Friday, September 9, 2016

Dren Quiz #1 (Days 17-18)

Today the eighth graders take their first Dren Quiz of the year. Recall that a Dren Quiz is my name for the basic multiplication quiz that I give from time to time in my classes.

Of course, I don't devote the entire 80-minute block to the Dren Quiz. After the Warm-Up and collecting the homework, I return to converting repeating decimals to fractions. But now I begin with a very famous problem:

Convert 0.999... to a rational number.

Let's use the method that I showed the students earlier this week:

x = 0.999...
10x = 9.999...
  -x = -0.999...
9x = 9
x = 1

And so we see that 0.999... is actually equal to 1. This counter-intuitive result has been discussed time and time again at various websites, for example, Cut the Knot:

and Metamath:

The proof that 0.999... = 1 is as rock-solid as the proof that 2+2 = 4 -- but according to the above link, 40% of the respondents to a certain poll disagreed with 0.999... = 1.

The famous YouTube star Vi Hart even created a video about 0.999... = 1:

Meanwhile, at Math Forum, debaters like to discuss 0.999... all the time. Here is a link to the most recent 0.999... debate:

Notice that the original dissenter created his own video on why 0.999... should not equal 1. I tell my students that people like John Gabriel, the video creator, and Wolfgang Muckenheim -- who is, believe it or not, a professor at a German university -- are called not "drens," but cranks due to their inability to accept a simple proof -- and that if they understand this proof, they are already smarter than a German professor!

After talking about 0.999... = 1, I sing my latest song at music break. Notice that this song has a separate verse for each grade level, in order to reflect what they are learning:


6th Grade:
What can we do with fractions?
What can we do with fractions?
As everyone knows
We write ratios.
You can do no worse
If you write the 1st one 1st.
In between write dots
Then the 2nd -- that's a lot
We can do with fractions!

7th Grade:
What can we do with fractions?
What can we do with fractions?
We see over there
With ratios we can compare.
The fractions to divide
Flip the 2nd & multiply.
Remember to simplify
And now you know why
We can use fractions!

8th Grade:
What can we do with fractions?
What can we do with fractions?
We can make them decimal
And that is not all.
Another major feat
Is that decimals repeat.
They go on forever.
Know that whenever
We can use fractions!

After the break, I like to lead into the Dren Quiz by discussing various traditionalists who emphasize the need to learn basic skills. I've said before that I want to emphasize the website of Barry Garelick especially since now he's a middle school teacher himself. His most recent post does discuss the need to learn basic math -- but as usual, it's SteveH who makes the specific comment:


In my son’s fifth grade class, many bright students still didn’t know the times table and some had to add 7+8 using their fingers.

Notice that SteveH doesn't blame the students for being drens, rather the school and its curriculum -- the U of Chicago elementary text that many traditionalists despise.

I say that part of the issue is about the generation gap:

1945 = End of World War II
1946-1964 = Baby Boomers
1964-1982 = Generation X
1982-2000 = Generation Y, or Millennials
2000-present = Generation Z
1978-2008 = The "Dumbest Generation" of Mark Bauerlein

Naturally my students don't like hearing a Boomer like Bauerlein tell them that they are in the "Dumbest Generation" -- but then again, I tell them that I myself am part of this generation as well. I try to be careful enough to use pronouns "we" and "us" in explaining why we're considered part of the Dumbest Generation.

By the way, the boundaries between the generations have been much debated. Next week, the CBS show Survivor will feature tribes made up of Gen X'ers and Millennials. The oldest player on the Gen X tribe is 52 and the youngest is 33, which matches well with the dates listed above. I always knew that I myself on the cusp of two generations. By this definition I'm a late Gen X'er, as I'm older than the youngest member of the Gen X tribe. Of course, I and the other youngest Gen X'ers are fully included in Bauerlein's "Dumbest Generation."

Earlier I said that I wouldn't quote Bill at the Joanne Jacobs site this year -- and I don't. I only read the quotes by two science teachers -- nothing by Bill:

Miller T. Smith:

I have honors chemistry students who have passed three high school level math classes who can’t do fractions, scienctfic notation, don’t kow the mutipliction tables, can’t graph on paper, can’t multiply fractions, on and on.

Physics Teacher:

I’ve had kids who can’t multiply by ten or divide by one.
How are these students supposed to grasp the idea of momentum if vector quantities are completely beyond there grasp?

I want to make sure that the students see some comments about the inability to multiply by ten right before their first Dren Quiz, which is on multiplying by ten. The idea is that the students should be motivated to prove people like the two science teachers and Bauerlein wrong.

As today is the first Dren Quiz, everyone should be starting with their tens today. But last week, two students had trouble logging into the computer system and so they couldn't use the online program that they were supposed to be using. These two girls were still upset that they hadn't fared well on the eighth grade test I'd given them earlier that day -- but then they found a stack of Dren Quizzes left over from the sixth grade lying on my desk. So they just took the quiz right then and there.

The top student in my class -- the girl I'm hoping to teach Algebra I this year -- is wondering why I'm insulting her by giving her a lowly Dren Quiz. On the other hand, the bottom student in my class enjoys the Dren Quiz and is hoping that I'll give another such quiz soon.

This is a two-day post, officially covering today and Monday. But Mondays are coding days with very little math taught, so there wouldn't be much for me to write about Monday anyway.

But notice that Monday -- Day 18 -- marks the end of the first 10% of the year, which I've designated as the Willis Unit (or Wong Unit). During the Willis Unit, I've learned the names of all my students as well as their strengths and weaknesses, while the students have learned all about me as a teacher.

I've made several changes to my plans during the Willis Unit. I've rotated my Dren Quizzes, general quizzes, and tests during the Willis Unit, but now my plan is to switch to giving them to all three grades on the same days.

I've also decided, after switching from local Participation Points to Scholar Dollars for the entire school, to return to Participation Points. What happened was that a few seventh graders ended up stealing a large number of Scholar Dollars, and my supply of them hasn't been replenished because I can't be trusted with them. So I'd rather return to Participation Points -- which I just record as marks on a sheet of paper.

By the way, speaking of the younger grades, I decided to let my seventh graders use their Foldables on the harder test, but the sixth graders couldn't use them on the easier quiz. But one sixth grader tries to use a Foldable anyway -- forcing me to give the student a zero and make a parent phone call.

My next post will be on Tuesday.

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