Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
What is the radius of the circle?
(Here are the givens from the diagram: inscribed in the circle is a pentagon with side lengths 32, 24, 20, 20, 20, with a right angle between the longest two sides.)
Unlike the trig problems from earlier this week, this is a true Geometry problem.
To solve this, we draw in a diagonal of the pentagon, from the far vertex of the 32 side to the far vertex of the 24 side. Then this diagonal becomes the hypotenuse of a right triangle with legs 24 and 32. From the Inscribed Angle Theorem, an arc subtended by a right angle is a semicircle, and so the hypotenuse of the right triangle is in fact the diameter of the circle. We now use the Pythagorean Theorem:
24^2 + 32^2 = c^2
576 + 1024 = c^2
1600 = c^2
c = 40
So the diameter of the circle is 40, and so its radius is 20. Therefore the desired radius is 20 -- and of course, today's date is the twentieth.
Notice that the three pentagon sides of length 20 have nothing to do with the radius being 20. We already know that the inscribed pentagon is hardly regular (since two of its sides aren't congruent to the other three, and it has a right angle instead of all angles being congruent). But now that we know that three of its sides are equal to the radius, we show that these three sides form half of a regular hexagon.
It's been a while since I've linked to the Sarah Carter website -- usually I only think about Carter when it's the first week of school, or when I'm teaching her DIXI-ROYD mnemonic. But there's something on her website that I just can't help bringing up:
https://mathequalslove.net/cross-solitaire-puzzle/
The goal is to pace the 32 markers in a cross pattern with the middle square unoccupied. Then, jump the markers horizontally and vertically so that a single marker remains in the center of the board.
Click on the link above -- does this puzzle look familiar?
That's right -- it's exactly the Peg Solitaire game that Arthur Benjamin mentions in Lecture 8 of The Mathematics of Games and Puzzles, which I featured in my January 22nd post. What Carter has done is create a version of this game that can be played in the classroom. She calls this version Cross Solitaire, since she doesn't use pegs (she uses bingo chips instead).
Readers who have seen Carter's version first can refer back to my January 22nd post for tips on how to solve this puzzle (or better yet, watch Benjamin's Lecture 8 if you can get the DVD). On the other hand, those who read my blog but don't own Benjamin's DVD and are confused because they can't visualize the puzzle (that I describe with words only) can click on the Carter link above and finally see it.
This is what I wrote two years ago about today's lesson:
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.
Both the girl and the guy from the Pre-Calc class correctly answer b^7 for this question.
Question 22 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 asked 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))
(f (b) - f (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.
Both the girl and the guy from the Pre-Calc have trouble with this question. In fact, the guy doesn't even attempt it -- he just leaves it blank.
The girl, on the other hand, tries to answer and struggles. There are many places where she writes something and then crosses it out. So let's follow her reasoning in more detail in this post.
By the way, recall that this post from a few years ago was one of my most popular posts by hit count. For some reason, I doubt that Question 21 and b^11/b^4 = b^7 is the reason for the hits. Teachers have been searching for my page because their students are unable to figure out Question 22. Chances are that students everywhere are making mistakes similar to those that this girl makes in my class.
Here's how she begins:
(21 - 18)/(18 - 9) = 15/9
She starts to simplify 15/9 by dividing the numerator and denominator by three, but then she crosses it out when she realizes that 21 - 18 is not 15. I suspect what happens is that instead of performing the subtraction 21 - 18, her mind thought of the arithmetic sequence 21, 18, 15. (I admit that sometimes I want to say 5 + 8 = 11 for the same reason.) So she crosses 15 out and continues:
3/9 = 1/3
And we know that 1/3 is indeed the correct answer -- but unfortunately, the girl doubts herself. So she writes something else on the next line:
21/18 - 18/9
that is, she breaks up (21 - 18)/(18 - 9) as two separate fractions 21/18 and 18/9 and then attempts to subtract them. She realizes that 21/18 < 18/9, so she crosses 21/18 out and rearranges the fractions so that the answer would be positive:
18/9 - 21/18
36/18 - 21/18 = 15/18
Then she divides the numerator and denominator by three to obtain 5/6. So she writes:
5/6 = 0.833
Notice that even if 5/6 is correct, the question asks to round to the nearest tenth. Instead, she has rounded it to the nearest thousandth.
Finally, the girl suddenly crosses 5/6 = 0.833 and just writes 3 as her answer. I'm not sure where she gets the 3 from, but notice that in some ways 3 has more in common with the correct answer. The correct average rate of change, after all, is 1/3 foot per hour -- that is, 3 hours per foot. Every three hours, the river rises by a foot. The problem is that we must specify rates as feet per hour, instead of hours per foot. Otherwise the girl's answer of 3 would be correct.
I'm not sure what to do about problems such as these. Even though "average rate of change" isn't a difficult concept (since all it means is "slope"), the phrasing is awkward. We've seen that many texts don't even use the phrase -- I myself never saw the phrase until I was in Calculus. I suspect that new Algebra I texts printed after the advent of the Common Core use that phrase. But this still doesn't mean that our students will be able to remember it, as we see with this Pre-Calc girl and guy today.
A Google search for average rate of change returns the following as the top two results:
http://home.windstream.net/okrebs/page201.html
This page points out that average rate of change is the slope of the secant line. This is contrasted with the derivative -- the slope of the tangent line.
https://www.khanacademy.org/math/algebra/algebra-functions/average-rate-of-change-word-problems/a/average-rate-of-change-review
It's interesting to read some of the comments left by the Khan Academy users. For example:
NCARalph:
Isn't teaching average rate of change like this pretty misleading? Sure, mathematically given 2 points you can get a rate of change, but for real world data you should do something like a least squares fit first. For non-linear equations, what does it tell you at all?
We teach it this way because this is what the Common Core demands. Thus Ralph's beef would be with the Common Core, not Khan Academy.
We teach it because it's something that's easy to calculate -- requiring Algebra, not Calculus -- and at least provides a crude approximation of how fast something is changing. If you're stuck in a flood, you might want to know how fast the water is rising so that you can estimate how much time you have to escape.
SBAC Practice Exam Question 22
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.
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