Functions & Equations:
Aren’t they the same
thing?
Susan Hull and Danielle Seabold
Mathematics Leadership
The Charles A. Dana Center
CAMT, 2005
Functions & Equations
Why are we here?
TEKS-based Resource:
Mathematics TEKS
Toolkit
www.mathtekstoolkit.org
An online resource hub for activities,
lessons, and assessments that are TEKS
based.
Look for the redesign to be launched March
2006.
Algebra I TEKS
Basic Understandings:
(3) Function concepts. A function is a fundamental
mathematical concept; it expresses a special kind of
relationship between two quantities. Students use
functions to determine one quantity from another, to
represent and model problem situations, and to
analyze and interpret relationships.
(4) Relationship between equations and functions.
Equations and inequalities arise as a way of asking
and answering questions involving functional
relationships. Students work in many situations to set
up equations and inequalities and use a variety of
methods to solve them.
Algebra I TEKS
Basic Understanding:
(5) Tools for algebraic thinking. Techniques for
working with functions and equations are essential
in understanding underlying relationships. Students
use a variety of representations (concrete, pictorial,
numerical, symbolic, graphical, and verbal), tools,
and technology (including, but not limited to,
calculators with graphing capabilities, data
collection devices, and computers) to model
mathematical situations to solve meaningful
problems.
Algebra I TEKS
(A.7) Linear functions. The student formulates equations and
inequalities based on linear functions, uses a variety of
methods to solve them, and analyzes the solutions in terms
of the situation.
(A) analyze situations involving linear functions and formulate
linear equations or inequalities to solve problems;
(B) investigate methods for solving linear equations and
inequalities using concrete models, graphs, and the
properties of equality, select a method, and solve the
equations and inequalities; and
(C) interpret and determine the reasonableness of solutions to
linear equations and inequalities.
Algebra I TEKS
(A.10) Quadratic and other nonlinear functions.
The student understands there is more than one way
to solve a quadratic equation and solves them using
appropriate methods.
(B) make connections among the solutions (roots)
of quadratic equations, the zeros of their related
functions, and the horizontal intercepts (xintercepts) of the graph of the function.
Algebra I TEKS and
beyond
What are the essential Basic
Understandings in Algebra I that are
also found in Algebra II and
Precalculus?
Cage the Cats
As your group works through the
Cage the Cats activity, record the
mathematical concepts that are
relevant to the ideas of functions and
equations.
Mathematical
Definition of Functions
Task #1:
How can this task be used to develop the
understanding of a ‘function’?
Using the list of mathematical concepts, which
are relevant to this task?
Mathematical
Definition of Functions
Task #2:
How do multiple representations reinforce the
idea of a ‘function’?
Mathematical
Definition of Functions
Task #3:
Which of these situations are functions? Why or
why not?
Now, reverse them! Are these functions?
Definition of Function
1. What information is important for you, as
teachers, to have in mind for an activity
such as this?
2. How can you use this activity to help
students focus on inverse functions?
3. What information is important for your
students to focus on during an activity such
as this?
Pattern Application
1. Where is the function represented:
•
•
•
on the graph?
in the table?
in the expression?
2. Where does the equation appear in each?
3. Describe the connection between each
representation of functions and of equations.
Boxes of Hexagons
1. What is surprising about this activity?
2. What information is important for you, as
teachers, to have in mind for an activity
such as this?
3. What information is important for your
students to focus on during an activity such
as this?
Functions vs. Equations
What is important for students to
understand?