Mathematics content standards and Key Elements

A. A student should understand mathematical facts, concepts, principles, and theories. A student who meets the content standard should:

1. understand and use numeration, including

A) numbers, number systems, counting numbers, whole numbers, integers, fractions, decimals, and percents; and
B) irrationals and complex numbers;

2. select and use appropriate systems, units, and tools of measurement, including estimation;
3. perform basic arithmetic functions, make reasoned estimates, and select and use appropriate methods or tools for computation or estimation including mental arithmetic, paper and pencil, a calculator, and a computer;
4. represent, analyze, and use mathematical patterns, relations, and functions using methods such as tables, equations, and graphs;
5. construct, draw, measure, transform, compare, visualize, classify, and analyze the relationships among geometric figures; and
6. collect, organize, analyze, interpret, represent, and formulate questions about data and make reasonable and useful predictions about the certainty, uncertainty, or impossibility of an event.

B. A student should understand and be able to select and use a variety of problem-solving strategies. A student who meets the content standard should:

1. use computational methods and appropriate technology as problem-solving tools;
2. use problem solving to investigate and understand mathematical content;
3. formulate mathematical problems that arise from everyday situations;
4. develop and apply strategies to solve a variety of problems;
5. check the results against mathematical rules;
6. use common sense to help interpret results;
7. apply what was learned to new situations; and
8. use mathematics with confidence.

C. A student should understand and be able to form and use appropriate methods to define and explain mathematical relationships. A student who meets the content standard should:

1. express and represent mathematical ideas using oral and written presentations, physical materials, pictures, graphs, charts, and algebraic expressions;
2. relate mathematical terms to everyday language;
3. develop, test, and defend mathematical hypotheses; and
4. clarify mathematical ideas through discussion with others.

D. A student should be able to use logic and reason to solve mathematical problems. A student who meets the content standard should:

1. analyze situations;
2. draw logical conclusions;
3. use models, known facts, and relationships to explain the student’s reasoning;
4. use deductive reasoning to verify conclusions, judge the validity of arguments, and construct valid arguments; and
5. use inductive reasoning to recognize patterns and form mathematical propositions.

E. A student should be able to apply mathematical concepts and processes to situations within and outside of school. A student who meets the content standard should:

1. explore problems and describe results using graphical, numerical, physical, algebraic, and verbal mathematical models or representations;
2. use mathematics in daily life; and
3. use mathematics in other curriculum areas.

Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

MATHEMATICS STANDARD A1: NUMERATION
CONTENT STANDARD A2: MEASUREMENT
CONTENT STANDARD A3: ESTIMATION AND COMPUTATION
CONTENT STANDARD A4: FUNCTIONS AND RELATIONSHIPS
CONTENT STANDARD A5: GEOMETRY
CONTENT STANDARD A6: STATISTICS/PROBABILITY
CONTENT STANDARD B: PROBLEM-SOLVING
CONTENT STANDARD C: COMMUNICATION
CONTENT STANDARD D: REASONING
CONTENT STANDARD E: CONNECTIONS

MATHEMATICS STANDARD A1: NUMERATION

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A1.1.1 Read, write, order, count, and model one-to-one correspondence with whole numbers to100. A1.2.1 Read, write, model, order, and count with positive whole numbers to 1,000,000 and negative whole numbers. A1.3.1 Read, write, model, and order real numbers, explaining scientific notation, exponents, and percents. A1.4.1 Read, write, model, order, and define real numbers and subsets.
A1.1.2 Use, model, and identify place value positions of 1’s, 10’s, and 100’s. A1.2.2 Use, model and identify place value positions from 0.001 to 1,000,000. A1.3.2 Model counting in a different base system. A1.4.2 Add in a different base system.
A1.1.3 Model and explain the processes of addition and subtraction, describing the relationship between the operations. A1.2.3 Model and explain the processes of multiplication and division. Describe the relationships among the four basic operations.   A1.4.3 Compare and contrast the relationship between various applications of the same operation.
A1.1.4 Select and use various representations of ordinal and cardinal numbers. A1.2.4 Identify and describe different uses for the same numerical representation. A1.3.4 Translate between equivalent representations of the same number. Select a representation that is appropriate for the situation. A1.4.4 Translate between equivalent representations of the same exponential expression.
A1.1.5 Identify, model, and label simple fractions, describing and defining them as equal parts of a whole, a region or a set. A1.2.5 Model and explain the process of adding and subtracting fractions with common denominators and decimals that represent money. A1.3.5 Describe and model the relationship of fractions to decimals, percents, ratios, and proportions.  
A1.1.6 Identify, describe and extend patterns inherent in the number system. Skip count by 2’s 5’s and 10’s. Add and subtract by 10. Identify even and odd numbers. A1.2.6 Identify and describe factors and multiples including those factors and multiples common to a pair or set of numbers. A1.3.6 Use, explain, and define the rules of divisibility, prime and composite numbers, multiples, and order of operations.  
A1.1.7 Demonstrate the commutative and identity properties of addition. A1.2.7 Demonstrate the commutative and identity properties of multiplication. A1.3.7 Use commutative, identity associative, and distributive properties with variables. A1.4.7 Recognize, describe, and use properties of the real number system.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD A2: MEASUREMENT

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A2.1.1 Compare and order objects by various measurable attributes including calendar, temperature, length, weight, capacity, area, and volume. A2.2.1 Estimate and measure weights, lengths, and temperatures to the nearest unit using the metric and standard systems. A2.3.1 Estimate and measure various dimensions to a specified degree of accuracy. A2.4.1 Evaluate measurements for accuracy, precision, and error with respect to the measuring tools, methods, and the computational process.
A2.1.2 Compare objects to standard and non-standard units to identify objects that are greater than, less than, and equal to a given unit . A2.2.2 Identify and use equivalent measurements (e.g. 60 minutes = 1 hour, 7 days = 1 week). A2.3.2 Estimate and convert measurements within the same system. A2.4.2 Estimate and convert measurements between different systems.
  A2.2.3 Use a variety of measuring tools; describe the attribute(s) they measure. A2.3.3 Use a variety of methods and tools to construct and compare plane figures. A2.4.3 Apply various measurement systems to describe situations and solve problems.
A2.1.4 Choose a unit of measure, estimate the length or weight of objects and then measure to check for reasonableness. A2.2.4 Estimate and measure the dimensions of geometric figures. A2.3.4 Describe and apply the relationships between dimensions of geometric figures to solve problems using indirect measurement; describe and apply the concepts of rate and scale. A2.4.4 Use indirect methods, including the Pythagorean Theorem and right triangle trigonometry, to find missing dimensions.
A2.1.5 Tell time to the nearest half hour, distinguishing between morning, afternoon, and evening. A2.2.5 Tell time using analog and digital clocks identifying AM and PM; find elapsed time. A2.3.5 Apply information about time zones and elapsed time to solve problems.  
A2.1.6 Identify coins, their value, and the value of given sets of coins. A2.2.6 Read, write, and use money notation, determining possible combinations of coins and bills to equal given amounts; count back change for any given situation.    

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD A3: ESTIMATION AND COMPUTATION

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A3.1.1 Make reasonable estimates of "how many" and "how much"; estimate the results of simple addition and subtraction problems. A3.2.1 Describe and use a variety of estimation strategies including rounding to the appropriate place value, multiply by powers of 10 and use front-end estimation to check the reasonableness of solutions. A3.3.1 Apply, explain, and assess the appropriateness of a variety of estimation strategies including truncating and rounding to compatible numbers. A3.4.1 Use estimation to solve problems and to check the accuracy of solutions; state whether the estimation is greater or less than the exact answer.
A3.1.2 Recall and use basic addition and subtraction facts orally and with paper and pencil without a calculator. A3.2.2 Recall and use basic multiplication and division facts orally, with paper and pencil without a calculator. A3.3.2 Apply basic operations efficiently and accurately, using estimation to check the reasonableness of results.  
A3.1.3 Add and subtract whole numbers to 100 using a variety of models and algorithms. A3.2.3 Add and subtract whole numbers and fractions with common denominators to 12 and decimals, including money amounts, using models and algorithms. A3.3.3 Add and subtract fractions, decimals, and percents. A3.4.3 Add and subtract real numbers using scientific notation, powers, and roots.
A3.1.4 Model multiplication as repeated addition and grouping objects; model division as "sharing equally" and grouping objects. A3.2.4 Multiply and divide multi-digit whole numbers by 2-digit numbers, limiting the 2-digit divisors to those that end in 0; multiply and divide decimals that represent money by whole numbers. A3.3.4 Multiply and divide rational numbers in various forms including fractions, decimals, and percents. A3.4.4 Multiply and divide real numbers in various forms including scientific notation, powers, and roots.
  A3.2.5 Find equivalent fractions. Convert between fractions and mixed numbers. A3.3.5 Convert between equivalent fractions, decimals, percents, and proportions. Convert from exact to decimal representations of irrational numbers. A3.4.5 Select, convert, and apply an equivalent representation of a number for a specified situation.
  A3.2.6 Develop and interpret scales and scale models. A3.3.6 Solve problems using ratios and proportions. A3.4.6 Use ratios and proportions to model and solve fraction and percent problems with variables.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD A4: FUNCTIONS AND RELATIONSHIPS

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A4.1.1 Recognize, describe, create and extend repeating and increasing patterns with a variety of materials including symbols, objects, and manipulatives. A4.2.1 Use patterns and their extensions to make predictions and solve problems; describe patterns found in the number system including those formed by multiples, factors, perfect squares, and powers of 10. A4.3.1 Identify numeric and geometric patterns, to find the next term and predict the nth term. A4.4.1 Identify, graph and describe the graphs of basic families of functions including linear, absolute value, quadratic, and exponential using a graphing calculator.
A4.1.2 Generate and solve simple functions by identifying and applying addition and subtraction patterns. A4.2.2 Generate and solve simple functions by identifying and applying multiplication and division patterns. A4.3.2 Identify and describe how a change in one variable in a function effects the remaining variables (e.g., how changing the length effects the area and volume of a rectangular prism). A4.4.2 Create and solve linear and quadratic equations and inequalities.
A4.1.3 Use a calculator to find and extend patterns in the number system. A4.2.3 Use a calculator to find a missing item in a number sequence. A4.3.3 Use a calculator to find a missing item in an arithmetic and a geometric sequence; predict the graph of each function. A4.4.3 Create and solve simple systems of equations, algebraically and graphically, using a graphing calculator.
  A4.2.4 Use words, lists, and tables to represent and analyze patterns. A4.3.4 Translate among and use tables of ordered pairs, graphs on coordinate planes, and linear equations as tools to represent and analyze patterns. A4.4.4 Use discrete structures, such as networks, matrices, sequences, and iterations as tools to analyze patterns, expressions, and equations.
A4.1.5 Complete open space sentences with missing numbers; use appropriate vocabulary including greater than, less than, and equal to; and use the correct symbols. A4.2.5 Explain the purpose of variables and use them in open sentences to express relationships and describe simple functions. A4.3.5 Find the value of a variable by evaluating formulas and algebraic expressions for given values. A4.4.5 Add, subtract, multiply, divide, and simplify rational expressions; add, subtract, and multiply polynomials.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD A5: GEOMETRY

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A5.1.1 Identify, sort, describe, model, and compare circles, triangles and rectangles including squares regardless of orientation. A5.2.1 Identify and compare various triangles and quadrilaterals according to their sides and/or angles. A5.3.1 Identify, classify, compare, and sketch regular and irregular polygons. A5.4.1 Identify and use the properties of polygons, including interior and exterior angles, and circles (including angles, arcs, chord, secants, and tangents) to solve problems.
A5.1.2 Identify, sort, describe, model, and compare solid figures including cubes, cylinders, and spheres. A5.2.2 Compare and contrast plane and solid figures (e.g., circle/sphere, square/cube, triangle/pyramid) using relevant attributes, including the number of vertices, edges, and the number and shape of faces. A5.3.2 Model, identify, draw and describe 3-dimensional figures including tetrahedrons, dodecahedrons, triangular prisms, and rectangular prisms. A5.4.2 Create 2-dimensional representations of 3-dimensional objects.
A5.1.3 Identify and create examples of line symmetry; compare and describe given circles, triangles, and rectangles as larger, smaller, or congruent. A5.2.3 Identify and model geometric figures that are congruent, similar, and/or symmetrical. A5.3.3 Apply the properties of equality and proportionality to solve problems involving congruent or similar shapes. A5.4.3 Identify congruent and similar figures using Euclidean and coordinate geometries; apply this information to solve problems.
A5.1.4 Demonstrate conservation of area using drawings or manipulatives. A5.2.4 Distinguish between area and perimeter; find both using a variety of methods including rulers, grid paper and tiles. A5.3.4 Estimate and determine volume and surface areas of solid figures using manipulatives and formulas; estimate and find circumferences and areas of circles.  

 

 

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A5.1.5 Describe and identify geometric transformations including slides, flips, and turns. A5.2.5 Identify and model transformations of geometric figures, describing the motions as slides, flips, or rotations. A5.3.5 Draw and describe the results of transformations including translations (slides), rotations (turns), reflections (flips), and dilations (shrinking or enlarging). A5.4.5 Use transformations to demonstrate geometric properties.
A5.1.6 Use comparative directional and positional words: above, below, inside, outside, on, in, right and left, horizontal, vertical, and middle. A5.2.6 Locate and describe objects in terms of their position with & without compass directions; identify coordinates for a given point or locate points of given coordinates on a grid. A5.3.6 Use coordinate geometry to represent and interpret relationships defined by equations and formulas including distance and midpoint. A5.4.6 Use coordinate geometry to graph linear equations, determine slopes of lines, identify parallel and perpendicular lines, and to find possible solutions to sets of equations.
A5.1.7 Draw and build familiar shapes. A5.2.7 Sketch and identify line segments, midpoint, intersections, parallel, and perpendicular lines. A5.3.7 Draw, measure and construct geometric figures including perpendicular bisectors, polygons with given dimensions and angles, circles with given dimensions, perpendicular and parallel lines. A5.4.7 Construct geometric models, transformations, and scale drawings using a variety of methods including paper folding, compass, straight edge, protractor, and technology.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD A6: STATISTICS/PROBABILITY

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A6.1.1 Collect, record, organize, display, and explain the classification of data. A6.2.1 Collect, organize and display data creating a variety of visual displays including tables, charts, and line graphs. A6.3.1 Collect, analyze and display data in a variety of visual displays including frequency distributions, circle graphs, box and whisker plots, stem and leaf plots, histograms, and scatter plots with and without technology. A6.4.1 Analyze and draw inferences from a wide variety of data sources that summarize data; constructing graphical displays with and without technology.
A6.1.2 Describe data from a variety of visual displays including tallies, tables, pictographs, bar graphs, and Venn diagrams. A6.2.2 Present the data using a variety of appropriate representations and explain the meaning of the data. A6.3.2 Interpret and analyze information found in newspapers, magazines, and graphical displays. A6.4.2 Determine the line of best fit and use it to predict unknown data values.
A6.1.3 Use the terms maximum and minimum when working with a data set. A6.2.3 Describe and interpret a data set using mean, median, mode, and range. A6.3.3 Determine and justify a choice of mean, median, or mode as the best representation of data for a practical situation. A6.4.3 Describe data, selecting measures of central tendencies and distribution, to convey information in the data.
    A6.3.4 Make projections based on available data and evaluate whether or not inferences can be made given the parameters of the data. A6.4.4 Analyze the validity of statistical conclusions and the use, misuse, and abuse of data caused by a wide variety of factors including choices of scale, inappropriate choices of measures of center, incorrect curve fitting, and inappropriate uses of controls or sample groups.

 

 

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
A6.1.5 Find and record the possibilities of simple probability experiments; explain differences between chance and certainty, giving examples. A6.2.5 Estimate whether a game is mathematically fair or unfair; analyze and present probability data using simple fractions. A6.3.5 Use tree diagrams and sample spaces to make predictions about independent events. A6.4.5 Analyze data from multiple events and predict theoretical probability; find and compare experimental and theoretical probability for a simple situation, discussing possible differences between two results.
A6.1.6

Conduct a survey and tally the results.

 A6.2.6 Conduct simple probability experiments using concrete materials and represent the results using fractions and probability. A6.3.6 Design and conduct a simulation to study a problem and communicate the results. A6.4.6 Design, conduct, analyze, and communicate the results of multi-stage probability experiments.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD B: PROBLEM-SOLVING

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
B1.1.1 Formulate problems from practical and mathematical activities. B1.2.1 Read and summarize a problem, using mathematical terms and symbols. B1.3.1 Analyze and summarize a problem using the relationships between the known facts and unknown information. B1.4.1 Recognize and formulate mathematical problems from within and outside the field of mathematics.
B1.1.2 Develop and apply strategies including guess and check, modeling and acting out, drawings, and extending patterns to solve a variety of problems. B1.2.2 Select and apply a variety of strategies including making a table, chart or list, drawing pictures, making a model, and comparing with previous experience to solve problems. B1.3.2 Select, modify and apply a variety of problem-solving strategies including graphing, inductive and deductive reasoning, Venn diagrams, and spreadsheets. B1.4.2 Apply multi-step, integrated, mathematical problem-solving strategies, persisting until a solution is found or it is clear no solution exits.
B1.1.3 Predict an answer before solving a problem and compare results to check for reasonableness. B1.2.3 Explain and verify results of the original problem and apply what was learned to new situations. B1.3.3 Evaluate, interpret, and justify solutions to problems. B1.4.3 Verify the answer by using an alternative strategy.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD C: COMMUNICATION

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
C1.1.1 Translate problems from everyday language into math language and symbols. C1.2.1 Use the mathematical vocabulary appropriate to the problem. C1.3.4 Use math vocabulary, symbols and notation to represent information in the problem. C1.4.1 Use appropriate technology to represent the information and ideas in a problem.
C1.1.2 Use manipulatives, models, pictures, and language to represent and communicate mathematical ideas. C1.2.2 Represent mathematical and practical situations using concrete, pictorial, and symbolic representation. C1.3.2 Represent a problem numerically, graphically, symbolically, and translate among these alternative representations. C1.4.2 Use numerical, graphic, and symbolic representations to support oral and written communication about math ideas.
C1.1.3 Use everyday language to explain thinking about problem solving strategies and solutions to problems. C1.2.3 Organize and communicate mathematical problem solving strategies and solutions to problems. C1.3.3 Use appropriate vocabulary, symbols and technology to explain, justify, and defend mathematical solutions. C1.4.3 Explain, justify and defend mathematical ideas, solutions, and methods to various audiences.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD D: REASONING

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
D1.1.1 Draw conclusions about mathematical problems. D1.2.1 Draw logical conclusions about mathematical situations. D1.3.1 Use informal deductive and inductive reasoning in both concrete and abstract contexts. D1.4.1 Follow and evaluate an argument, judging its validity using inductive or deductive reasoning and logic.
D1.1.2 Find examples that support or refute mathematical statements. D1.2.2 Given a rule or generalization, determine whether the example fits. D1.3.2 State counterexamples to disprove statements. D1.4.2 Make and test conjectures.
D1.1.3 Explain why a prediction, estimation, or solution is reasonable. D1.2.3 Justify answers and mathematical strategies as reasonable. D1.3.3 Justify and defend the validity of mathematical strategies and solutions using examples and counterexamples. D1.4.3 Use methods of proofs including direct, indirect, and counterexamples, to validate conjectures.

Return to Standards
Return to top of page.
Return to Main Page.

MATHEMATICS PERFORMANCE STANDARDS PROFICIENT LEVEL DESCRIPTORS

CONTENT STANDARD E: CONNECTIONS

Between ages 5-7, students: Between ages 8-10, students know and are able to do everything required at earlier ages and: Between ages 11-14, students know and are able to do everything required at earlier ages and: Between ages 15-18, students know and are able to do everything required at earlier ages and:
E1.2.3 Apply mathematical skills and processes to literature. E1.2.3 Apply mathematical processes to social studies. E1.3.3 Apply mathematical skills and processes to science and humanities. E1.4.3 Apply mathematical skills and processes to global issues.
E1.2.4 Apply mathematical skills and processes to situations with self and family. E1.2.4 Apply mathematical skills and processes to situations with friends and school. E1.3.4 Apply mathematical skills and processes to situations with peers and community. E1.4.4 Describe how mathematics can be used in knowing how to prepare for careers.

Return to Standards
Return to top of page.
Return to Main Page.