Mat111 Week 1 notes

Course Administration and Attendance Roll Call

• Roll call status for recorded students:

  • Present: Destiny, Anna, Erin, Ash, Mackenzie, Lauren, Toby, Bernie, Mia, Dash, Rosalina, AJ, Evangel, Kelly, Kate, Lily, Josh, Chris, Brianna, Brad, Miley, Brody.

  • Absent: Amy.

• Attendance system nuances:

  • Different administrative systems process attendance entries independently.

  • Mackenzie was pre-marked as present in the official system record prior to manual call.

• Name pronunciations and adjustments:

  • Evangel: Pronunciation clarified with the intent to master the precise phonetic delivery by the end of the semester.

  • Kaylee / Kylie: Discussion regarding student first names starting with the letter K.

Traffic Engineering, Commuting Route Analyses, and Infrastructure Delays

• Instructor Commute Timeline and Route Logistics:

  • Origin: Manheim Township.

  • School District Context: Hanfield.

  • Scheduled arrival time at destination: $9:00\,\text{AM}$.

  • Actual arrival time at destination: $9:30\,\text{AM}$.

  • Child bus boarding time: $7:36\,\text{AM}$ (bus driver was punctual).

  • Home departure time: $8:00\,\text{AM}$.

  • Standard expected travel duration: $40\,\text{minutes}$ (expected arrival at $8:40\,\text{AM}$).

  • Actual travel time: $1\,\text{hour } 30\,\text{minutes}$ (an unexpected delay of $50\,\text{minutes}$).

• Arterial and Parallel Road Network Dynamics:

  • Route 222: Major north-south corridor experiencing extreme traffic congestion.

  • Parallel Bypass Routes: Oregon Pike and Butter Road (located directly off Butter Road). Both parallel routes were completely gridlocked due to spillover traffic attempting to bypass Route 222.

  • Route 30 and Columbia Bridge Incident: Traffic was severe bumper-to-bumper back-up caused by a vehicle entrapment incident on the opposite side of the Columbia Bridge (reported via social media/Facebook).

• Student Commute Benchmarking (Mackenzie):

  • Origin: Denver, PA.

  • Standard travel duration without traffic: $1\,\text{hour}$.

  • Departure time: $8:00\,\text{AM}$.

  • Arrival time: $9:20\,\text{AM}$ (total ride duration $1\,\text{hour } 20\,\text{minutes}$, representing a $20\,\text{minute}$ delay).

• Infrastructure Construction Efficiency Comparisons:

  • Route 222 and Route 30 Interchange Construction: Project has been ongoing for approximately $3\,\text{years}$, though perceived by local commuters as lasting $20$ to $25\,\text{years}$.

  • New Jersey Turnpike Expansion Case Study:

  • Historical comparison from over $10\,\text{years}$ ago along the route from Exit 6 to Exit 13 (connecting to Staten Island Expressway and Verrazano Bridge).

  • Bottleneck transition at Exit 8: Highway transitioned from $6\,\text{lanes}$ down to $3\,\text{lanes}$ northbound, creating severe structural congestion.

  • Opening transition ( expanding from $3\,\text{lanes}$ to $6\,\text{lanes}$ southbound) also induced localized congestion due to rapid vehicle lane shifting.

  • Infrastructure project completion: The state of New Jersey constructed $3\,\text{additional lanes}$ on both sides over multiple miles within a span of just a couple of years.

Platform Logistics, Course Management, and Student Inquiries

• Student On-Campus Predicaments:

  • Ray: Required to unlock dorm room twice in a single morning to assist a roommate who locked himself out on both occasions.

  • Campus Distance & Scheduling Dilemma: A student with a gap between classes until $1:00\,\text{PM}$ living in Northside dorms faced a choice between a $10\,\text{minute}$ walk back to the dorm room or spending the hour in the library.

• Course Materials and Software Platforms:

  • Homework Requirements: Obtain a physical or digital copy of the required textbook and successfully log into Pearson MyLab and Mastering.

  • Software Ecosystem: Content is managed across Pearson, Cengage, and WebAssign/Blocks platforms.

  • Course Rollover Configuration Issue: Automatically rolled-over course dates preserved past default due dates in May; instructor is manually adjusting individual assignment due dates line-by-line.

Scientific Calculator Specifications, Pricing, and Pedagogical History

• Recommended Device Specifications:

  • Model: Texas Instruments TI-30XS MultiView (TI-30XS).

  • Features: Two-line display, highly user-friendly interface, MathPrint ("Pretty Print") capability (displays structural fractions with numerator directly over denominator).

  • Alternative models noted: Graphing calculators (e.g., TI-84) and secondary scientific models (TI-30X II S).

  • Policy on Cell Phone Calculators: Mobile phones are strictly prohibited as calculation tools during formal examinations and graded assessments.

• Retail and Cost Analysis:

  • Historical baseline cost: Approximately $\$12.00$.

  • Target retail price: Standard MSRP $\$20.00$, listed on sale for $\$16.99$.

  • Amazon Prime purchase price: $\$20.00$.

• Historical Impact of Scientific Calculators in Mathematics Pedagogy:

  • In $1987$, the College Board first permitted students to utilize scientific calculators on the Advanced Placement (AP) Calculus examination.

  • Empirical Outcome: Student examination performance and average scores unexpectedly declined.

  • Root Cause: Students lacked operational familiarity with the technology, leading to order-of-operations input errors, misinterpreting output, and failing to verify calculated results manually.

Mathematical Principles: Idiomatic Time Expressions vs. Order of Operations

• Linguistic Idiom vs. Mathematical Operation ("Quarter of 12"):

  • Common English idiom meaning: $15\,\text{minutes}$ prior to $12:00$, corresponding to $11:45$.

  • Strict mathematical interpretation: The word "of" translates directly to the multiplication operator ($\times$).

  • Formal evaluation: $\frac{1}{4} \times 12 = 3$.

• Junior Driver License Regulations (State Comparative Context):

  • Pennsylvania Junior License Curfew: Restricted from driving past $11:00\,\text{PM}$.

  • Age Requirements: Junior license eligible at age $16.5$; full regular license granted at age $18$.

  • Passenger restrictions: First $6\,\text{months}$ restricted to maximum $1\,\text{passenger}$.

  • Legal Consequences: Driving past $11:00\,\text{PM}$ on a junior license carries the penalty of official license suspension.

  • Comparison with New York State: Different cut-off ages and driver licensing operational rules (e.g., driver from Poughkeepsie, NY).

Section 1.1 Exercise Solutions: Fraction and Decimal Evaluations

• Textbook Reference: Page $101$, Problem $14$ ("Evaluate each expression")

• Problem 14(a): $\frac{1}{3} + \frac{1}{5}$

  • Exact Fractional Answer: $\frac{8}{15}$

  • Repeating Decimal Answer: 0.5\begin{dots}3\text{ or } 0.5\bar{3}

  • Method: Common denominator is $15$; $\frac{5}{15} + \frac{3}{15} = \frac{8}{15}$. On TI-30XS, input using the n/d key yields $\frac{8}{15}$; standard division yields $0.53333333$.

  • Precision Note: Writing $0.53$ without repeating notation (bar above the $3$) is mathematically inaccurate due to truncation error.

• Problem 14(b): $\frac{10}{7}$

  • Exact Improper Fraction: $\frac{10}{7}$

  • Mixed Number Equivalent: $1\frac{3}{7}$

  • Repeating Decimal Answer: $1.\bar{428571}$

  • Rounded Decimal Answer: $1.43$ (rounded to the nearest hundredth).

• Problem 14(c): $\frac{5}{8}$

  • Exact Decimal Answer: $0.625$

  • Exact Fractional Answer: $\frac{5}{8}$

• Problem 14(d): $\frac{3}{4} + \frac{2}{3} + \frac{1}{2}$

  • Common Denominator Evaluation: $\frac{9}{12} + \frac{8}{12} + \frac{6}{12} = \frac{23}{12}$

  • Mixed Number Answer: $1\frac{11}{12}$

  • Repeating Decimal Answer: $1.916\bar{6}$

• Problem 14(e): $\frac{2}{3} + \frac{4}{5}$

  • Exact Fractional Answer: $\frac{22}{15}$

  • Mixed Number Answer: $1\frac{7}{15}$

  • Repeating Decimal Answer: $1.46\bar{6}$

• Problem 14(f): $\frac{6}{35}$

  • Exact Fractional Answer: $\frac{6}{35}$

  • Decimal Answer: $0.17142857...$ (repeating sequence $0.1\bar{714285}$)

• Problem 14(g): $\frac{1}{3} + \frac{13}{6}$

  • Calculation: $\frac{2}{6} + \frac{13}{6} = \frac{15}{6} = \frac{5}{2}$

  • Simplified Improper Fraction: $\frac{5}{2}$

  • Mixed Number Answer: $2\frac{1}{2}$

  • Exact Decimal Answer: $2.5$

• Problem 14(h): Product simplification containing factors of $3$, $5$, $10$, and $2$

  • Simplification result: $3$

Section 1.1 Exercise Solutions: Converting Decimals to Common Fractions

• Textbook Reference: Page $101$, Problem $16$ ("Write each of the following as a common fraction")

• Problem 16(a): Convert $2.75$ to a common fraction

  • Expanded Form: $2 + \frac{75}{100} = 2\frac{3}{4}$

  • Improper Fraction Form: $\frac{275}{100} = \frac{11}{4}$

  • TI-30XS Key Execution: Input $275$, press n/d, input $100$, press ENTER to return simplified $\frac{11}{4}$. Toggle decimal/fraction using the f<->d key (located above table).

• Problem 16(b): Convert $0.45$ to a common fraction

  • Unsimplified Fraction: $\frac{45}{100}$

  • Fully Simplified Fraction: $\frac{9}{20}$

• Problem 16(c): Convert $0.005$ to a common fraction

  • Unsimplified Fraction: $\frac{5}{1000}$

  • Fully Simplified Fraction: $\frac{1}{200}$

• Problem 16(d): Convert $1.16$ to a common fraction

  • Unsimplified Fraction: $\frac{116}{100}$

  • Simplified Improper Fraction: $\frac{29}{25}$

  • Mixed Number Form: $1\frac{4}{25}$

• Problem 16(e): Convert $6.5$ to a common fraction

  • Unsimplified Fraction: $\frac{65}{10}$

  • Simplified Improper Fraction: $\frac{13}{2}$

  • Mixed Number Form: $6\frac{1}{2}$

• Problem 16(f): Very small decimal values and Scientific Notation

  • Scientific Notation Representation: $3 \times 10^{-4}$

  • Calculator Screen Display: 3E-4

• Problem 16(g): Convert $0.0003$ to a common fraction

  • Four decimal places corresponds to four zeros in denominator: $\frac{3}{10000}$

  • Exponential/Scientific Notation: $3 \times 10^{-4}$

Classroom Discussion and Q&A

• Question: How do improper fractions differ from unsimplified fractions?

  • Answer: An improper fraction (where numerator is greater than or equal to denominator, such as $\frac{29}{25}$ or $\frac{13}{2}$) is mathematically fully simplified if the numerator and denominator share no common factors other than $1$. Conversely, an unsimplified fraction (such as $\frac{6}{4}$) contains shared common factors that must be factored out (e.g., reducing $\frac{6}{4}$ to $\frac{3}{2}$).

• Question: What is truncation error in calculator operations?

  • Answer: Truncation occurs when a long or infinite repeating decimal string is arbitrarily chopped off at a specific decimal position without rounding or proper repeating bar notation, creating potential propagation errors in multi-step calculations.

• Question: How do you perform direct fraction-to-decimal conversions on the TI-30XS calculator?

  • Answer: Use the n/d key to enter structural numerator and denominator values, or press the 2nd function f<->d toggle key above the table button to instantly convert existing display values between decimal and fraction outputs.