Ohio Resource Center
Lessons
Hanging Chains
Discipline
Mathematics
9, 10, 11, 12
Professional Commentary
Both ends of a small chain are attached to a board with a grid on it to form a catenary curve, similar to a parabola. Students choose three points along the curve and use them to identify an equation. Repeating the process, students discover how the equation changes when the chain is shifted. Overhead masters, discussion questions, lesson extensions, suggestions for assessment, and prompts for teacher reflection are included. (author/sw)

Common Core State Standards for Mathematics
High School - Functions
Interpreting Functions
Analyze functions using different representations
HSF-IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
2. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
3. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
4. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
5. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Building Functions
Build a function that models a relationship between two quantities
HSF-BF.A.1
Write a function that describes a relationship between two quantities.
1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
2. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
3. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Build new functions from existing functions
HSF-BF.B.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Ohio Mathematics Academic Content Standards (2001)
Patterns, Functions and Algebra Standard
Benchmarks (8–10)
D.
Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.