Today I subbed in a junior English class. It's in my LA County district. Since it's a high school class that isn't math, there's no need for "A Day in the Life" today.
This time, the students have a video to watch -- not a movie about a novel they've read, but a TED video on the need to belong to a group. The speaker tells the story of a girl who moved from foster home to foster home until she was finally placed in a permanent home.
I have trouble choosing a song for today. The closest I can get to today's topic is "Mousetrap Car Song," only because the last verse is on learning to communicate with a group. (When I first wrote this song at the old charter school, it was a separate song, until I rewrote it as a verse of the mousetrap car song.)
Today is Sevenday on the Eleven Calendar:
Resolution #7: We sing to help us learn math.
Well, I definitely sing "Mousetrap Car Song" today, even though this isn't really a math class. Singing is the easiest resolution for me to follow whenever I find myself in an English or any other non-math class.
This is what I wrote two years ago about today's lesson:
Question 9 of the SBAC Practice Exam is on irrational numbers:
Cheryl claims that any irrational number squared will result in a rational number.
Part A
Drag an irrational number into the first response box that when squared will result in a rational number.
Part B
Drag an irrational number into the second response box that when squared will result in an irrational number.
Here are irrationals that can be dragged: cbrt(2)/sqrt(3), sqrt(3)/sqrt(2), cbrt(2), sqrt(2), pi, sqrt(pi).
This is a tricky one to place. In the Common Core Standards, rational and irrational numbers appear in the eighth grade, and so it's arguably not a high school topic at all. If they do appear in an Algebra I text, it's likely to be in the context of quadratic equations, thus it's a second semester topic.
It's also the first question in the calculator section. Then again, calculators won't really help students with rational and irrational numbers.
Cheryl's idea that the square of an irrational number must be rational is an attractive one. After all, the first irrational numbers we encounter are numbers like sqrt(3), which when squared produces the rational number 3.
But, as Georg Cantor shows us (discussed in old posts), most numbers are irrational. Therefore, the square of most irrational numbers is still irrational. The list of choices includes cube roots, so their squares are still irrational. In fact, Cheryl's conjecture that the square of an irrational must be rational is just like a claim that doubling any fraction produces a whole number (presumably because the most commonly used fractions like 1/2 and 1 1/2 indeed have that property).
The choices involving pi are tricky. We know that the square of sqrt(pi) is pi, which is clearly irrational, and so sqrt(pi) would be dragged into the second box. But to which box should the number pi itself be dragged? I doubt that any high school text explains that pi^2 is irrational. We know that pi is transcendental, and so no integer power of pi can be rational -- but high school students wouldn't be expected to know this. Of course, students can forget about pi and just drag sqrt(pi) or one of the cube roots into the second box, since only one number needs to be dragged there.
So here is a complete answer: only sqrt(3)/sqrt(2) or sqrt(2) can be dragged into the first box. All other numbers are possibilities for the second box.
Both the girl and the guy from the Pre-Calc class correctly answer this question, even though they choose different answers. The girl uses sqrt(3)/sqrt(2) for the first box and cbrt(2) for the second, while the guy uses sqrt(2) for the first box and pi for the second.
Question 10 of the SBAC Practice Exam is on building equations:
A train travels 250 miles at a constant speed (x), in miles per hour.
Enter an equation that can be used to find the speed of the train, if the time to travel 250 miles is 5 hours.
The guiding equation is d = rt, rate times time equals distance. The rate of speed is x, the time is 5, and the distance is 250. Therefore the equation is 5x = 250.
I consider this to be a first-semester Algebra I problem. While we might avoid the formula in middle school (or perhaps even mention dimensional analysis), by the time the students reach Algebra I, I should teach them the formula, the guiding equation.
Both the girl and the guy from the Pre-Calc class correctly answer this question, but I wonder whether the SBAC computer would mark them as correct. The guy writes 250 = x * 5 (and gives the solution as 50 mph). This is correct, but mathematicians usually write 5x not x * 5. The girl writes her equation as s = 250/5. Technically this equation can be used to find the speed, but I suspect SBAC is looking for a multiplication equation, not division with the variable isolated. (Actually, now I definitely think SBAC will mark it as wrong because she uses the variable s when the directions plainly state to use the variable x.)
SBAC Practice Exam Question 9
Common Core Standard:
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
SBAC Practice Exam Question 10
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: Lesson 12-6 of the U of Chicago Algebra I text is on Rational and Irrational Numbers, and the Exploration Question there is on rational solutions of quadratics. Lesson 4-4 of the U of Chicago text is on Solving ax = b, with d = rt mentioned as an example. Students should have no problem with this question if they know the guiding equation.
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