Lesson 4.3.1: Solving Two-Step Equations (Day 68)

Today is Giving Tuesday, the day after Cyber Monday. Back in May, a second Giving Tuesday was declared, but the real one is today. It's the last day of the special Thanksgiving week where almost everyday has a special name.

Since it's Tuesday, this is a seventh grade post. The title of this post is "Solving Two-Step Equations," but in reality, today wasn't anywhere nearly as simple as doing the Texas two-step.

For starters, we had to squeeze in the Performance Task portion of the Benchmarks today. So once again, it's a balancing act between giving the Benchmarks and teaching the new lesson on APEX.

In the end, I decided to go 30-40-30 for consistency. That is, the first half-hour of class is for last-minute review to make sure that the students understand what they are to do on the Benchmarks. Then the middle 40 minutes of class are for the Benchmarks proper. The last half-hour of class is the time into which I must squeeze the entire APEX lesson. (In Canvas, the test is set to unlock and relock at these particular times.)

You might think that 30 minutes would be enough time to show the students two-step equations -- especially considering that I already introduced two-step equations a few weeks ago in order to prepare them for the multiple choice Benchmarks. But the problem is that the APEX Lesson 4.3.1 isn't just about two-step equations. First, it discusses how to convert real-world phrases into math (the usual "total," "ten less than," and all that). Second, problems are solved by working backwards rather than traditional algebraic manipulation of equations. Third, two-step equations are actually solved. And finally, equations of the form p(x + q) = r are solved by first dividing by p, not by distributing p.

Time is wasted when APEX decides to slow down -- for some reason, at certain times it doesn't let me advance to the next page of the lesson. Then I almost tore my hair out trying to decide which parts of the lesson to cover and which parts to throw out.

Of course, I'll fix these problems and improve the lesson throughout the week. But this isn't necessarily fair to the Tuesday cohort. They always get the short end of the stick with an unpolished lesson as I figure out the best things to teach as the week proceeds. I suppose that even before the pandemic, this has always been a problem with secondary teachers -- first period gets the rough version of the lesson, while the last class gets the polished version as teachers adjust throughout the day. (To this end, period rotations are useful because the same group of students isn't always first.)

And indeed, I'm still trying to figure out what to teach tomorrow. The last section of the lesson, on solving p(x + q) = r, will certainly be thrown out. (By the way, the APEX method is influenced by the way it appears in the Common Core Standards.) But then the department head tells me that I should replace this with solving equations px = r where p is a fraction -- this isn't adequately covered by APEX at all. And so there's still lots for me to cover. One way to save some time will be to have the students fill out the chart for the first part (listing all the addition phrases, subtraction, etc.) as they individually finish the test.

OK, let's get to the songs for this week. The eighth graders didn't start Benchmarks until this week, and so I start with the "Benchmark Tests" song. After the test, I sing the main song (that is, the ones whose lyrics I post up in the room) -- Square One TV's "Nine, Nine, Nine." Here is that old YouTube video:

Here are the lyrics to this song:

Nine Nine Nine

Lead vocals by Reg E. Cathey

Backup vocals by Cynthia Darlow

Nine nine nine
Fantastic number nine
It’s perfectly consistent
It works out every time
Nine nine nine
That crazy number nine
Times any number you can find
It all comes back to nine

Two times nine is eighteen
Eight and one is nine
Three times nine is twenty-seven
Seven and two is nine
Four times nine is thirty-six
Six and three is nine
Five times nine is forty-five
Five and four is nine

Six times nine is fifty-four
Five and four is nine
Seven times nine is sixty-three
Six and three is nine
Eight times nine is seventy-two
Seven and two is nine
Nine times nine is eighty-one
Eight and one is nine

Nine nine nine
Fantastic number nine
It’s perfectly consistent
It works out every time
Nine nine nine
That crazy number nine
Times any number you can find
It all comes back to nine

Nine times ten is ninety
Just drop the zero sign
Nine times eleven is ninety-nine
Makes me rhyme with another line
Then nine and nine is eighteen
And eight and one is nine
Times any number you can find
It all comes back to nine

This work for bigger numbers, too?
Yup.
Let’s try this’un: three thousand four hundred’n eighty seven
Hmm … nine times three thousand four hundred eighty seven is
Thirty-one thousand three hundred eighty-three
Three plus one plus three plus eight plus three is eighteen;
One plus eight is nine.
It always works!

Nine nine nine
Fantastic number nine
It’s perfectly consistent
And it works out every time
Nine nine nine
That crazy number nine
Times any number you can find
It all comes back to nine
It all comes back to nine
It all comes back to nine

The reason I choose this song is that this is my ninth week of teaching at this school. Thus the Hero Quiz for fifth period Math Skills will be on the 9's times tables. Even though I don't sing songs during fifth period, just having the lyrics posted in the room should help them with their 9's.

The songs for seventh grade are another matter. First of all, this group already got "Benchmark Tests," and so they need another song for me to sing. It turns out that one of the questions on their Benchmarks is related to toy trains, and so I choose the song "Old Toy Trains." It is one of the Christmas songs that I sang as a young elementary school student for the annual concert -- this was the song that my fourth grade class chose that year. (I explained those concerts in my Thanksgiving and Black Friday posts.) I perform this song today since it's related to a Benchmark question, and it just happens to be December, when singing Christmas tunes is justified.

After the Benchmarks, they also get "Nine, Nine, Nine," as that's the posted song (and there are seventh graders in my fifth period as well). But then one of the tasks that the department head has prepared for seventh grade happens to be in the format of a murder mystery -- and I just happened to write a song almost exactly one year ago about a murder mystery:

WHODUNNIT?

First Verse:
Was Mrs. fixing snacks in the kitchen?
Was Dr. hearing songs in atrium?
Professor's iPad by the pool now?
Did Miss kill the victim?

Second Verse:
A cougar's mauling was the method?
Or had a green snake's bite begun it?
Or object's falling was the weapon?
Just tell me... Whodunnit!

And so I sing this song at the end of the third period Math 7 class.

Believe it or not, I could have performed a fourth song as well -- one of the APEX methods for solving problems is to work backwards, and there's a "Working Backwards" song from Square One TV as well (first performed by the Fat Boys). But I had to leave something out -- otherwise I'd just be singing the entire period and never teaching anything.

With all of these songs, I should say something about the guitar and musical tunings -- especially since I wrote "Whodunnit?" last year using a 12EDL scale (before I started thinking about a hypothetical 18EDL fretting -- indeed, playing this song on the guitar was the last thing on my mind at the time).

Since there's already so many lyrics in this post, I won't drag this out and start worrying about 12EDL and 18EDL scales right now. As for tuning, I kept my guitar in EACGAE rather than tune it to EGCGAE as I mentioned in my Thanksgiving post. This is because none of the songs I play today are in the key of G.

Right now, I need to worry more about how I'm going to improve my teaching for tomorrow's class, not endless discussion about hypothetical EDL scales.

END