In Part 1 of this series, I worked through how I established my rating scale and associated grade scale using the Algebra 1 CST performance level cut score percentages as a benchmark. I ended by revealing the resultant, preliminary end of semester grade distribution, as shown in table 1. Each grade was determined by equally weighting a student’s assignment score and assessment score. The preliminary grade did not reflect any changes I may make based on proximity to a cut score, specific student circumstances, or etcetera.
Table 1: Distribution for Egalitarian Grading System
Grading System Revisions
After contemplating a colleague’s comments, and reflecting on what a final grade for a student meant to me, I decided to explore a variety of algorithms for determining a grade within the framework I use for rating individual student work, as shown in table 2, as well as the scale I use to convert weighted average percentage scores from the rating scales into final grades, as shown in table 3. 
Table 2: Individual Work Rating Scale Table 3: Grade Reporting Scale
Egalitarian Grading System
As a point of comparison, the grade distribution in table 1 is based on weighting the average percentage score for the semester of all assignment ratings by 50% as well as the average percentage score of all assessment ratings by 50%. I refer to this as the “Egalitarian Grading System” since every student’s grade depends equally upon their assignment and assessment score. The upside for students with this approach typically inures to the benefit of students who do not fare as well on assessments, but maximize their assignment grade, especially since I rate assignments on completion only. The downside falls mostly upon students who score high on assessments, but for a variety of reasons, do not rate as high on assignments.
Progressive Grading System
To compensate for the downside in the Egalitarian distribution, the following grading scale assigns a student the highest of the weighted average score or the average assessment score. In this way, the upside discussed above continues, while the downside is removed. I named this the Progressive grading system, as shown in Table 4.
Table 4: Distribution for Progressive Grading System
Note that the number of C and above grades increased by six-percentage points with this system.
Ultra Progressive Grading System
After creating the Progressive grading system, I wondered how the grade distribution might shift if the final grade was the highest of a student’s weighted average score, their average assessment score, or their average assignment score. Table 5 shows the distribution for what I call the Ultra Progressive grading system.
Table 5: Distribution for Ultra Progressive Grading System
With this system, note that the percentage of students receiving a C or higher reaches 69%, much higher than the Egalitarian system, which is 58%, and five percentage points higher than the Progressive system, which is 64%. However, I was not comfortable basing a student’s grade solely on their average assignment score, especially in cases where their assessment score significantly differed from their assignment score. While this system increases the percentage of students receiving C or higher grades, it provides a grade to students that does not align with their performance level on the content standards, which I believe is a primary purpose for the grade.
Super Progressive Grading System
Moving back to the Progressive system, and building upon it, I created the Super Progressive grading system, where a student’s score on the final exam, which is cumulative, is included in the decision criteria for their final grade. If a student fared better on the final exam than either their weighted average score or their average assessment score, they received their final exam score. I felt this was the most fair system to implement for my students. It definitely benefited one student who otherwise would have failed the course. It also materially affected the final ratings of one-third of the class in a beneficial way. Table 6 shows the Super Progressive grading scale’s grade distribution.
Table 6: Distribution for Super Progressive Grading System
While the Super Progressive grading system reduces the overall number of students who receive a C or higher, it ends up lowering the number of students who otherwise would have received an F. Of course, I could create a Super Ultra Progressive grading system to deliver both benefits, however, it would keep the problem I discuss above with the Ultra Progressive grading system, notably that a student’s grade does not align with their performance level on the content standards.
Final Grading System
Tiring of the lengthy adjectives defining the various grading systems, I named the system I ultimately selected the Final grading system, which consisted of taking the Super Progressive grading system and making manual increases to various student grades mostly based on proximity to a grade cut score. Table 7 shows the grade distribution for the final grading system.
Table 7: Distribution for Final Grading System
Note that the combined percentage for students receiving a C or higher is the greatest yet at 71%, while the number of students receiving an F dropped to the lowest level of any system. Eight students, who otherwise would have received a failing grade, are able to receive credit for their first semester of algebra 1. To a one, they demonstrated to me that they possessed a ‘below basic,’ but passing, level of understanding of the algebra 1 content standards. For this, they deserve to receive credit for the semester.
To what extent these eight students are able to pass the second semester is yet to be seen. Many of them have behavioral issues, which interfere with our classroom learning environment. At the same time, this is the primary chance in their life to learn this material, and I plan to help them do so, in spite of themselves. Hopefully, in time, they will recognize how this benefitted them.
Comparing Grading System Distributions
As you can see, grade distributions vary considerably in the various grading systems I explored, most of which varied in progressiveness. At the other end of the spectrum, a grading system based solely on assessment, yields the distribution shown in table 8.
Table 8: Distribution for an Assessment Only Grading System
This system delivers the lowest overall percentage of students receiving a C or higher grade, and hence the highest percentage of students receiving a D or F. As such, it is the most punitive of all systems, in my opinion, and unworthy of implementation in any high school today.
What is not clear to me, as of this writing, is the extent to which assignment scores, participation scores, and assessment scores factored into grades over the years. There is a dearth of information on specific grade distributions at any level. I was able to find grade distributions from the year 1934 and 1982 for comparative purposes. 
1943 Grades and Distribution
The following data were taken from a JSTOR site for the article written by Norman E. Rutt, originally published in National Mathematics Magazine in 1943. I placed each data set into the same tabular format as used above for ease in comparing the distributions. Table 9 and table 10 contain this data.
Note that the grouped data for C grades and above, and for D and F grades, in table 9 matches exactly that of table 6, the Super Progressive grading system. This is entirely coincidental, however, of interest for future discussion. Mr. Rutt’s intent in publishing his data was to explore to what extent grades follow a normal distribution. I have not assessed his data, or mine, for normality. My data were not normalized.
Table 9: Grade Distribution from 1943 (near normal distribution)
Table 10: Grade Distribution from 1943
The data in table 10 do not seem to follow a normal distribution. Surprisingly, the data in this table also match the grouped data for C grades and above, and for D and F grades, in table 7, the final grading system.
1982 Seniors’ Distribution of Grades in Mathematics
Table 11 shows data released by the National Center for Education Statistics (“NCES”) in 1984 estimating seniors’ grades in mathematics in 1982. Somewhat of note, I graduated from high school in 1982; however, I did not take any mathematics courses my senior year, not that I would impact data from all graduating seniors in the U.S.
Table 11: Distribution of Mathematics Grades for Seniors in 1982
These data show the most favorable distribution of C and above grades versus D and F grades. Why this is the case is unknown. It could be due to the differences in student socioeconomic and other demographic status at that time. The extent to which this data exhibits normality has not been determined either, at least from my first read of the NCES bulletin.
The results of the final grading system leave me feeling quite satisfied. In fact, I was elated last night when I settled upon using it. I find it to be the most fair and justified system for assigning final grades to students. I believe they will feel similarly. I have not explored how my AP Calculus students grades will fare in this system. However, due to its very nature, students should receive a grade reflective of their performance level with the content, which is the primary goal for assigning grades from my perspective.
 Explaining how table 2 and table 3 relate to each other in any detail will take another post. Suffice it to say that individual assignments are scored using a 0-100% scale which are then reported to students online using the rating scale in table 2. Once in the online system, cumulative ratings are converted to percentages where the system uses table 3 as a lookup table to convert those percentages to the A-F grades required by the district.
 It is not clear to me why there is so little data published on grade distributions, since it can only benefit our educational system if we publish and explore data such as this to help develop the most effective systems possible for educating our nation’s future. There is no personally identifiable data, either directly or indirectly, associated with such data, although I can understand why many might be reluctant to share data like this, or even have the time to do so if they wanted to share.