"To give us the science of motion God and Nature have joined hands and created the intellect of Galileo." -- Fra Paolo Sarpi, contemporary of Galileo.
Kline begins:
"Galileo had formulated his program and he proceeded to put it into effect. Whereas scientists of Greek and medieval times had tried to embrace the whole of man and nature, Galileo, 'with the restraint that shows the master,' decided to select...the subject of motion."
So as we can see, Kline is still writing about Galileo's work. But now we've finally reached the great Italian scientist's most famous experiment:
"Hence, if air resistance is neglected, all bodies take the same time to fall a given distance. This is the lesson Galileo is supposed to have learned by dropping objects from the leaning tower of Pisa."
Galileo showed that all falling objects accelerate at 32 feet per second squared. From this, Kline is able to derive the equation d = 16t^2 for the distance an object falls in a given time. He does this without Calculus, but simply by finding the average velocity and multiplying by the time. This is the source of those -16t^2 problems that often appear in Algebra I courses (usually in the chapter on solving quadratics).
Question 12 of the PARCC Practice Exam is about segments and partitions:
The diagram shows
(Here are the endpoints: M(-4, 4), N(6, -2).)
Point P lies on
As it turns out, the correct answer is (7/2, -1/2) or (3.5, -0.5). Just by looking at the answer, we can tell that students will have trouble with this problem since the answer is not a lattice point. I hope that they wouldn't have trouble entering the answer in -- I assume that simply pressing the period key produces the decimal point and pressing the dash key produces the negative sign, and those are the only keys they need in addition to the digits.
But of course, we must figure out how the students are supposed to solve this problem. I wrote extensively about this type of question last year. This is what I wrote last year -- but there's one major change that I need to point out. (I also changed the discussion so that it refers to this year's problem, not last year's.):
We've discussed this type of problem here on the blog before. I pointed out that, while Lesson 11-4 of the U of Chicago text is on the Midpoint Formula, this sort of question where we are finding a point that divides the segment into a ratio other than 1:1 doesn't appear in the U of Chicago. Dr. Franklin Mason includes some questions like this in his text (his Lesson 13.3), and I added a quick activity on this blog, because both of us knew that this is mentioned in the Common Core Standards.
There are three ways to solve this problem. One of them is to start by using the Midpoint Formula to find O, the midpoint of
But the problem with this method is that it only works if we're dividing the segment into fourths or eighths (to find ratios such as 1:7 or 3:5), but not thirds (1:2) or fifths (1:4, 2:3). So we must find another method that works for general divisions of a segment. This is a lot of work, but dividing segments into ratios is important -- for example, in acoustics and music.
Here's one way I might teach this lesson. We note that the U of Chicago text introduces midpoints by discussing the center of gravity. We learned that the center of gravity of a set of points is the point whose x- and y-coordinates are the average, or mean, of the all of the respective coordinates of the points in that set. The center of gravity of two points is just the midpoint of the segment joining them.
(By the way, speaking of gravity, no -- Kline hasn't mentioned gravity yet. Galileo only calculated how fast objects fall, and not what causes objects to fall. We're still a few chapters away from gravity and Newton.)
Now as it turns out, we can divide the segment into other ratios by taking the mean of a list of coordinates with one of the points repeated as many times as indicated by the ratio. Since we want a 1:3 ratio here, we repeat one of the points three times:
Mean of MMMN: ((-4 + 6 + 6 + 6)/4, (4 - 2 - 2 - 2)/4) = (3.5, -0.5)
Notice that the resulting point P is closer to M or N depending on which point is repeated. In general, P will be closer to whichever point is repeated more often.
The U of Chicago text doesn't discuss finding the center of gravity when one or more points are repeated -- but then again, it doesn't cover the division of a segment into ratios adequately at all. Still, this is the best way to get from what appears on the U of Chicago to what appears on the PARCC.
The tricky part is making sure that the chosen points divide the segment in the correct ratio. A huge problem will be that the points that divide the segment into the ratio 1:3 are actually a quarter of the way on the segment. We see that when MP = 3 PN, we have MN = 4 PN. This sort of confusion often occurs in similarity problems as well -- the dilation mapping PN to MN has scale factor 4 (and if there are similar triangles with sides PN and MN, the sides of the latter would be 4 times those of the former), even though MP is only 3 times PN.
Depending on the wording of the problem, my center of gravity method minimizes this sort of error. To find the points dividing the segment into 1:3, we list one point once and the other point thrice. So the numbers in the ratio tell us how many times to list each point. Of course, the number 4 is still involved, as we must divide by 4 to find the mean, but at least we see where the 1 and 3 come from.
Another method that often appears is a vector method. We consider the coordinates of M(-4, 4) and N(6, -2) to be the vectors m and n, and we find the vector n + 1/4 (m - n) -- that is, we start at one point and add 1/4 of the vector that gets us from one point to the other. But this would be very confusing -- first of all the number 3 doesn't appear at all (unless we change 1/4 to 3/4 to show 3/4 of the way from one to the other, which gives us m + 3/4 (n - m), and also this is prone to sign errors as it's not as obvious when we want m - n and when we want n - m. If we use the wrong vector difference, then our point will still end up on line MN, but it won't be between M and N.
The vector method might be preferable if the question was stated as "1/4 of the way from J to K." But since the PARCC test uses the ratio 1:3, I like my center of gravity method better.
...that is, I liked the center of gravity method better last year. That method made sense last year because last year's question was worded as "divides
But actually, as I think about it more and more, I'm wondering whether the best method to solve both last year's and this year's questions is just to find O as the midpoint of
Not only that, but think about the following. Last year, we were asked to find M, the point that divides
That settles it! On today's worksheet, students will repeatedly use the Midpoint Formula to divide the segment into common ratios like quarters, since this will be the easiest way for the students to get the PARCC questions correct. (But with our luck, the PARCC will probably throw in a question where we must divide a segment into thirds or fifths.)
PARCC Practice Test Question 12
U of Chicago Correspondence: Lesson 11-4, The Midpoint Formula
Key Theorem: Midpoint Formula
If a segment has endpoints (a, b) and (c, d), its midpoint is ((a + c)/2, (b + d)/2).
If a segment has endpoints (a, b) and (c, d), its midpoint is ((a + c)/2, (b + d)/2).
Common Core Standard:
Commentary: The U of Chicago only focuses on midpoints -- that is, points that divide the segment into a 1:1 ratio. But we can apply this formula repeatedly to divide a segment into quarters and eighths.
CCSS.MATH.CONTENT.HSG.GPE.B.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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