NCTM
Presents
“Higher Standards
for Our Students...
Higher Standards
for Ourselves”
1
An Overview of the
Algebra Standard
for School
Mathematics?
2
Algebra Standard
Instructional programs from prekindergarten
through grade 12 should enable all students to—
• Understand patterns, relations, and
functions;
• Represent and analyze mathematical
situations and structures using algebraic
symbols;
• Use mathematical models to represent and
understand quantitative relationships;
• Analyze change in various context.
3
Algebra Standard
for Grades 6 – 8.
4
Understand patterns, relations, and functions;
• Represent, analyze, and generalize a variety
of patterns with tables, graphs, words, and,
when possible, symbolic rules;
• Relate and compare different forms of
representation for a relationship;
• Identify functions as linear or nonlinear and
contrast their properties from tables, graphs,
or equations.
5
Represent and analyze mathematical
situations and structures using algebraic
symbols;
• Develop an initial conceptual understanding
of different uses of variables;
• Explore relationships between symbolic
expressions and graphs of lines, paying
particular attention to the meaning of
intercept and slope;
continued
6
Represent and analyze mathematical situations
and structures using algebraic symbols;
• Use symbolic algebra to represent situations
and to solve problems, especially those that
involve linear relationships;
• Recognize and generate equivalent forms for
simple algebraic expressions and solve linear
equations.
7
Use mathematical models to represent and
understand quantitative relationships;
• Model and solve contextualized problems
using various representations such as
graphs, tables, and equations.
8
Analyze change in various context.
• Use graphs to analyze the nature of
changes in quantities in linear relationships.
9
Algebra Standard
for Grades 9 – 12.
10
Understand patterns, relations, and functions;
• 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;
continued
11
Understand patterns, relations, and functions;
• understand and perform transformations
such as arithmetically combining, composing,
and inverting commonly used functions, using
technology to perform such operations on
more-complicated symbolic expressions;
• understand and compare the properties of
classes of functions, including exponential,
polynomial, rational, logarithmic, and
periodic functions;
• interpret representations of functions of two
variables
12
Represent and analyze mathematical situations and
structures using algebraic symbols;
• understand the meaning of equivalent forms of
expressions, equations, inequalities, and
relations;
• write equivalent forms of equations,
inequalities, and systems of equations and
solve them with fluency—mentally or with
paper and pencil in simple cases and using
technology in all cases;
• use symbolic algebra to represent and explain
mathematical relationships;
continued
13
Represent and analyze mathematical situations
and structures using algebraic symbols;
• use a variety of symbolic representations,
including recursive and parametric equations,
for functions and relations;
• judge the meaning, utility, and reasonableness
of the results of symbol manipulations,
including those carried out by technology.
14
Use mathematical models to represent and
understand quantitative relationships;
• identify essential quantitative relationships in
a situation and determine the class or classes
of functions that might model the
relationships;
• use symbolic expressions, including iterative
and recursive forms, to represent
relationships arising from various contexts;
• draw reasonable conclusions about a
situation being modeled.
15
Analyze change in various context.
• approximate and interpret rates of change
from graphical and numerical data.
16
Assignment
Read:
• NCTM Principles and
Standards p 60-63,
268-73, and 348-52.
17
Reasoning in Algebra
It’s not only a good idea it’s the law!
TEACHER’S GUIDE
Grade 8
First Year Algebra
Algebra and Problem Solving.
Materials: Activity sheets; graph
paper.
Objective: Students will use algebra
and problem solving to represent a
real life situation that involves a linear
relationship and use graphing as
another way to represent that
relationship.
Prerequisite: Students should know
how to graph linear relationships on
the Cartesian coordinate system.
Directions: Form groups of three to
five students. Pass out the
worksheet and have each group
meet and solve one of the problems
by the described method. Each
group is to prepare a presentation
of their findings to the class. The
class should then work on a
generalization for all of the cases.
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Reasoning in Algebra
It’s not only a good idea it’s the law!
Activity Sheet
Group A
Automobile speeding fines in Pennsylvania are calculated as $35 plus $2 per
mile in excess of five miles per hour of the speed limit. Suppose that you were
caught exceeding the speed limit in a school zone with a limit of 15 miles per
hour. Graph the speed on the x-axis (0 to 75 mph) verses the cost of the fine in
dollars on the y-axis.
ALL GROUPS
Can you generalize the above situations? Let x be the number of miles per
hour over the speed limit. Let y be the cost of the fine. Graph the relationship
between x and y. Discuss how this graph relates to the previous problems.
Can you develop an equation to show the relationship between x and y?
19
Reasoning in Algebra
It’s not only a good idea it’s the law!
PROBLEM SOLVING STANDARD
Instructional Programs from prekindergarten through grade 12
should enable students to –
• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other context;
• Apply and adapt a variety of appropriate strategies to solve
problems;
• Monitor and reflect on the process of mathematical problem
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solving.
Reasoning in Algebra
It’s not only a good idea it’s the law!
ALGEBRA STANDARD
Instructional Programs from prekindergarten through grade 12
should enable students to –
Understand patterns, relations, and functions:
• Represent, analyze, and generalize a variety of patterns with
tables, graphs, words, and, when possible, symbolic rules;
• Relate and compare different forms of representation for a
relationship;
• Identify functions as linear or nonlinear and contrast their
properties from tables, graphs, or equations.
21
Reasoning in Algebra
It’s not only a good idea it’s the law!
ALGEBRA STANDARD
Instructional Programs from prekindergarten through grade 12
should enable students to –
Represent and analyze mathematical situations and structures
using algebraic symbols:
• Develop an initial conceptual understanding of different uses of
variables;
• Explore relationships between symbolic expressions and
graphs of lines, paying particular attention to the meaning of
intercept and slope;
• Use symbolic algebra to represent situations and to solve
problems, especially those that involve linear relationships;
• Recognize and generate equivalent forms for simple algebraic
22
expressions and solve linear equations.
Reasoning in Algebra
It’s not only a good idea it’s the law!
ALGEBRA STANDARD
Instructional Programs from prekindergarten through grade 12
should enable students to –
Use mathematical models to represent and understand
relationships:
• Model and solve contextualized problems using various
representations, such as graphs, tables, and equations.
Analyze change in various context:
•Use graphs to analyze the nature of changes in quantities in
linear relationships.
23