 (Day 173)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

What is the 10th (base two) root of 10000 (base five)?

Well, 10 in base two is just two, so we're asking about the square root of 10000. Notice that the square root of 10000 in any base is 100 in that base -- that's because 10000 is just b^4 in base b, while 100 is just b^2, and sqrt(b^4) = b^2. Thus sqrt(10000) = 100 (base five) = 25. Therefore the answer is 25 -- and of course, today's date is the 25th.

Hold on a minute -- that's not a Geometry question, so why am I doing a Pappas question today? As these are the final weeks of school, I always celebrate by revisiting my most popular posts during the past year. My #1 post in hit count was a summer post on number bases. And so this is a good excuse to have another post on bases -- one of my favorite topics.

(By the way, today I subbed in an eighth grade U.S. History class. Because all classes just watch a video -- Civil War by Ken Burns -- and take notes, I won't do a "Day in the Life" today.)

The two bases that appear in this Pappas problem are two (binary) and five (quinary). Even though binary is ubiquitous in computers, it's so small that it's considered well below the human scale -- large numbers take too long to write in binary.

Quinary, at least, has its own thread at the Dozens Online website. (Recall that I have Dozens Online to thank for making number bases my most popular post of the year.)

At this thread, it's pointed out that humans could have developed quinary as the dominant base over decimal, since we do have five fingers on a hand. But quinary might still be a bit too small to be considered human-scale -- and besides, it's odd. Since even bases are generally preferred, we count two hands and use decimal as the dominant base.

When I was young, my first scientific calculator, the TI-34, came with a manual. This calculator can convert numbers to binary as well as two binary-related bases, octal and hexadecimal. But in order to introduce the concept of number bases, the manual uses base five, the "handal" system.

{5} (default quinary/handal)

In this base, 10 is simply called "hand." The count goes as follows: one, two , three, four, hand, handy-one, handy-two, handy-three, handy-four, twandy (20). The other numbers also sound like decimal numbers with the syllable "hand" thrown in: thrandy (30), frandy (40), and of course, we must have one handred (100).

The manual didn't reach 1000. I suppose the easiest way to get my "hand" in is "one thouhand." Then 10000, following the decimal pattern, would be "hand thouhand." Perhaps it's better to change the Greek word "myriad" instead -- "one myrihand." So today's Pappas problem becomes:

The square root of one myrihand is one handred.

{a} (default decimal)

Since handal is such as small base, it has many simple divisibility rules. Divisibility by 2 and 4 is covered by the omega, while 3 and 6 are covered by the alpha. The SPD rule -- used for divisibility by 5 in dozenal, applies to 13 (twandy-three) in handal, since 26 = 2 * 13. Here SPD is based on one more than the square of the base (square-alpha). There is also SPD based on square-omega. It's useless in even bases since any factor of square-omega is available in either alpha or omega. But in odd bases, an extra power of two (beyond alpha or omega) is available in square-omega. In this case, square-omega gives us 8 (handy-three), since 24 = 3 * 8.

Quinary is small enough that SPD based on cube-alpha is also available. One more than 5^3 is 126, which factors as 2 * 7 * 9, so this gives us 7 (handy-two) and 9 (handy-four). For 7, we need to memorize 18 multiples of seven. This is fewer than the 29 multiples of five needed to do 5 in dozenal, but each multiple of seven has three quinary digits, so it's a wash. For 9, we only need to memorize 14 multiples of nine.

So quinary has feasible divisibility rules for factors 2-10. The useful numbers 2-6 have easier rules (divisor, alpha, omega) while the less useful 7-9 have more obscure rules (SPD).

But quinary has no simple rule for 11 (twandy-one). In fact, we can prove that 11 can't have a simple rule based on SPD for squares or cubes -- 11 can only have divisor, alpha, or omega rules. Thus decimal and dozenal are the smallest bases with an easy rule for 11 (along with undecimal of course).

The only possible SPD rule for 11 would be based on fifth powers. Since fifth powers are so large, this is infeasible for all bases -- except:

{2} (default binary)

It's possible to use SPD for eleven (1011) in binary. Only three multiples need to be memorized, and all are trivial: 00000, 01011, and 10110.

{a} (default decimal)

In ternary, 23 multiples of eleven must be memorized. This is fewer than 5 in dozenal, but the multiples contain five digits each, so it's difficult. Binary is below human-scale, but we can do 11 in bases 4, 8, and 16 by converting them to binary first.

OK, let's get back to music. Today we wish to code Mocha for 16EDL music:

The 16EDL scale:
Degree     Ratio     Note
16            1/1         white E
15            16/15     green F
14            8/7         red F#
13            16/13     ocher G
12            4/3         white A
11            16/11     amber B
10            8/5         green C
9              16/9       white D
8              2/1         white E

In 12EDL, it's easy to alternate between the 12:10:8 (that is, 6:5:4) and 11:9:7 chords. For 16EDL, we must get a little more creative. Our main chord is 16:13:11, and so our secondary chord might end up being something like 15:12:10 (a just major triad). Here's a possibility:

10 CLS
20 N=16
30 FOR A=0 TO 6
40 B=4
50 X=A-INT(A/2)*2
60 IF X=0 THEN D=16 ELSE D=15
80 L=RND(B)
90 SOUND 261-N*D,4*L
100 IF L>1 THEN FOR I=1 TO L-1:PRINT "   ";:NEXT I
110 B=B-L
120 IF B>0 THEN D=17-RND(9):GOTO 70
140 NEXT A
150 PRINT 16
160 SOUND 261-N*16,16

So in this song, each measure starts with either 16 or 15 to represent the root of the 16:13:11 and 15:12:10 triads, and then random notes fill out each measure.

Yesterday, I wrote about converting the "Dren Song" to EDL. We might program it like this:

10 N=20
20 FOR V=1 TO 2
30 FOR X=1 TO 12
40 FOR Y=1 TO 7
60 IF V=1 THEN SOUND 261-N*D,4
70 IF V=2 THEN SOUND 261-N*(20-D),4
90 FOR I=1 TO 400
100 NEXT I,X
120 NEXT V
130 DATA 12,11,10,12,13,12,11
140 DATA 13,12,11,13,12,11,10
150 DATA 12,11,10,12,13,12,11
160 DATA 13,12,11,13,12,11,12
170 DATA 10,9,8,10,11,10,9
180 DATA 12,11,10,12,13,12,11
190 DATA 10,9,8,10,11,10,9
200 DATA 11,10,9,11,10,9,10
210 DATA 10,9,8,10,9,8,7
220 DATA 11,10,9,11,10,9,8
230 DATA 10,9,8,10,9,8,7
240 DATA 9,8,7,9,8,7,8

This song begins with N=20, which gives us F minor. Another alternative is N=13, which would give us ocher C minor (to represent C# minor, the key in which I originally wrote the song).

The song consists of 24 lines, with the second 12 lines as the inversion of the first 12. The inversion is played by replacing D with 20 - D depending on whether it's the first (V=1) or second (V=2) verse.

Each line consists of seven quarter notes followed by a quarter rest (the FOR I loop).

What scale should we declare the song to be in. Here 12 is the root note, but it's not quite 12EDL because 13 is used. It's sort of like Bart Hopkin's mode 12 of the 16EDL scale. Since 15 isn't used, perhaps it's more accurate to call it mode 12 of the 14EDL scale. Then again, maybe we can simply call it 12EDL, add 13.

This just goes to show us that sometimes rather than having a span of one octave beginning and ending on the tonic, it's better to use a different sort of range. In this song, the range is 13-7, which 12 as the tonic.

As usual, with each EDL we play, I like to find EDO's we can convert it to. Let's run the EDO program I wrote a few years back, and input 16EDL. Here is the list of EDO's it outputs:

1, 2, 4, 6, 7, 10, 24, 31, 41, 53, 87, 130, ...

The first non-trivial EDO we see here is 24EDO. Unlike 19EDO and 27EDO which are tricky to play (especially the latter), 24EDO is easier because it's closely related to 12EDO.

According to this link, 24EDO is a contorted 12EDO in the 5-limit -- that is, all 5-limit intervals (white, green, yellow) are the same in 24EDO as in 12EDO. The twelve new notes are quarter tones (half of a semitone). The quarter tones are used to give us 11 and 13. As it turns out, using the quarter tones for 7 (subminor and supermajor) is slightly more accurate than simply using 12EDO.

Here is some more information about 24EDO from the link, including Kite's colors. (As usual, all links will be dead after July.)


The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as quarter-tone tuning, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in Arabic music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - see this page

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
qualitycolormonzo formatexamples
downminorblue{a, b, 0, 1}7/6, 7/4
minorfourthward white{a, b}, b < -132/27, 16/9
"green{a, b, -1}6/5, 9/5
midjade{a, b, 0, 0, 1}11/9, 11/6
"amber{a, b, 0, 0, -1}12/11, 18/11
majoryellow{a, b, 1}5/4, 5/3
"fifthward white{a, b}, b > 19/8, 27/16
upmajorred{a, b, 0, -1}9/7, 12/7

The 11th harmonic, and most intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.

Alternatively, ups and downs notation can be used. Here are the blue, green, jade, yellow and red triads:
color of the 3rdJI chordnotes as edostepsnotes of C chordwritten namespoken name
blue6:7:90-5-14C Ebv GC.vmC downminor
green10:12:150-6-14C Eb GCmC minor
jade18:22:270-7-14C Ev GC~C mid
yellow4:5:60-8-14C E GCC major or C
red14:18:270-9-14C E^ GC.^C upmajor or C dot up

Even though we've been using "d" for quarter-flat, the quarter-sharp symbol isn't easy for me to write in ASCII format. So let's follow the suggestion and use "^" for up/quarter-sharp, and so "v" will be down/"quarter-flat."

Here is the 16EDL scale in E, converted to 24EDO:

The 16EDL scale, converted to 24EDO:
Degree     Ratio     Note
16            1/1         E
15            16/15     F
14            8/7         Gv
13            16/13     G^
12            4/3         A
11            16/11     Bv
10            8/5         C
9              16/9       D
8              2/1         E

Since 24EDO agrees with 12EDO in the 5-limit, 24EDO tempers out both the syntonic comma and the Pythagorean comma. By tempering out the syntonic comma, 24EDO is a meantone tuning in which C-E is a major third. By tempering out the Pythagorean comma, G# and Ab are enharmonic, and B# and C are also enharmonic.

A YouTube search for 24EDO reveals the following video of a quarter-tone guitar. (No, I will not be playing my 16EDL songs on the 24EDO guitar this summer.)

OK, since I'm posting videos anyway, let me post the "Be Sharps" video from The Simpsons:

Here are the key parts I wish to emphasize in this video. The song "Coney Island Baby" begins at the 0:24 mark of the video. It starts with Seymour (bass), Homer (baritone), Clancy (lead), and Apu (tenor) singing a barbershop 7th chord, with each singing the word "Goodbye." The notes of this chord follow a 4:5:6:7 (otonal) pattern. The root of the chord is F, and so in septimal meantone, the notes are F-A-C-D#.

Then at 0:48, the quartet sings a D chord (the lyric here is "again"). In septimal meantone, a D barbershop chord is D-F#-A-B#, and so the "Be Sharps" really do sing the note B#. (Later Barney replaces Clancy as the lead, and D-F#-A-B# occurs near the middle of "Baby on Board," but there is no clean cut of this chord.)

Again 19EDO and 31EDO are good EDO's for septimal meantone. And 12EDO also supports septimal meantone, even though B# sounds as C. But 24EDO, while still meantone, isn't a septimal meantone EDO, since a 7/4 above D is now B quarter-sharp, not B#.

There are also higher EDO's that approximate the 13-limit. In fact, I can tie this to number bases -- you see, there was a debate on Dozens Online about the best EDO for 13-limit:

The original poster, Tony, asks, "Dare I admit good musical things about decimal?" He then points out that half of 100 in decimal is 50, and 50EDO approximate the 13-limit well. Indeed, 50EDL is also a septimal meantone EDO.

But notice that 50EDO doesn't appear in the list of good EDO's for 16EDL. Instead, another EDO emerges as a good EDO for 13-limit -- 72EDO. (Admittedly 72EDO doesn't appear in the 16EDL list, but it does appear in the list for 14EDL, which is also 13-limit.)

Another poster, Ebbe, explains that 72EDO, unlike 50EDO, isn't a meantone scale (much less septimal meantone). Indeed, the step size for 50EDO is about the size of the syntonic comma, and so it's not surprising that the best EDO's in this range are no longer meantone. (But 72EDO nonetheless tempers out the Pythagorean comma, so we still have enharmonics G#=Ab and B#=C.) And 72 is half of the dozenal hundred (gross), and so it's "dozenal for the win again."

In 72EDO, six steps make a semitone and twelve steps make a whole tone. Thus each step of the scale can be called a twelfth-tone.

The major third above C is now one step below E -- that is, the scale maps the syntonic comma to a single step. The septimal comma is now two steps. The jade quarter-tone is now three steps (as 3/12 equals 1/4), and the ocher third-tone is now four steps (as 4/12 = 1/3). So we can see how ocher fits 72EDO better than 24EDO.

In fact, here are some of the notes of 72EDO, in sequence (listed as thirds above C)

Step     Note     Color
16        Eb-2     blue Eb (subminor 3rd above C)
17        Eb-1     yellow Eb
18        Eb        white Eb
19        Eb+1    green Eb (minor 3rd above C)
20        Eb+2    red Eb (emerald E)
21        E-3       jade Eb (amber E, neutral 3rd above C)
22        E-2       blue E (ocher Eb)
23        E-1       yellow E (major 3rd above C)
24        E          white E
25        E+1      green E
26        E+2      red E (supermajor 3rd above C)

Within each classical name, the rainbow is emerald-amber-blue-yellow-white-green-red-jade-ocher.

Here's one more recent music-related post at the Dozens Online website:

The original poster Dan is writing about Phi, the Golden Ratio. In this post he ties Phi to music:

One practical use of "almost rational" irrational numbers is musical tuning.  For example, the equal-tempered major third has a frequency ratio of the cube root of two (1.259921 decimal, 1.315188 dozenal), approximating the just intonation ratio 5/4.  So, would a tuning based on setting the minor sixth to φ (or equivalently, setting the major fifth to (8/φ)^(1/4), or 691.7274 cents) sound dissonant?

(It's too bad this wasn't posted before Phi Day of the Century.) Anyway, we know that the ratio of consecutive Fibonacci numbers approaches Phi, so let's start with these:

Ratio     Name
1/1         unison
2/1         octave
3/2         perfect fifth
5/3         major sixth
8/5         minor sixth
13/8       large tridecimal neutral sixth
21/13     small tridecimal neutral sixth

The largest Fibonacci numbers within Mocha range are 144 and 233, so playing Degree 233 (Sound 28) and Degree 144 (Sound 117) best approximates Phi in Mocha.

Amond EDO's, the aforementioned 72EDO approximates Phi well. If the root note is C, then Phi appears two steps above Ab (about 833 cents). We can refer to this note as either emerald A (to represent 13/8) or ocher-blue Ab (21/13).

The Xenharmonic site has its own page about applying Phi to music:

Phi taken as a musical ratio (ϕ*f where f=1/1) is about 833.1 cents. This is sometimes called "acoustical phi".
As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 cents.

"Logarithmic phi", or 1200*ϕ cents = 1941.6 cents (or, octave-reduced, 741.6 cents) is also useful as a generator, for example in Erv Wilson's "Golden Horagrams".

("Acoustical Phi" is to "Logarithmic Phi" as EDL is to EDO. Thus Acoustical Phi appears in Fibonacci EDL's like 34EDL, while Logarithmic Phi appears in Fibonacci EDO's like 34EDO.)

Question 19 of the SBAC Practice Exam is on solving equations:

Consider this solution to a problem:

Problem: -4(6 - y) + 4 = -4
Step 1: -24 - 4y + 4 = -4
Step 2: -20 - 4y = -4
Step 3: -4y = 16
Step 4: y = -4

In the first response box, enter the number of the step where the mistake is made.
In the second response box, enter the correct solution to the problem.

This is a strong first semester Algebra I problem. The original problem has lots of negative signs, and so the obvious error to search for is a sign error. And sure enough, there is a sign error right away -- -4 times -y should be 4y, not -4y. So the step that contains the mistake is step 1.

The solution is easy to correct. All we have to do is change all the signs on y:

Problem: -4(6 - y) + 4 = -4
Step 1: -24 + 4y + 4 = -4
Step 2: -20 + 4y = -4
Step 3: 4y = 16
Step 4: y = 4

Question 20 of the SBAC Practice Exam is on solving equations:

Consider a sequence whose first five terms are: -1.75, -0.5, 0.75, 2, 3.25

Which function (with domain all integers n > 1) could be used to define and continue this sequence?

A) f (n) = (7/4)(n - 1) - 5/4
B) f (n) = (5/4)(n - 1) - 7/4
C) f (n) = (7/4)n - 5/4
D) f (n) = (5/4)n - 7/4

Because this is an arithmetic sequence, this is also considered to be first semester Algebra I. The first thing we notice is that the terms are listed as decimals, but the choices are all fractions. So the students must convert between decimals and fractions. There is an embedded calculator available, but that assumes that the students know how to use it to make the conversion. I used the calculator (powered by Desmos!) to divide to convert fractions to decimals. Some calculators can convert decimals to fractions, but I don't see that option on this calculator. So the first step would be to divide to convert 5/4 (a major 3rd!) and 7/4 (a barbershop 7th!) to fractions.

We find out that the first term -1.75 is -7/4 and the common difference is -1.25 or 5/4. But we can't use the first term unless we use the (n - 1) version of the formula, f (n) = f (1) + (n - 1)d. Plugging in to this formula, we obtain f (n) = -7/4 + (n - 1)5/4, which is rewritten as f (n) = (5/4)(n - 1) - 7/4. So the correct answer is B).

Today is an activity day. Here the activities come from Chapter 6 of the U of Chicago Algebra I text, since this chapter matches these questions the best. Lesson 6-9 of the U of Chicago text is called "Subtracting Quantities" and emphasizes use of the Distributive Property with negatives, while Lesson 6-3 contains arithmetic sequences. The Exploration Question in the latter lesson is on programming in BASIC. Erase that music song using NEW so we can program Mocha to answer the activity question, or just skip that part of the activity.

SBAC Practice Exam Question 19
Common Core Standard:
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

SBAC Practice Exam Question 20
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Commentary: Distributing a negative sign correctly appears in Lesson 6-9 of the U of Chicago Algebra I text. But the method of finding explicit formulas in Lesson 6-3 -- finding the phantom "zeroth term," doesn't work for Question 10. Instead, students must know and be able to use the formula f (n) = f (1) + (n - 1)d, which most likely appears in later texts based on the Common Core, such as Glencoe and Edgenuity.

Monday is Memorial Day, and so my next post will be on Tuesday.