Mathematics Makes a Splash: Evaluating Hand Timing Systems

Mathematics plays a role in virtually every sport practiced by modern man. Be it the simple arithmetic used in score keeping, the calculation of statistics for teams and players, the determination of odds in favor of winning, or the precision of measurements needed to guarantee the standardization of competitive facilities, mathematics is a common thread in the fabric of the sports world. Sports officials have the opportunity to invest in very accurate electronic timing systems. For this reason, it is important to determine the significance of errors when measurements are done by less expensive hand timing systems.

Note: The information below was created with the assistance of AI.

Level of Mathematics
This lesson is ideal for:

High school students (grades 10–12), especially those in:

Algebra II

Precalculus

Math Modeling or Math & Science Integrated Courses

Also well-suited for introductory college-level quantitative reasoning or applied mathematics courses.

The mathematics combines algebra, geometry, data analysis, and basic physics.

Application Areas
The module highlights real-world applications of mathematics in:

Sports Science & Competitive Athletics:

Precision timing in swimming competitions.

Effects of human error in manual vs. electronic timing systems.

Physics & Engineering:

Application of distance, time, and speed relationships.

Modeling sound propagation as a function of temperature.

Computer Science & Programming:

Implementation of a BASIC program to simulate timing advantage scenarios.

Input/output data interpretation from simulations.

Mathematical Modeling:
Translating real-world conditions into mathematical expressions and computational models.

STEM Fair or Project-Based Learning:

Encourages students to run simulations, interpret data, and report findings.

Prerequisites
Students should be proficient in:

Algebraic formulas and expressions

Distance formula in the Cartesian coordinate system

Unit conversion and precision with measurement

Basic understanding of functions and equations

Introductory programming concepts (understanding structured code in BASIC)

Subject Matter
Key Mathematical Topics

Cartesian Geometry:
Use of coordinate plane to model swimmer, timer, and starter positions in a 50×22 meter pool.

Distance Formula: 
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Applied to calculate distance from starter to swimmers and timers.

Speed of Sound (Temperature-Dependent):

Timing Advantage Analysis:

Difference in time for sound to reach timers vs. swimmers, used to determine potential unfairness.

Programming Integration:
Students interact with and interpret a BASIC program (“Program Pool”) to simulate and calculate timing advantages.

Data Analysis & Interpretation:
Reviewing sample program outputs and interpreting time advantages by lane and starter position.

Correlation to Mathematics Standards
Common Core State Standards (CCSS) – High School

Algebra

HSA-CED.A.2: Create equations in two or more variables to represent relationships.

HSA-REI.D.11: Interpret and solve equations involving distance and time.

Geometry

HSG-GPE.B.7: Use coordinates to compute perimeters, areas, and distances.

Functions & Modeling

HSF-IF.B.6: Calculate and interpret the average rate of change.

HSF-LE.A.1c: Recognize growth patterns and model with exponential or linear functions.

HSM-MA.A.1–3: Analyze mathematical models and use simulations to predict behavior.

Mathematical Practices

MP2: Reason abstractly and quantitatively.

MP4: Model with mathematics.

MP5: Use appropriate tools strategically (e.g., BASIC programming, calculators).

MP6: Attend to precision.