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GRADE 6

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 and numeration

1.1.1 Understand the concept of integers as the set of natural numbers (1, 2, 3 …), their opposites (-1, -2, -3 …), and 0. W

· Illustrate integer values using models and pictures (e.g., temperature, elevators, net worth/debt, riding a bus or subway). [CU]

· Apply rules of divisibility to show if a quotient is an integer. [RL]

· Explain the meaning of integers and give examples.

· Identify the opposite of a given integer.

1.1.2 Understand the relative values of integers and non-negative rational numbers. W

· Compare different representations of non-negative rational numbers by implementing strategies (e.g., like denominators, changing to the same form). [RL, CU, MC]

· Identify equivalence between non-negative integers, fractions, percents, and decimals. [MC]

· Compare and order integer values and explain which is greater and why (e.g., place the integers on a number line). [CU]

· Represent and identify integers on a model (e.g., number line, fraction line, or decimal grid). [RL, CU]

1.1.3 Apply properties of addition and multiplication to non-negative rational numbers. W

· Illustrate and explain the commutative and associative properties and why they work (e.g., use physical models, pictures). [CU]

· Use addition and multiplication properties to assist in computations (e.g., 5 · 7 · 6 can be rewritten as 5 · 6 · 7 which is 30 · 7 or 210).

· Determine whether a solution is accurate based on application commutative, associative, and identity properties of addition and/or multiplication. [RL]

1.1.4 Understand the concepts of ratio and percent. W

· Write ratios 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, or symbols. [CU]

· Represent equivalent percentages using objects, pictures, and symbols. [CU]

· Identify percent as 100 equal size parts of a set (e.g., 1% of 200 items is 2 items).

· Explain ratio and percents and give examples of each. [CU]

Computation

1.1.5 Understand the meaning of multiplication and division on non-negative rational numbers. W

· Explain the meaning of multiplying and dividing non-negative fractions and decimals using words, visual, or physical models (e.g., sharing a restaurant bill, cutting a board into equal-sized pieces, drawing a picture of an equation or situation). [CU, MC]

· Explain why multiplication of fractions can be done by multiplying denominators while addition of fractions requires finding common denominators. [CU]

· Use technology to demonstrate how multiplication and division with decimals affects place value.

1.1.6 Apply computational procedures with fluency for addition and subtraction on non-negative rational numbers. W

· Find the sums or differences of non-negative fractions or decimals.

· Write and solve real-world problem situations to find sums or differences of decimals or fractions. [CU, MC]

· Use the least common multiple and the greatest common factor of whole numbers to solve problems with fractions (e.g., to find a common denominator, to add two fractions, or to find the simplified form for a fraction). [MC]

· Use addition and subtraction to solve real-world problems involving non-negative rational numbers. [SP]

· Solve multiple-step computations requiring one, two, or more different operations. [MC]

1.1.7 Understand and apply strategies and tools to complete tasks involving addition and subtraction on non-negative rational numbers.

· Select and justify the selection of appropriate strategies and tools (e.g., mental computation, estimation, calculators, and paper and pencil) to compute in a problem situation. [SP, CU]

· Describe strategies for mentally solving problems involving fractions and decimals. [CU]

· Use calculators to add and subtract with decimal numbers with precision to the thousandths place and beyond.

Estimation

1.1.8 Apply estimation strategies to predict or determine the reasonableness of answers in situations involving addition and subtraction on non-negative rational numbers. W

· Identify when an approximation is appropriate. [MC]

· Apply estimation strategies prior to computation on whole numbers, decimals, and fractions to approximate an answer. [RL]

· Use estimation to verify the reasonableness of calculated results. [RL]

· Identify appropriate estimated answers for a given situation.

· Describe various strategies used during estimation involving fractions and decimals. [CU]

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

Attributes, units, and systems

1.2.1 Understand the concepts of volume and extend the concept of area to surface area of rectangular prisms. W

· Compare the relative capacity of two containers and explain the differences (e.g., paper cylinders formed horizontally and vertically and filled with popcorn). [RL]

· Represent the volume for given rectangular prisms using pictures or models. [CU]

· Compare the surface area of two different rectangular prisms.

· Describe and provide examples for surface area measurement (e.g., gift wrapping, painting a room, amount of material needed to build a box). [MC]

· Explain and give examples of how the area and surface area are related (e.g., surface area is the sums of the areas of all the sides of a rectangular prism). [CU, MC]

· Describe and compare the use of area and volume (e.g., covering and filling). [CU]

1.2.2 Understand the differences between square and cubic units. W

· Identify cubic units to measure volume (e.g., linking cubes, cubic centimeter).

· Identify and read incremental units for capacity (e.g., milliliters, cups, ounces).

· Use the appropriate units when describing a situation (e.g., five square meters of carpet, five cubic meters of water). [MC]

· Explain why volume is measured in cubic units. [CU, MC]

· Explain how the selected unit of length affects the size of cubic units (e.g., centimeter versus inch). [CU]

Procedures, precision, and estimation

1.2.4 Understand and apply systematic procedures to measure volume and capacity for solid shapes. W

· Identify the attribute to be measured in the situation (e.g., volume or capacity).

· Choose the appropriate standard unit for measuring volume or capacity (e.g., cubic inches vs. cubic feet, cups vs. gallons).

· Select and use tools that match the unit.

· Count or compute to obtain the volume or capacity and label the measurement.

· Use volume and capacity to describe and compare figures (e.g., fill containers with cubes to find which has a greater volume). [RL, CU]

· Measure the capacity of containers using appropriate tools and label (e.g., graduated cylinders, measuring cups, tablespoons). [CU]

· Evaluate whether measurement has been done correctly. [RL]

1.2.6 Understand and apply strategies to obtain reasonable estimates of volume or capacity. W

· Identify situations in which estimated measures are sufficient.

· Estimate volume or capacity.

· Use estimation to justify reasonableness of a volume of a rectangular prism. [RL]

· Estimate a measurement of volume or capacity using standard or non-standard units (e.g., estimate the capacity of a bowl in cups and handfuls). [SP]

· Use or describe a process to find a reasonable estimate of volume or capacity (e.g., fill a container with rice or popcorn). [CU]

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

Properties and relationships

1.3.1 Understand the characteristics of circles and rectangular prisms. W

· Name and sort circles or rectangular prisms according to their attributes (faces, edges, radii, base, parallel faces). [RL]

· Draw a figure with given characteristics (e.g., the set of points equidistant from a given point). [CU]

· Identify lines of symmetry in rectangular prisms.

· Explain lines of symmetry for circles. [CU]

· Describe the relationship between the diameter and the radius of a circle. [CU]

1.3.2 Apply understanding of angles and polygons. W

· Identify geometric figures and concepts in nature and art (e.g., triangle in architecture, rhombus in beadwork, culturally relevant textiles, quilts). [MC]

· Combine polygons to create given two-dimensional figures and represent them on grid paper (e.g., use all pieces of tangrams to create a square). [SP, RL, CU]

· Create a three-dimensional shape given its net or draw the net of a given three-dimensional shape. [RL]

· Find the missing measure of an angle using the properties of parallel lines, perpendicular lines, vertical and corresponding angles.

· Find the missing angle given all but one of the angles of a polygon. [RL]

Locations and transformations

1.3.3 Understand the relative location of integers on a number line. W

· Show the order of a given set of integers on a number line. [CU]

· Identify the point of final destination given directions for movement on a number line including positive and negative numbers (vertical or horizontal) (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. [RL]

· Describe relative location of points and objects on a number line with both positive and negative numbers. [CU]

· Identify objects on a number line based on given numeric locations.

1.3.4 Apply understanding of rotations (turns) to two-dimensional figures. W

· Apply rotations (turns) of 900 or 1800 to a simple two-dimensional figure.

· Create a design using (900, 1800, 2700, 3600) rotations (turns) of a shape. [SP, MC]

· Show how a shape has been rotated by 900 or 1800. [CU]

· Describe a rotation so that another person could draw it. [CU]

· Identify the coordinates of objects that have been rotated 90°, 180°, or 270° on a coordinate grid.

· Determine whether an object has been translated or rotated on a coordinated grid.

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 0 and 1. W

· Determine whether a real-life event has zero probability, 50% probability, or 100% probability of occurring. [MC]

· Express probabilities as fractions or decimals between 0 and 1 and percents between 0 and 100. [CU]

· Translate between representations of probability (e.g., translate a probability of 6 out of 16 to 3/8 or 37.5%). [MC]

1.4.2 Understand various ways to determine outcomes of events or situations. W

· Determine and use the probabilities of the outcome of a single event.

· Represent or describe all possible outcomes of experiments (e.g., an organized list, a table, a tree diagram, or a sample space). [RL, CU]

· Calculate probability for an event (e.g., pulling colored or numbered balls from a bag, drawing a card, rolling a six on a number cube, spinning a spinner, etc.).

· Determine all possible outcomes (sample space) of an experiment or event (e.g., all different choices a person has to wear one top and one skirt from three different tops and two different skirts). [CU]

Statistics

1.4.3 Analyze how data collection methods affect the data collected. W

· Evaluate how a question or data collection method may affect the data. [RL]

· Determine whether a sampling method will result in a representative sample.

· Describe a data collection method that will provide an unbiased sample. [CU]

· 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]

· Identify different ways of selecting a sample (e.g., convenience sampling, response to a survey, random sampling) and explain which method makes a sample more representative for a population. [SP, MC]

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

· Determine when it is appropriate to use mean, median, or mode and why a specific measure provides the most useful information in a given context. [RL, CU]

· 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, MC]

· Find the missing number given a mean for a data set with a missing element (e.g., given a set of homework scores and the desire to earn an average score of 80%, determine what score the student must earn on the next assignment). [SP, RL]]

1.4.5 Understand how to organize, display, and interpret data in text from single line graphs and scatter plots. W

· Justify a choice of a graph type for a given situation using information about the type of data. [RL, CU, MC]

· Read and interpret data from single line graphs and scatter plots and determine when the use of these graphs is appropriate. [RL, CU]

· Use an appropriate representation to display data (e.g., table, graphs) given a particular situation and audience. [MC, CU]

· Make inferences based on a set of data. [RL]

· Use data from a table, graph, or chart to support an interpretation. [RL, CU]

· Use technology to generate bar graphs, line graphs, and scatter plots from tables of data. [MC

1.4.6 Evaluate a data set to determine how it can be, or has been, used to support a point of view. W

· Compare graphs to data sets (e.g., given unlabeled graphs and data sets, match the appropriate data to a graph). [RL]

· Judge the appropriateness of inferences made from a set of data and support the judgment. [CU, MC]

· Identify claims based on statistical data and assess the validity of the claims. [CU, RL]

· Explain whether the scale on a graph accurately represents the data. [CU]

· Compare or evaluate two or more interpretations of the same set of data for accuracy.

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

Patterns, functions, and other relations

1.5.1 Apply rules for number patterns based on two arithmetic operations. W

· Recognize or extend patterns and sequences using operations that alternate between terms. [RL]

· Create, explain, or extend number patterns involving two related sets of numbers and two operations including addition, subtraction, multiplication, or division. [CU]

· Use rules for generating number patterns (e.g., Fibonacci sequence, bouncing ball) to model real-life situations. [MC]

· Use technology to generate patterns based on two arithmetic operations. [SP]

· Supply missing elements in a pattern based on two operations.

· Select or create a pattern that is equivalent to a given pattern.

1.5.2 Apply understanding of patterns involving two arithmetic operations to develop a rule. W

· Describe the rule for a pattern with combinations of two arithmetic operations in the rule.

· Identify patterns involving combinations of operations in the rule, including exponents (e.g., 2, 5, 11, 23). [RL, MC]

· Represent a situation with a rule involving a single operation (e.g., presidential elections occur every four years; when will the next three elections occur after a given year). [CU, MC]

· Create a pattern involving two operations using a given rule.

Symbols and representations

1.5.3 Apply understanding of equalities and inequalities to interpret and represent relationships between quantities. W

· Express relationships between quantities (decimals, percents, and integers) using =, ≠, <, >, ≤, and ≥. [CU]

· Match a given situation to the correct inequality or equality. [MC]

· Express relationships between non-negative rational numbers using symbols.

· Write an inequality with a single variable to match a particular situation. [RL, CU]

1.5.4 Apply understanding of tables, graphs, expressions, equations, or inequalities to represent situations involving two arithmetic operations. W

· Translate a situation involving multiple arithmetic operations into algebraic form using equations, tables, and graphs. [RL, CU, MC]

· Identify or describe a situation involving two arithmetic operations that matches a given graph. [CU, MC]

· Represent an equation, expression, or inequality using a variable in place of an unknown number. [CU]

· Represent or evaluate algebraic expressions involving a single variable. [RL, CU]

· Represent an equation or expression using a variable in place of an unknown number. [RL, CU]

· Identify a situation that corresponds to a given equation or expression.

Evaluating and solving

1.5.5 Understand and apply procedures to evaluate expressions and formulas. W

· Evaluate simple expressions and formulas using pictures and/or symbols. [RL]

· Represent and evaluate algebraic expressions involving a single variable. [RL, CU]

· Evaluate an expression by substituting non-negative values for variables (e.g., find the value of 3y + 2 when y=3). [RL, MC]

· Determine the expression that represents a given situation. [MC, CU]

· Describe a situation that fits with a given expression. [RL, MC, CU]

1.5.6 Understand and apply a variety of strategies to solve one-step equations. W

· Solve one-step equations using pictures and symbols.

· Solve one-step single variable equations using any strategy (e.g., what number goes in the mystery box).

· Solve real-world situations involving single variable equations. [CU, MC]

· Explain a strategy for solving a single variable equation. [CU]

· Write and solve one-step single variable equations for a given situation. [MC]

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

Component 2.1: Understand problems

Example: A gardener living in Yakima has 100 feet of fencing material. Find the dimensions of the largest rectangular area that he could enclose using all of the fencing material.

2.1.1 Analyze a situation to define a problem. W

· Use strategies to become informed about the situation (e.g., listing information, asking questions).

· Summarize the situation (e.g., there is 100 feet of fencing and we want to enclose as much land, in the shape of a rectangle, as possible).

· Determine whether enough information is given to find a solution (e.g., list what is needed to find the area of a rectangle and compare to the list of known things).

· Determine whether information is missing or extraneous (e.g., compare the list of known things to the list of needed things to see if there are things that are not needed).

· Define the problem (e.g., find the rectangle with largest area with a perimeter of 100 feet).

Component 2.2: Apply strategies to construct solutions

2.2.1 Apply strategies, concepts, and procedures to devise a plan to solve the problem. W

· Organize relevant information from multiple sources to devise a plan (e.g., create a list of known and unknown information; create a table of values for length, width, and area of rectangles with perimeter of 100).

· Select and apply appropriate mathematical tools for a situation (e.g., guess and check, creating tables of values [with or without technology], examine relationships between sides of a rectangle and area).

2.2.2 Apply mathematical tools to solve the problem. W

· Implement the plan devised to solve the problem (e.g., in a table of values of lengths, widths, and areas find the one that shows the largest area; check smaller increments to see if this is the largest that works).

· Identify when an approach is unproductive and modify or try a new approach (e.g., while guess and check may give some sense of a neighborhood of values, it is less efficient than a more organized method).

· Check the solution to see if it works (e.g., if the solution gives a perimeter that is not 100, it makes no sense in the given problem).

EALR 3: The student uses mathematical reasoning.

Component 3.1: Analyze information.

3.1.1 Analyze information from a variety of sources to interpret and compare information. W

· Identify claims based on statistical data and evaluate the validity of the claims. [1.4.5]

· Read and interpret data from single line graphs and scatter plots and determine when the use of these graphs is appropriate. [1.4.5]

· Use volume and capacity to describe and compare figures (e.g., fill containers with cubes to find which has a greater volume). [1.2.4]

Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.

3.2.1 Apply prediction and inference skills to make or evaluate conjectures. W

· Identify claims based on statistical data and evaluate the validity of the claims. [1.4.5]

· Predict a future element in a relation (e.g., find the fifteenth term in a pattern). [1.5.1]

3.2.2 Apply the skill of drawing conclusions and support those conclusions using evidence. W

· Draw conclusions from displays, texts, or oral discussions and justify those conclusions with logical reasoning or other evidence (e.g., read a newspaper article or ad; draw a conclusion and support that conclusion with evidence from the article or elsewhere).

3.2.3 Analyze procedures and results in various situations. W

· Represent and interpret all possible outcomes of experiments (e.g., an organized list, a table, a tree diagram, or a sample space). [1.4.2]

Component 3.3: Verify results.

3.3.1 Analyze procedures and information used to justify results using evidence. W

· Find and compare rectangular prisms that have a given volume (e.g., if two rectangular prisms have the same volume and one has twice the height of the other, determine how the areas of their bases compare). [1.2.5]

· Apply estimation strategies prior to computation of whole numbers, decimals, and fractions to determine reasonableness of answers. [1.1.8]

· Identify different ways of selecting a sample (e.g., convenience sampling, response to a survey, random sampling) and which method makes a sample more representative for a population. [1.4.3]

3.3.2 Analyze thinking and mathematical ideas using models, known facts, patterns, relationships, or counter examples. W

· Identify claims based on statistical data and evaluate the validity of the claims. [1.4.5]

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

Component 4.1: Gather information.

4.1.1 Apply a planning process to collect information for a given purpose. W

· 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). [1.4.4]

· Decide on information needed to create a report on a mathematical topic (e.g., compare the predicted rainfall in a given period with the actual rainfall).

4.1.2 Understand how to extract information from multiple sources using reading, listening, and observation. W

· Use mean, median, and mode to explain situations (e.g., the heights of students in the class; hair color of students in the class; favorite movie of students in the class; most watched movie in a specific time frame). [1.1.4]

Component 4.2: Organize, represent, and share information.

4.2.1 Apply organizational skills for a given purpose. W

· Show the order of the set of integers on a number line with both positive and negative numbers (e.g., organize the given birth years of the following Arabic kings on a number line). [1.3.3]

4.2.2 Apply communication skills to clearly and effectively express or present ideas and situations using mathematical language or notation. W

· Articulate various strategies used during estimation involving fractions and decimals. [1.1.8]

· Clearly explain, describe, or represent mathematical information in a pictorial, tabular, graphical, two- or three-dimensional drawing, or other form as appropriate for the mathematical information (e.g., time, distance, categories), audience, and/or purpose, such as to perform or persuade, with notation and labels as needed.

· Use an appropriate representation to display data (e.g., table, graphs) given a particular situation and audience. [1.4.5]

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

Component 5.1: Relate concepts and procedures within mathematics

5.1.1 Apply concepts and procedures from a variety of mathematical areas in a given problem or situation. W

· Translate a situation involving multiple arithmetic operations into algebraic form using equation, table, and graphs. [1.5.4]

· Given a set of data, compare various representations (e.g., table, graph, rule) for a given situation. [1.4.5]

5.1.2 Apply different mathematical models and representations to the same situation. W

· Represent equivalent ratios or given percentages using objects, pictures, and symbols. [1.1.4]

· Match a graph with a data set. [1.5.4]

Component 5.2: Relate mathematical concepts and procedures to other disciplines.

5.2.1 Analyze mathematical patterns and ideas to extend mathematical thinking and modeling to other disciplines. W

· Identify geometric figures and concepts in nature and art (e.g., triangle in architecture, rhombus in beadwork). [1.3.2]

· Show the order of the set of integers on a number line with both positive and negative numbers (e.g., organize and graph on a number line the given birth years of the given Arabic kings). [1.3.3]

· Read a micrometer to the nearest hundredth of an inch or centimeter, depending on the tool. [1.2.4]

· Create a physical activity plan that results in 2500 calories expended over the week.

· Calculate the ratio of various parts of an artwork (length of eyes to ears).

· Discuss the difference between ¾ time and 6/8 time and how it relates to a model.

5.2.2 Know the contributions of individuals and cultures to the development of mathematics.

· Recognize the contributions of a variety of people to the development of mathematics (e.g., research the concept of the golden ratio).

Component 5.3: Relate mathematical concepts and procedures to real-world situations.

5.3.1 Understand that mathematics is used in daily life and extensively outside the classroom.

· Write and solve real-world problem situations to find sums or differences of decimals or fractions (e.g., explain how to find the change received from a $50.00 bill when a given amount of CD’s and tapes with prices are bought). [1.1.6]

· Calculate the ratio of bicycle gears.

5.3.2 Understand that mathematics is used within many occupations or careers.

· Explain or describe the mathematics necessary to get and perform in a particular job (e.g., complete a project that researches how mathematics is used in careers or occupations of interest).

· Identify where in a particular career mathematics is used (e.g., police work ─ looking for patterns in fingerprints or crimes).

GRADE 7

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 and numeration

1.1.1 Understand the concept of rational numbers (integers, decimals, fractions). W

· Create a model when given a symbolic representation of a rational number. [CU, MC]

· Write the rational number when given a model (e.g., number line, area model, situation, diagram, picture). [CU, MC]

· Identify and convert between equivalent forms of rational numbers (e.g., fractions to decimals, percents to fractions). [MC]

· Identify prime, square, or composite numbers. [CU]

· Explain the meaning of rational numbers and give examples. [CU]

1.1.2 Understand the relative values of rational numbers. W

· Compare and order rational numbers using physical models or implementing strategies (e.g., like denominators, changing to the same form). [RL, MC]

· Locate symbolic representations of rational numbers on a model (e.g., a number line, fraction line, decimal grid, and circle graph). [MC]

· Explain the value of a given digit in a rational number (e.g., 2.3 is 2 ones and 3 tenths). [CU]

1.1.3 Apply properties of addition and multiplication including inverse properties to the rational number system. W

· Use the inverse relationships between multiplication and division to simplify computations and solve problems. [SP, RL]

· Use the inverse properties of addition and multiplication to simplify computations with integers, fractions, and decimals. [SP, RL, MC]

· Identify the inverse elements when using the additive inverse and the multiplicative inverse properties (e.g., 8 + -8 = 0; 2 x ½ = 1).

· Use the additive inverse property to solve problems. [RL]

· Illustrate or explain the additive and multiplicative inverse properties and why they work. [CU]

1.1.4 Understand the concept of direct proportion. W

· Express proportional relationships using objects, pictures, and symbols. [CU]

· Explain the meaning of a proportion. [CU]

· Represent a new relationship from a given ratio (e.g., height of a totem pole, May pole). [MC]

· Represent percentages less than 1% or greater than 100% using objects, pictures, and symbols. [CU]

· Complete or write a proportion for a given situation. [CU]

· Solve problems involving proportions (e.g., determine the number and kinds of baked goods to bring to a bake sale based on proportions of different goods sold at previous bake sales). [SP, MC]

· Use ratios to make predictions about proportions in a future situation. [RL, MC]

Computation

1.1.5 Understand the meaning of addition and subtraction on integers. W

· Explain the meaning of addition and subtraction of integers using real-world models (e.g., reducing debt, temperature increase or decrease, yards gained and lost, movement of a hot-air balloon). [CU, MC]

· Create a problem situation involving addition or subtraction of integers. [CU, MC]

· Explain or show the meaning of addition or subtraction of integers. [CU]

· Use technology to demonstrate addition and subtraction with integers.

1.1.6 Apply computational procedures with fluency for multiplication and division on non-negative rational numbers. W

· Find the product or quotient using non-negative decimals and fractions with unlike denominators.

· Apply percentages to solve a problem in a variety of situations (e.g., taxes, discounts, interest). [SP, MC]

· Use multiplication and division to solve real-world problems involving non-negative rational numbers. [SP]

· Multiply non-negative decimal numbers to the hundredths place.

· Divided non-negative decimals numbers to the thousandths place by non-negative decimal numbers to the hundredths place.

1.1.7 Understand and apply strategies and tools to complete tasks involving addition and subtraction on integers and the four basic operations on non-negative rational numbers.

· Select and justify the selection of appropriate strategies and tools (e.g., mental computation, estimation, calculators, and paper and pencil) to compute in a problem situation. [SP, RL]

· Convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator. [MC]

· Use calculators to add and subtract with integers of two or more digits.

· Use calculators to compute with decimal numbers with precision from the thousandths place and beyond.

Estimation

1.1.8 Apply estimation strategies to predict or determine the reasonableness of answers in situations involving addition and subtraction of integers and the four basic operations on non-negative rational numbers. W

· Identify when an approximation is appropriate in situations. [MC]

· Use estimation strategies prior to operations on non-negative rational numbers to approximate an answer. [RL]

· Justify why estimation would be used rather than an exact computation. [CU]

· Describe a situation where estimation is sufficient in real life contexts. [CU, MC]

· Use estimation to verify the reasonableness of calculated results. [RL]

· Evaluate the appropriateness of estimation in a situation and support the evaluation. [RL]

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

Attributes, units, and systems

1.2.1 Analyze how a change in a linear dimension affects other linear measurements (perimeter, circumference) and area measurements. W

· Describe the relationships among linear dimensions (e.g., radius of a circle, length of a side or base, changes in the diameter affects the amount of deer hide needed to cover a drum face) and area of the figure (e.g., change the radius or length of a side, and check the change in area; describe that change). [CU]

· Explain changing one, two, or three dimensions in a rectangular prism and how it affects the surface area and volume; give three examples.

· Solve problems involving the effects of changes in one dimension on area (e.g., given a garden with certain dimensions, make the area of the garden x square units by changing only one dimension of the garden). [SP]

1.2.3 Understand how the unit of measure affects the precision of measurement. W

· Select the appropriate measurement tool to match the precision needed (e.g., if needing measurement to the nearest 1/16 inch, select a ruler that has 1/32 increments).

· Explain how the unit selected for a situation can affect the precision of the measurement (e.g., when you have a ruler that has only 1/10 increments, you cannot measure something to the nearest hundredth with confidence of precision).

· Explain how measurement systems allow for different levels of precision (e.g., millimeters give more precise measurement than centimeters). [CU]

Procedures, precision, and estimation

1.2.5 Apply formulas to find measurements of circles, triangles, and rectangular prisms. W

· Apply formulas to determine missing measurements for circles, rectangular prisms, and triangles.

· Explain how to use a formula for finding the area and circumference of a circle (e.g., calculate the area needed to cover a drum face). [CU]

· Find and compare the volumes of rectangular prisms that have a given volume (e.g., if two rectangular prisms have the same volume and one has twice the height of the other, determine how the areas of their bases compare). [RL]

· Justify the standard formula for finding the area of a right triangle (e.g., 1/2 of a rectangle). [CU]

· Use given dimensions to determine surface area and volume.

1.2.6 Understand and apply strategies to obtain reasonable estimates of circle measurements, right triangles, and surface area for rectangular prisms. W

· Identify situations in which estimated measures are sufficient. [MC]

· Estimate circle and triangle measurements.

· Use common approximations of pi (3.14; 22/7) to calculate the approximate circumference and the area of circles.

· Use or describe a process to find a reasonable estimate of circle measurements (e.g., wrap a string around it). [RL]

· Explain why estimation or precise measurement is appropriate in a given situation. [CU]

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

Properties and relationships

1.3.1 Understand the concept of similarity. W

· Identify corresponding sides and angles of two similar figures.

· Determine and justify if two figures are similar using the definition of similarity. [CU, RL]

· Differentiate between similar and congruent figures, either geometric figures or real-world objects, and justify the conclusion. [RL, MC]

· Explain how a scale drawing is an example of similarity. [CU]

1.3.2 Apply understanding of the characteristics of rectangular prisms and circles. W

· Identify, describe, compare, and sort figures.

· Draw rectangular prisms and circles with specified properties (e.g., circumference of an 18 centimeter quadrilateral having equal sides but no right angles; a triangle with no equal sides). [CU]

· Use the properties of rectangular prisms and circles to solve problems (e.g., determine which of two rectangular prism-shaped boxes will hold the most cans of food at the food drive and explain how the geometric characteristics affect capacity). [SP, RL, CU, MC]

· Compare two rectangular prisms based on their characteristics (e.g., compare the geometric characteristics of two rectangular prisms with different dimensions and the same volume). [RL]

Locations and transformations

1.3.3 Understand the location of points on a coordinate grid in any of the four quadrants. W

· Identify the coordinates of the fourth point to make a rectangle given three points. [RL]

· Plot and label ordered pairs in any of the four quadrants. [CU]

· Name the coordinates of a given point in any of the four quadrants.

· Identify objects or the location of objects on a coordinate grid using coordinates or labels.

· Use technology to locate objects on a two-dimensional grid.

· Use ordered pairs to describe the location of objects on a grid.

1.3.4 Understand and apply combinations of translations (slides) and reflections (flips) to two-dimensional figures. W

· Identify and explain whether a shape has been translated (slid) or reflected (flipped) with or without a grid. [RL, CU]

· Use transformations to create congruent figures and shapes in multiple orientations.

· Find the coordinate pairs for a translation or a reflection across an axis given a shape on a coordinate grid. [RL]

· Match a shape with its image following one or two transformations (sliding or flipping). [RL]

· Use combinations of translations and reflections to draw congruent figures. [RL]

· Use ordered pairs to describe the location of an object on a coordinate grid after a translation and reflection. [CU]

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

Probability

1.4.1 Understand the concepts of complementary, independent, and mutually exclusive events. W

· Determine and explain when events are mutually exclusive (e.g., your grade on a test is an A, B, or C). [CU, MC]

· Determine and explain when events are complementary (e.g., a person awake or asleep, you pass or fail a test, coin throw ─ heads or tails). [CU, MC]

· Identify or explain when events are complementary, mutually exclusive, or neither (e.g., spinning a 4 or a 5 but with the possibility of spinning 1, 2, 3, or 6) and explain. [CU]

1.4.2 Understand and apply the procedures for determining the probabilities of multiple trials. W

· Calculate the probabilities of independent or mutually exclusive outcomes or events.

· Calculate the probability of an event given the probability of its complement.

· Create a game that has an equal probability for all players to win. [SP, MC]

· Revise a game with unequal probabilities for all players and make it a fair game. [SP, MC]

· Determine, interpret, or express probabilities in the form of a fraction, decimal, or percent. [CU, MC]

· Predict the probability of outcomes of experiments and test the predictions. [RL]

· Predict the probability of future events based on empirical data. [RL]

Statistics

1.4.3 Apply data collection processes to inform, persuade, or answer questions. W

· Formulate a question and collect data from a population, describing how the questions, collection method, and sample population affect the results. [CU]

· Present collected data to support an opinion to inform or persuade an identified audience. [CU, MC]

· Determine whether given data provides useful information for a situation (e.g., given a set of data, decide whether all of the information provided is necessary). [SP]

· Determine whether data support a given opinion and explain the decision. [CU]

· Identify a sample relevant to a given question and population.

· Determine and use range and measures of central tendency to describe a set of data.

1.4.4 Understand how variations in data may affect the choice of data analysis techniques used. W

· Describe the effects of extreme values on means in a population. [CU, MC]

· Explain the difference between median or mean as a measure of central tendency in a given situation (e.g., when an extreme value skews the mean). [RL, CU, MC]

· Describe how additional data added to data sets may affect the result of measures of central tendency. [SP, CU]

· Find the range of a set of data.

· Explain what the range adds to measures of central tendency. [CU]

1.4.5 Understand and apply various data display techniques including box-and-whisker plots. W

· Read and interpret various data displays.

· Determine the appropriate representation for given data. [RL, CU]

· Construct bar graphs, circle graphs, line graphs, box-and-whisker and scatter plots using collected data. [CU, MC]

· Use scatter plots to describe trends and interpret relationships. [RL, CU]

· Read and interpret data from box-and-whisker plots and determine when using this type of graph is appropriate. [RL, CU]

· Describe statistical information given a box-and-whisker plot (e.g., median, range, interquartile range). [CU]

· Compare different graphical representations of the same data. [RL, MC]

· Make and justify an inference drawn from a sample. [RL, CU, MC]

1.4.6 Evaluate how different representations of the same set of data can support different points of view. W

· Critique the use of data and data displays for univariate data.

· Judge the reasonableness of conclusions drawn from a set of data and support that position with evidence (e.g., from newspapers, web sites, opinion polls). [MC, RL]

· Determine the accuracy and completeness of the data in a table or graph. [RL, CU]

· Explain how different representations of the same set of data can support different points of view. [RL, CU]

· Describe how statistics or graphics have been used or misused to support a point of view.

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

Patterns, functions, and other relations

1.5.1 Apply understanding of linear relationships to analyze patterns, sequences, and situations. W

· Identify patterns that are linear relations and provide missing terms. [RL]

· Describe the relationship between the terms in a sequence and their positions in the sequence. [CU]

· Identify, extend, or represent patterns and sequences using tables, graphs, or expressions. [RL, MC]

· Use technology to generate graphic representations of linear relationships. [SP]

· Make predictions using linear relationships in situations. [RL]

· Identify a linear relationship that has the same pattern as another linear relationship.

· Create a representation of a linear relationship given a rule. [MC]

1.5.2 Apply understanding of linear patterns in a table, graph, or situation to develop a rule. W

· Describe the rule and/or construct a table to represent a pattern with combinations of two arithmetic operations in the rule.

· Write an expression or equation with a single variable representing a situation or real-world problem. [CU, MC]

· Write a story about a situation that represents a given linear equation, expression, or graph. [CU, MC]

· Describe the rule or construct a table to represent a pattern with combinations of two arithmetic operations in the rule. [RL, CU]

· Use technology to determine the rule for a linear relationship. [SP, RL]

Symbols and representations

1.5.3 Understand relationships between quantities using squares and square roots. W

· Represent relationships between quantities using exponents (squares) and radicals (roots). [CU]

· Simplify square roots of square numbers (e.g., the square root of 9 is 3). [RL]

· Demonstrate understanding of square roots with physical models and examples. [CU]

· Use exponents (squares) and radicals (square roots) to represent relationships (e.g., finding the area of a square with a side of 5 could be represented by 52). [CU]

1.5.4 Apply understanding of equations, tables, and graphs to represent situations involving linear relationships. W

· Represent linear relationships through expressions, equations, tables, and graphs of situations involving non-negative rational numbers.

· Graph data to demonstrate relationships in familiar contexts (e.g., conversions, perimeter, area, volume, and scaling). [CU, MC]

· Develop a situation that corresponds to a given equation or expression. [CU, MC]

· Create a table or graph given a description of, or an equation for, a situation involving a linear relationship. [CU, MC]

· Describe a situation involving a linear or non-linear relationship that matches a given graph (e.g., time-distance, time-height). [CU, MC]

· Explain the meaning of a variable in a formula, expression, or equation. [CU]

Evaluating and solving

1.5.5 Understand and apply procedures to evaluate expressions and formulas considering order of operations. W

· Substitute non-negative rational values for variables in order to evaluate expressions and formulas (e.g., length x width when length=3 and width=4)

· Explain the simplification of expressions and equations using order of operations. [CU]

· Evaluate expressions and formulas considering order of operations. [RL]

· Determine the expression that represents a given situation. [MC, CU]

· Describe a situation that fits with a given expression. [RL, MC, CU]

· Write expressions or equations for a situation.

1.5.6 Understand and apply a variety of strategies to solve two-step equations with one variable. W

· Explain and justify the solution to a problem in a given context. [RL, CU, MC]

· Solve two-step equations with one variable on only one side of the equal sign (e.g., 2x + 4 = 12).

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

Component 2.1: Understand problems.

Example: On the playground, Juan made 13 free throws out of 18 tries. If Bonita shoots 25 free throws, what is the lowest number she has to make in order to have a better free throw percentage than Juan?

2.1.1 Analyze a situation to define a problem. W

· Use strategies to become informed about the situation (e.g., listing information, asking questions).

· Summarize the situation (e.g., two people are shooting free throws, one shot 18, the other 25; we are trying to find the percentage made for each).

· Determine whether enough information is given to find a solution (e.g., list what is needed to find the percentage of free throws made).

· Determine whether information is missing or extraneous (e.g., compare the list of known things to the list of needed things to see if there are things that are not needed ─ names, location).

· Define the problem (e.g., find the smallest number of free throws Bonita needs to make out of 25 attempts in order to top Juan’s percentage).

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 and numeration

1.1.1 Understand the concept of rational numbers including whole number powers and square roots of square numbers. W

· Explain the meaning of a whole number exponent. [CU]

· Read and use exponential notation to represent large numbers (e.g., 2500 = 502). [MC]

· Identify a square number and find its root.

· Identify different representations of rational numbers and select the best representation in the situation (e.g., percent for sales discount or sales tax, fraction for probability, and decimals for money, distance [4.35 kilometers], batting averages).

· Write a squared number.

1.1.2 Understand the relative values of rational numbers including whole number powers and square roots of square numbers. W

· Compare and order rational numbers using models or implementing strategies. [RL]

· Order different representations of rational numbers. [RL]

· Place symbolic representations of rational numbers on a number line including whole number powers and square roots of square numbers. [CU]

1.1.3 Apply properties of addition, multiplication, and the distributive property to the rational number system. W

· Illustrate and explain the distributive property of multiplication over addition (e.g., using an area model or picture). [CU]

· Use the distributive property to simplify expressions including those using integers. [RL]

· Use the distributive property to factor expressions (e.g., 3▪9+3=3▪(9+1)). [RL]

· Identify the multiplicative inverse of a number.

1.1.4 Apply ratio, percent, and direct proportion in situations. W

· 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, MC]

· Solve problems involving percentages (e.g., percent increase/decrease, tax, commission, discount). [SP, MC]

· Explain advantages and disadvantages of different representations of ratios or percents in a given situation (e.g., using 1/8 versus 12½ %). [CU, MC]

· Determine an unknown value for a dimension or a number of events or objects using ratio or proportion.

· Complete a proportion in a situation.

Computation

1.1.5 Understand the meaning of operations on rational numbers (including square roots of square numbers and whole number powers). W

· Create a problem situation to match a given rational number equation. [CU, MC]

· Explain the meaning of negative and zero exponents. [CU]

· Demonstrate or describe the meaning of multiplication and division of integers using words, visual, or physical models. [CU]

· Create a problem situation involving multiplication or division of integers. [CU, MC]

· Explain solutions when dividing by fractions (e.g., when dividing by a number between 0 and 1, the result is larger than the dividend). [CU]

1.1.6 Apply computational procedures with fluency on rational numbers including whole number powers and square roots of square numbers. W

· Compute with rational numbers using order of operations.

· Compute fluently with rational numbers in all forms except exponential.

· Write and solve problems that involve computation with rational numbers. [CU, MC]

· Solve problems using rational numbers with whole number powers. [SR]

· Solve problems using rational numbers with square roots of perfect squares (e.g., given a square garden with an area of nine square meters, how much fence would be needed to encompass a garden twice the size of the original garden). [SR]

1.1.7 Understand and apply strategies and tools to complete tasks involving computation on rational numbers.

· Select and justify appropriate strategies and tools (e.g., mental computation, estimation, calculators, and paper and pencil) to compute in a problem situation. [SP, RL]

· Describe strategies for mentally solving problems involving integers and exponents. [CU]

· Use calculators to compute with whole number powers beyond the cubed numbers.

· Use calculators to compute square roots of perfect squares greater than 100.

Estimation

1.1.8 Apply estimation strategies to predict or determine the reasonableness of answers in situations involving computation on rational numbers in any form including whole number powers and square roots of square numbers. W

· Identify when an approximation is appropriate. [MC]

· Explain situations involving rational 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]

· Describe various strategies used during estimation involving integers. [CU]

· Use estimation to predict or to verify the reasonableness of calculated results. [RL]

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

Attributes, units, and systems

1.2.1 Analyze how a change in a linear dimension affects volume and surface area of rectangular prisms and right cylinders. W

· Compare the impact that a change in one dimension has on volume and surface area in right cylinders and rectangular prisms. [SP, RL]

· Describe the relationships among linear dimensions, volume, and surface area (e.g., changing the length of a side affects the surface area and volume). [CU]

· Solve problems involving the effects of changes in one dimension on area (e.g., given a box with certain dimensions, make the volume of the box y cubic units by changing only one dimension of the box). [SP]

1.2.2. Understand and apply derived units of measurement. W

· Explain the concept of a rate. [CU]

· Explain how division of measurements produces a derived unit of measurement (e.g., miles traveled divided by hours traveled yields the derived unit [miles per hour]). [CU]

· Find a rate of change in a situation (e.g., increase per year in stamp cost) and label the results. [SP, RL, MC]

· Use unit analysis to find equivalent rates (e.g., miles per hour to feet per second). [MC]

· Use rate to determine a measured outcome.

1.2.3 Understand why different situations require different levels of precision. W

· Explain the relationships among units within both the customary and metric system (e.g., kilograms to grams, feet to inches).

· Justify the use of a unit of measure (e.g., measuring to order fencing requires a different precision than if one is selling land and needs to be precise about borders). [CU, MC]

· Compare situations for the level of precision needed. [RL]

· Explain and give examples of situations that require more and less precision. [CU]

Procedures, precision, and estimations

1.2.5 Understand and apply formulas including the Pythagorean Theorem to right prisms, right cylinders, and triangles. W

· Explain how to use a formula for finding the surface area and volume of a solid. [CU]

· Find missing sides or area of right triangles (e.g., use the Pythagorean Theorem to find any of the missing values).

· Calculate measures of objects for which no direct information is given (e.g., apply ratio, proportion, and scale to determine the area, surface area, and/or volume of a similar figure or solid). [SP, MC]

· Compare surface areas of shapes with given volumes (e.g., compare cost of material to make various right cylinder and right prism containers with a given volume). [RL, MC]

1.2.6 Apply strategies to obtain reasonable estimates of volume and surface area measurements for right cylinders, right prisms, and of the lengths of sides of right triangles. W

· Estimate volume and surface area for right cylinders and right prisms.

· Estimate the length of the remaining side of a right triangle given the lengths of two sides.

· Approximate distance or height in a problem situation using similar triangles or Pythagorean relationships (e.g., height of a flagpole using proportional reasoning, distance across a lake using Pythagorean relationship). [SP]

· Use or describe a process for finding area of a right triangle.

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

Properties and relationships

1.3.1 Apply understanding of characteristics and relationships among one-dimensional, two-dimensional, and three-dimensional figures to solve problems. W

· Identify and label rays, lines, end points, line segments, vertices, and angles. [CU]

· Match or draw three-dimensional objects from different perspectives using the same properties and relationships (e.g., match to the correct net, draw the top view). [RL]

· Draw and label with names and symbols, nets of prisms, and cylinders. [RL, CU]

· Describe everyday objects in terms of their geometric characteristics. [CU]

· Identify the two-dimensional components of three-dimensional figures.

1.3.2 Apply understanding of similarity to two-dimensional figures. W

· Use properties of similarity to draw, describe, and compare two-dimensional figures.

· Find the length of a missing side or the measure of a missing angle of one of the figures, given two similar figures. [SP, RL]

· Create symmetrical, congruent, or similar figures using a variety of tools (e.g., ruler, pattern blocks, geoboards). [RL, CU]

· Draw a similar shape to a given shape. [RL, CU, MC]

· Use properties of circles, cylinders, and figures with rotational symmetry to compare figures. [RL, CU]

· Create a scale drawing and label the scale and the dimensions. (SP, CU, MC).

Locations and transformations

1.3.3 Understand and apply procedures to find distance between points in two-dimensional representations. W

· Locate a missing vertex given the coordinates of the vertices of a regular polygon. [RL]

· Apply the Pythagorean Theorem to find the length of a side of a right triangle or distance between two points.

· Explain a method for finding the missing side of a triangle in a real-world setting (e.g., the height of a totem pole or building). [CU]

· Describe the relationship of any two or more points on a coordinate grid. [CU]

· Find the distance between two points on a coordinate grid including lines that are non-parallel with either axis (oblique). [RL, MC]

1.3.4 Understand and apply transformations to figures. W

· Identify and explain how a shape has been translated, reflected, or rotated with or without a grid (e.g., location of the North Star, rotate the Big Dipper). [CU]

· Use transformations (rotations, reflections, and translations) to draw or locate congruent two-dimensional figures. [RL, CU]

· Find the image of a given shape after a combination of transformations. [RL]

· Tessellate a plane by using transformations. [RL, MC]

· Create a design using a combination of two or more transformations with one or two two-dimensional figures. [SP, RL]

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

Probability

1.4.1 Understand the concept of compound events. W

· Determine and explain when events are compound. [CU]

· Explain the difference between compound events involving ‘and’ and ‘or’ (e.g., rolling a six and rolling an odd number vs. rolling a six or rolling an odd number). [CU]

1.4.2 Understand and apply the procedures for comparing theoretical probability and empirical results for independent or compound events. W

· 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 empirical probability of compound events. [CU]

· Predict the probability of outcomes of experiments and compare the predictions to empirical results. [RL]

· Design or create a situation that would produce a given probability (e.g., how many of each colored marble would it take to have a given probability of selecting one particular color). [SP, MC]

· Design a game using compound probabilities with equal chances of winning for all players. [SP, MC]

Statistics

1.4.3 Analyze how different samples of a population affect the data. W

· Identify sources of sampling bias given a situation (e.g., interviewing only girls, only a certain age group, or too few people). [CU, MC]

· Describe a procedure for selecting an unbiased sample. [CU, MC]

· Compare the results of a survey given two different sample groups. [RL, CU]

· Identify the appropriate population for a given research question.

· Describe how sampling may have affected the resulting data. [CU]

1.4.4 Analyze variations in data to determine the effect on the measures of central tendency. W

· Identify clusters and outliers and determine how clusters or outliers may affect measures of central tendency. [RL]

· Alter a set of data so that the median is a more reasonable measure than the mean. [RL, CU, MC]

· Use and interpret the most appropriate measure of central tendency and the range to describe a given set of data (e.g., the model hourly wage earned by eighth graders is $5.75 per hour and the range is $5.00 to $6.50; therefore, there are very small differences in hourly wages for eighth graders). [RL, CU, MC]

1.4.5 Understand and apply data techniques to interpret bivariate data. W

· Interpret graphic and tabular representations of bivariate data.

· Use a line of best fit to predict a future value of a variable. [ RL]

· Use a line of best fit to interpolate between existing data values. [RL]

· Draw trend lines with or without technology and make predictions about real-world situations (e.g., population trends, socio-economic trends). [CU, MC, RL]

· Examine data in a two-column table to interpolate or extrapolate additional values. [RL]

· 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, racial/ethnic distributions). [RL, MC, CU]

1.4.6 Evaluate how statistics and graphic displays can be used to support different points of view. W

· Critique the use of data and data displays for bivariate data. [RL]

· Judge the reasonableness of conclusions drawn from a set of data and support that position with evidence (e.g., from newspapers, web sites, opinion polls). [MC, RL]

· Determine whether a prediction is reasonable based on a trend line and explain the rationale. [RL]

· Determine whether claims made about results are based on biased representations of data (e.g., whether a scale has been intentionally used to support a point of view).

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

Patterns, functions, and other relations

1.5.1 Apply understanding of linear and non-linear relationships to analyze patterns, sequences, and situations. W

· Extend, represent, or create linear and non-linear patterns and sequences using tables and graphs. [RL]

· Explain the difference between linear and non-linear relationships. [CU]

· Predict an outcome given a linear relationship (e.g., from a graph of profit projections, predict the profit). [RL]

· Use technology to generate linear and non-linear relationship. [SP, RL]

1.5.2 Analyze a pattern, table, graph, or situation to develop a rule. W

· Use technology to help develop a table or graph from an iterative definition (e.g., the number of cells doubles every hour starting with one cell at noon). [CU, MC]

· Explain the nature of changes in quantities in linear relationships using graphs, tables, or expressions. [CU, MC]

· Develop recursive equations that describe linear relations in terms of current and previous values (e.g., start = 7; Current = Previous + 5 would give a set of values (1,7),(2,12),(3,17) …).

· 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]

Symbols and representations

1.5.3 Understand relationships between quantities including whole number exponents, square roots, and absolute value. W

· Represent relationships between quantities using exponents (squares) and radicals (roots). [CU]

· Explain the placement of numbers including square roots and exponents on a number line. [CU]

· Model or describe a real-life situation using absolute value (e.g., the taxi-cab distance from one point to another can be represented by the sum of two absolute values). [CU, MC]

· Use relational symbols to express relationships between rational numbers including percents, square roots, absolute value, and exponents. [CU]

1.5.4 Apply understanding of concepts of algebra to represent situations involving single-variable relationships. W

· Represent variable quantities, through expressions, linear equations, inequalities, tables, and graphs of situations. [CU]

· Write an expression, equation, or inequality with a single variable representing a situation or real-world problem. [SP, RL, MC]

· Identify and use variables to read and write relationships involving rational numbers.

· Model a given description or situation involving relationships with a graph or table. [CU, MC]

· Describe a situation involving relationships that matches a given graph. [CU, MC]

· Create a table or graph given a description of, or an expression for, a situation involving a linear or non-linear relationship. [CU, MC]

Evaluating and solving

1.5.5 Understand and apply the procedures for simplifying single-variable expressions. W

· Simplify expressions and evaluate formulas involving integers. [RL, MC]

· Match expressions to equivalent simplified expressions. [MC]

· Explain a simplification of an expression involving integers. [CU]

· Simplify expressions by combining like terms.

· Simplify expressions using mathematical properties (distributive, commutative, associative, etc.). [RL]

· Determine the expression that represents a given situation. [MC, CU]

· Describe a situation that fits with a given expression. [RL, MC, CU]

1.5.6 Understand and apply a variety of strategies to solve multi-step equations and one-step inequalities with one variable. W

· Solve multi-step equations and one-step inequalities with one variable.

· Solve single variable equations involving parentheses, like terms, or variables on both sides of the equal sign.

· Solve one-step inequalities (e.g., 2x<6, x+4>10).

· Solve real-world situations involving single variable equations and proportional relationships and verify that the solution is reasonable for the problem. [SP, RL, CU]

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

Component 2.1: Understand problems.

Example:The following information was provided to a group of students. They were asked to interpret this information for someone that has a speed of 19 feet per second and also for someone who takes 5 steps per second. How would you answer these questions?

Speed (ft/s) Steps per second

15.86 3.05

16.88 3.12

17.50 3.17

18.62 3.25

19.97 3.36

21.06 3.46

22.11 3.55

2.1.1 Analyze a situation to define a problem. W

· Use strategies to become informed about the situation (e.g., listing information, asking questions).

· Summarize the problem (e.g., we have information about the relationship between the number of steps per second and the speed in feet per second; we wish to find approximate speed or stride rates).

· Determine whether enough information is given to find a solution (e.g., list what is needed to find the relationship between stride rate and speed; list known and unknown information).

· Determine whether information is missing or extraneous (e.g., compare the list of known things to the list of needed things to see if there are things that are not needed ─ names, location).

· Define the problem (e.g., find the relationship between the steps per second and speed).

Component 2.2: Apply strategies to construct solutions.

2.2.1 Apply strategies, concepts, and procedures to devise a plan to solve the problem. W

· Organize relevant information from multiple sources.

· Select and apply appropriate mathematical tools for a situation (e.g., plot steps per second vs. speed; check to see if model is linear; calculate successive differences or quotients to see if a pattern emerges; find an equation for a line that approximates the relationship or extend the pattern to approximate the speed at 5 steps per second).

2.2.2 Apply mathematical tools to solve the problem. W

· Implement the plan devised to solve the problem or answer the question posed (e.g., in a table of values of lengths, widths, and areas find the one that shows the largest area; check smaller increments to see if this is the largest that works).

· Identify when an approach is unproductive and modify or try a new approach (e.g., if an additive model didn’t work, try a multiplicative model).

· Check the solution to see if it works (e.g., if the solution for a speed of 19 feet per second is 5 steps per second, perhaps the assumption of linearity was incorrect).

EALR 3: The student uses mathematical reasoning.

Component 3.1: Analyze information.

3.1.1 Analyze information from a variety of sources to interpret and compare information. W

· Predict the probability of outcomes of experiments and compare the predication to empirical results. [1.4.2]

· Predict an outcome given a linear relationship and a particular input (e.g., from a graph of profit projections, predict the profit in 2005). [1.5.1]

Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.

3.2.1 Apply prediction and inference skills to make or evaluate conjectures. W

3.2.2 Apply the skill of drawing conclusions and support those conclusions using evidence. W

· Draw conclusions from displays, texts, or oral discussions and justify those conclusions with logical reasoning or other evidence (e.g., read an editorial or ad, draw a conclusion and support that conclusion with evidence in the article or elsewhere).

3.2.3 Analyze procedures and results in various situations. W

· Critique conclusions drawn from a set of data and support with evidence (e.g., from magazines, newspapers, web sites, opinion polls). [1.4.6]

GRADES 9/10

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 and numeration

1.1.1 Understand and apply scientific notation. W

· Read and use scientific and exponential notation. [MC, RL]

· Identify a real-life situation to match a particular number written in scientific or exponential notation and justify the answer. [MC, RL]

· Use scientific or exponential notation to simplify a problem. [RL, MC]

· Illustrate the meaning of scientific notation using pictures, diagrams, or numbers. [CU]

· Read and translate numbers represented in scientific notation from calculators and other technology, tables, and charts.

1.1.4 Apply understanding of direct and inverse proportion to solve problems. W

· Explain a method for determining whether a real-world problem involves direct proportion or inverse proportion. [SP, CU, MC]

· Explain a method for solving a real-world problem involving direct proportion. [CU, MC]

· Explain a method for solving a real-world problem involving inverse proportion. [CU, MC]

· Solve problems using direct or inverse models (e.g., similarity, age of car vs. worth). [SP, MC]

· Explain, illustrate, or describe examples of direct proportion. [CU]

· Explain, illustrate, or describe examples of inverse proportion. [CU]

· Use direct or inverse proportion to determine a number of objects or a measurement in a given situation.

Computation

1.1.6 Apply strategies to compute fluently[c1] with rational numbers in all forms including whole number exponents. W

· Complete multi-step computations using order of operations in situations involving combinations of rational numbers including whole number exponents and square roots of square numbers. [MC]

· Calculate using order of operations on all forms of rational numbers (e.g., (3·2+5)2-8, 22+ 32).

· Use properties to reorder and rearrange expressions to compute more efficiently. [ RL]

Estimation

1.1.8 Apply estimation strategies to determine the reasonableness of [c2] results in situations involving multi-step computations with rational numbers including whole number powers and square and cube roots. W

· Identify when an approximation is appropriate. [MC]

· 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]

· Describe various strategies used during estimation involving integers, rational numbers. [CU]

· Use estimation to predict or to verify the reasonableness of calculated results. [RL]

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

Attributes, units, and systems

1.2.1 Analyze how changes in one or two dimensions of an object affect perimeter, area, surface area, and volume. W

· Describe and compare the impact that a change in one or more dimensions has on objects (e.g., how doubling one dimension of a cube affects the surface area and volume). [CU, MC]

· Describe how changes in the dimensions of objects affect perimeter, area, and volume in real world situations (e.g., how does the change in the diameter of an oil drum affect the area and volume). [CU, MC]

· Solve problems by deriving the changes in two dimensions necessary to obtain a desired surface area and/or volume (e.g., given a box with certain dimensions, make the volume of the box y cubic units by changing two dimensions of the box). [SP]

· Compare a given change in one or two dimensions on the perimeter, area, surface areas, or volumes of two objects.

· Determine the change in one dimension given a change in perimeter, area, volume, or surface area.

1.2.3 Understand how to convert units of measure within systems (U.S. or metric). W

· Understand how to convert units of measure within U.S. or within metric systems to achieve an appropriate level of precision.

· Convert within a system to a unit size appropriate to a given situation.

· Convert to a larger unit within a system while maintaining the same level of precision (e.g., represent 532 centimeters to 5.32 meters).

· Convert to a smaller unit within a system to increase the precision of a derived unit of measurement.

Procedures, precision, and estimation

1.2.5 Apply formulas to calculate measurements of right prisms or right circular cylinders. W

· Explain how to use a formula for finding the volume of a prism or cylinder. [CU, MC]

· Use a formula to find the volume of a prism or cylinder. [RL, MC]

· Use a formula to derive a dimension of a right prism or right cylinder given other measures.

· Use formulas to describe and compare the surface areas and volumes of two or more right prisms and/or right cylinders. [RL]

· Use formulas to obtain measurements needed to describe a right cylinder or right prism.

1.2.6 Understand and apply strategies to obtain reasonable measurements at an appropriate level of precision. W

· Identify situations in which approximate measurements are sufficient.

· Estimate a reasonable measurement at an appropriate level of precision. [MC]

· Estimate quantities using derived units of measure (e.g., distance or time using miles per hour, cost using unit cost). [MC]

· Estimate derived units of measure (e.g., miles per hour, people/year, grams/cubic centimeters). [MC]

· Apply a process that can be used to find a reasonable estimate for the volume of prisms, pyramids, cylinders, and cones.

· Estimate volume and surface area for right cylinders and right prisms.

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

Properties and relationships

1.3.1 Understand the relationship among characteristics of one-dimensional, two-dimensional, and three-dimensional figures. W

· Identify and label one- and two-dimensional characteristics (rays, lines, end points, line segments, vertices, and angles) in three-dimensional figures. [CU]

· Match or draw three-dimensional objects from different perspectives using the same properties and relationships (e.g., match to the correct net, draw the top view). [RL]

· Draw and label with names and symbols nets of right prisms and right cylinders. [RL, CU]

· Describe everyday objects in terms of their geometric characteristics. [CU]

· Describe or classify various shapes based on their characteristics.

· Make and test conjectures about two-dimensional and three-dimensional shapes and their individual attributes and relationships using physical, symbolic, and technological models (e.g., diagonal of a rectangle or prism is the longest interior segment; what figures make up cross-sections of a given three-dimensional shape). [SP, RL, CU, MC]

1.3.2 Apply understanding of geometric properties and relationships. W

· Use geometric properties and relationships to describe, compare, and draw two-dimensional and three-dimensional shapes and figures.

· Construct geometric figures using a variety of tools and technologies (e.g., angle bisectors, perpendicular bisectors, triangles given specific characteristics). [MC]

· Draw a plane shape and justify the answer given a set of characteristics. [RL, CU]

· Use the properties of two-dimensional and three-dimensional shapes to solve mathematical problems (e.g., find the width of a river based on similar triangles; given a set of parallel lines, a transversal, and an angle, find the other angles). [SP, RL, CU, MC]

· Compare two-dimensional and three-dimensional shapes according to characteristics including faces, edges, and vertices, using actual and virtual modeling. [RL, CU]

· Use technology to generate two and three dimensional models of geometric figures with given geometric characteristics (e.g., generate a two-dimensional animation using pentagons with fixed coordinates for one edge). [RL, SP]

· Create a three-dimensional scale drawing with particular geometric characteristics. [SP, CU, MC]

Locations and transformations

1.3.3 Apply understanding of geometric properties and location of points to figures. W

· Use coordinates to describe or identify the location of objects on coordinate grids.

· Describe geometric characteristics of two-dimensional objects using coordinates on a grid. [MC]

· Describe the location of points that satisfy given conditions (e.g., the set of points equidistant from a given point; a point equidistant from a given set of points). [CU]

· Represent situations on a coordinate grid or describe the location of points that satisfy given conditions (e.g., locate a gas station to be equidistant from given cities; locate a staking point to maximize the grazing area of a tethered goat). [MC, SP, RL]

· Use tools and technology to draw objects on a coordinate grid based on given conditions. [CU]

· Identify, interpret, and use the meaning of slope of a line as a rate of change using physical, symbolic, and technological models. [SP, RL, MC]

1.3.4 Apply understanding of multiple transformations. W

· Apply multiple transformations to create congruent and similar figures in any or all of the four quadrants.

· Use multiple transformations (combinations of translations, reflections, or rotations) to draw an image. [RL]

· Use dilation (expansion or contraction) of a given shape to form a similar shape. [RL, CU]

· Determine the final coordinates of a point after a series of transformations. [RL, CU]

· Examine figures to determine rotational symmetry about the center of the shape. [RL, MC]

· Define a set of transformations that would map one onto the other given two similar shapes. [SP, RL]

· Create a design with or without technology using a combination of two or more transformations with one or two two-dimensional figures. [SP, RL]

· Use technology to create two- and three-dimensional animations using combinations of transformations. [MC, SP, RL]

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

Probability

1.4.1 Understand the concept of conditional probability. W

· Compare the probabilities of dependent and independent events. [CU, MC]

· Determine and justify whether the outcome of a first event affects the probability of a later event (e.g., drawing cards from a deck with or without replacement). [CU]

· Explain the difference between dependent and independent events. [CU]

· Explain and give examples of compound events. [CU]

1.4.2 Apply understanding of dependent and independent events to calculate probabilities. W

· Determine probabilities of dependent and independent events. [SP]

· Generate the outcomes and probability of multiple independent and dependent events using a model or procedure (e.g., tree diagram, area model, counting procedures).

· Generate the outcomes and probability of events using a counting procedure (e.g., the number of license plates that can be made with three letters and three numbers; winning the lottery). [MC]

· Explain the relationship between theoretical probability and empirical frequency of dependent events using simulations with and without technology. [CU]

· Create a simple game based on independent probabilities wherein all players have an equal probability of winning. [MC, SP]

· Create a simple game based on compound probabilities. [MC, SP]

· Determine the sample space for independent or dependent events.

Statistics

1.4.3 Apply appropriate methods and technology to collect data or evaluate methods used by others for a given research questions. W

· Identify sources of bias in data collection questions, samples, and/or methods and describe how such bias can be controlled. [RL, CU]

· Evaluate methods and technology used to investigate a research question. [CU, MC]

· Collect data using appropriate methods.

· Use technology appropriately to collect data. [RL, MC]

· Identify appropriate data collection methods that might impact the accuracy of the results of a given situation. [RL, CU]

· Determine whether the sample for a given study was from a representative sample.

· Determine whether the methods of data collection used were appropriate for a given question or population. [RL]

1.4.4 Understand and apply techniques to find the equation for a reasonable linear model. W

· Determine the equation for a reasonable line to describe a set of bivariate data. [RL, MC]

· Determine the equation of a line that fits the data displayed on a scatter plot. [SP, RL]

· Use technology to determine the line of best fit for a set of data. [MC]

· Match an equation with a set of data. [MC]

· Match an equation with a graphic display. [MC]

· Create a graph based on the equation for the line.

1.4.5 Analyze a linear model to judge its appropriateness for a data set. W

· Determine whether a straight line is an appropriate way to describe a trend in a set of bivariate data. [MC, RL]

· Determine whether the underlying model for a set of data is linear. [RL, MC]

· Decide and explain whether it is appropriate to extend a given data set following a line of best fit. [RL, MC]

· Determine whether a linear prediction from a given set of data is appropriate for the data and support the decision with data. [MC].

· Determine whether an equation for a line is appropriate for a given set of data and support the judgment with data. [RL, MC]

· Use technology to generate data to fit a linear model. [SP, MC]

1.4.6 Apply understanding of statistics to make, analyze, or evaluate a statistical argument. W

· Identify trends in a set of data in order to make a prediction based on the information. [CU, MC]

· Justify a prediction or an inference based on a set of data. [CU, MC]

· State possible factors that may influence a trend but not be reflected in the data (e.g., population growth of deer vs. availability of natural resources or hunting permits). [MC, CU, RL]

· Use statistics to support different points of view. [RL]

· Analyze a set of statistics to develop a logical point of view. [RL. CU, MC]

· Justify or refute claims and supporting arguments based on data. [CU, MC]

· Determine whether statistics have been used or misused to support a point of view or argument and support the evaluation with data. [RL]

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

Patterns, functions, and other relations

1.5.1 Apply processes that use repeated addition (linear) or repeated multiplication (exponential). W

· Recognize, extend, or create a pattern or sequence between sets of numbers and/or linear patterns. [RL, CU, MC]

· Identify, extend, or create a geometric or arithmetic sequence or pattern. [RL, CU]

· Translate among equivalent numerical, graphical, and algebraic forms of a linear function. RL, MC]

· Make predictions based on a pattern or sequence.

1.5.2 Analyze a pattern, table, graph, or model involving repeated addition (linear) or repeated multiplication (exponential) model to write an equation or rule. W

· Find the equation of a line in a variety of ways (e.g., from a table, graph, slope-intercept, point-slope, two points). [RL, MC]

· Generate and use rules for a pattern to make predictions about future events (e.g., population growth, future sales, growth of corn stalks, future value of savings account). [SP, RL, MC]

· Identify or write an equation or rule to describe a pattern, sequence, and/or a linear function. [RL, CU, MC]

· Write an equation for a line given a set of information (e.g., two points, point-slope, etc.). [CU, MC]

· Write a recursive definition of a geometric pattern (e.g., Start and New = Old * Number). [CU, MC]

· Represent systems of equations and inequalities graphically. [RL, MC]

· Write a story that represents a given linear equation or expression. [CU, MC]

· Write an expression, equation, or inequality with two variables representing a linear model of a real-world problem. [CU, MC]

Symbols and representations

1.5.4 Apply understanding of equations, tables, or graphs to represent situations involving relationships that can be written as repeated addition (linear) or repeated multiplication (exponential). W

· Represent variable quantities through expressions, equations, inequalities, graphs, and tables to represent linear situations involving whole number powers and square and cube roots. [CU, MC]

· Identify and use variable quantities to read and write expressions and equations to represent situations that can be described using repeated addition (e.g., models that are linear in nature). [CU, MC]

· Identify and use variable quantities to read and write expressions and equations to represent situations that can be described using repeated multiplication (e.g., models that are exponential such as savings accounts and early stages of population growth). [CU, MC]

· Recognize and write equations in recursive form for additive models (e.g., starting value, New=Old + some number). [CU, MC]

· Recognize and write equations in recursive form for additive models (e.g., starting value, New=Old × some number). [CU, MC]

· Select an expression or equation to represent a given real world situation. [MC]

Evaluating and solving

1.5.5 Apply procedures to simplify expressions. W

· Simplify expressions and evaluate formulas involving exponents.

· Justify a simplification of an expression involving exponents. [RL, CU]

· Use multiple mathematical strategies and properties to simplify expressions.

1.5.6 Apply procedures to solve equations and systems of equations. W

· Rearrange formulas to solve for a particular variable (e.g., given, solve for h). [MC, CU]

· Solve real-world situations involving linear relationships and verify that the solution makes sense in relation to the problem. [SP, RL, CU, MC]

· Find the solution to a system of linear equations using tables, graphs, and symbols. [CU, MC]

· Interpret solutions of systems of equations. [CU, MC]

· Solve multi-step equations. [SP, RL]

· Use systems of equations to analyze and solve real-life problems. [SP, CU, MC]

· Determine when two linear options yield the same outcome (e.g., given two different investment or profit options, determine when both options will yield the same result).

· Use systems of equations to determine the most advantageous outcome given a situation (e.g., given two investment options, determine under what conditions each will yield the best result.). [MC, SP]

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

Component 2.1: Investigate situations.

Example: The following are the times (in seconds) of the Olympics in the given years. Using this information, is it reasonable to believe that the women will run as fast as the men in this event? Justify your answer using this data:

Year Men’s Women’s Year Men’s Women’s

1948 10.3 11.9 1976 10.06 11.08

1952 10.4 11.5 1980 10.25 11.06

1956 10.5 11.5 1984 9.99 10.97

1960 10.2 11.0 1988 9.92 10.54

1964 10 11.4 1992 9.96 10.82

1968 9.95 11.0 1996 9.84 10.94

1972 10.14 11.07 2000 9.87 10.75

2.1.1 Analyze a situation to define a problem. W

· Use strategies to become informed about the situation (e.g., listing information; examine the table for patterns; create a scatter plot to look for patterns; asking questions).

· Summarize the problem (e.g., there are Olympic winning times over the past 50 years; both men’s and women’s times are decreasing; will there come a time when women run faster than men).

· Determine whether enough information is given to find a solution (e.g., list what is needed to be found; extend the pattern to see if women’s times will be less).

· Determine whether information is missing or extraneous (e.g., compare the list of known things to the list of needed things to see if there are things that are not needed).

· Define the problem (e.g., if the pattern continues in the same fashion, will women run faster than men and, if so, when will that occur).

Component 2.2: Apply strategies to construct solutions.

2.2.1 Apply strategies, concepts, and procedures to devise a plan to solve the problem. W

· Organize relevant information from multiple sources (e.g., create a list of known and unknown information; create a scatter plot of men’s and women’s times vs. time on the same coordinate axis to analyze the patterns).

· Select and apply appropriate mathematical tools to devise a strategy in a situation (e.g., if the data, in either tabular or graphical form, suggest a linear relationship, plan to find a linear equation for each set of data; solve those equations simultaneously [or use technology to find the intersection of the two lines] to answer the question). If the data pattern suggests a non-linear model, plan to project what the pattern is and extend that pattern.

2.2.2 Apply mathematical tools to solve the problem. W

· Implement the plan devised to solve the problem (e.g., solve the set of simultaneous equations to arrive at a time where the two times are the same).

· Use mathematics to solve the problem (e.g., use algebra to write equations for the two linear models, solve the system of equations using either symbols or technology).

· Identify when an approach is unproductive and modify or try a new approach (e.g., if the result does not make sense in the context, return to the plan to see if something has gone wrong and adjust accordingly).

· Check the solution to see if it works (e.g., the solution may be a partial year [i.e., 2003.6]; decide how to deal with this and also if the year is reasonable [i.e., 1925 does not make sense given the context]).

EALR 3: The student uses mathematical reasoning.

Component 3.1: Analyze information.

3.1.1. Synthesize information from multiple sources in order to answer questions. W

· Use the properties of two-dimensional and three-dimensional figures to solve mathematical problems (e.g., find the width of a river based on similar triangles; given a set of parallel lines, a transversal, and an angle, find the other angles).

Component 3.2: Make predictions, inferences, conjectures, and draw conclusions.

3.2.1 Apply skill of conjecturing and analyze conjectures by formulating a proof or constructing a counter example. W

· Make and test conjectures about two-dimensional and three-dimensional figures and their individual attributes and relationships using physical, symbolic, and technological models (e.g., diagonal of a rectangle or prism is the longest interior segment; what figures make up cross-sections of a given three-dimensional shape). (1.3.1)

3.2.2 Analyze information to draw conclusions and support them using inductive and deductive reasoning. W

· Compare and describe the volume of cylinders, cones, and prisms when an attribute is changed (e.g., the area of the base, the height of solid). (1.2.4)

· Draw a plane shape of a given set of characteristics and justify the answer. (1.3.2)

· Identify trends in a set of data in order to make a prediction based on the information. (1.4.6)

· Use statistics to support different points of view. (1.4.6)

3.2.3 Analyze procedures to determine appropriateness of claims and arguments. W

· Examine claims and supporting arguments based on data and make needed revisions. (1.4.6)