Algebra 1.1 notes

Class Logistics and Tech Setup

  • Ensure access to Wiley course resources as the textbook: log in to Wiley as a student and load the textbook to view chapter 1, section 1.3.
  • Two students reported the page loading to a blank white screen ("white screen of doom"). Meeting with Wiley representative planned to troubleshoot.
  • If you’re trying to access instructor files, you must be logged into your AppState account (AppState Google accounts). This is for security; avoid using private/personal accounts.
  • Expectation: pre-read 1.3 (functions) before class; finish 1.1 and start on 1.2 during class today, then 1.1, 1.2, 1.3 after class tomorrow.
  • Checkpoint/update rhythm: checkpoint two will cover mean calculations; keep up with additions and be prepared to show work, not just final answers.
  • Instructor will be addressing questions, comments, concerns; live Q&A and neighbor discussions encouraged.
  • Metaphor for representations: multiple representations (graphs, tables, equations) help highlight different features and show connections between representations; Desmos is a key tool to see equation, table, and graph together.

1.1 Means and Medians (Review of Measures of Center)

  • Key ideas:
    • Means and medians are two measures of center used to summarize a data set.
    • Mean (average): sum of all data values divided by the number of values.
    • Median: middle value when data are ordered; for an even number of values, the median is the average of the two middle values (the two central data values).
  • Core definitions in compact form:
    • Mean: mean=∑<em>i=1nx</em>in.\text{mean} = \frac{\sum<em>{i=1}^n x</em>i}{n}.
    • Median: if data ordered as x<em>(1)≤⋯≤x</em>(n)x<em>{(1)} \le \cdots \le x</em>{(n)},
      median={x<em>(n+12),n oddx</em>(n2)+x(n2+1)2,n even.\text{median}=\begin{cases}x<em>{(\tfrac{n+1}{2})}, & n \text{ odd} \\ \frac{x</em>{(\tfrac{n}{2})}+x_{(\tfrac{n}{2}+1)}}{2}, & n \text{ even} \end{cases}.
  • Practical notes:
    • Repeated addition is multiplication (e.g., summing many equal values can be done via multiplication).
    • Mean is sensitive to outliers; median is robust to outliers and may better represent a typical value in skewed distributions.
    • When data are ordered, the median is the central point; with an even number of data values, the median may lie between two data values (not necessarily equal to a data value).
  • Quick recap of setup from class discussion:
    • Last class ended with a discussion of means and meanings (measures of center).
    • Today’s plan: work on examples that emphasize means and medians, including a case with an added data value and a tabular data set.

Example 7 (data set from Example 6 plus an added value)

  • Setup:
    • The data set from Example 6 is augmented by an additional value 102. The numbers are arranged in order to facilitate discussion of the median.
    • There are 10 data values after adding 102.
  • Mean calculation:
    • Mean is computed as the sum of the values divided by the number of values:
      mean=∑<em>i=1nx</em>in.\text{mean} = \frac{\sum<em>{i=1}^{n} x</em>i}{n}.
    • Efficient approach: count repeated values via multiplication when many values are the same (e.g., if a value appears many times, use that as a multiplier).
    • In the class example, adding 102 makes the mean become 49.2 (the instructor notes this as the observed mean after the addition).
    • Note: The instructor also emphasizes computing by adding all numbers and then dividing by the total count (10 in this case).
  • Median discussion (even $n$):
    • For $n=10$ (even), the median is the average of the 5th and 6th ordered values.
    • The instructor notes that with even $n$, the median lies between two central values, not necessarily equal to one of the data values.
    • They illustrate this by noting the middle values fall between two given values, and the resulting median may be between them (e.g., the value between 35 and 45 is 40 in the example context).
    • Takeaway: the median can be a value that is not itself in the data set when $n$ is even.

Example: Local scholarship distribution and summary statistics

  • Scenario: A local scholarship committee distributed scholarships as follows: 24 scholarships of $1{,}000$, 10 scholarships of $5{,}000$, and 3 scholarships of $24{,}000$.
  • Data interpretation: Instead of listing all 37 scholarship amounts, represent them as frequencies:
    • $1{,}000$ appears 24 times, $5{,}000$ appears 10 times, $24{,}000$ appears 3 times.
  • Mean calculation via weighted sums (repeated addition is multiplication):
    • Numerator: 24×1,000+10×5,000+3×24,000=24,000+50,000+72,000=146,000.24\times 1{,}000 + 10\times 5{,}000 + 3\times 24{,}000 = 24{,}000 + 50{,}000 + 72{,}000 = 146{,}000.
    • Denominator: n=24+10+3=37.n = 24 + 10 + 3 = 37.
    • Mean: mean=146,00037≈3,945.95.\text{mean} = \frac{146{,}000}{37} \approx 3{,}945.95.
    • Note the instructor’s initial miscalculation (writing 14,600) was corrected to 146,000, yielding the mean above.
  • Median calculation for the same distribution:
    • There are 37 scholarships; the middle value is the 19th value when ordered from least to greatest.
    • Since the first 24 values are $1{,}000$, the 19th value is within the $1{,}000$ group.
    • Therefore, the median is median=1,000.\text{median} = 1{,}000.
  • Key takeaway:
    • The mean (≈3,945.95\approx 3{,}945.95) is heavily influenced by the few large scholarships, while the median is $1{,}000$.
    • This illustrates how skewed distributions affect mean vs. median.
  • Quick practical note from the discussion:
    • When representing data with many identical values, you can write as a frequency table rather than listing every value; the sum and mean are computed from the frequencies.

Example 9: Weekly spending and solving for an unknown day

  • Setup: Track spending over five days; the average daily spending is $42. Daily values given: Monday $38, Tuesday $47, Wednesday $35, Thursday $50, Friday $x.
  • Goal: Find $x$ so that the mean is $42$.
  • Algebraic solution:
    • Sum known days: 38+47+35+50=170.38 + 47 + 35 + 50 = 170.
    • Equation for the mean: 170+x5=42.\frac{170 + x}{5} = 42.
    • Multiply both sides by 5: 170+x=210.170 + x = 210.
    • Solve for $x$: x=210−170=40.x = 210 - 170 = 40.
  • Alternative method: a guess-and-check approach (write down a guess for Friday and check if the resulting mean is 42). If you guess and check, write down your guess so the thought process is visible.
  • Algebraic check with Desmos (optional): Solve the equation
    • 170+x5=42\frac{170 + x}{5} = 42
    • In Desmos, you can input this equation and observe the solution for $x$; the instructor emphasizes that tests require showing work by hand, not just using Desmos for the final answer.
  • Takeaway: practice two-step equation solving in the context of averages; both methods yield the same $x$ (Friday spending $40$).

Frequency and Relative Frequency from a Histogram (1.1 recap exercise)

  • Task: Given a histogram, create a table with bins, frequencies, and relative frequencies.
  • Provided bin counts (example values discussed):
    • 0–5: 6
    • 5–10: 8
    • 10–15: 6
    • 15–20: 1
    • 20–25: 0
    • 25–30: 2
  • Total observations: N=6+8+6+1+0+2=23.N = 6 + 8 + 6 + 1 + 0 + 2 = 23.
  • Relative frequencies (fractions, decimals, or percentages):
    • 0–5: 623≈0.2609(≈26.1%)\frac{6}{23} \approx 0.2609 \quad (\approx 26.1\%)
    • 5–10: 823≈0.3478(≈34.8%)\frac{8}{23} \approx 0.3478 \quad (\approx 34.8\%)
    • 10–15: 623≈0.2609(≈26.1%)\frac{6}{23} \approx 0.2609 \quad (\approx 26.1\%)
    • 15–20: 123≈0.0435(≈4.3%)\frac{1}{23} \approx 0.0435 \quad (\approx 4.3\%)
    • 20–25: 023=0\frac{0}{23} = 0
    • 25–30: 223≈0.0870(≈8.7%)\frac{2}{23} \approx 0.0870 \quad (\approx 8.7\%)
  • Takeaway: Frequencies are counts per bin; relative frequencies express those counts as proportions of the total, useful for comparing distributions even when totals differ.
  • Reminder: fractions vs percentages; percentages are relative frequencies times 100%.

Graphs and Multiple Representations (Tables, Graphs, Equations)

  • Central idea: Equations, tables, graphs, and mappings are different representations of the same relationships; they highlight different features and reveal connections among features.
  • Desmos capability: You can view equation, table, and graph simultaneously to see how they tell the same story from different perspectives.
  • The Lion King analogy (to motivate multiple viewpoints):
    • The same story (Lion King) is told via different formats (book Hamlet, animation storyboard, movie, musical, script), each highlighting different aspects yet connected.
    • This mirrors how tables, graphs, and equations each emphasize different features of a relationship.
  • Quick activity: With a neighbor, interpret a given table and discuss what would make the context clearer (e.g., units, scale, what the rows/columns represent).
  • Key takeaways about representations:
    • Tables show the data values and order.
    • Graphs show trends, relationships, and patterns visually.
    • Equations describe the rule governing the relationship.
    • Mappings and ordered pairs emphasize the input-output correspondence.
  • Example discussion prompts:
    • A candy-table: consider sugar vs. calories; a table lists values but may lack context (e.g., unit amounts, serving sizes). A scatter plot would reveal the relationship between sugar and calories more clearly.
    • A scatter plot can show a positive relationship (as sugar increases, calories tend to increase), though there may be variability where the same sugar amount corresponds to different calories depending on candy type.
  • Real-world graphs shown:
    • App State enrollment (2012–2023): generally increasing over time with noticeable dips in spring semesters due to new admissions and graduation timing.
    • A global-decline graph (with a note that India remains relatively on track): highlights that global trends vary by region.
  • Practical note on graph interpretation:
    • When faced with multiple representations, use them together to understand the data more completely; each representation is a different lens on the same phenomenon.

Celsius–Fahrenheit Conversions: Formulas and Practice

  • Core conversion formula:
    • F=95 C+32=1.8 C+32.F = \frac{9}{5}\,C + 32 = 1.8\,C + 32.
  • Quick rule of thumb (approximate): double the Celsius value and add 30 to estimate Fahrenheit.
  • Example 1 (Celsius to Fahrenheit): if C=16,C = 16, then
    • F=1.8×16+32=28.8+32=60.8 ∘F.F = 1.8\times 16 + 32 = 28.8 + 32 = 60.8\,^{\circ}\mathrm{F}.
  • Example 2 (Fahrenheit to Celsius) using algebra:
    • Given F=85,F = 85, solve for C.C.
    • Start from 85=1.8 C+32.85 = 1.8\,C + 32.
    • Subtract 32: 53=1.8 C.53 = 1.8\,C.
    • Divide by 1.8: C=531.8≈29.4 ∘C.C = \frac{53}{1.8} \approx 29.4\,^{\circ}\mathrm{C}.
  • Substitution discipline:
    • When substituting values into an equation, use parentheses to ensure proper order of operations, especially when multiple variables are involved.
  • Practice tip: for a two-step equation in conversions, isolate the variable by performing inverse operations in reverse order (subtract/add, then multiply/divide).
  • Quick recap: the two-step equation approach confirms the relationship between Celsius and Fahrenheit and allows solving for an unknown temperature in either unit.

Practical Reflections and Tips

  • Ethical/practical implications:
    • Always pay attention to context and units when interpreting data; a table may hide scale or units that a graph clarifies.
    • Be mindful of skewness in data; mean can be misleading when outliers are present; median can provide a better sense of the typical value.
  • Study strategies:
    • Practice computing mean and median with both simple and tabular data (as in the scholarship example).
    • Practice converting between Celsius and Fahrenheit both with and without a calculator, reinforcing the algebraic steps for unknowns.
    • Use multiple representations (table, graph, equation) to understand a relationship and to cross-check results.
  • Homework and upcoming content:
    • Finish 1.1 (means, medians, and basic frequency analysis) and begin 1.2 (graphs and multiple representations).
    • Pre-read 1.3 (functions) to be prepared for class discussion tomorrow.
    • Be ready to discuss and practice how to