Arc length etc

Distinguishing Between Sector Area and Arc Length

• Conceptual Overview of Errors
      * A frequent error observed in the assessment was the transposition of methodologies between Problem 1 and Problem 3.
      * Students mistakenly applied the process required for Problem 3 to Problem 1 and vice-versa.
      * The instructor emphasizes the need to check for "giveaway words" to determine the correct formula to apply.

• Problem 1: Sector Area
      * Keyword Identification: The primary keywords for this problem are "Sector" and "Area."
      * Mathematical Concept: Finding the area of a sector requires taking a fraction of the total area of the circle.
      * Formula Components:
          * Fraction: The fraction is determined by comparing the central angle to the total degrees in a circle: $\frac{\text{Central Angle}}{360}$.
          * Specifics for Problem 1: The central angle is given as $120^{\circ}$, making the fraction $\frac{120}{360}$.
          * Area Formula: The area of a circle is calculated as $\pi \times r^2$.
      * Common Error Identified: Many students mistakenly calculated the fraction of the circumference instead of the area. Specifically, students were found doing $2 \times \pi \times 19$, applying the circumference formula when the area formula was required.

• Problem 3: Arc Length
      * Keyword Identification: The primary keyword for this problem is "Arc length."
      * Mathematical Concept: Finding the arc length involves taking a fraction of the circle's circumference.
      * Formula Components:
          * Fraction: Determined by the central angle over $360^{\circ}$.
          * Circumference Formula: The circumference is calculated as $2 \times \pi \times r$.
          * Specifics for Problem 3: The radius ($r$) given for this problem is $9$. The correct calculation involves $\frac{\text{Central Angle}}{360} \times 2 \times \pi \times 9$.
      * Common Error Identified: Students reversed the calculations, using the area formula for an arc length problem.

• Scoring Implications
      * The instructor noted that as many as $2$ points could be recovered by students who simply swapped these methods. For instance, a score of $6/10$ could have potentially been an $8/10$ if the calculation was correct but applied to the wrong question number.

Area of Regular Polygons: Problem 2

• Problem Description
      * Problem 2 required finding the area of a regular hexagon.
      * The problem provided all necessary dimensions directly, meaning students did not need to use the tangent function to find missing components like the apothem or side length.

• Given Numerical Values
      * Shape: Hexagon ($n = 6$).
      * Apothem ($a$): $16$.
      * Side Length ($s$): $18.5$.
      * Number of Sides ($n$): $6$.

• Calculation and Formula
      * Regular Polygon Area Formula: $\text{Area} = \frac{1}{2} \times a \times s \times n$
      * Substitution: $\text{Area} = \frac{1}{2} \times 16 \times 18.5 \times 6$
      * The final calculation for this problem was intended to be performed using Desmos.

• Academic Integrity and Misplaced Formulas
      * The instructor noted suspicious work on several papers involving the formula $\frac{\sqrt{3}}{3}$.
      * It was explicitly stated that this formula was never taught in class.
      * The instructor hypothesized that students used AI or their phones while in a lockdown browser to find external formulas, leading to a warning that phones will be locked up in the future to prevent cheating.

Calculating Arc Measure: Problem 4

• Problem Goal
      * Problem 4 was identified as the most frequently missed question on the assessment.
      * The objective was to find the measure of the arc starting at point $v$ and moving around the circle to point $x$.
      * Important Distinction: Finding the "measure" of an arc is different from finding the "arc length." Measure is determined by the central angle in degrees.

• Step-by-Step Geometrical Analysis
      1. Identify Known Values:
          * There is a $70^{\circ}$ angle given in a specific segment.
          * The arc from $x$ to $w$ is $48^{\circ}$, which means its corresponding central angle is also $48^{\circ}$.
      2. Identify the Semicircle:
          * The line extending from $u$ to $x$ acts as a diameter, cutting the circle in half and creating a semicircle.
          * A semicircle has a total measure of $180^{\circ}$.
      3. Calculate the Missing Central Angle:
          * To find the central angle for the arc segment in question (designated as the "question mark"), subtract the known parts of the semicircle from the total.
          * Calculation: $180^{\circ} - 70^{\circ} - 48^{\circ} = 62^{\circ}$.
      4. Determine Final Arc Measure:
          * The measure of the arc from $v$ to $x$ is the sum of the calculated central angle ($62^{\circ}$) and the adjacent central angle ($48^{\circ}$).
          * Final Calculation: $62^{\circ} + 48^{\circ} = 110^{\circ}$.

Logistics and Classroom Announcements

• Schedule Notes
      * Today is an "Advisory Day."
      * Review of the remaining problems (the back page of the assessment) will occur on Friday, likely following the STAR test.

• Classroom Procedures
      * Students were instructed to leave their papers on their desks at the end of the meeting for collection.