Exams in Branching Forms (Theory)

On the page,  I showed how Microsoft Forms can be used for test exams. This type of exams is especially useful if a teacher has too many examination papers for personal grading. Despite electronic distant control, it is possible that students may communicate to distribute correct answers. It is true that Microsoft Forms questions can be distributed randomly. The problem is that students get the whole exam in which all questions are just reordered. If a university has also a Moodle LMS, then the questions can be randomly distributed with blocked access to already answered questions. However, this option does not solve the problem since some students can just wait and do nothing until somebody sends them correct answers. It is possible to create a bank of questions in Moodle. This bank may be only effective if it includes at least five time more questions than the number of questions in the real final exam. In practice, this means that an examinator must have at least 200 questions or more to distribute 40 questions randomly to students.

I couldn’t find any example of time restricted questions neither in Moodle, nor in Microsoft 365. Here I present another solution to this problem. For a final exam I prepare 42=3x14 questions of two types. My exam has 28 relatively simple questions one point for each, and 14 more difficult ones two points per each. Simple questions are of multiple-choice type, whereas more difficult questions require some calculations. They are of the type shown on pic. 486. So, the maximal total grade available for this exam is 28+2x14=56 points. Notice that 70% of this grade equals 39.2, which is approximately 40. The value of 70% goes back to Advanced Placement exams in Calculus running by College Board. This organization has a long-term experience in running such exams world-wide. The maximum possible grade in Calculus AP is 108. A paper is graded with the maximal possible grade 5 if a student gets at least 70-75 points of 108 available.

 Every difficult question is followed by two simple ones, so that there are 4 points in each block. Hence the exam consists of 14 blocks, which in total accumulate 56 points. I make several copies of the form obtained. The idea is to make a few variants of the exam just by rearranging the blocks properly. When a version is ready, I may introduce branching. I just click on the ellipsis in the form at the first question and add the option “Add branching,” see pic. 483. Usually, I provide at least four different options for every question. Then to the right of each option the “Branching” inserts the menu to select the numbers of the question to follow. I prefer to have four different questions to follow even if the number of options in the initial question is greater than four. Some options may be mapped to a one question to follow.

Let me show on examples how this system works. Below I list the first twelve blocks of my exam with indication of the grades for each question. A student may fail the first question for 2 points.


 It is necessary to list the options with wrong answers in the order of the level of mistakes. At the first row I place the correct answer. Then a student must go the next question in the list. Higher the level of the mistake, the more questions a student must miss. If a small mistake is made, then the student misses only one question, and the loss is 2+1=3 points. A mistake of a higher level moves the student to the second difficult question, which results in 4 points loss. Finally, if this is a mistake of the highest level, then the loss will be 2+1+1+2=6 points. If the student makes a mistake at the second question, then the possible losses are 2, 4, 5 points. Finally, if the third question is answered wrong, then the possible losses are 3, 4, 5. In average, one wrong answer to a question leads to a loss of

points depending on the question position in the list. We may assume that in average the loss is 4 points for every wrongly answered question. Similarly, each wrong answer leads to a miss in average of two question.

To pass the exam a student must get at least 20 points. If the result is 20, then it means that 56-20=36 points are lost. It follows that a student in average fails the exam if more than 36/4=9 questions were answered wrong. If a student fails 9 questions, then extra 18 questions were missed. It follows that a student answered correct only about 42-27=15 questions. Of these 15, one third, i.e., 5 are for 2 points, and the rest 10 are for one point each, which gives 5x2+10=20 in total.  

Let us consider the opposite case. A student got 36 points, implying that 56-36=20 points are missed. This means that 20/4=5 questions were answered wrong. Then extra 10 were missed. Hence this student answered correctly only 42-15=27 questions. This means that this student got 9x2+18=36 points.

In general, if a student gets X points, then this student loses 56-X points, implying that the number of wrong answered questions is approximately 0.25(56-X). Since in average one wrong answered question results in the miss of two others, the number of missed questions is 0.5(56-X). It follows that the number of questions with correct answers is about



It follows that a student with the result of X points saw



 questions. Then the function


evaluates the percentage of correctly solved questions in a student attempt.

Notice, that for X in the interval [40, 56] the values of f(X) correspond to A. The first impression is that 04f(X) is a good candidate for the grade of students who got X points. Let us observe that the quotient

is a decreasing function in X. It follows that 0.4f(X) is less or equal X for X not less than 32 and X is less or equal 0.4f(X) otherwise. Since the idea of any exam is to stimulate students to solve as many problems as possible and since students with X points solve more problems correct, it is natural to upgrade 0.4f(X) to X if X>32, and downgrade 0.4f(X) to X if X<32. These arguments show that the right grade function for branching exams is

An important item is timing. I suggest only one minute for a simple question and 3 minutes for a difficult one. Therefore, the total duration of this exam is: 28x1+14x3=70 minutes.

Since the exam includes 14 difficult questions, theoretically up to 14 versions of this exam are possible. If you reduce the number of versions to four, then you have enough freedom to variate other difficult questions so that the first difficult question in each version will occur in other versions close to the end of the exam. The exam questions in the branching forms are shown to students one by one. Unfortunately, Microsoft has not yet provided blocking of the return button for these types of forms. The problem is that students may try to extend the length of their exams by accidental marking of different options. The picture they see is messy but still they can get some points using such methods. Such attempts contradict to the academic honesty policy. At present, I solve this problem with the screen recording. The screen recordings in the corresponding form folder can be watched one by one relatively fast. If I find such places, then the grade for this exam will be reduced.