Problem 58 - Applied Calculus

Question

If a grade of 90 through 100 is an \(\mathrm{A},\) at least 80 but less than 90 is a \(\mathrm{B}\), at least 70 but less than 80 a \(C\), at least 60 but less than 70 a \(D\), and below 60 an \(F\), write these grade levels in interval form (ignoring rounding). [Hint: \(\mathrm{F}\) would be \([0,60) .]\)

Step-by-Step Solution

Answer

\([90, 100]\) is A, \([80, 90)\) is B, \([70, 80)\) is C, \([60, 70)\) is D, \([0, 60)\) is F.

Step 1: Understanding the Problem

We need to express the ranges of grades as intervals for different letter grades based on the given criteria. Each letter grade corresponds to a specific range of numerical values, which we can express using interval notation. The intervals state the starting and ending points of a range, and we use brackets or parentheses to indicate whether the endpoints are included or excluded.

Step 2: Grade A Interval

According to the problem, a grade between 90 and 100 inclusively is an 'A'. Therefore, in interval notation, this is represented as \([90, 100]\), where both 90 and 100 are included in the interval.

Step 3: Grade B Interval

A grade that is at least 80 but less than 90 is a 'B'. This is represented in interval notation as \([80, 90)\), where 80 is included but 90 is not included.

Step 4: Grade C Interval

For grades that are at least 70 but less than 80, we use the interval \([70, 80)\), where 70 is included and 80 is not included.

Step 5: Grade D Interval

Grades that are at least 60 but less than 70 correspond to a 'D'. In interval notation, this is \([60, 70)\), where 60 is included, but 70 is not.

Step 6: Grade F Interval

Grades below 60 are labeled as 'F'. The interval for this would be \([0, 60)\), where 0 is included, but 60 is not, marking it as open-ended at 60 exactly as the hint suggested.

Key Concepts

Grade Intervals

Grade intervals are a way to express a range of grades using a mathematical set notation. They help us quickly understand who falls within which grade category. With interval notation:

• Square brackets \( [ ] \) indicate that the endpoints are included in the interval.
• Parentheses \( ( ) \) indicate that the endpoints are excluded.

For example, an "A" grade is from 90 to 100 points, expressed as \[ [90, 100] \]. This means both 90 and 100 are included. The ease of using intervals allows us to describe grade bands clearly and succinctly.
Examples include:

• An "A" is \([90, 100]\).
• A "B" is \([80, 90)\), including 80 but excluding 90.

Understanding these intervals helps when assessing or setting grading standards.

Letter Grades

Letter grades are a common way of representing a student's performance in a course. They break down numerical scores into meaningful categories, making it easier to evaluate and compare performance.
Common letter grades include:

• "A" for exemplary performance.
• "B" for above-average performance.
• "C" for average competence.
• "D" for below-average understanding.
• "F" for failing or insufficient grasp of the material.

These grades can provide quick visual cues on a student's achievement level. They often help educators and students to interpret scores in an understandable way, reflecting both strengths and areas needing improvement.

Mathematical Notation

Mathematical notation is a system used to write concepts and calculations succinctly and clearly. In the context of expressing grade intervals, it allows us to relay information quickly. By converting grades into intervals:

• The notation \( [90, 100] \) concisely shows the range of an "A" grade.
• The use of symbols such as brackets or parentheses helps in understanding whether endpoints are included.

Using this notation helps avoid ambiguity and makes it easier to grasp the grading scales employed.
For example:

• A grade of "F" is represented as \( [0, 60) \), immediately showing the boundary where a passing grade begins.

Overall, the mathematical notation is a universal language in mathematics that compactly demonstrates information, which is particularly handy in education for clarity and uniformity.