# Course Syllabus for Beginning Algebra

**Algebra Standard**

**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;

▪ 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.

**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;

▪ 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.

**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.

**Analyze change in various contexts
**▪ approximate and interpret rates of change from graphical and numerical
data.

**Geometry Standard**

**Analyze characteristics and properties of two- and
three-dimensional geometric shapes and develop mathematical arguments about
geometric relationships
**▪ analyze properties and determine attributes of two- and three-dimensional
objects;

▪ explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;

▪ establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others; • use trigonometric relationships to determine lengths and angle measures.

**Specify locations and describe spatial relationships
using coordinate geometry and other representational systems
**▪ use Cartesian coordinates and other coordinate systems, such as
navigational, polar, or spherical systems, to analyze geometric situations;

▪ investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.

**Apply transformations and use symmetry to analyze
mathematical situations
**▪ understand and represent translations, reflections, rotations, and
dilations of objects in the plane by using sketches, coordinates, vectors,
function notation, and matrices;

▪ use various representations to help understand the effects of simple transformations and their compositions.

**Use visualization, spatial reasoning, and geometric
modeling to solve problems
**▪ draw and construct representations of two- and three-dimensional
geometric objects using a variety of tools;

▪ visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections;

▪ use vertex-edge graphs to model and solve problems;

▪ use geometric models to gain insights into, and answer questions in, other areas of mathematics;

▪ use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.

**Measurement Standard**

**Understand measurable attributes of objects and the
units, systems, and processes of measurement
**▪ make decisions about units and scales that are appropriate for problem
situations involving measurement.

**Apply appropriate techniques, tools, and formulas to
determine measurements.
**▪ analyze precision, accuracy, and approximate error in measurement
situations;

▪ understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders;

▪ apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations;

▪ use unit analysis to check measurement computations.

**Data Analysis and Probability Standard**

**Formulate questions that can be addressed with data and
collect, organize, and display relevant data to answer them
**▪ understand the differences among various kinds of studies and which types
of inferences can legitimately be drawn from each;

▪ know the characteristics of well-designed studies, including the role of randomization in surveys and experiments;

▪ understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable;

▪ understand histograms, parallel box plots, and scatterplots and use them to display data;

▪ compute basic statistics and understand the distinction between a statistic and a parameter.

**Select and use appropriate statistical methods to
analyze data
**▪ for univariate measurement data, be able to display the distribution,
describe its shape, and select and calculate summary statistics;

▪ for bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools;

▪ display and discuss bivariate data where at least one variable is categorical;

▪ recognize how linear transformations of univariate data affect shape, center, and spread;

▪ identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.

**Develop and evaluate inferences and predictions that
are based on data
**▪ use simulations to explore the variability of sample statistics from a
known population and to construct sampling distributions;

▪ understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference;

▪ evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions;

▪ understand how basic statistical techniques are used to monitor process characteristics in the workplace.

**Understand and apply basic concepts of probability
**▪ understand the concepts of sample space and probability distribution and
construct sample spaces and distributions in simple cases;

▪ use simulations to construct empirical probability distributions;

▪ compute and interpret the expected value of random variables in simple cases;

▪ understand the concepts of conditional probability and independent events;

▪ understand how to compute the probability of a compound event.

**Problem Solving Standard
**▪ Build new mathematical knowledge through problem solving

▪ Solve problems that arise in mathematics and in other contexts

▪ Apply and adapt a variety of appropriate strategies to solve problems

▪ Monitor and reflect on the process of mathematical problem solving

**Reasoning and Proof Standard
**▪ Recognize reasoning and proof as fundamental aspects of mathematics

▪ Make and investigate mathematical conjectures

▪ Develop and evaluate mathematical arguments and proofs

▪ Select and use various types of reasoning and methods of proof

**Communication Standard
**▪ Organize and consolidate their mathematical thinking through
communication

▪ Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

▪ Analyze and evaluate the mathematical thinking and strategies of others;

▪ Use the language of mathematics to express mathematical ideas precisely.

**Connections Standard
**▪ Recognize and use connections among mathematical ideas

▪ Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

▪ Recognize and apply mathematics in contexts outside of mathematics

**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