EALR: 1. The student understands and applies the concepts and procedures of mathematics.

Component 1:1
Understands and applies concepts and procedures from number sense (number and numeration, computation, and estimation).

1. Uses pictures and symbols to demonstrate understanding of fractions, decimals, percents, place value in non-negative decimals, and properties of the rational number system including integers.
2. Compares and orders whole numbers, fractions, decimals, and integers.
3. Understands the concepts of prime and composite numbers, factors and multiples, and divisibility rules.
4. Understands the concepts of ratio and direct proportion.
5. Understands operations on rational numbers including powers and square roots.
6. Adds, subtracts, multiplies, and divides fractions and decimals using rules for order of operation.
7. Uses mental arithmetic, pencil and paper, calculator, or computer as appropriate to the task involving rational numbers.
8. Uses the proportion method for solving percent problems.
9. Identifies situations involving rational numbers in which estimation is sufficient and computation is not required.
10. Uses estimation to predict computation results and to determine the reasonableness of answers involving rational numbers -- for example, estimating a tip.

Component 1:2
Understands and applies concepts and procedures from measurement (attributes and dimensions, approximation and precision, and systems and tools).

1. Understands the relationship among perimeter, area, and volume.
2. Measures objects and events directly or using indirect methods such as finding the area of a rectangle given its length and width.
3. Understands the concept of rate and how to calculate rates and determine units.
4. Calculates volume, area and perimeters of various 2- and 3-dimensional shapes.
5. Understands that precision is related to the unit of measurement used and the calibration of the measurement tool.
6. Uses estimation to obtain reasonable approximations -- for example, estimating the length and width of the athletic field to approximate its area.
7. Understands the benefits of standard units of measurement for both direct and indirect measurement.
8. Understands the relationship among units within both the US and metric systems.
9. Selects and uses tools that will provide an appropriate degree of precision -- for example, using meters vs. kilometers.

Component 1:3
Understands and applies concepts and procedures from geometric sense (shape and dimension, and relationships and transformations).

1. Uses multiple attributes to describe geometric shapes.
2. Identifies and describes objects in the surrounding environment in geometric terms -- for example, describe the triangles that make up a bridge structure.
3. Describes location of objects on coordinate grids.
4. Understands and identifies properties and relationships of plane geometry including ray, isosceles, equilateral, and degrees in a circle, triangle, or quadrilateral.
5. Constructs symmetric, congruent, and similar figures.
6. Understands and constructs simple geometric transformations using combinations of slides, flips, or turns.
7. Uses a compass and straightedge, and/or computer software to perform geometric constructions.

Component 1:4
Understands and applies concepts and procedures from probability and statistics (probability, statistics, and prediction and inference).

1. Knows how to calculate probability of uncertainty for simple events.
2. Understands procedures for counting outcomes to determine probabilities.
3. Knows how to conduct experiments and simulations and to compare results with mathematical expectations.
4. Identifies how statistics can be used to support different points of view.
5. Collects a random sample of data that represents a described population.
6. Organizes and displays data in appropriate forms such as frequency tables, circle graphs, line graphs, bar graphs, and stem-and-leaf graphs.
7. Calculates and uses mean, median, mode, and range as appropriate in describing a set of data..
8. Predicts outcomes of experiments and simulations and compares the predictions to experimental results.
9. Understands and makes inferences based on experimental results.

Component 1:5
Understands and applies concepts and procedures from algebraic sense (relations and representations, and operations).

1. Recognizes, creates, and extends patterns and sequences.
2. Represents number patterns with tables, graphs, and rulers.
3. Represents equalities and inequalities symbolically using = , ¹ , < , > , ³ , ó .
4. Understands and uses variables in simple equations, inequalities, and formulas -- for example, 3x > 18.
5. Evaluates simple expressions.
6. Sets up and solves single-variable equations.

EALR: 2. The student uses mathematics to define and solve problems.

Component 2:1
Investigates situations (by searching for patterns and exploring a variety of approaches).

1. Searches systematically for patterns in simple situations.
2. Develops and uses a variety of strategies and approaches.
3. Identifies missing or extraneous information.
4. Recognizes the need to modify or abandon an unproductive approach.

Component 2:2
Formulates questions and defines the problem.

1. Identifies questions to be answered in new situations.
2. Defines problems in new situations.
3. Identifies the unknowns in new situations.

Component 2:3
Constructs solutions (by choosing the necessary information and using the appropriate mathematical tools).

1. Organizes relevant information from multiple sources.
2. Selects and uses appropriate mathematical tools.
3. Applies appropriate methods, operations, and processes.

EALR: 3. The student uses mathematical reasoning.

Component 3:1
Analyzes information (from a variety of sources; uses models, known facts, patterns and relationships to validate thinking).

1. Interprets, compares, and contrasts information from a variety of sources.
2. Validates thinking and mathematical ideas using models, known facts, patterns, relationships, and counter-examples.

Component 3:2
Predicts results and makes inferences (and makes conjectures based on analysis of problem situations).

1. Makes conjectures and inferences based on analysis of new problem situations.

Component 3:3
Draws conclusions and verifies results (supports mathematical arguments, justifies results, and checks for reasonableness of solutions).

1. Tests conjectures and inferences and explains why they are true or false.
2. Supports arguments and justifies results using inductive reasoning.
3. Checks for reasonableness of results.
4. Reflects and evaluates procedures and results in new problem situations.

EALR: 4. The student communicates knowledge and understanding in both everyday and mathematical language.

Component 4:1
Gathers information (reads, listens, and observes to access and extract mathematical information).

1. Develops a plan for collecting information.
2. Uses reading, listening, and observation skills to access and extract mathematical information from multiple sources such as pictures, diagrams, physical models, oral narratives, and symbolic representations.
3. Chooses appropriate available technology to browse, select, and retrieve relevant mathematical information from a variety of sources.

Component 4:2
Organizes and interprets information.

1. Organizes and clarifies mathematical information in at least one way – reflecting, verbalizing, discussing, or writing.

Component 4:3
Represents and shares information (shares, explains, and defends mathematical ideas using terms, language, charts, and graphs that can be clearly understood by a variety of audiences).

1. Expresses ideas using mathematical language and notation in multiple forms, such as, physical and pictorial models, tables, charts, graphs, etc.
2. Expresses mathematical ideas with clarity using both everyday and mathematical language appropriate to audience.

EALR: 5. The student understands how mathematical ideas connect within mathematics, to other subject areas, and to                          real-life situations.

Component 5:1
Relates concepts and procedures within mathematics (recognizes relationships among mathematical ideas and topics).

1. Connects conceptual and procedural understandings among different mathematical content areas.
2. Relates and uses different mathematical models and representations for the same situation.

Component 5:2
Relates mathematical concepts and procedures to other disciplines (identifies and applies mathematical thinking and notation in other subject areas).

1. Identifies mathematical patterns and ideas in other disciplines.
2. Uses mathematical thinking and modeling in other disciplines.
3. Describes examples of contributions to the development of mathematics such as the contributions of women, men, and different cultures.

Component 5:3
Relates mathematical concepts and procedures to real-life situations (understands the connections between mathematics and problem-solving skills used every day at work and at home).

1. Recognizes the extensive use of mathematics outside the classroom -- for example, in banking or sports statistics.
2. Investigates the use of mathematics within several occupational/career areas of interest.