# Illinois Learning Standards

## Stage I - Math

### Descriptors

#### 6A —

##### Students who meet the standard can demonstrate knowledge and use of numbers and their many representations in a broad range of theoretical and practical settings. (Representations)
1. Illustrate the relationship between second and third roots and powers of a number.
2. Organize problem situations using matrices.
3. Represent, order, and compare real numbers.
4. Place real numbers on a number line.

#### 6B —

##### Students who meet the standard can investigate, represent and solve problems using number facts, operations, and their properties, algorithms, and relationships. (Operations and properties)
1. Compare and contrast the properties of numbers and number systems, including the rational and the real numbers. **
2. Determine an appropriate numerical representation of a problem situation, including roots and powers, if applicable.
3. Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities. *
4. Solve problems using simple matrix operations (addition, subtraction, scalar multiplication).
5. Develop fluency in operations with real numbers using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. **
6. Judge the reasonableness of numerical computations and their results. *

#### 6C —

##### Students who meet the standard can compute and estimate using mental mathematics, paper-and-pencil methods, calculators, and computers. (Choice of method)
1. Develop fluency in operations with real numbers and matrices using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. **
2. Determine and explain whether exact values or approximations are needed in a variety of situations.
3. Determine an appropriate number of digits to represent an outcome.

#### 6D  —

##### Students who meet the standard can solve problems using comparison of quantities, ratios, proportions, and percents.
1. Explain how ratios and proportions can be used to solve problems of percent, growth, and error tolerance.
2. Set up and solve proportions for direct and inverse variation of simple quantities.

#### 7A —

##### Students who meet the standard can measure and compare quantities using appropriate units, instruments, and methods. (Performance and conversion of measurements)
1. Select units and scales that are appropriate for problem situations involving measurement. **
2. Convert between the U.S. customary and metric systems given the conversion factor.

#### 7B —

##### Students who meet the standard can estimate measurements and determine acceptable levels of accuracy. (Estimation)
1. Estimate the magnitude and directions of physical quantities (e.g., velocity, force, slope).
2. Determine answers to an appropriate degree of accuracy using significant digits.

#### 7C —

##### Students who meet the standard can select and use appropriate technology, instruments, and formulas to solve problems, interpret results, and communicate findings. (Progression from selection of appropriate tools and methods to application of measurements to solve problems)
1. Solve problems using indirect measurement by choosing appropriate technology, instruments, and/or formulas.
2. Check measurement computations using unit analysis. **
3. Describe the general trends of how the change in one measure affects other measures in the same figure (e.g., length, area, volume).
4. Determine linear measures, perimeters, areas, surface areas, and volumes of similar figures using the ratio of similitude.
5. Determine the ratio of similar figure perimeters, areas, and volumes using the ratio of similitude.
6. Calculate by an appropriate method the length, width, height, perimeter, area, volume, surface area, angle measures, or sums of angle measures of common geometric figures, or combinations of common geometric figures.
7. Solve problems involving multiple rates, measures, and conversions.

#### 8A—

##### Students who meet the standard can describe numerical relationships using variables and patterns. (Representations and algebraic manipulations)
1. Write equivalent forms of equations, inequalities, and systems of equations. **
2. Represent and explain mathematical relationships using symbolic algebra. **
3. Model and describe slope as a constant rate of change.
4. Explain the difference between constant and non-constant rate of change.
5. Create an equation of a line of best fit from a set of ordered pairs or set of data points.
6. Simplify algebraic expressions using a variety of methods, including factoring.
7. Justify the results of symbol manipulations, including those carried out by technology. **
8. Identify essential quantitative relationships in a situation and determine the class or classes of functions (e.g., linear, quadratic) that might model the relationships. **
9. Represent relationships arising from various contexts using algebraic expression.
10. Rewrite absolute value inequalities in terms of two separate equivalent inequalities with the appropriate connecting phrase of "AND" or "OR".

#### 8B—

##### Students who meet the standard can interpret and describe numerical relationships using tables, graphs, and symbols. (Connections of representations including the rate of change)
1. Describe the relationships of the independent and dependent variables from a graph.
2. Interpret the role of the coefficients and constants on the graph of linear and quadratic functions given a set of equations.
3. Relate the effect of translations on linear graphs and equations.
4. Create and connect representations that are tabular, graphical, numeric, and algebraic from a set of data.
5. Recognize and describe the general shape and properties of the graphs of linear, absolute value, and quadratic functions.
6. Approximate and interpret rates of change from graphical and numerical data. *
7. Identify slope in an equation and from a table of values.
8. Graph absolute values of linear functions on the Cartesian plane.
9. Recognize direct variation, inverse variation, linear, and exponential curves from their graphs, a table of values, or equations. **
10. Interpret and use functions as a geometric representation of linear and non-linear relationships.

#### 8C—

##### Students who meet the standard can solve problems using systems of numbers and their properties. (Problem solving; number systems, systems of equations, inequalities, algebraic functions)
1. Describe and compare the properties of linear and quadratic functions. **
2. Solve problems by recognizing how an equation changes when parameters change.
3. Interpolate and extrapolate to solve problems using systems of numbers.
4. Solve problems using translations and dilations on basic functions.

#### 8D—

##### Students who meet the standard can use algebraic concepts and procedures to represent and solve problems. (Connection of 8A, 8B, and 8C to solve problems)
1. Solve equivalent forms of equations, inequalities, and systems of equations with fluency-mentally or with paper-and-pencil in simple cases and using technology in all cases. **
2. Create word problems that meet given conditions and represent simple power or exponential relationships, or direct or inverse variation situations.
3. Solve simple quadratic equations using algebraic or graphical representations.
4. Solve problems of direct variation situations using a variety of methods.

#### 9A—

##### Students who meet the standard can demonstrate and apply geometric concepts involving points, lines, planes, and space. (Properties of single figures, coordinate geometry and constructions)
1. Describe and apply properties of a polygon or a circle in a problem-solving situation.
2. Classify angle relationships for two or more parallel lines crossed by a transversal.
3. Analyze geometric situations using Cartesian coordinates. **
4. Represent transformations of an object in the plane using sketches, coordinates, and vectors.
5. Design a net that will create a given figure when folded.
6. Solve problems using constructions.
7. Gain insights into, and answer questions in, other areas of mathematics using geometric models. **
8. Calculate distance, midpoint coordinates, and slope using coordinate geometry.
9. Visualize a three-dimensional object from different perspectives and describe their cross sections. **
10. Identify and apply properties of medians, altitudes, angle bisectors, perpendicular bisectors, and midlines of a triangle.

#### 9B—

##### Students who meet the standard can identify, describe, classify and compare relationships using points, lines, planes, and solids. (Connections between and among multiple geometric figures)
1. Solve problems using triangle congruence and similarity of figures.
2. Extend knowledge of plane figure relationships to relationships within and between geometric solids.
3. Identify relationships among circles, arcs, chords, tangents, and secants.
4. Solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture using geometric ideas. **
5. Analyze and describe the transformations that lead to successful tessellations of one or more figures.

#### 9C—

##### Students who meet the standard can construct convincing arguments and proofs to solve problems. (Justifications of conjectures and conclusions)
1. Create and critique arguments concerning geometric ideas and relationships such as properties of circles, triangles and quadrilaterals.
2. Develop a formal proof for a given geometric situation on the plane.
3. Provide a counter-example to disprove a conjecture.
4. Develop conjectures about geometric situations with and without technology.
5. Justify constructions using geometric properties.
6. Describe the difference between an inductive argument and a deductive argument.

#### 9D—

##### Students who meet the standard can use trigonometric ratios and circular functions to solve problems.
1. Determine distances and angle measures using indirect measurement and properties of right triangles.
2. Solve problems using 45°-45°-90° and 30°-60°-90° triangles.

#### 10A—

##### Students who meet the standard can organize, describe and make predictions from existing data. (Data analysis)
1. Describe the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable. **
2. Display a scatter plot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients for bivariate measurement data using technological tools.
3. 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. *
4. Analyze two-variable data for linear or quadratic fit.
5. Make decisions based on data, including the relationships of correlation and causation.

#### 10B—

##### Students who meet the standard can formulate questions, design data collection methods, gather and analyze data and communicate findings. (Data Collection)
1. Describe the characteristics of well-designed studies, including the role of randomization in surveys and experiments. **
2. Discuss informally different populations and sampling techniques.
3. Decide if a survey was "successful" in gathering intended data and justify the decision.

#### 10C—

##### Students who meet the standard can determine, describe and apply the probabilities of events. (Probability including counting techniques)
1. Determine geometric probability based on area.
2. Calculate probability using Venn diagrams.
3. Determine simple probabilities using frequency tables.
4. Construct empirical probability distributions using simulations.**
5. Describe the concepts of conditional probability.
6. Develop an understanding of permutations and combinations as counting techniques. *

* National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, Va: National Council of Teachers of Mathematics, 2000.
** Adapted from: National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, Va: National Council of Teachers of Mathematics, 2000.