"Only a few of the many thousands of mathematicians and ideas are mentioned in this chapter. Those I discuss in no way reflect their superiority or stature over those not appearing. I urge you to use these sections as spring boards for further study and to enhance your understanding of the magic of the past."
[emphasis the author's]
This is last page of the chapter introduction. Pappas provides a long list of six dozen famous mathematicians (and scientists) on this page. Here are some of them:
Kepler, Galois, Einstein, Zeno, Jacobi, Godel, Lobachevsky, Laplace, Legendre, Bolyai, Eudoxus, Poincare, Thales, Riemann, Kronecker, etc.
I chose these names from her list at random. So her disclaimer applies -- the names I chose don't reflect their superiority either. As far as further study is concerned, notice that I've already devoted several posts to some of these mathematicians:
-- I mentioned Kepler and Thales while reading Morris Kline's Mathematics and the Physical World.
-- I mentioned Galois while reading Mario Livio's The Equation That Couldn't Be Solved.
-- I'm currently watching Einstein on National Geographic's Genius.
-- I mentioned Zeno's and Godel's paradoxes while watching David Kung's Mind Bending Math.
-- Lobachevsky, Bolyai, and Poincare are the fathers of hyperbolic geometry. (I mentioned Poincare.)
-- Legendre and Riemann are the fathers of spherical geometry. (I did mention these two of course.)
-- Kronecker was quoted by Stephen Hawking, the subject of the movie The Theory of Everything.
Well, Pappas, it appears that my readers and I already have a spring board for further study of the famous mathematicians -- namely this blog.
Question 3 of the SBAC Practice Exam is on irrational numbers:
3. Select True or False to indicate whether each comparison is true:
-- sqrt(37) < 5 1/4
-- 3pi > 3sqrt(3)
-- sqrt(5) > 5/7
-- 15/sqrt(10) > 8.38
This is tricky. In theory, there's supposed to be an online calculator, but I haven't been able to figure out how to access it.
Fortunately, there's a trick to answering this without a calculator. Notice that every single radicand in this question is either one more or one less than a perfect square. This makes it easy to estimate the values of the square roots -- for example sqrt(37) is a little more than 6. So rounding off gives us:
sqrt(37) < 5 1/4
6 < 5 1/4
which is clearly false. And notice that sqrt(37) is even greater than 6, so if 6 isn't less than 5 1/4, sqrt(37) can't be less than 5 1/4 either. Notice that had this been sqrt(35) rather than sqrt(37), we couldn't easily compare it to 5 1/4, since sqrt(35) is less than 6, but we don't know how much less than 6 it is, so it could be less than 5 1/4. (In reality, we must drop all the way to sqrt(27) before it becomes less than 5 1/4.) So the first answer is False.
Fortunately, all of the rounding is in the correct direction to answer all of these problems. We see that sqrt(5) is more than 2, which is definitely greater than 5/7, so sqrt(5) > 5/7 is True. We also see that sqrt(10) is greater than 3, so 15/sqrt(10) must be less than 15/3 or 5 (as increasing the denominator makes the value smaller). Since 5 isn't greater than 8.38, neither can 15/sqrt(10) be greater than 8.38, so the answer to the last part is False.
This leaves us with 3pi > 3sqrt(3), which contains pi as well as sqrt(3). We see that sqrt(3) is less than 2, so the right hand side is less than 6. We estimate pi as 3, so the left hand side is more than 9. So we have that something more than 9 is greater than something less than 6, which is clearly True. Thus the correct answer is False, True, True, False.
Question 4 of the SBAC Practice Exam is also on irrational numbers:
Select all possible values of x in the equation x^3 = 375.
-- 5cbrt(3)
-- cbrt(375)
-- 75cbrt(5)
-- 125cbrt(3)
The obvious answer is cbrt(375). It turns out that this can be simplified since 375 is thrice 125, the cube of 5. So the other correct answer is 5cbrt(3). Notice that both wrong answers involve removing factors from the radicand without taking their cube root -- both would be correct if they were given as cbrt(75)cbrt(5) and cbrt(125)cbrt(3).
SBAC Practice Exam Question 3
Common Core Standard:
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
SBAC Practice Exam Question 4
Common Core Standard:
Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Commentary: These questions frustrated my eighth graders. Earlier in the year, I talked a little about how sqrt(37) is slightly more than six, but only briefly. Notice that the Illinois State square root arts project might have helped, as could a Square Root Approximation Day. As for the second, I'd only made brief mention of how to simplify roots, so sqrt(27) = 3sqrt(3) -- and no mention of simplifying cube roots at all. Now I know what to emphasize more in the future.
Monday is Memorial Day, so my next post will be on Tuesday.
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