May 20, 1998

Dear Dr. Granger,

I hope this letter finds you in good health and spirits. I am writing to you today to enclose an A level calculus exam for one of my students, Septima Vector (a fitting surname for a math major, I must say).

As you know, Septima has been working diligently in my calculus class throughout the year, and I am confident that she is well-prepared to excel on this exam. I believe that she has the potential to achieve a high grade and further her studies in mathematics.

I have ensured that the exam paper follows the guidelines set by the exam board, and all the questions have been thoroughly tested and reviewed to ensure their accuracy and fairness. I am confident that this exam will provide an accurate assessment of Septima's understanding of the subject matter.

I appreciate your willingness to proctor this exam for my student, and I trust that you will administer it with the same level of professionalism and care that you have demonstrated in the past. If you have any questions or concerns regarding this exam, please do not hesitate to contact me.

Thank you for your assistance in this matter. I look forward to sharing Septima's results with you.

Sincerely,

Dr. Tremaine

oooooooooooooooooooooooooo

''Well Septima, today is the big day: your A level calculus exam. You have been doing very well in your math classes, as I have heard,'' said Hermione Granger.

Septima Vector had been the Hogwarts arithmancy professor, but being tortured last year by the Carrows with the Cruciatus curse had resulted in her having a head injury and needing to relearn math and arithmancy. With the help of her former student and close friend Granger (or more accurately, Dr. Granger, who also has a Muggle master's degree in math and so is authorized to proctor Muggle university math exams), Vector had enrolled in a Muggle correspondence university in order to retrain as a mathematician and get a math degree.

''I remember when I proctored your linear algebra exam. You were attending the same Muggle correspondence university back then. I guess it is my turn now,'' Vector said with a smile.

''Very well, Ms. Vector. I will go over the rules of the exam, which you have 3 hours to write. A calculator is allowed. Let me know if you have any questions,'' Granger said with a fake stern voice.

''In all seriousness though, I am sure you will do well. You will have your PhD in math in a few year. I am sure. You may start the exam now. Good luck,'' Granger said in an encouraging tone.

''Very well,'' Vector said with a grin.

ooooooooooooooooooooooooooooo

May 25, 1998

Dear Dr. Granger,

I hope this letter finds you well. I am writing to express my sincerest gratitude for your hard work and dedication in proctoring the Calculus A Level Exam. I understand that it is a demanding task that requires great attention to detail and unwavering commitment, and I am truly grateful for your efforts.

I would also like to take this opportunity to extend my congratulations to one of your students, Septima Vector, who achieved an outstanding grade of 98% on the exam. I had the pleasure of teaching Septima in my calculus class and I must say that she has consistently demonstrated exceptional aptitude and dedication to her studies.

Septima is an outstanding student who has shown great interest and passion for mathematics. She has consistently been an active participant in class discussions, regularly asking insightful questions and providing well-reasoned answers. Her work ethic and dedication to her studies have been truly inspiring, and it is clear that she has put in a tremendous amount of effort and hard work to achieve such a high grade.

I believe that Septima has a bright future ahead of her and that her achievements are a testament to her natural talent and her unwavering commitment to her studies. I have no doubt that she will continue to excel in her academic pursuits, and I look forward to hearing about her future accomplishments.

Once again, thank you for all of your hard work and dedication in proctoring the Calculus A Level Exam. It is truly appreciated. Please convey my heartfelt congratulations to Septima and let her know how proud I am of her.

Sincerely,

Dr. Tremaine