Video summary

High School Math Story Problems

Main summary

Key takeaways

Educational

Main ideas / lessons conveyed

  • Ticket pricing modeled with a system of linear equations

    • Children ticket price = (C)
    • Adult ticket price = (A)
    • Each “day” gives a linear total equation of the form:
      • ((\text{# children})\cdot C + (\text{# adults})\cdot A = \text{total revenue})

  • Solving for (C) and (A) using linear algebra methods

    • Methods mentioned and used:
      • RREF (reduced row echelon form) / calculator-based matrix methods
      • Elimination
      • Matrix inverse approach (using inverse to solve)
    • Key observation:
      • Using more equations than necessary (e.g., 3 equations for 2 unknowns) can introduce an extra row in RREF output, but the underlying prices can still match.

  • Property of equality / maintaining balance

    • If you perform operations (like doubling or tripling) on both sides of an equation/system consistently, the solution does not change.
    • The class connects this directly to the “Property of equality”.
    • Nuance tested:
      • The wording may say “double the first day” and “triple the second day,” so the scaling may differ per equation/day—but must follow the instructions correctly.

  • Interpreting negative values

    • A negative value (e.g., (-5) adult tickets) is treated as something that must be explained by a story that makes sense with the math output.
    • Example story explanations students brainstormed:
      • Reserved tickets never picked up → the theater is “out” tickets (negative representation)
      • Refunds / returned money scenario
      • Free/prize tickets given to students (adult contribution interpreted as negative)
      • Raffle/prize tickets combined with paying children scenario

  • Geometric interpretation

    • Ticket-price functions are linear.
    • Lines for different days intersect at the same equilibrium point:
      • Equilibrium point at ((C, A) = (4, 10))
    • Even when lines tilt (due to transforming equations), the intersection point stays the same if the system is constructed correctly.

  • Creativity + verification

    • Students make posters and explain their work and reasoning.
    • They verify solutions by substitution/checking totals when appropriate.

  • Hands-on extension: create your own data and test solvability

    • Students create a new dataset for April 1–8 using chosen (C) and (A).
    • They compute ticket prices from their own table.
    • Some datasets may lead to:
      • No meaningful solution (inconsistent results)
      • Problematic/degenerate cases (examples discussed included “zero” and inconsistent matrix forms)

  • Homework direction

    • Next task extends the model to three categories:
      • children price, adult price, and senior price
    • Students determine what system/method is needed for three unknowns.

Classroom workflow for the original two-price problems (children (C), adult (A))

  1. Build one linear equation per day

    • For each day:
      • Multiply the number of children by (C)
      • Multiply the number of adults by (A)
      • Add them to equal that day’s total revenue

  2. Solve the system for (C) and (A)

    • Use one of:
      • RREF / matrix method (possibly on a calculator)
      • Elimination
      • Matrix inverse (when applicable)

  3. Understand how many equations are needed

    • Since there are 2 unknowns ((C) and (A)):
      • At least 2 independent equations are sufficient
      • 3 equations may still work, but can produce an “extra row” in RREF output

  4. If using altered equations (doubling/tripling)

    • Apply the transformation to the appropriate equations/days exactly as instructed (e.g., first day doubled, second day tripled).
    • Confirm operations preserve the solution using the property of equality.

  5. Verify

    • Substitute the solved prices back into the day equations to ensure totals match.

  6. If the solution includes negatives

    • Write a story explanation consistent with what the negative term represents (refunds, free/prize tickets, unused reservations, ticket losses, etc.).

Property of equality guidance tested in the video

  • Do multiply/divide/add/etc. both sides of an equation by the same amount.
  • Don’t transform only one side (that changes the solution).
  • Allowed examples:
    • Doubling both sides
    • Tripling both sides
  • Consequence:
    • The system’s equilibrium/intersection point (the solution) remains unchanged.

Poster / presentation expectations

  • Make a poster that includes:
    • What system of equations you formed
    • What method you used (RREF/inverse/elimination)
    • The resulting ticket prices
    • Verification logic (e.g., totals match)

Final extension task instructions

  • Choose new ticket prices (child and adult) and write them on paper (not directly in the table).
  • Create a table for April 1–8 with nonzero-ish numbers of attendees (as instructed).
  • Compute totals for each day using the chosen prices.
  • Determine how many days of data are needed to solve the system.
  • Solve the system using any method:
    • function/linear equation setup
    • elimination
    • matrix approaches

Speakers / sources featured (identified in subtitles)

  • Teacher / instructor (unnamed)
  • Students (by name as spoken)
    • Ashlyn
    • Roy
    • McKayla
    • Josh
    • Carly
    • Lauren
    • Maddie
    • Logan
    • Sequoia
    • Mason
    • Bailey
    • Fisher
    • Ren (mentioned as “Almost hit Ren” / passing representatives)

Original video