Last year at this point, I posted what I officially called the "Chapter 4 Test," and it was all about rotations and parallel lines. Now I've changed our Rotation Unit so that it covers congruent triangles instead of parallel lines.

So I must change the test. I could keep the questions about rotations and replace the parallel line questions with questions from the Chapter 7 Test. But now there's another problem -- I never actually posted a Chapter 7 Test last year!

This is because last year, Chapter 7 was the last chapter I covered before the first semester final. For the last chapter of a semester, it's sometimes tough to decide whether to give a test for that last unit or just roll into the final. When I was a student teacher, I taught Algebra I and II -- and we gave our Algebra II students a separate test for the last chapter, but not the Algebra I kids. Last year I decided to follow my Algebra I class and incorporate the Chapter 7 test questions into the final.

But this year, Chapter 7 appears at the end of the

*quarter*, not the semester, so there is no first semester final to give now. So instead, I will post what I will now call the "Chapter 7 Test" -- and it will contain rotation questions from the old test as well as Chapter 7 material. As usual, all questions on today's review worksheet come from the SPUR section of the U of Chicago text.

The quarter mark is a good time for me to discuss my philosophy regarding grades. Last year I wrote about various grading scales.

A grade represents a percentage, a scale from 0 to 100. One of the most commonly used grading scales in schools is something like this:

90-100 = A

80-90 = B

70-80 = C

60-70 = D

0-60 = F

Now let's return to school grades. For batting averages .300 is excellent, but in school 30% is surely a failing grade. For winning percentages .600 is one of the best teams in the majors, but in school 60% is the lowest passing grade. All of the grade boundaries -- from B to A, C to B, and so on -- occur in the upper half of the scale. The lower half of the scale -- from 0 to 50 percent -- is irrelevant as far as determining letter grades is concerned.

To the extent that C is average, the average student is earning between 70% and 80%. So we expect the average score in the class to be around the midpoint of this range -- 75%. So the average batter gets

*one*hit every four at-bats, the average team wins

*two*out of every four games, and the average student gets

*three*out of four questions right.

And so, unlike the batter who gets a hit then makes an out and sees that overall his average has risen, the student who gets 100% on one test and 0% the next can never have a higher letter grade -- only a lower letter grade is possible. Therefore, a grade is much more likely to drop very rapidly than it is to rise very rapidly. Since the average grade is around 75%, a student would have to receive 150% on a test in order for the grade to rise as rapidly as a 0% drops it. (And all of this is assuming that the student isn't at one of those schools that has abolished D grades and makes 70% the lowest passing score -- that 10% difference means that a student would have to earn 160% on a test to have the same impact on the grade as a 0%!)

I've seen students come up to me and ask me why their grade has dropped so quickly -- especially when their grades have dropped from B to D seemingly overnight, while no one's grade is rising from D to B as quickly. They think that I'm a mean teacher and a harsh grader -- when the true reason is the nature of the grading scale, where the F grade takes up three-fifths of the scale.

Now it's the end of the quarter, and grades are about to come out. Let's assume for simplicity that there are equal numbers of points possible in the first and second quarters (in reality, the second quarter may have more points because that quarter has a final exam while the first quarter doesn't).

Let's say a student is earning 10% at the quarter. Then this student will surely fail the class -- even if 100% is earned the second quarter, the average grade is only 55%, an F. This is akin to a baseball team who is 11 games back with only 10 to play -- that team has been mathematically eliminated from winning the division.

A student earning 20% at the quarter can still pass the class -- but only if 100% is earned the second quarter, which isn't very likely. And a student who gets 30% the first quarter would still need to earn 90% (an A) the second quarter to pass the class. The type of student who would earn only 30% in a quarter is not the type of student who would earn an A the second quarter -- which typically covers more difficult material than the first quarter. This is akin to a baseball team who is nine games back with only 10 to play -- although the team hasn't been

*mathematically*eliminated, they are

*realistically*eliminated from the division. It would take a miracle for the team to come back and claim the division, and a similar miracle is needed for the student to get any semester grade other than F.

In fact, in some ways, any quarter grade below 50% would realistically eliminate a student from receiving a grade higher than F at the semester. A 50% quarter grade would require a second quarter of 70%, a C, to pass the class -- and this might be doable, if the student works a little harder during the second quarter.

(And that, of course, assumes that 60% is the lowest passing grade. If the 50% student is at a no-D school, then that student will need 90%, an A, in the second semester to get any letter other than F on the semester report card -- once again, realistic elimination.)

When I am teaching class, what I want is for

*as few students as possible to be*I have no sympathy for a student who is realistically eliminated when there are only nine or ten

**mathematically or realistically eliminated**from getting any semester grade other than F, when there are still plenty of weeks separating them from the end of the semester.*days*left in the semester, but nine or ten

*weeks*, that's another matter.

A player on a baseball team that is realistically eliminated from the playoffs may be traded to a team that has a shot of winning. And even if the player isn't traded, he will still play out the rest of the season, no matter how little his heart is in it, because he's under contract to do so. Even a college student realistically eliminated from passing a class can drop out and take a W instead. But for high school students, there is no choice but for the student to take the F.

By the way, another complaint about grades is that it seems so difficult to get an A. A student with an 88% late in the semester asks me whether it's possible to get an A in the class. The truth be told, it's virtually impossible to raise that grade the two percentage points necessary to get an A -- one would essentially have to earn 100% on every remaining assignment. This is because ninety percent is just a very, very difficult target to reach if one is currently below it. It's much easier to raise a 68% two points up to 70% than it is to go from 88 to 90, or even 89 to 90. After all, if you have a 68%, you can get an 85% on the next assignment and raise it up to 70% easily. The equivalent for our 88% student would be to get a 105% to raise it up to 90%, but the highest possible score is 100%. In other words, there are plenty of possible scores above 70 to help raise your grade to a C, but there just aren't enough grades between 90 and 100 to help raise your grade to an A. Students think I'm just being mean or stingy with the A's, but in reality, it's the percentage scale that assigns three-fifths of it to the lowest grade that makes it difficult to raise one's grades, especially from F to D or from B to A.

The reason that I'm bringing up grades again is that in the news, there is a school district up in Northern California that is implementing a controversial new grading system:

http://www.pressdemocrat.com/news/4640349-181/rohnert-park-cotati-schools-rethink

I mentioned last year that one way to avoid realistic elimination would be to allow students to retake the tests on which they received a low score. But in this article, there is mentioned of two separate -- and I repeat,

*separate*-- grading scales that help students avoid realistic elimination.

The first is to record scores of zero as equaling 50%. After all, it's the devastating effect of 0% on the grades that eliminate students from passing the course. The average of 0% and 100% is 50%, an F, but the average of 50% and 100% is 75%, a C.

Back when I was an Algebra I student, my teacher did something similar. When she recorded test scores, she only recorded a letter grade, so if one received an F, one can't tell whether the F was really a score of 1% or 59%. Then to determine the average at the quarter or semester, she converted the grades back into percentages by letting each letter represent the midpoint of the interval. So an A was 95%, a B was 85%, a C was 75%, and a D was 65%. But an F was converted into 55%. So this represents a similar scheme to the 0 -> 50 system, since a score of 5% instantly became 55%.

Now this California district had already implemented the 0 -> 50 system, but now it is switching to yet another system. The grading scale will now look like this:

80-100 = A

60-79 = B

40-59 = C

20-39 = D

0-19 = F

Notice that this equal-interval scale has the same effect as the 0 -> 50 scale -- the average of 0% and 100% is still 50%, but that is now a C. Both systems are designed to lessen the effect of a zero so that fewer students are eliminated from passing the class with many weeks left in the semester. So because of this, there is no need to do

*both*. It's makes

*no*sense to change zeros to 50%

*and then*say that 50% is a C.

I was under the impression that in converting to the equal-interval scale, the district would eliminate the 0 -> 50 conversion once and for all. But some comments to this and related articles makes it sound as if the district really is giving out C's for missing assignments. I agree with the commenters who say that doing both is a terrible idea.

A quote from the article:

*Lowery, who has tried to retain the traditional grading system, said he has a student who early in the semester had a grade in the 80s range and another in the 25 percent range, but then stopped coming to class. Because his missed assignments all received a 50 percent, the student has a 49 percent average, instead of the 3 percent average he would have under the previous grading system, Lowery said.*

Notice that under the traditionalist scale, 49% is still an F, So 0 -> 50 has no effect on this student's letter grade -- it would be F no matter what. If we use the equal-interval scale, then we should

*not*do the 0 -> 50 conversion on top of that. So the student's grade would still be 3%, which is still an F even under the equal-interval scale.

But the rationale for either 0 -> 50 or equal-interval is so that students can raise their grades with plenty of time left in the semester. So let's say that it's today -- the end of the first

*quarter*-- and the student who has missed so many days in the first quarter has returned to class and wants to work hard during the second quarter in order to get a passing grade at the semester.

Under the 0 -> 50 system, the student has 49% at the quarter. So that student only needs a 71% the second quarter to raise the grade to D. He even has an outside chance -- working extremely hard -- at getting 91% the second quarter to raise the grade to C.

Under the equal-interval system, the student now has a 3% He only needs 37% the second quarter to get a D, and has an outside chance at getting 77% the second quarter to get a C.

On the other hand, under traditionalist grading with neither 0 -> 50 or equal-interval, even getting 100% the second semester only raises the grade to 51.5%. It is mathematically impossible for the student to receive any grade other than F at the semester. So the student, who might have considered working hard the second quarter to raise his grade, instead has no incentive to attend class the second quarter and might as well skip class the second quarter, just as he did the first quarter.

Under either 0 -> 50 or equal-interval, the student can work hard for a D or even a C. Under the traditionalist scale, the student stays at home. So, ironically, more

*actual learning*can occur under the 0 -> 50 or equal-interval scales than under the traditionalist scale.

In a related article, a teacher from this district defends the alternative grading scales:

http://www.pressdemocrat.com/opinion/4665017-181/close-to-home-in-defense?page=2

*Now let’s look at student Y. Through no fault of her own and for myriad possible reasons, she comes to my English class with a deficit of skills in reading, writing and even staying organized. Her early assignments earn abysmal grades, and she determines to do better. Through our school’s various interventions — and especially through her own hard work — she masters the skills being taught in my class.*

*By the end of the semester, she is scoring in the high 80s on assessments similar to the ones she failed earlier.*

*Under what logic should her early, low grades be counted against her?*

*Is it not right to give her a grade reflective of her mastery of the class material?*

Well, let's find out what the problem is. Let's assume that the D, C, and B grades mentioned by the commenter are the highest possible grades in their respective ranges, but the F here is a zero.

Under the 0 -> 50 system:

F: 50%

D: 69%

C: 79%

B: 89%

Semester Average: 71.75%

Grade: C

Under the equal-interval system:

F: 0%

D: 39%

C: 59%

B: 79%

Average: 44.25%

Grade: C

Under the traditionalist system:

F: 0%

D: 69%

C: 79%

B: 89%

Average: 59.25%

Grade: F

So even though this girl has done such amazing things -- going from hardly knowing any English at all to being one of the top students in the class -- she just can't quite make it to sixty percent for the overall semester average, so under the traditionalist system, the only letter that can appear on her report card is F. In other words, she was

*realistically*eliminated from passing with two months left in the semester.

The same commenter later on wrote:

Another person responded to say that no, cheaters get an F no matter what the scale is (provided, of course, that they are

*caught*cheating.) On the other hand, one sometimes reads about students who do no work the entire semester yet still get a perfect score on the final -- of course, they could've earned 100% on the final the very first day of class, because they already knew the material beforehand. In this case, they are commenting on a bureaucracy that won't let students who already know the material challenge or skip the course -- but that's a separate matter.

This takes us to other issues such as retention -- which is said not tot work -- or using other schemes to divide the students into classes, such as the Path Plan mentioned earlier on the blog. But then one must be careful when dividing students into classes, since then it could turn into tracking, which has several political, economic, and demographic implications.

Indeed, here's another commenter:

As for that last question, suppose an employer decided to use the following pay schedule -- instead of paying by the hour, let's say employees have to work at least 60 hours a week to get any money at all (just as students have to score 60% to receive any credit at all, under the traditionalist plan). And if one works fewer than 60 hours, then one has to make up the hours the following week such that the average number of hours worked is 60. So someone who falls behind will have trouble catching up and thus ever making any money at all. And let's say you have to wait 20 weeks before the earliest weeks aren't included in the average. Are employers going to pay employees by this system? No, but that is how the traditionalist grading system appears to many students.

Notice that this was also previously mentioned here on the blog -- it takes us to the Raenbo Certificate of Academic Proficiency proposal.

Here's what I say about 0 -> 50 and equal-interval grading. If -- and I repeat,

*if*-- one were to use equal-interval grading, then here's how I would recommend doing it:

-- Don't use 0 -> 50 grading at the same time.

-- Take full advantage of the use of the equal-interval scale. In a higher Algebra I or Algebra II class, add all those topics that are missing from Common Core that help prepare for Calculus. Under the traditionalist grading scale, if there are too many advanced questions on the test, too many students will score below 60% and fail. But if it takes only 20% to pass, then there's room to include more advanced questions to separate the A's from the B's without too many students getting F's. Sure, one only needs 80% to get an A, but they should have to work to get that 80% by answering questions that prepare them for Calculus.

If one were to use a 0 -> 50 type of grading scale, then here's what I recommend:

-- Don't convert zeros or low scores on the final to 50%. The whole point of 0 -> 50 is to protect students when there is plenty of time left in the semester. There's no need to protect them by the time they reach the final.

Well, here's today's review worksheet. You can use whatever grading scale you want when it's time to grade the actual test.

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