*not*the same as what I call a "quaver," which is the midpoint of the

*quarter*. Although the name "hexter" is used mostly in Colorado middle schools, any secondary school that issues progress reports at the midpoint of the trimester -- or one-third and two-thirds of the way through the semester -- are in effect observing a hexter calendar.

And speaking of midpoints, Question 5 of the PARCC Practice Test is all about midpoints and dividing segments into parts:

Line segment

*JK*in the

*xy*-coordinate plane has endpoints with coordinates (-4, 11) and (8, -1). What are

**two**possible locations for point

*M*so that

*M*divides

Indicate

**both**locations.

(A) (-2, 9)

(B) (-1, 8)

(C) (0, 7)

(D) (1, 6)

(E) (3, 4)

(F) (4, 3)

(G) (5, 2)

(H) (6, 1)

We've discussed this type of problem here on the blog before. I pointed out that, while Section 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 Section 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.

Notice that any question where a segment is divided into a ratio other than 1:1 has an inherent ambiguity -- do we want to find

*M*such that

*KM*= 3

*JM*, or

*JM*= 3

*KM*? The answer for this problem is

*both*. The students are directly to mark both locations.

There are two ways to solve this problem. One of them is to start by using the Midpoint Formula to find

*L*, the midpoint of

*JK*, as ((-4 + 8)/2, (11 - 1)/2) = (2, 5). Then we use the Midpoint Formula to find the midpoints of

*JL*, ((-4 + 2)/2, ((11 + 5/2) = (-1, 8), and

*KL*, ((8 + 2)/2, ((-1 + 5)/2) = (5, 2) -- we find that these points divide the segment into fourths, so the three-quarter segment is indeed thrice as long as the quarter segment. So the answers are (B) and (G).

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 will seem like a lot of work, but dividing segments into ratios is important. After all, just last week we were discussing how dividing strings into ratios produces musical intervals. If

*J*is at the nut of the guitar and

*K*at the bridge, and

*M*is the point dividing

*JK*into a 1:3 ratio, then

*KM*would be 3/4 as long as the string

*JK*. The ratio 4/3 gives us a perfect fourth, so if this were the E string, a perfect fourth above E gives the note A.

Here's how I would 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.

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

*JJJK*: ((-4 - 4 - 4 + 8)/4, (11 + 11 + 11 - 1)/4) = (-1, 8)

Mean of

*JKKK*: ((-4 + 8 + 8 + 8)/4, (11 - 1 - 1 - 1)/4) = (5, 2)

Notice that the resulting point

*M*is closer to

*J*or

*K*depending on which point is repeated. In general,

*M*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.

How well will students do on this problem? For one thing, many students may mark only one answer instead of two -- no matter how many times the words

**two**and

**both**appear in boldface.

Notice that all eight answer choices lie on

*quarter*of the way on the segment. We see that when

*KM*= 3

*JM*, we have

*JK*= 4

*JM*. This sort of confusion often occurs in similarity problems as well -- the dilation mapping

*JM*to

*JK*has scale factor 4 (and if there are similar triangles with sides

*JM*and

*JK*, the sides of the latter would be 4 times those of the former), even though

*KM*is only 3 times

*JM*.

I believe that 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

*J*(-4, 11) and

*K*(8, -1) to be the vectors

**j**and

**k**, and we find the vectors

**j**+ 1/4 (

**k**-

**j**) and

**k**+ 1/4 (

**j**-

**k**) -- 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), and also this is prone to sign errors as it's not as obvious when we want

**k**-

**j**and when we want

**j**-

**k**. If we use the wrong vector difference, then our point will still end up on line

*JK*, but it won't be between

*J*and

*K*.

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. This method gives the shortest path from the ratio mentioned on PARCC (1:3) to a formula for the answer (add one point once to the other point thrice and divide to find the mean). Dr. M worded his question as "1/4th of the distance from..." (and called them "Quarter Points") and I had the foresight to mention both the fraction of the way and the ratio. Of course, on this worksheet I will use the ratio, since that is how it will appear on the PARCC.

With all of this discussion about hexters and dividing things up -- be it school years or segments -- I want to reveal my ideal academic calendar. Do I prefer semesters subdivided into quarters, or trimesters divided into hexters? Notice that semesters are most common at the high school level. I point that trimesters are most frequent in elementary school, still fairly common in middle school, and rare but still occasionally seen in high school.

Well, my personal preference is actually not for 2, 3, 4, or 6 divisions of the year, but

*five*. Notice what happens if we were to divide the year into five -- for lack of a better name -- "quinters." We see a close correspondence between quinters and months:

Quinter #1: September and October

Quinter #2: November and December

Quinter #3: January and February

Quinter #4: March and April

Quinter #5: May and June

Two quinters would be close to the time period between Labor Day and Christmas, so we can end the second quinter before winter break without having to start in August. And then the fourth quinter would end around testing time -- not only the Common Core exams such as PARCC EOY and SBAC, but also the AP, which is the first full week in May. Here in California, a full year class is considered to be 10 credits, so the quinter system would allow each quinter class to be two credits.

One reason that schedules other than semesters are rare at high school -- even in districts that have trimesters for grades K-8 -- is to avoid confusing the college admissions officers. Under this proposal, I would treat two quinters to be equal to one semester for admissions purposes. Finals week would therefore be at the end of the second quinter (just before winter break, just as in the Early Start calendar) and fourth quinter (around the Common Core and AP tests -- that is, when students are studying for tests anyway).

Of course, this proposal isn't perfect. In years when Labor Day is late (i.e., September 7th rather than the 1st), one may still have to start school a few days before Labor Day in order to complete 72 days (2/5 of the year) by Christmas -- but then again, September 3rd is better than August 3rd. There are also problems when Easter is late. There may end up being only one week of school between the return from spring break and the start of AP testing. Should that week be for taking finals, or Common Core tests, or what? Then again, late Easters cause problems with testing with or without my quinter proposal.

I'd much rather have Common Core testing be closer to the last day of school, but since that isn't happening, this proposal has an entire quinter of school left after the testing. This fifth quinter can be reserved for "fun" stuff -- you know, the things that teachers wouldn't mind teaching earlier in the year but can't because of having to prepare for Common Core. Schools might even set up elective classes to take place during fifth quinter. (In some ways, this blog is already following a quinter calendar, as 4/5 of 180 is 144. Day 144 is when I posted the last test, and after that is when I began the review worksheets.)

Notice that a quinter is approximately eight weeks long, and two quinters is around 16 weeks. As it turns out, many colleges have 16-week semesters, so it may be possible to have the quinters line up with college semesters, especially in states where the college spring semester ends around the last week of April or first week of May. (California is

*not*such a state -- Cal States on the semester system tend to have a one-month January term and the 16-week spring semester run from February to May, so it wouldn't line up with the quinter system as described here.)

On this worksheet, I include a grid for graphing, which may make the problems easier. Unfortunately, no grid would be available on the actual PARCC exam.

**PARCC Practice Test Question 5**

**U of Chicago Correspondence: Section 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).

**Common Core Standard:**

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.

**Commentary: The U of Chicago only focuses on midpoints -- that is, points that divide the segment into a 1:1 ratio. But we can use the U of Chicago's center of gravity analogy to partition the segment into any other ratio, such that the numerator and denominator of the ratio become the weights at each point.**

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