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. Compares and renames fractions, decimals, and percent.
2. Understands place value in decimals to the thousandths.
3. Recognizes the application of properties of the system of non-negative rational numbers.
4. Understands and explains the magnitude of numbers by comparing, sequencing, and ordering whole numbers, simple/common fractions, and decimals.
5. Understands and applies concepts involving prime and composite numbers, divisibility, factors, and multiples.
6. Recognizes square numbers and their roots.
7. Recognizes the relationship between simple/common ratios and percent.
8. Applies whole number operations of addition, subtraction, multiplication, and division.
9. Understands multiplication and division of fractions and decimals at concrete and pictorial level.
10. Adds and subtracts non-negative rational numbers.
11. Multiplies and divides non-negative rational numbers at the concrete and pictorial level.
12. Understands, represents, and uses numbers in a variety of forms (fractions, decimals, percents, exponents).
13. Uses mental arithmetic, pencil and paper, calculator, or computer as appropriate to the task involving rational numbers.
14. Understands there are situations in which estimation is sufficient and computation is not required.
15. Uses estimation to predict computation results.
16. Uses estimation to determine reasonableness of answers.

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

1. Understands the structure and use of systems of measurement.
2. Understands and uses different ways in which various objects and events can be measured directly or indirectly.
3. Understands the concepts of linear dimensions and area.
4. Understands how dimensional changes affect perimeter and/or area.
5. Develops an understanding of common angle measures.
6. Understands the concept of rate and how to calculate rates and determine units.
7. Understands that precision is a function of the unit of measurement and the calibration of the measurement tool.
8. Uses estimation to obtain reasonable approximations -- for example, estimating the length and width of the playground to approximate its area.
9. Understands the benefits of standard and non-standard units of measurement.
10. Knows relationships among units within each system (US/metric).
11. 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 attributes of shape and size to describe geometric shapes.
2. Visualizes and represents geometric figures.
3. Uses compass and straightedge and/or computer software to perform geometric constructions.
4. Describes the location of objects on coordinate grids.
5. Understands and constructs simple geometric transformations and symmetric and congruent figures.
6. Represents and solves problems using geometric models.
7. Understands that objects in nature, architecture, and other items in the environment can be identified and described in geometric terms.
8. Understands and identifies properties and relationships of plane geometry including angle, ray, isosceles, equilateral, and degrees in a circle, triangle, or quadrilateral.

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

1. Calculates numerical measures of certainty for simple events.
2. Conducts experiments and simulations to compare results with mathematical expectations.
3. Understands procedures for counting outcomes to determine probabilities.
4. Understands that there can be different interpretations of the same set of data and how statistics can be used and misused to support different points of view and arguments.
5. Describes a population and collects a random sample of data that represents it.
6. Organizes, displays, and interprets data in simple tables, charts, and graphs as appropriate.
7. Understands, calculates, and uses mean, median, mode, and range appropriately to describe a set of data.
8. Uses theoretical probabilities to predict outcomes of experiments and simulations to compare the predictions to experimental results.
9. Understands and makes inferences or predictions based on experimental results.

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

1. Analyzes relationships to recognize how a change in one quantity results in a change in another.
2. Recognizes, creates, and extends series and sequence.
3. Represents and analyzes number patterns with tables, graphs, and rules.
4. Symbolically represents equalities and inequalities symbolically using = , ¹ , < , > , ³ .
5. Understands and uses variables in simple equations.
6. Understands how the basic arithmetic operations are related.
7. Evaluates simple expressions.
8. Solves simple linear equations.
9. Applies algebraic methods to solve a variety of problems.
10. Represents situations and number patterns with tables, graphs, verbal rules, and equations and explores the interrelationships of these representations.

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. Conducts systematic and open-ended explorations.
2. Searches systematically for patterns.
3. Develops and uses a variety of strategies.
4. Identifies what information is missing or extraneous.
5. Recognizes when an attempted approach is unproductive and tries to modify it or tries a new 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 collected from a variety of sources.
2. Selects and uses tools appropriate for situation and degree of required precision.
3. Uses appropriate strategies.

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 information from a variety of sources.
2. Compares and contrasts information accessed from different sources.
3. Finds examples that contradict a mathematical argument.
4. Validates thinking and mathematical ideas using models, known facts, patterns, and relationships.

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 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 discusses why they are true or not true.
2. Supports arguments and justifies results using inductive reasoning.
3. Reflects on and evaluates procedures and results.
4. Check for reasonableness of results.

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 system 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 technology to browse, select, and retrieve mathematical information from a variety of sources.

Component 4:2
Organizes and interprets information.

1. Organizes mathematical information within given parameters.
2. Clarifies mathematical understandings in multiple ways – reflecting, verbalizing, and discussing.

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 or pictorial models, tables, charts, graphs, and algebraic notation.
2. Expresses mathematical ideas with clarity in ways 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. Links conceptual and procedural understandings among a variety of mathematical content areas.
2. Uses equivalent mathematical models and representations.
3. Sees relationships among parts of mathematics.

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 the contributions of various cultures to the historical development of number systems, measurement, and patterns.
4. Describes examples of the contributions of both women and men to the development of mathematics.

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.