MATHEMATICS 
The Illinois Learning Standards for Mathematics were developed by Illinois teachers for Illinois schools. These goals, standards and benchmarks are an outgrowth of the 1985 Illinois State goals for Learning influenced by the latest thinking in school mathematics. This includes the National council of Teachers of Mathematics; Curriculum and Evaluation Standards for School Mathematics; ideas underlying recent local and national curriculum projects; results of state, national, and international assessment findings; and the work and experiences of Illinois school districts and teachers. Mathematics is a language we use to identify, describe and investigate the patterns and challenges of everyday living. It helps us to understand the events that have occurred and to predict and prepare for events to come so that we can more fully understand our world and more successfully live in it. Mathematics encompasses arithmetic, measurement, algebra, geometry, trigonometry, statistics, probability and other fields. It deals with numbers, quantities, shapes and data, as well as numerical relationships and operations. Confronting, understanding and solving problems is at the heart of mathematics. Mathematics is much more than a collection of concepts and skills; it is a way of approaching new challenges through investigating, reasoning, visualizing and problem solving with the goal of communicating the relationships observed and problems solved to others. All students in Illinois schools need to have the opportunity to engage in learning experiences that foster mastery of these goals and standards. Knowledge of mathematics and the ability to apply math skills to solve problems can be an empowering force for all students—both while in school and later in their lives. Students reaching these goals and standards will have an understanding of how numbers are used and represented. They will be able to use basic operations (addition, subtraction, multiplication, division) to both solve everyday problems and confront more involved calculations in algebraic and statistical settings. They will be able to read, write, visualize and talk about ways in which mathematical problems can be solved in both theoretical and practical situations. They will be able to communicate relationships in geometric and statistical settings through drawings and graphs. These skills will provide all Illinois students with a solid foundation for success in the workplace, a basis for continued learning about mathematics, and a foundation for confronting problem situations arising throughout their lives. 
APPLICATIONS OF LEARNING 
Through Applications of Learning, students demonstrate and deepen their understanding of basic knowledge and skills. These applied learning skills cross academic disciplines and reinforce the important learning of the disciplines. The ability to use these skills will greatly influence students' success in school, in the workplace and in the community. 
SOLVING PROBLEMS 

Recognize and investigate problems; formulate and propose solutions supported by reason and evidence. 
The solving of problems is at the heart of "doing mathematics." When people are called on to apply their knowledge of numbers, symbols, operations, measurement, algebraic approaches, geometric concepts and relationships, and data analysis, mathematics' power emerges. Sometimes problems appear well structured, almost like textbook exercises, and simply require the application of an algorithm or the interpretation of a relationship. Other times, particularly in occupational settings, the problems are nonroutine and require some imagination and careful reasoning to solve. Students must have experience with a wide variety of problemsolving methods and opportunities for solving a wide range of problems. The ability to link the problemsolving methods learned in mathematics with a knowledge of objects and concepts from other academic areas is a fundamental survival skill for life. 
COMMUNICATION 

Express and interpret information and ideas. 
Everyone must be able to read and write technical material to be competitive in the modern workplace. Mathematics provides students with opportunities to grow in the ability to read, write and talk about situations involving numbers, variables, equations, figures and graphs. The ability to shift between verbal, graphical, numerical and symbolic modes of representing a problem helps people formulate, understand, solve and communicate technical information. Students must have opportunities in mathematics classes to confront problems requiring them to translate between representations, both within mathematics and between mathematics and other areas; to communicate findings both orally and in writing; and to develop displays illustrating the relationships they have observed or constructed. 
USING TECHNOLOGY 

Use appropriate instruments, electronic equipment, computers and networks to access information, process ideas and communicate results. 
Technology provides a means to carry out operations with speed and accuracy; to display, store and retrieve information and results; and to explore and extend knowledge. The technology of paper and pencil is appropriate in many mathematical situations. In many other situations, calculators or computers are required to find answers or create images. Specialized technology may be required to make measurements, determine results or create images. Students must be able to use the technology of calculators and computers including spreadsheets, dynamical geometry systems, computer algebra systems, and data analysis and graphing software to represent information, form conjectures, solve problems and communicate results. 
WORKING ON TEAMS 

Learn and contribute productively as individuals and as members of groups. 
The use of mathematics outside the classroom requires sharing expertise as well as applying individual knowledge and skills. Working in teams allows students to share ideas, to develop and coordinate group approaches to problems, and to share and learn from each other in communicating findings. Students must have opportunities to develop the skills and processes provided by team problemsolving experiences to be prepared to function as members of society and productive participants in the workforce. 
MAKING CONNECTIONS 

Recognize and apply connections of important information and ideas within and among learning areas. 
Mathematics is used extensively in business; the life, natural and physical sciences; the social sciences; and in the fine arts. Medicine, architecture, engineering, the industrial arts and a multitude of occupations are also dependent on mathematics. Mathematics offers necessary tools and ways of thinking to unite the concepts, relationships and procedures common to these areas. Mathematics provides a language for expressing ideas across disciplines, while, at the same time, providing connections linking number and operation, measurement, geometry, data and algebra within mathematics itself. Students must have experiences which require them to make such connections among mathematics and other disciplines. They will then see the power and utility that mathematics brings to expressing, understanding and solving problems in diverse settings beyond the classroom. 
STATE GOAL 6: Demonstrate and apply a knowledge
and sense of numbers, including numeration and
operations (addition, subtraction, multiplication,
division), patterns, ratios and proportions.
As a result of their schooling, students will be able to:
LEARNING STANDARD 
HIGH SCHOOL 
A. Demonstrate knowledge and use of numbers and their representations in a broad range of theoretical and practical settings. 
6.A.4 Identify and apply the associative, commutative, distributive and identity properties of real numbers, including special numbers (pi and square roots) 6.A.5 Perform addition, subtraction and multiplication.

B. Investigate, represent and solve problems using number facts, operations (addition, subtraction, multiplication, division) and their properties, algorithms and relationships. 
6.B.4 Select and use appropriate arithmetic operations in practical situations including calculating wages after taxes, developing a budget and balancing a checkbook.

C. Compute and estimate using mental mathematics, paperandpencil methods, calculators and computers. 
6.C.4 Determine whether exact values or approximations are appropriate (e.g. bid a job, determine gas mileage for a trip) 6.C.5 Determine the level of accuracy needed for computations involving measurement and irrational numbers.

D. Solve problems using comparison of quantities, ratios, proportions and percents. 
6.D.4 Solve problems involving recipes or mixtures, financial calculations and geometric similarity using ratios, proportions, and percents.

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STATE GOAL 7: Estimate, make and use measure
ments of objects, quantities and relationships and
determine acceptable levels of accuracy.
As a result of their schooling, students will be able to:
LEARNING STANDARD 
HIGH SCHOOL 
A. Measure and compare quantities using appropriate units, instruments and methods. 
7.A.4a Apply units and scales to describe and compare numerical data and physical objects. 7.A.4b Apply formulas in a wide variety of theoretical and practical realworld measurement applications involving perimeter, area, volume, angle, time, temperature, mass, speed, distance, density and monetary values.

B. Estimate measurements and determine acceptable levels of accuracy. 
7.B.4 Estimate and Measure the magnitude and directions of physical quantities (velocity, force, slope) using rulers, protractors, and other scientific instruments including timers, calculators, and computers. 7.B.5 Estimate perimeter, area, volume, and capacity of irregular shapes, regions and solids and explain the reasoning supporting the estimate. 
C. Select and use appropriate technology, instruments and formulas to solve problems, interpret results and communicate findings. 
7.C.5a Use dimensional analysis to determine units and check answers in applied measurement problems. 7.C.5b Determine how changes in one measure may affect other measures (e.g., what happens to the volume and surface area of a cube when the side of the cube is halved).

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STATE GOAL 8: Use algebraic and analytical methods
to identify and describe patterns and relationships
in data, solve problems and predict results.
As a result of their schooling, students will be able to:
LEARNING STANDARD 
HIGH SCHOOL 
A. Describe numerical relationships using variables and patterns. 
8.A.4a Use algebraic methods to convert decimals to fractions. 8.A.4b Represent mathematical patterns and describe their properties using variables and mathematical symbols.

B. Interpret and describe numerical relationships using tables, graphs and symbols. 
8.B.4a Represent algebraic concepts with physical materials, words, diagrams, tables, graphs, equations and inequalities, and use appropriate technology. (Rule of 4)

C. Solve problems using systems of numbers and their properties. 

D. Use algebraic concepts and procedures to represent and solve problems. 

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STATE GOAL 9: Use geometric methods to analyze,
categorize and draw conclusions abut points, lines,
planes and space.
As a result of their schooling, students will be able to:
LEARNING STANDARD 
HIGH SCHOOL 
A. Demonstrate and apply geometric concepts involving points, lines, planes and space. 
9.A.5 Use geometric figures and their properties to solve problems in the arts, the physical and life sciences and the building trades, with and without the use of technology. 9.A.4a Construct a model of a three dimensional figure from a twodimensional pattern. 9.A.4b Make perspective drawings, tesselations and scale drawings, with and without the use of technology.

B. Identify, describe, classify and compare relationships using points, lines, planes and solids. 
9.B.5 Construct and use two and threedimensional models of objects that have practical applications (e.g., blueprints, topographical maps, scale models).

C. Construct convincing arguments and proofs to solve problems. 
9.C.5a Perform and describe an original investigation of a geometric problem and verify the analysis and conclusions to an audience. 9.C.5b Apply physical models, graphs, coordinate system, networks and vectors to develop solutions in applied contexts (e.g., bus routing, areas of irregular shapes, describing forces and other physical quantities). 
D. Use trigonometric ratios and circular functions to solve problems. 
9.D.4 Analyze and solve problems involving triangles (e.g., distances which cannot be measured directly) using trigonometric ratios. (Pythagorean theorem is another method of indirect measurement)

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STATE GOAL 10: Collect, organize and analyze data
using statistical methods; predict results; and
interpret uncertainty using concepts of probability.
As a result of their schooling, students will be able to:
LEARNING STANDARD 
HIGH SCHOOL 
A. Organize, describe and make predictions from existing data. 
10.A.4a Construct, read and interpret tables, graphs, scatterplots, boxplots and charts to organize and represent data. 10.A.4b Analyze the data using the mean, median, mode and range, and standard deviations with and without the use of technology. 10.A.3c Test the reasonableness of an argument based on data and communicate their findings. 10.A.4c Predict from data using interpolation, extrapolation and trendlines.

B. Formulate questions, design data collection methods, gather and analyze data and communicate findings. 
10.B.4 Design and execute surveys or experiments, gather data to answer relevant questions, and communicate results and conclusions to an audience using traditional methods and contemporary technology.

C. Determine, describe and apply the probabilities of events. 
10.C.4a Solve problems of chance using the principles of probability including conditional settings. 10.C.4b Design and conduct simulations (e.g., waiting times at restaurant, probabilities of births, likelihood of game prizes), with and without the use of technology. 10.C.4c Propose and interpret discrete probability distributions, with and without the use of technology.

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