Comparison Benchmarks 6,7,8

EALR Comparison -Grade Level Expectations: Grades 6-8

[For a seach of the EALRs click ]HERE

SP=Solves Problems CU=Communicates Understanding RL=Reasons Logically MC=Makes Connections
8-8-2003 Jeff LeMieux

Grade 6 Grade 7 Grade 8

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

Component 1.1: Understand and apply concepts and procedures from number sense - number, numeration, computation, and estimation.

Number and Numeration
1.1.1 Understand the concepts of integers.

Represent and identify integers on a number line.
Identifies, compares, and orders non-negative whole numbers, fractions, and decimals
1.1.2 Understand the relative values of integers and non-negative rational numbers.

Compare different representations of non-negative rational numbers by implementing strategies (e.g., like denominators, benchmark fractions). [SP, CU, MC]
Identify equivalence between common fractions and decimals. [MC]
Compare integer values and explain which is greater and why. [RL, CU]
1.1.3 Understand the additive inverse property.

Illustrate the additive inverse property using physical models and pictures. [CU]
Explain the additive inverse property and why it works. [CU]
1.1.4 Understand the concepts of ratio and percent.

Write ratios and proportions in part/part and part/whole relationships using objects, pictures, and symbols (e.g., using /, :, or to as representations for ratios). [CU]
Represent equivalent ratios using objects, pictures, and symbols. [CU, MC]
Identify percent as 100 equal size sets (e.g., 1% of 200 items is 2 items). [RL]
Represent a given percentage with a physical model. [CU]
Find a percentage of a given set of items other than 100.
Computation
1.1.5 Understand the meaning of addition and subtraction of integers and multiplication and division on non-negative rational numbers.

Explain meaning of multiplying and dividing non-negative common fractions and decimals using visual and physical models. [CU]
Explain the meaning of addition and subtraction of integers using models (e.g., reducing debt, temperature increase or decrease, yards gained and lost). [CU]
1.1.6 Understand and apply procedures for addition and subtraction on non-negative decimals and fractions with fluency.

Find the sums and/or differences of common fractions and decimals.
Write and solve problem situations to find sums and/or differences of decimals or fractions. [SP, RL, CU, MC]
1.1.7 Apply strategies for mental arithmetic, pencil and paper, or calculator as appropriate to the task involving non-negative decimals and simple fractions.

This indicator should be incorporated in all instruction and applies to all grade levels.

Estimation
1.1.8 Understand situations in which estimation is appropriate and apply estimation strategies to determine the reasonableness of answers involving content appropriate to the grade level.

Apply estimation strategies prior to computation of whole numbers, decimals, and common fractions to determine reasonableness of answers.
Use estimation to predict or check answers.
Identify appropriate estimated answers for a given situation.
Articulate various strategies used during estimation. [CU]

Number and Numeration
1.1.1 Understand the concepts of rational numbers.

Create a model when given a symbolic representation of a rational number. [SP, CU, MC]
Write the rational number when given a model (e.g., number line, area model, situation, diagram, picture). [SP, CU, MC]
Convert between equivalent forms of rational numbers.
Explain the value of a given digit in a rational number. [CU]
1.1.2 Understand the relative values of rational numbers.

Compare and order non-negative rational numbers using concrete models or implementing strategies (e.g., like denominators, benchmark fractions, conversions). [SP, CU, MC]
Order different representations of rational numbers. [MC]
Locate symbolic representations of rational numbers on a number line including fractions, decimals, and integers.
1.1.3 Understand and apply the distributive and multiplicative inverse properties and use order of operations.

Understand and use the inverse relationships of multiplication and division to simplify computations and solve problems. [SP]
Use order of operations to simplify computations and solve problems.
Illustrate the distributive property of multiplication over addition using an area model or picture. [CU, MC]
Identify errors in the application of order of operations. [RL]
1.1.4 Understand and apply the concepts of ratio, percent, and direct proportion.

Write a ratio or proportion for a given situation.
Express proportional relationships using objects, pictures, and symbols. [CU, MC]
Explain the meaning of a ratio or proportion. [CU]
Represent a new relationship from a given ratio (e.g., part/part to part/whole; given a ratio of girls to boys, find the ratio of girls to class). [CU]
Represent percentages less than 1% or greater than 100% using objects, pictures, and symbols. [CU, MC]

Computation
1.1.5 Understand the meaning of multiplication and division of integers.

Explain the meaning of multiplication and division of integers using visual and physical models. [CU]
Create a problem situation involving multiplication or division of integers. [SP, CU, RL, MC]
1.1.6 Understand and apply the procedures of addition and subtraction for integers and multiplication and division on non-negative rational numbers with fluency.

Find the sum, difference, product, and/or quotient of decimals and common fractions without common denominators.
Use order of operations to solve problem situations with decimals and common fractions.
Find the sums and differences of integers.
1.1.7 Apply strategies for mental arithmetic, pencil and paper, or calculator as appropriate to the task involving non-negative decimals and simple fractions. · This indicator should be incorporated in all instruction and applies to all grade levels.
Estimation
1.1.8 Understand situations in which estimations is appropriate and apply estimation strategies to determine the reasonableness of answers involving addition and subtraction of integers and operations on non-negative rational numbers

Apply estimation strategies prior to computing addition and subtraction of integers and operations on non-negative rational numbers to determine reasonableness of answers.
Justify why estimation would be used rather than an exact computation. [RL, CU]
Describe a situation where estimation is sufficient in real life contexts. [CU, MC]

Number and Numeration
1.1.1 Understand and apply symbolic representations of rational numbers including whole number powers and roots of square numbers.

Read and use scientific notation to represent large numbers. [MC, RL, SP]
Explain the meaning of a whole number exponent. [CU]
Read and use exponential notation to represent large numbers. [MC, RL, SP]
Identify a square number and find its root.
1.1.2 Understand the relative values of rational numbers including whole number powers and roots of square numbers.

Compare and order rational numbers using concrete models or implementing strategies. [RL]
Order different representations of rational numbers including large numbers in scientific and exponential notation.
Locate symbolic representations of rational numbers on a number line including whole number powers and roots of square numbers.

1.1.3:
1.1.4 Apply ratio, percent, and direct proportion in complex situations.

Solve problems involving ratio and proportion (e.g., similar figures, scale drawings, rates, find unit pricing, increase or decrease a recipe, find the portions for a group converting between different units of measure, or finding medicinal dosages). [SP, RL, CU, MC]
Solve problems involving percentages in complex situations (e.g., percent increase/decrease, tax, commission, discount). [SP, RL, CU, MC]
Explain advantages and disadvantages of different representations in a given situation (e.g., using 1/3 versus 33 1/3 %). [CU]

Computation
1.1.5 Understand the meaning of operations on rational numbers.

Compare and contrast operations on rational numbers using pictures and symbols. [RL, CU]
Create a problem situation to match a given rational number equation. [SP, RL, CU, MC]
Identify a rational number equation to match a given situation. [CU, MC]
1.1.6 Understand and apply computations on rational numbers with fluency.

Compute fluently with rational numbers in all forms except exponential.
Write and solve problems that involve computation with rational numbers. [SP, RL, CU, MC]
1.1.7 Apply strategies for mental arithmetic, pencil and paper, calculator or computer as appropriate to the task involving rational numbers.

Select and justify appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation.

Estimation
1.1.8 Understand situations in which estimation is appropriate and apply estimation strategies to determine reasonableness of answers involving non-negative rational numbers.

Explain situations involving real numbers where estimates are sufficient and others for which exact value is required. [CU]
Justify why an estimate would be used rather than an exact answer in a given situation. [CU]

Component 1.2: Understand and apply concepts and procedures from measurement.

Attributes, Units, and Tools
1.2.1 Understand the attributes of volume and capacity.

Given two rectangular solids, compare their volume.
Compare the relative capacity of two containers (e.g., paper cylinders formed horizontally and vertically and filled with popcorn). [SP, CU, MC]
1.2.2: Understand and apply standard units to measure the attributes of volume and capacity.

Identify cubic units to measure volume (e.g., linking cubes, cubic centimeter).
Identify and read incremental units for capacity (e.g., milliliters, cups, ounces).
1.2.3 Apply appropriate tools to measure volume/capacity using both standard and metric systems.

Measure volume of rectangular prisms and label appropriately (e.g., cubic units). [SP, CU]
Measure the capacity of containers using appropriate tools and label (e.g., graduated cylinders, measuring cups, tablespoons). [SP, CU]

1.2.4:

Procedures, Precision, and Estimation
1.2.5 Understand that precision is related to the size of the unit of the measurement used.

Compare the appropriateness of standard to nonstandard units in measuring volume or capacity. [CU]
Choose the appropriate standard unit for measuring volume or capacity (e.g., cubic inches vs. cubic feet, cups vs. gallons). [SP]
1.2.6 Understand when approximate measurements are sufficient and apply estimation strategies to obtain reasonable measurements of volume.

Identify situations when approximate measurements are sufficient.
Recognize when a measurement is approximate rather than exact. [SP]
Use estimation to justify reasonableness of a measurement. [RL, CU]
Estimate a measurement using standard or nonstandard units. [SP]

Attributes, Units, and Tools
1.2.1 Understand the attribute of surface area.

Compare and describe the surface area of a variety of rectangular solids. [RL, CU]
Apply surface area to real world situations (e.g., paint or wallpaper for a room). [SP, MC]

1.2.2:
1.2.3 Apply tools and formulas to determine the area of quadrilaterals and triangles.
Find the area of triangles and quadrilaterals using formulas. [MC]
Justify the formula of a triangle or quadrilateral using pictures or manipulatives. [RL, CU]
Use appropriate tools to measure linear dimensions to determine surface area. [SP]
Find the surface area of a rectangular solid. [SP]
Procedures, Precision, and Estimation

1.2.4:
1.2.5 Understand that different situations require different levels of precision.

Justify the use of a unit of measure (e.g., meters or kilometers, inches or feet). [CU, RL]
Select and use appropriate unit of measurement according to type of unit (standard vs. metric) or size of unit (millimeters vs. centimeters). [MC, RL]
1.2.6 Understand when approximate measurements are sufficient and apply estimation strategies to obtain reasonable measurements of surface area.

Use estimation to find reasonable approximation of surface area.
Justify the reasonableness of an estimate. [RL]
Decide when it is appropriate to estimate the volume and/or surface area of a solid and explain why. [CU, RL, SP, MC]

Attributes , Units, and Tools
1.2.1 Understand and apply right triangle relationships.

Demonstrate the meaning of the Pythagorean Theorem with drawings or manipulatives. [CU]
Use simple trigonometric relationships to find indirect measurements of length. (e.g., height of a tree or flagpole, side of a triangle using sine, cosine, or tangent). [SP,RL,MC]
Calculate measures of objects for which no direct information is given (e.g., similar figures, ratio, proportion, scale). [SP,RL,MC]

1.2.2:
1.2.3 Apply tools and formulas to find the area of circles, the volume and surface area of cubes, right cylinders, and rectangular prisms.

Calculate the area of circles and volume and surface area of cubes, right cylinders, and rectangular prisms.
Describe a strategy for finding the area of a figure (e.g., formulate a means for finding the area of an irregular figure). [SP,RL,CU]
Given the perimeter, circumference, or area, determines linear dimensions.
Justify the formula for surface area of a rectangular prism or right cylinder with pictures or manipulatives. [RL, CU]
Measure objects directly using appropriate tools to find area of circles, and volumes and surface area of right cylinders and rectangular prisms (e.g., rulers to find dimensions).

1.2.4:

Procedures, Precision, and Estimation
1.2.5 Understand that precision and accuracy of measurement are affected by units, tools, and their applications.

Determine the appropriate unit of measurement in a problem situation (e.g., the height of a tree, surface area of a pond). [MC, RL, SP]
Justify the use of a particular tool in a measurement. [RL, CU]
Convert within systems (e.g., centimeters to meters, inches to feet).
1.2.6 Understand when approximate measurements are sufficient and apply estimation strategies to obtain reasonable measurements of right triangle relationships.

Identify situations when approximate measurements are sufficient.
Estimate distance or height in a problem situation using right triangle or Pythagorean relationships (e.g., height of a flagpole using tangent, distance across a lake using Pythagorean relationship). [MC,CU,SP]

Component 1.3: Understand and apply concepts and procedures from geometric sense.

Properties and Relationships
1.3.1 Understand properties of angles, triangles, and quadrilaterals.

Explain the difference between a regular and irregular polygon. [CU, RL]
Identify, sort, classify, and explain the properties of specific quadrilaterals including squares, rectangles, parallelograms, and trapezoids based on attributes. [CU, RL]
Identify angles as acute, right, obtuse, or straight.
Classify triangles as right, equilateral, isosceles, and/or scalene.
1.3.2 Apply understanding of the properties of 3-D figures and shapes.

Identify, name, compare, and sort 3-D shapes and figures.
Identify geometric figures and concepts in nature and art (e.g. triangle in architecture).
Given two 3-D solids, explain how they are alike and different in terms of their attributes (e.g., using a Venn diagram). [RL, CU]
1.3.3 Understand the relative location of integers on a number line.

Show the order of a given set of integers on a number line with both positive and negative numbers.
Given directions for movement on a number line, including positive and negative numbers (vertical and horizontal), identify the point of final destination (e.g., temperature variation at different times of the day, bank accounts, gain and loss of weight). [MC]
Determine the distance between any two integers on a number line.
1.3.4 Understand and analyze rotations (turns) of a 2-D figure about the center of the figure.

Create a design using rotational symmetry of a shape. [MC]
Match a figure with its image following one transformation. [RL]

Properties and Relationships
1.3.1 Understand the similarity relationship in 2-D shapes and figures.

Identify corresponding sides and angles of two similar figures.
Determine and justify if two figures are similar using the definition of similarity. [RL, CU]
1.3.2 Understand and apply properties of polygons and circles.

Identify the attributes of a circle including diameter, radius, and circumference.
Given all but one of the angles of a polygon, find the missing angle. [RL]
Draw polygons and circles with specified properties (e.g., circumference of 18 cm). [CU]
Use the properties of polygons and circles to solve mathematical problems. [SP, MC, CU]
1.3.3 Understand location of points on coordinate grids in any of the four quadrants.

Given three points, identify the coordinates of the fourth point to make a rectangle. [RL]
Plot ordered pairs in any of the four quadrants.
Identify the coordinates of a given point in any of the four quadrants.
1.3.4 Apply and analyze combinations of transformations.

Use transformations to create congruent shapes and figures in multiple orientations.
Given a figure on a coordinate grid, find the coordinate pairs for a translation and/or a reflection across an axis.
Match a figure with its image following two transformations. [RL]

Properties and Relationships

1.3.1:
1.3.2 Understand and analyze properties and relationships of 1-D, 2-D, and 3-D objects.

Identify and label rays, lines, end points, line segments, vertices, and angles. [CU]
Given a figure, write a verbal description of its geometric properties. [CU]
Given two similar figures find the length of a missing side, or the measure of a missing angle of one of the figures. [RL]
Create symmetrical, congruent, and/or similar figures using a variety of tools (e.g., grid, ruler, pattern blocks, geo-boards).
Given a simple figure, appropriate tools and a scale, draw a similar figure (e.g., graph paper, geo-board, dot paper). [RL]
Match or draw 3-D objects from different perspectives using the same properties and relationships (e.g., match to the correct net, draw the top view). [RL]

Locations and transformations
1.3.3 Understand the relationships between two or more points on a coordinate grid.

Find length of horizontal and vertical lines on a coordinate plane.
Find the length of a line segment on a coordinate grid. [RL]
Given the coordinates of the vertices of a regular polygon, locate the missing vertex. [RL]
Use the Pythagorean Theorem or trigonometric ratios to find the missing side of a triangle in a real world setting (e.g., the height of a ladder). [SP, RL, MC]

1.3.4:

Component 1.4: Understand and apply concepts and procedures from probability and statistics.

Probability
1.4.1 Understand probability as a ratio between and including zero and one.

Calculate probability for a simple event.
Express probability as a ratio, decimal, and percent.
1.4.2 Understand the procedures for determining all possible outcomes of simple experiments.

Represent and interpret all possible outcomes of simple experiments (e.g., an organized list, a table, or a tree diagram). [RL, CU]

1.4.3:

Statistics
1.4.4 Apply measures of central tendency to interpret a set of data.

Determine when it is appropriate to use mean, median, or mode. [RL]
Use mean, median, and mode to explain familiar situations (e.g., the heights of students in the class, the hair color of students in the class). [CU, RL, MC]
1.4.5 Understand how to evaluate a question or data collection method for fairness.

Compare data collection methods for a given situation to determine fairness of the method (e.g., compare a phone survey, a web survey, and a personal interview survey). [RL, MC]
Judge a data collection method to consider limitations that could affect interpretations (e.g., to examine battery life, compare how long batteries last in a flashlight vs. a portable CD player). [SP, RL, MC]
1.4.6 Analyze and evaluate data appropriate to the grade level.

Make a conjecture about an entire group or population from a sample (e.g., sample the classroom to find the favorite school lunch). [MC, CU, RL]
Evaluate and explain conclusions drawn from data (e.g., from newspapers or web sites). [RL, MC, SP]

Probability
1.4.1 Understand the concepts of complementary and mutually exclusive events.

Determine when events are mutually exclusive (e.g., you're grade on a test is an A, B, or C).
Determine when events are complementary (e.g., a person awake or asleep, you pass or fail a test).
Identify events that are complementary or mutually exclusive or neither and explain. [RL, CU, MC]
1.4.2 Understand the procedures for determining the probabilities for simple experiments.

Calculate the probabilities of simple experiments (e.g., pulling colored balls from a bag, drawing a card, rolling a 6 on a number cube, spinning a spinner, etc.).
Calculate the probability of an event given the probability of its complement.

Statistics
1.4.3 Understand how to organize, display, and interpret data from line graphs, box-and- whisker, and scatter plots.

Construct line graphs, box-and-whisker, and scatter plots from collected data. [CU, MC]
Describe population characteristics from a box-and-whisker plot. [CU]
Use scatter plots to describe trends and interpret relationships. [RL]
Draw trend lines with or without technology and make predictions about real-world situations. [RL, CU, MC, SP]
1.4.4 Understand the effects of outliers and clusters on measures of central tendency and select the appropriate measure of central tendency for a set of data.

Describe the effects of extreme values on means in a population. [CU, RL, MC]
Explain the use of median or mean as a measure of central tendency in a given situation (e.g., when an extreme value skews the mean). [RL, SP, CU, MC]

1.4.5 :
1.4.6 Analyze data and evaluate the inferences that can be drawn from data.

Make and justify an inference drawn from a sample. [CU, RL, MC] Evaluate and explain conclusions drawn from data (e.g., from newspapers, web sites, opinion polls). [MC, RL, SP]

Probability
1.4.1 Understand the concept of compound events.

Determine and explain when events are compound. [CU, RL]
Explain the difference between compound events involving 'and' and 'or' (e.g., rolling a six and drawing an ace of spades vs. rolling a six or drawing an ace spades). [CU, RL]
1.4.2 Understand and apply the procedures for determining theoretical and experimental probabilities for simple and compound events.

Design and conduct a simulation, with and without technology, to determine the probability of an uncertain event occurring. [MC, RL, SP, CU]
Calculate the probability of two independent events occurring simultaneously using various methods (e.g., organized list, tree diagram, counting procedures, and area model).
Explain the relationship between theoretical and experimental probability of simple and compound events. [CU, RL]

1.4.3:
1.4.4 Understand and find average deviation from a set of data.

Calculate the average deviation from a set of data.
Explain the meaning of average deviation in real-world situations (e.g., foot size in a class). [MC, RL, CU]
1.4.5 Understand how different samples of a population affect the data.

Identify sources of sampling bias, given a situation (e.g,. interviewing only girls, a certain age group, or too small a sample). [MC, RL, CU, SP]
Describe a procedure for selecting an unbiased sample. [CU, RL, MC]
Compare the results of a survey given two different sample groups. [CU,RL]
1.4.6 Analyze data and evaluate the inferences that can be drawn from data.

Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken (e.g., age groups, regions of the U.S., genders). [SP, RL, MC, CU]
Critique conclusions drawn from a set of data and support with evidence (e.g., from newspapers, web sites, opinion polls). [MC, RL, SP]

Component 1.5 Understand and apply concepts and procedures from algebraic sense.

Patterns and Relationships
1.5.1 Understand and apply rules for generating number patterns based on two arithmetic operations.

Create, explain, and/or extend number patterns involving two related sets of numbers and two operations, including addition, subtraction, multiplication, and/or division. [CU]
Use rules for generating number patterns (e.g., Fibonacci sequence, bouncing ball) to model real-life situations. [MC]
Predict a future element in a simple relation (e.g., find the fifteenth term).
Identify patterns involving combinations of operations in the rule, including simple exponents (e.g., 2, 5, 11, 23).

Symbols and Representations
1.5.2 Understand situations involving multiple arithmetic operations.

Represent and evaluate algebraic expressions involving a single variable.
Translate a situation involving multiple arithmetic operations into algebraic form using equations, tables, and graphs.
Identify or describe a situation which may be modeled by a graph. [CU, SP]
Represent an equation or expression using a variable in place of an unknown number.

Evaluating and Solving
1.5.3 Understand and apply the procedures for solving two-step equations with one variable that include adding and subtracting positive rational numbers.

Express relationships between non-negative rational numbers using symbols.
Evaluate an expression by substituting values for variables (e.g., 3y + 2, for y=3).

1.5.4:

Patterns and Relationships
1.5.1 Understand and analyze the characteristics of linear and non-linear relationships.

Identify patterns that are linear functions and provide missing terms.
Justify whether the relationship represented in a table, graph, or written situation is linear or non-linear. [CU, RL]
Explain the difference between linear and non-linear relationships. [CU, RL]

Symbols and Representations
1.5.2 Understand and apply equations, tables, and graphs to represent situations involving simple linear and non-linear relationships.

Given an equation for a situation, create a table and/or graph.
Given a description of a situation, model the relationship with a graph and/or table. [RL,MC,CU]
Describe a situation that matches a given graph (e.g., time-distance, time-height). [CU,MC,RL]
Write simple equations, expressions, or inequalities with a single variable to match a particular situation. [SP]
Match a given situation to the correct inequality or equality.
1.5.3 Understand the procedures for evaluating simple expressions and formulas involving positive rational numbers.

Substitute values for variables in order to evaluate expressions and formulas (e.g., LxW=Area, RxT=Distance).
Simplify expressions that involve the distributive property (e.g., 3(y+2) for y=2).

Evaluating and Solving
1.5.4 Understand and apply the procedures for solving equations and inequalities with one variable involving positive rational numbers.

Solve two-step single variable equations (e.g., .2x + .4 = 12).
Solve one-step single variable inequalities (e.g., 2x<6, x+4>10).
Solve real-world situations involving single variable equations. [SP, CU]
Explain and justify the solution to a contextualized problem. [RL, CU, MC]

Patterns and Relationships
1.5.1 Understand rules for iterative and linear relationships and patterns.

Use words or algebraic symbols to describe a rule for an iterative pattern. [CU]
Use words or algebraic symbols to describe a rule for a linear relationship between two sets of numbers (e.g., given a table, describe a rule). [CU]
Use words or algebraic symbols to describe a rule for a given iterative pattern. (e.g., Fibonacci sequence, 2,4,8,16. . .). [CU]

Symbols and Representations
1.5.2 Understand, apply, and analyze linear and iterative situations using tables, graphs and equations.

Graph the solution set of an inequality or equality on a number line.
Graph compound inequalities on a number line (e.g., –2Develop a table or graph from an iterative definition (e.g., the number of cells doubles every hour starting with 1 cell at noon). [MC]
Identify an expression or equation with two variables that represents a given linear situation or pictorial representation. [MC]
Write an expression, equation, and/or inequality with a single variable representing a situation or real-world problem. [SP, MC]
Write a story about a situation that represents a given linear equation, expression, or graph. [MC, SP]
Analyze the nature of changes in quantities in linear relationships using graphs. [RL,MC]

Evaluating and Solving
1.5.3 Understand and apply the procedures for simplifying single-variable expressions.

Simplify expressions and evaluate formulas involving integers.
Simplify expressions that involve the distributive property and/or combining like terms.
Match expressions to equivalent simplified expressions
Justify a simplification of an expression involving integers. [CU, RL]
1.5.4 Understand and apply the procedures for solving multi-step equations and inequalities with one variable.

Solve single variable equations involving parentheses, like terms, and/or variables on both sides of the equal sign. [SP]
Solve two-step single variable inequalities (e.g., 2x+3<6).
Solve real-world situations involving single variable equations and proportional relationships and interpret the solution. [SP, CU, RL]

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

Component 2.1: Investigate and Analyze Situations

2.1.1 Analyze situations to determine known and unknown information in new situations.

2.1.2 Analyze situations to determine when information is missing or extraneous.

2.1.1 Analyze situations to determine known and unknown information in new situations.

2.1.2 Analyze situations to determine when information is missing or extraneous.

2.1.1 Analyze situations to determine known and unknown information in complex situations.

2.1.2 Analyze situations to determine when information is missing or extraneous.

Component 2.2: Formulate Questions and Define the Problem

2.2.1 Understand the problem to be solved involving number sense, measurement, geometric sense, and probability and statistics.

2.2.2 Generate questions to be answered in new situations.

2.2.1 Understand the problem to be solved involving number sense, measurement, geometric sense, algebraic sense, and probability and statistics.

2.2.2 Generate questions to be answered in new situations.

2.2.1 Understand the problem to be solved in complex situations.

2.2.2 Generate questions to be answered in complex situations.

Component 2.3: Construct Solutions

2.3.1 Apply a variety of strategies and approaches to problem situations from number sense, measurement, geometric sense, and probability and statistics to construct a solution.

2.3.1 Apply a variety of strategies and approaches to problem situations from number sense, measurement, geometric sense, algebraic sense, and probability and statistics to construct a solution.

2.3.1 Apply a variety of strategies and approaches to problem situations from number sense, measurement, geometric sense, and statistics to construct a solution.

Component 2.4: Draw Conclusions

2.4.1 Understand how to make conjectures and support or contradict them with evidence.

2.4.2 Analyze solutions to draw conclusions and support them with evidence.

2.4.1 Understand how to make conjectures and support or contradict them with evidence.

2.4.2 Analyze solutions to draw conclusions and support them with evidence.

2.4.1 Understand how to make and test conjectures by formulating proofs or constructing counter-examples.

2.4.2 Analyze solutions to draw conclusions and support them with evidence.

Component 2.5: Evaluate and Verify Results

2.5.1 Evaluate strategies and procedures for accuracy and appropriateness.

2.5.2 Evaluate results for reasonableness.

2.5.3 Evaluate conclusions using evidence.

2.5.1 Evaluate strategies and procedures for accuracy and appropriateness.

2.5.2 Evaluate results for reasonableness.

2.5.3 Evaluate conclusions using evidence.

2.5.1 Evaluate strategies and procedures for accuracy and appropriateness.

2.5.2 Evaluate results for reasonableness.

2.5.3 Evaluate conclusions using inductive and deductive reasoning.

EALR 3 The student communicates knowledge and understanding in both everyday and mathematical language.

Component 3.1: Gather Information

3.1.1 Apply a plan for collecting information for a given purpose, which requires using number sense, measurement, geometric sense, or probability and statistics.

3.1.2 Analyze mathematical information for a given purpose, requiring number sense, measurement, geometric sense, or probability and statistics from multiple sources using reading, listening, and observation.

3.1.1 Apply a plan for collecting information for a given purpose, which requires using number sense, measurement, geometric sense, algebraic sense, or probability and statistics.

3.1.2 Analyze mathematical information for a given purpose, requiring number sense, measurement, geometric sense, algebraic sense, or probability and statistics from multiple sources using reading, listening, and observation.

3.1.1 Apply an efficient system for collecting information for a given purpose.

3.1.2 Analyze mathematical information for a given purpose, from multiple self-selected sources using reading, listening, and observation.

3.2 predict results and make inferences Component 3.2: Organize and Interpret Information

3.2.1 Understand how to organize and interpret information for a given purpose (reflecting, verbalizing, discussing, and writing).

3.2.1 Understand how to organize and interpret information for a given purpose (reflecting, verbalizing, discussing, and writing).

3.2.1 Understand how to organize, clarify, and refine mathematical information for a given purpose in multiple ways (reflecting, verbalizing, discussing, or writing).

Component 3.3: Represent and Share Information

3.3.1 Understand how to express and present ideas using mathematical language and notation.

3.3.2 Understand how to represent ideas and information from number sense, measurement, geometric sense, and probability and statistics in ways appropriate to audience and purpose.

3.3.1 Understand how to express and present ideas using mathematical language and notation.

3.3.2 Understand how to represent ideas and information from number sense, measurement, geometric sense, algebraic sense, and probability and statistics in ways appropriate to audience and purpose.

3.3.1 Understand how to express ideas using mathematical language and notation.

3.3.2 Understand how to represent complex mathematical ideas and information in ways appropriate for audience and purpose.

EALR 4: The student understands how mathematical ideas connect within mathematics, other subject areas, and real-world situations.

Component 4.1: Relate Concepts and Procedures within Mathematics

4.1.1 Apply concepts and procedures from a variety of content strands (number sense, measurement, geometric sense, and probability and statistics) in a given problem or situation.

4.1.2 Analyze relationships between equivalent mathematical models and representations.

4.1.1 Apply concepts and procedures from a variety of content strands (number sense, measurement, geometric sense, algebra and probability and statistics) in a given problem or situation.

4.1.2 Analyze relationships between equivalent mathematical models and representations.

4.1.1 Apply concepts and procedures from multiple mathematics content strands in a given problem or situation.

4.1.2 Analyze multiple mathematical models and representations to determine equivalence.

Component 4.2: Relate Mathematical Concepts Procedures to Other Disciplines

4.2.1 Analyze the concepts, strategies, and procedures from other disciplines.

4.2.2 Apply mathematical thinking and modeling in other disciplines.

4.2.3 Understand the importance of contributions to the development of mathematics such as the contributions of women, men, and different cultures.

4.2.1 Analyze the concepts, strategies, and procedures from other disciplines.

4.2.2 Apply mathematical thinking and modeling in other disciplines.

4.2.3 Understand the importance of contributions to the development of mathematics such as the contributions of women, men, and different cultures.

4.2.1 Analyze the concepts, strategies, and procedures from other disciplines to extend mathematical patterns and concepts in new situations.

4.2.2 Apply mathematical thinking and modeling in other disciplines.

4.2.3 Understand the importance of contributions to the development of mathematics such as the contributions of women, men, and different cultures.

Component 4.3: Relate Mathematical Concepts and Procedures to Real-World Situations

4.3.1 Understand the extensive uses of mathematics outside the classroom.

4.3.2 Understand how mathematics is used in several occupations/careers of interest.

4.3.1 Understand the extensive uses of mathematics outside the classroom.

4.3.2 Understand how mathematics is used in several occupations/careers of interest.

4.3.1 Understand how mathematics is used to solve problems in local, national or international situations.

4.3.2 Understand how mathematics is used in career settings.

Grade 6	Grade 7	Grade 8
EALR 1: The student understands and applies the concepts and procedures of mathematics.
Component 1.1: Understand and apply concepts and procedures from number sense - number, numeration, computation, and estimation.
Number and Numeration 1.1.1 Understand the concepts of integers. Represent and identify integers on a number line. Identifies, compares, and orders non-negative whole numbers, fractions, and decimals 1.1.2 Understand the relative values of integers and non-negative rational numbers. Compare different representations of non-negative rational numbers by implementing strategies (e.g., like denominators, benchmark fractions). [SP, CU, MC] Identify equivalence between common fractions and decimals. [MC] Compare integer values and explain which is greater and why. [RL, CU] 1.1.3 Understand the additive inverse property. Illustrate the additive inverse property using physical models and pictures. [CU] Explain the additive inverse property and why it works. [CU] 1.1.4 Understand the concepts of ratio and percent. Write ratios and proportions in part/part and part/whole relationships using objects, pictures, and symbols (e.g., using /, :, or to as representations for ratios). [CU] Represent equivalent ratios using objects, pictures, and symbols. [CU, MC] Identify percent as 100 equal size sets (e.g., 1% of 200 items is 2 items). [RL] Represent a given percentage with a physical model. [CU] Find a percentage of a given set of items other than 100. Computation 1.1.5 Understand the meaning of addition and subtraction of integers and multiplication and division on non-negative rational numbers. Explain meaning of multiplying and dividing non-negative common fractions and decimals using visual and physical models. [CU] Explain the meaning of addition and subtraction of integers using models (e.g., reducing debt, temperature increase or decrease, yards gained and lost). [CU] 1.1.6 Understand and apply procedures for addition and subtraction on non-negative decimals and fractions with fluency. Find the sums and/or differences of common fractions and decimals. Write and solve problem situations to find sums and/or differences of decimals or fractions. [SP, RL, CU, MC] 1.1.7 Apply strategies for mental arithmetic, pencil and paper, or calculator as appropriate to the task involving non-negative decimals and simple fractions. This indicator should be incorporated in all instruction and applies to all grade levels. Estimation 1.1.8 Understand situations in which estimation is appropriate and apply estimation strategies to determine the reasonableness of answers involving content appropriate to the grade level. Apply estimation strategies prior to computation of whole numbers, decimals, and common fractions to determine reasonableness of answers. Use estimation to predict or check answers. Identify appropriate estimated answers for a given situation. Articulate various strategies used during estimation. [CU]	Number and Numeration 1.1.1 Understand the concepts of rational numbers. Create a model when given a symbolic representation of a rational number. [SP, CU, MC] Write the rational number when given a model (e.g., number line, area model, situation, diagram, picture). [SP, CU, MC] Convert between equivalent forms of rational numbers. Explain the value of a given digit in a rational number. [CU] 1.1.2 Understand the relative values of rational numbers. Compare and order non-negative rational numbers using concrete models or implementing strategies (e.g., like denominators, benchmark fractions, conversions). [SP, CU, MC] Order different representations of rational numbers. [MC] Locate symbolic representations of rational numbers on a number line including fractions, decimals, and integers. 1.1.3 Understand and apply the distributive and multiplicative inverse properties and use order of operations. Understand and use the inverse relationships of multiplication and division to simplify computations and solve problems. [SP] Use order of operations to simplify computations and solve problems. Illustrate the distributive property of multiplication over addition using an area model or picture. [CU, MC] Identify errors in the application of order of operations. [RL] 1.1.4 Understand and apply the concepts of ratio, percent, and direct proportion. Write a ratio or proportion for a given situation. Express proportional relationships using objects, pictures, and symbols. [CU, MC] Explain the meaning of a ratio or proportion. [CU] Represent a new relationship from a given ratio (e.g., part/part to part/whole; given a ratio of girls to boys, find the ratio of girls to class). [CU] Represent percentages less than 1% or greater than 100% using objects, pictures, and symbols. [CU, MC] Computation 1.1.5 Understand the meaning of multiplication and division of integers. Explain the meaning of multiplication and division of integers using visual and physical models. [CU] Create a problem situation involving multiplication or division of integers. [SP, CU, RL, MC] 1.1.6 Understand and apply the procedures of addition and subtraction for integers and multiplication and division on non-negative rational numbers with fluency. Find the sum, difference, product, and/or quotient of decimals and common fractions without common denominators. Use order of operations to solve problem situations with decimals and common fractions. Find the sums and differences of integers. 1.1.7 Apply strategies for mental arithmetic, pencil and paper, or calculator as appropriate to the task involving non-negative decimals and simple fractions. · This indicator should be incorporated in all instruction and applies to all grade levels. Estimation 1.1.8 Understand situations in which estimations is appropriate and apply estimation strategies to determine the reasonableness of answers involving addition and subtraction of integers and operations on non-negative rational numbers Apply estimation strategies prior to computing addition and subtraction of integers and operations on non-negative rational numbers to determine reasonableness of answers. Justify why estimation would be used rather than an exact computation. [RL, CU] Describe a situation where estimation is sufficient in real life contexts. [CU, MC]	Number and Numeration 1.1.1 Understand and apply symbolic representations of rational numbers including whole number powers and roots of square numbers. Read and use scientific notation to represent large numbers. [MC, RL, SP] Explain the meaning of a whole number exponent. [CU] Read and use exponential notation to represent large numbers. [MC, RL, SP] Identify a square number and find its root. 1.1.2 Understand the relative values of rational numbers including whole number powers and roots of square numbers. Compare and order rational numbers using concrete models or implementing strategies. [RL] Order different representations of rational numbers including large numbers in scientific and exponential notation. Locate symbolic representations of rational numbers on a number line including whole number powers and roots of square numbers. 1.1.3: 1.1.4 Apply ratio, percent, and direct proportion in complex situations. Solve problems involving ratio and proportion (e.g., similar figures, scale drawings, rates, find unit pricing, increase or decrease a recipe, find the portions for a group converting between different units of measure, or finding medicinal dosages). [SP, RL, CU, MC] Solve problems involving percentages in complex situations (e.g., percent increase/decrease, tax, commission, discount). [SP, RL, CU, MC] Explain advantages and disadvantages of different representations in a given situation (e.g., using 1/3 versus 33 1/3 %). [CU] Computation 1.1.5 Understand the meaning of operations on rational numbers. Compare and contrast operations on rational numbers using pictures and symbols. [RL, CU] Create a problem situation to match a given rational number equation. [SP, RL, CU, MC] Identify a rational number equation to match a given situation. [CU, MC] 1.1.6 Understand and apply computations on rational numbers with fluency. Compute fluently with rational numbers in all forms except exponential. Write and solve problems that involve computation with rational numbers. [SP, RL, CU, MC] 1.1.7 Apply strategies for mental arithmetic, pencil and paper, calculator or computer as appropriate to the task involving rational numbers. Select and justify appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation. Estimation 1.1.8 Understand situations in which estimation is appropriate and apply estimation strategies to determine reasonableness of answers involving non-negative rational numbers. Explain situations involving real numbers where estimates are sufficient and others for which exact value is required. [CU] Justify why an estimate would be used rather than an exact answer in a given situation. [CU]
Component 1.2: Understand and apply concepts and procedures from measurement.
Attributes, Units, and Tools 1.2.1 Understand the attributes of volume and capacity. Given two rectangular solids, compare their volume. Compare the relative capacity of two containers (e.g., paper cylinders formed horizontally and vertically and filled with popcorn). [SP, CU, MC] 1.2.2: Understand and apply standard units to measure the attributes of volume and capacity. Identify cubic units to measure volume (e.g., linking cubes, cubic centimeter). Identify and read incremental units for capacity (e.g., milliliters, cups, ounces). 1.2.3 Apply appropriate tools to measure volume/capacity using both standard and metric systems. Measure volume of rectangular prisms and label appropriately (e.g., cubic units). [SP, CU] Measure the capacity of containers using appropriate tools and label (e.g., graduated cylinders, measuring cups, tablespoons). [SP, CU] 1.2.4: Procedures, Precision, and Estimation 1.2.5 Understand that precision is related to the size of the unit of the measurement used. Compare the appropriateness of standard to nonstandard units in measuring volume or capacity. [CU] Choose the appropriate standard unit for measuring volume or capacity (e.g., cubic inches vs. cubic feet, cups vs. gallons). [SP] 1.2.6 Understand when approximate measurements are sufficient and apply estimation strategies to obtain reasonable measurements of volume. Identify situations when approximate measurements are sufficient. Recognize when a measurement is approximate rather than exact. [SP] Use estimation to justify reasonableness of a measurement. [RL, CU] Estimate a measurement using standard or nonstandard units. [SP]	Attributes, Units, and Tools 1.2.1 Understand the attribute of surface area. Compare and describe the surface area of a variety of rectangular solids. [RL, CU] Apply surface area to real world situations (e.g., paint or wallpaper for a room). [SP, MC] 1.2.2: 1.2.3 Apply tools and formulas to determine the area of quadrilaterals and triangles. Find the area of triangles and quadrilaterals using formulas. [MC] Justify the formula of a triangle or quadrilateral using pictures or manipulatives. [RL, CU] Use appropriate tools to measure linear dimensions to determine surface area. [SP] Find the surface area of a rectangular solid. [SP] Procedures, Precision, and Estimation 1.2.4: 1.2.5 Understand that different situations require different levels of precision. Justify the use of a unit of measure (e.g., meters or kilometers, inches or feet). [CU, RL] Select and use appropriate unit of measurement according to type of unit (standard vs. metric) or size of unit (millimeters vs. centimeters). [MC, RL] 1.2.6 Understand when approximate measurements are sufficient and apply estimation strategies to obtain reasonable measurements of surface area. Use estimation to find reasonable approximation of surface area. Justify the reasonableness of an estimate. [RL] Decide when it is appropriate to estimate the volume and/or surface area of a solid and explain why. [CU, RL, SP, MC]	Attributes , Units, and Tools 1.2.1 Understand and apply right triangle relationships. Demonstrate the meaning of the Pythagorean Theorem with drawings or manipulatives. [CU] Use simple trigonometric relationships to find indirect measurements of length. (e.g., height of a tree or flagpole, side of a triangle using sine, cosine, or tangent). [SP,RL,MC] Calculate measures of objects for which no direct information is given (e.g., similar figures, ratio, proportion, scale). [SP,RL,MC] 1.2.2: 1.2.3 Apply tools and formulas to find the area of circles, the volume and surface area of cubes, right cylinders, and rectangular prisms. Calculate the area of circles and volume and surface area of cubes, right cylinders, and rectangular prisms. Describe a strategy for finding the area of a figure (e.g., formulate a means for finding the area of an irregular figure). [SP,RL,CU] Given the perimeter, circumference, or area, determines linear dimensions. Justify the formula for surface area of a rectangular prism or right cylinder with pictures or manipulatives. [RL, CU] Measure objects directly using appropriate tools to find area of circles, and volumes and surface area of right cylinders and rectangular prisms (e.g., rulers to find dimensions). 1.2.4: Procedures, Precision, and Estimation 1.2.5 Understand that precision and accuracy of measurement are affected by units, tools, and their applications. Determine the appropriate unit of measurement in a problem situation (e.g., the height of a tree, surface area of a pond). [MC, RL, SP] Justify the use of a particular tool in a measurement. [RL, CU] Convert within systems (e.g., centimeters to meters, inches to feet). 1.2.6 Understand when approximate measurements are sufficient and apply estimation strategies to obtain reasonable measurements of right triangle relationships. Identify situations when approximate measurements are sufficient. Estimate distance or height in a problem situation using right triangle or Pythagorean relationships (e.g., height of a flagpole using tangent, distance across a lake using Pythagorean relationship). [MC,CU,SP]
Component 1.3: Understand and apply concepts and procedures from geometric sense.
Properties and Relationships 1.3.1 Understand properties of angles, triangles, and quadrilaterals. Explain the difference between a regular and irregular polygon. [CU, RL] Identify, sort, classify, and explain the properties of specific quadrilaterals including squares, rectangles, parallelograms, and trapezoids based on attributes. [CU, RL] Identify angles as acute, right, obtuse, or straight. Classify triangles as right, equilateral, isosceles, and/or scalene. 1.3.2 Apply understanding of the properties of 3-D figures and shapes. Identify, name, compare, and sort 3-D shapes and figures. Identify geometric figures and concepts in nature and art (e.g. triangle in architecture). Given two 3-D solids, explain how they are alike and different in terms of their attributes (e.g., using a Venn diagram). [RL, CU] 1.3.3 Understand the relative location of integers on a number line. Show the order of a given set of integers on a number line with both positive and negative numbers. Given directions for movement on a number line, including positive and negative numbers (vertical and horizontal), identify the point of final destination (e.g., temperature variation at different times of the day, bank accounts, gain and loss of weight). [MC] Determine the distance between any two integers on a number line. 1.3.4 Understand and analyze rotations (turns) of a 2-D figure about the center of the figure. Create a design using rotational symmetry of a shape. [MC] Match a figure with its image following one transformation. [RL]	Properties and Relationships 1.3.1 Understand the similarity relationship in 2-D shapes and figures. Identify corresponding sides and angles of two similar figures. Determine and justify if two figures are similar using the definition of similarity. [RL, CU] 1.3.2 Understand and apply properties of polygons and circles. Identify the attributes of a circle including diameter, radius, and circumference. Given all but one of the angles of a polygon, find the missing angle. [RL] Draw polygons and circles with specified properties (e.g., circumference of 18 cm). [CU] Use the properties of polygons and circles to solve mathematical problems. [SP, MC, CU] 1.3.3 Understand location of points on coordinate grids in any of the four quadrants. Given three points, identify the coordinates of the fourth point to make a rectangle. [RL] Plot ordered pairs in any of the four quadrants. Identify the coordinates of a given point in any of the four quadrants. 1.3.4 Apply and analyze combinations of transformations. Use transformations to create congruent shapes and figures in multiple orientations. Given a figure on a coordinate grid, find the coordinate pairs for a translation and/or a reflection across an axis. Match a figure with its image following two transformations. [RL]	Properties and Relationships 1.3.1: 1.3.2 Understand and analyze properties and relationships of 1-D, 2-D, and 3-D objects. Identify and label rays, lines, end points, line segments, vertices, and angles. [CU] Given a figure, write a verbal description of its geometric properties. [CU] Given two similar figures find the length of a missing side, or the measure of a missing angle of one of the figures. [RL] Create symmetrical, congruent, and/or similar figures using a variety of tools (e.g., grid, ruler, pattern blocks, geo-boards). Given a simple figure, appropriate tools and a scale, draw a similar figure (e.g., graph paper, geo-board, dot paper). [RL] Match or draw 3-D objects from different perspectives using the same properties and relationships (e.g., match to the correct net, draw the top view). [RL] Locations and transformations 1.3.3 Understand the relationships between two or more points on a coordinate grid. Find length of horizontal and vertical lines on a coordinate plane. Find the length of a line segment on a coordinate grid. [RL] Given the coordinates of the vertices of a regular polygon, locate the missing vertex. [RL] Use the Pythagorean Theorem or trigonometric ratios to find the missing side of a triangle in a real world setting (e.g., the height of a ladder). [SP, RL, MC] 1.3.4:
Component 1.4: Understand and apply concepts and procedures from probability and statistics.
Probability 1.4.1 Understand probability as a ratio between and including zero and one. Calculate probability for a simple event. Express probability as a ratio, decimal, and percent. 1.4.2 Understand the procedures for determining all possible outcomes of simple experiments. Represent and interpret all possible outcomes of simple experiments (e.g., an organized list, a table, or a tree diagram). [RL, CU] 1.4.3: Statistics 1.4.4 Apply measures of central tendency to interpret a set of data. Determine when it is appropriate to use mean, median, or mode. [RL] Use mean, median, and mode to explain familiar situations (e.g., the heights of students in the class, the hair color of students in the class). [CU, RL, MC] 1.4.5 Understand how to evaluate a question or data collection method for fairness. Compare data collection methods for a given situation to determine fairness of the method (e.g., compare a phone survey, a web survey, and a personal interview survey). [RL, MC] Judge a data collection method to consider limitations that could affect interpretations (e.g., to examine battery life, compare how long batteries last in a flashlight vs. a portable CD player). [SP, RL, MC] 1.4.6 Analyze and evaluate data appropriate to the grade level. Make a conjecture about an entire group or population from a sample (e.g., sample the classroom to find the favorite school lunch). [MC, CU, RL] Evaluate and explain conclusions drawn from data (e.g., from newspapers or web sites). [RL, MC, SP]	Probability 1.4.1 Understand the concepts of complementary and mutually exclusive events. Determine when events are mutually exclusive (e.g., you're grade on a test is an A, B, or C). Determine when events are complementary (e.g., a person awake or asleep, you pass or fail a test). Identify events that are complementary or mutually exclusive or neither and explain. [RL, CU, MC] 1.4.2 Understand the procedures for determining the probabilities for simple experiments. Calculate the probabilities of simple experiments (e.g., pulling colored balls from a bag, drawing a card, rolling a 6 on a number cube, spinning a spinner, etc.). Calculate the probability of an event given the probability of its complement. Statistics 1.4.3 Understand how to organize, display, and interpret data from line graphs, box-and- whisker, and scatter plots. Construct line graphs, box-and-whisker, and scatter plots from collected data. [CU, MC] Describe population characteristics from a box-and-whisker plot. [CU] Use scatter plots to describe trends and interpret relationships. [RL] Draw trend lines with or without technology and make predictions about real-world situations. [RL, CU, MC, SP] 1.4.4 Understand the effects of outliers and clusters on measures of central tendency and select the appropriate measure of central tendency for a set of data. Describe the effects of extreme values on means in a population. [CU, RL, MC] Explain the use of median or mean as a measure of central tendency in a given situation (e.g., when an extreme value skews the mean). [RL, SP, CU, MC] 1.4.5 : 1.4.6 Analyze data and evaluate the inferences that can be drawn from data. Make and justify an inference drawn from a sample. [CU, RL, MC] Evaluate and explain conclusions drawn from data (e.g., from newspapers, web sites, opinion polls). [MC, RL, SP]	Probability 1.4.1 Understand the concept of compound events. Determine and explain when events are compound. [CU, RL] Explain the difference between compound events involving 'and' and 'or' (e.g., rolling a six and drawing an ace of spades vs. rolling a six or drawing an ace spades). [CU, RL] 1.4.2 Understand and apply the procedures for determining theoretical and experimental probabilities for simple and compound events. Design and conduct a simulation, with and without technology, to determine the probability of an uncertain event occurring. [MC, RL, SP, CU] Calculate the probability of two independent events occurring simultaneously using various methods (e.g., organized list, tree diagram, counting procedures, and area model). Explain the relationship between theoretical and experimental probability of simple and compound events. [CU, RL] 1.4.3: 1.4.4 Understand and find average deviation from a set of data. Calculate the average deviation from a set of data. Explain the meaning of average deviation in real-world situations (e.g., foot size in a class). [MC, RL, CU] 1.4.5 Understand how different samples of a population affect the data. Identify sources of sampling bias, given a situation (e.g,. interviewing only girls, a certain age group, or too small a sample). [MC, RL, CU, SP] Describe a procedure for selecting an unbiased sample. [CU, RL, MC] Compare the results of a survey given two different sample groups. [CU,RL] 1.4.6 Analyze data and evaluate the inferences that can be drawn from data. Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken (e.g., age groups, regions of the U.S., genders). [SP, RL, MC, CU] Critique conclusions drawn from a set of data and support with evidence (e.g., from newspapers, web sites, opinion polls). [MC, RL, SP]
Component 1.5 Understand and apply concepts and procedures from algebraic sense.
Patterns and Relationships 1.5.1 Understand and apply rules for generating number patterns based on two arithmetic operations. Create, explain, and/or extend number patterns involving two related sets of numbers and two operations, including addition, subtraction, multiplication, and/or division. [CU] Use rules for generating number patterns (e.g., Fibonacci sequence, bouncing ball) to model real-life situations. [MC] Predict a future element in a simple relation (e.g., find the fifteenth term). Identify patterns involving combinations of operations in the rule, including simple exponents (e.g., 2, 5, 11, 23). Symbols and Representations 1.5.2 Understand situations involving multiple arithmetic operations. Represent and evaluate algebraic expressions involving a single variable. Translate a situation involving multiple arithmetic operations into algebraic form using equations, tables, and graphs. Identify or describe a situation which may be modeled by a graph. [CU, SP] Represent an equation or expression using a variable in place of an unknown number. Evaluating and Solving 1.5.3 Understand and apply the procedures for solving two-step equations with one variable that include adding and subtracting positive rational numbers. Express relationships between non-negative rational numbers using symbols. Evaluate an expression by substituting values for variables (e.g., 3y + 2, for y=3). 1.5.4:	Patterns and Relationships 1.5.1 Understand and analyze the characteristics of linear and non-linear relationships. Identify patterns that are linear functions and provide missing terms. Justify whether the relationship represented in a table, graph, or written situation is linear or non-linear. [CU, RL] Explain the difference between linear and non-linear relationships. [CU, RL] Symbols and Representations 1.5.2 Understand and apply equations, tables, and graphs to represent situations involving simple linear and non-linear relationships. Given an equation for a situation, create a table and/or graph. Given a description of a situation, model the relationship with a graph and/or table. [RL,MC,CU] Describe a situation that matches a given graph (e.g., time-distance, time-height). [CU,MC,RL] Write simple equations, expressions, or inequalities with a single variable to match a particular situation. [SP] Match a given situation to the correct inequality or equality. 1.5.3 Understand the procedures for evaluating simple expressions and formulas involving positive rational numbers. Substitute values for variables in order to evaluate expressions and formulas (e.g., LxW=Area, RxT=Distance). Simplify expressions that involve the distributive property (e.g., 3(y+2) for y=2). Evaluating and Solving 1.5.4 Understand and apply the procedures for solving equations and inequalities with one variable involving positive rational numbers. Solve two-step single variable equations (e.g., .2x + .4 = 12). Solve one-step single variable inequalities (e.g., 2x<6, x+4>10). Solve real-world situations involving single variable equations. [SP, CU] Explain and justify the solution to a contextualized problem. [RL, CU, MC]	Patterns and Relationships 1.5.1 Understand rules for iterative and linear relationships and patterns. Use words or algebraic symbols to describe a rule for an iterative pattern. [CU] Use words or algebraic symbols to describe a rule for a linear relationship between two sets of numbers (e.g., given a table, describe a rule). [CU] Use words or algebraic symbols to describe a rule for a given iterative pattern. (e.g., Fibonacci sequence, 2,4,8,16. . .). [CU] Symbols and Representations 1.5.2 Understand, apply, and analyze linear and iterative situations using tables, graphs and equations. Graph the solution set of an inequality or equality on a number line. Graph compound inequalities on a number line (e.g., –2Develop a table or graph from an iterative definition (e.g., the number of cells doubles every hour starting with 1 cell at noon). [MC] Identify an expression or equation with two variables that represents a given linear situation or pictorial representation. [MC] Write an expression, equation, and/or inequality with a single variable representing a situation or real-world problem. [SP, MC] Write a story about a situation that represents a given linear equation, expression, or graph. [MC, SP] Analyze the nature of changes in quantities in linear relationships using graphs. [RL,MC] Evaluating and Solving 1.5.3 Understand and apply the procedures for simplifying single-variable expressions. Simplify expressions and evaluate formulas involving integers. Simplify expressions that involve the distributive property and/or combining like terms. Match expressions to equivalent simplified expressions Justify a simplification of an expression involving integers. [CU, RL] 1.5.4 Understand and apply the procedures for solving multi-step equations and inequalities with one variable. Solve single variable equations involving parentheses, like terms, and/or variables on both sides of the equal sign. [SP] Solve two-step single variable inequalities (e.g., 2x+3<6). Solve real-world situations involving single variable equations and proportional relationships and interpret the solution. [SP, CU, RL]
EALR 2: The student uses mathematical reasoning to define and solve problems.
Component 2.1: Investigate and Analyze Situations
2.1.1 Analyze situations to determine known and unknown information in new situations. 2.1.2 Analyze situations to determine when information is missing or extraneous.	2.1.1 Analyze situations to determine known and unknown information in new situations. 2.1.2 Analyze situations to determine when information is missing or extraneous.	2.1.1 Analyze situations to determine known and unknown information in complex situations. 2.1.2 Analyze situations to determine when information is missing or extraneous.
Component 2.2: Formulate Questions and Define the Problem
2.2.1 Understand the problem to be solved involving number sense, measurement, geometric sense, and probability and statistics. 2.2.2 Generate questions to be answered in new situations.	2.2.1 Understand the problem to be solved involving number sense, measurement, geometric sense, algebraic sense, and probability and statistics. 2.2.2 Generate questions to be answered in new situations.	2.2.1 Understand the problem to be solved in complex situations. 2.2.2 Generate questions to be answered in complex situations.
Component 2.3: Construct Solutions
2.3.1 Apply a variety of strategies and approaches to problem situations from number sense, measurement, geometric sense, and probability and statistics to construct a solution.	2.3.1 Apply a variety of strategies and approaches to problem situations from number sense, measurement, geometric sense, algebraic sense, and probability and statistics to construct a solution.	2.3.1 Apply a variety of strategies and approaches to problem situations from number sense, measurement, geometric sense, and statistics to construct a solution.
Component 2.4: Draw Conclusions
2.4.1 Understand how to make conjectures and support or contradict them with evidence. 2.4.2 Analyze solutions to draw conclusions and support them with evidence.	2.4.1 Understand how to make conjectures and support or contradict them with evidence. 2.4.2 Analyze solutions to draw conclusions and support them with evidence.	2.4.1 Understand how to make and test conjectures by formulating proofs or constructing counter-examples. 2.4.2 Analyze solutions to draw conclusions and support them with evidence.
Component 2.5: Evaluate and Verify Results
2.5.1 Evaluate strategies and procedures for accuracy and appropriateness. 2.5.2 Evaluate results for reasonableness. 2.5.3 Evaluate conclusions using evidence.	2.5.1 Evaluate strategies and procedures for accuracy and appropriateness. 2.5.2 Evaluate results for reasonableness. 2.5.3 Evaluate conclusions using evidence.	2.5.1 Evaluate strategies and procedures for accuracy and appropriateness. 2.5.2 Evaluate results for reasonableness. 2.5.3 Evaluate conclusions using inductive and deductive reasoning.
EALR 3 The student communicates knowledge and understanding in both everyday and mathematical language.
Component 3.1: Gather Information
3.1.1 Apply a plan for collecting information for a given purpose, which requires using number sense, measurement, geometric sense, or probability and statistics. 3.1.2 Analyze mathematical information for a given purpose, requiring number sense, measurement, geometric sense, or probability and statistics from multiple sources using reading, listening, and observation.	3.1.1 Apply a plan for collecting information for a given purpose, which requires using number sense, measurement, geometric sense, algebraic sense, or probability and statistics. 3.1.2 Analyze mathematical information for a given purpose, requiring number sense, measurement, geometric sense, algebraic sense, or probability and statistics from multiple sources using reading, listening, and observation.	3.1.1 Apply an efficient system for collecting information for a given purpose. 3.1.2 Analyze mathematical information for a given purpose, from multiple self-selected sources using reading, listening, and observation.
3.2 predict results and make inferences		Component 3.2: Organize and Interpret Information
3.2.1 Understand how to organize and interpret information for a given purpose (reflecting, verbalizing, discussing, and writing).	3.2.1 Understand how to organize and interpret information for a given purpose (reflecting, verbalizing, discussing, and writing).	3.2.1 Understand how to organize, clarify, and refine mathematical information for a given purpose in multiple ways (reflecting, verbalizing, discussing, or writing).
Component 3.3: Represent and Share Information
3.3.1 Understand how to express and present ideas using mathematical language and notation. 3.3.2 Understand how to represent ideas and information from number sense, measurement, geometric sense, and probability and statistics in ways appropriate to audience and purpose.	3.3.1 Understand how to express and present ideas using mathematical language and notation. 3.3.2 Understand how to represent ideas and information from number sense, measurement, geometric sense, algebraic sense, and probability and statistics in ways appropriate to audience and purpose.	3.3.1 Understand how to express ideas using mathematical language and notation. 3.3.2 Understand how to represent complex mathematical ideas and information in ways appropriate for audience and purpose.
EALR 4: The student understands how mathematical ideas connect within mathematics, other subject areas, and real-world situations.
Component 4.1: Relate Concepts and Procedures within Mathematics
4.1.1 Apply concepts and procedures from a variety of content strands (number sense, measurement, geometric sense, and probability and statistics) in a given problem or situation. 4.1.2 Analyze relationships between equivalent mathematical models and representations.	4.1.1 Apply concepts and procedures from a variety of content strands (number sense, measurement, geometric sense, algebra and probability and statistics) in a given problem or situation. 4.1.2 Analyze relationships between equivalent mathematical models and representations.	4.1.1 Apply concepts and procedures from multiple mathematics content strands in a given problem or situation. 4.1.2 Analyze multiple mathematical models and representations to determine equivalence.
Component 4.2: Relate Mathematical Concepts Procedures to Other Disciplines
4.2.1 Analyze the concepts, strategies, and procedures from other disciplines. 4.2.2 Apply mathematical thinking and modeling in other disciplines. 4.2.3 Understand the importance of contributions to the development of mathematics such as the contributions of women, men, and different cultures.	4.2.1 Analyze the concepts, strategies, and procedures from other disciplines. 4.2.2 Apply mathematical thinking and modeling in other disciplines. 4.2.3 Understand the importance of contributions to the development of mathematics such as the contributions of women, men, and different cultures.	4.2.1 Analyze the concepts, strategies, and procedures from other disciplines to extend mathematical patterns and concepts in new situations. 4.2.2 Apply mathematical thinking and modeling in other disciplines. 4.2.3 Understand the importance of contributions to the development of mathematics such as the contributions of women, men, and different cultures.
Component 4.3: Relate Mathematical Concepts and Procedures to Real-World Situations
4.3.1 Understand the extensive uses of mathematics outside the classroom. 4.3.2 Understand how mathematics is used in several occupations/careers of interest.	4.3.1 Understand the extensive uses of mathematics outside the classroom. 4.3.2 Understand how mathematics is used in several occupations/careers of interest.	4.3.1 Understand how mathematics is used to solve problems in local, national or international situations. 4.3.2 Understand how mathematics is used in career settings.