Ohio Resource Center
Lessons
Discipline
Mathematics
9, 10, 11
Professional Commentary

How many toothpicks does it take to make an n x n square composed of 1 x 1 squares? The lesson begins with a review of transformations of quadratic functions--vertical and horizontal shifts, and stretches and shrinks. First, students match the symbolic form of the function to the appropriate graph, then given the graphs, students analyze the various transformations and determine the equation for the functions. This review is followed by an activity in which students explore a mathematical pattern that emerges as they build larger and larger squares composed of 1 x 1 squares. The pattern is quadratic, and students determine the mathematical model in several different forms. Students examine the recursive nature of the relationship, and using the model, extend the domain to negative integers. An explicit model for the relation is developed by examining the scatterplot and determining the equation from the transformations. Finally, the class uses graphing calculators to develop another model and verify that all of the models are equivalent. In addition to the lesson plan, the site includes activity sheets, selected answers, ideas for assessment, teacher discussion, extensions of the lesson, a discussion of the mathematical content, and video clips illustrating lesson procedures. Some computers may not be able to view these video clips as they require Windows Media Player. (author/pk/jk).

Common Core State Standards for Mathematics
High School - Functions
Interpreting Functions
Understand the concept of a function and use function notation
HSF-IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context
HSF-IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
HSF-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
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)
A.
Generalize and explain patterns and sequences in order to find the next term and the nth term.
C.
Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.
D.
Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.
E.
Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros.
J.
Describe and interpret rates of change from graphical and numerical data.
Benchmarks (11–12)
A.
Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
C.
Use recursive functions to model and solve problems; e.g., home mortgages, annuities.
1.
Relate the various representations of a relationship; i.e., relate a table to graph, description and symbolic form.
2.
Generalize patterns and sequences by describing how to find the nth term.
7.
Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems.
8.
Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.
2.
Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations.
3.
Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations.
5.
Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, rate of change, maximum or minimum.
15.
Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs.
4.
Use algebraic representations and functions to describe and generalize geometric properties and relationships.
4.
Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.
2.
Translate between the numeric and symbolic form of a sequence or series.
Mathematical Processes Standard
Benchmarks (8–10)
E.
Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.
F.
Use precise mathematical language and notations to represent problem situations and mathematical ideas.
Principles and Standards for School Mathematics
Algebra Standard
Understand patterns, relations, and functions
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
Represent and analyze mathematical situations and structures using algebraic symbols
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
use symbolic algebra to represent and explain mathematical relationships;
use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
Use mathematical models to represent and understand quantitative relationships
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
use symbolic algebra to represent and explain mathematical relationships;
use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
draw reasonable conclusions about a situation being modeled.
Analyze change in various contexts
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;
understand relations and functions and select, convert flexibly among, and use various representations for them;
analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
use symbolic algebra to represent and explain mathematical relationships;
use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
draw reasonable conclusions about a situation being modeled.
approximate and interpret rates of change from graphical and numerical data.
Representation Standard
Create and use representations to organize, record, and communicate mathematical ideas
Select, apply, and translate among mathematical representations to solve problems
Use representations to model and interpret physical, social, and mathematical phenomena