Appendix II: The Grade Levels Compared: Mathematics
 The elementary school standards are set at a level of performance approximately equivalent to the end of fourth grade. The middle school standards are set at a level of performance approximately equivalent to the end of eighth grade. The high school standards are set at a level of performance approximately equivalent to the end of tenth grade or the end of the common core. It is expected that some students might achieve these levels earlier and others later than these grades.

 Arithmetic and Number Concepts/Number and Operation Concepts

### Elementary School

The student produces evidence that demonstrates understanding of arithmetic and number concepts; that is, the student:

a Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is:
• adds, i.e., joins things together, increases;
• subtracts, i.e., takes away, compares, finds the difference;
• multiplies, i.e., uses repeated addition, counts by multiples, combines things that come in groups, makes arrays, uses area models, computes simple scales, uses simple rates;
• divides, i.e., puts things into groups, shares equally; calculates simple rates;
• analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide;
• solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another;
• computes answers mentally, e.g., 27 + 45, 30 x 4;
• uses simple concepts of negative numbers, e.g., on a number line, in counting, in temperature, “owing.”
b Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is:
• counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000;
• uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.
c Estimates, approximates, rounds off, uses landmark numbers, or uses exact numbers, as appropriate, in calculations.
d Describes and compares quantities by using concrete and real world models of simple fractions; that is:
• finds simple parts of wholes;
• recognizes simple fractions as instructions to divide, e.g., ¼ of something is the same as dividing something by 4;
• recognizes the place of fractions on number lines, e.g., in measurement;
• uses drawings, diagrams, or models to show what the numerator and denominator mean, including when adding like fractions, e.g., 1/8 + 5/8, or when showing that 3/4 is more than 3/8;
• uses beginning proportional reasoning and simple ratios, e.g., “about half of the people.”
e Describes and compares quantities by using simple decimals; that is:
• adds, subtracts, multiplies, and divides money amounts;
• recognizes relationships among simple fractions, decimals, and percents, i.e., that is the same as 0.5, and is the same as 50%, with concrete materials, diagrams, and in real world situations, e.g., when discovering the chance of a coin landing on heads or tails.
f Describes and compares quantities by using whole numbers up to 10,000; that is:
• connects ideas of quantities to the real world, e.g., how many people fit in the school’s cafeteria; how far away is a kilometer;
• finds, identifies, and sorts numbers by their properties, e.g., odd, even, multiple, square.

### Middle School

The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:

a Consistently and accurately adds, subtracts, multiplies, and divides rational numbers using appropriate methods (e.g., the student can add + mentally or on paper but may opt to add + on a calculator) and raises rational numbers to whole number powers. (Students should have facility with the different kinds and forms of rational numbers, i.e., integers, both whole numbers and negative integers; and other positive and negative rationals, written as decimals, as percents, or as proper, improper, or mixed fractions. Irrational numbers, i.e., those that cannot be written as a ratio of two integers, are not required content but are suitable for introduction, especially since the student should be familiar with the irrational number .)
b Uses and understands the inverse relationships between addition and subtraction, multiplication and division, and exponentiation and root-extraction (e.g., squares and square roots, cubes and cube roots); uses the inverse operation to determine unknown quantities in equations.
c Consistently and accurately applies and converts the different kinds and forms of rational numbers.
d Is familiar with characteristics of numbers (e.g., divisibility, prime factorization) and with properties of operations (e.g., commutativity and associativity), short of formal statements.
e Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.
f Uses ratios and rates to express “part-to-part” and “whole-to-whole” relationships, and reasons proportionally to solve problems involving equivalent fractions, equal ratios, or constant rates, recognizing the multiplicative nature of these problems in the constant factor of change.
g Orders numbers with the > and < relationships and by location on a number line; estimates and compares rational numbers using sense of the magnitudes and relative magnitudes of numbers and of base-ten place values (e.g., recognizes relationships to “benchmark” numbers 1/2 and 1 to conclude that the sum 1/2 + 5/6 must be between 1 and 1 1/2 (likewise, 13/24 + ¼)).

### High School

The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:

a Uses addition, subtraction, multiplication, division, exponentiation, and root-extraction in forming and working with numerical and algebraic expressions.
b Understands and uses operations such as opposite, reciprocal, raising to a power, taking a root, and taking a logarithm.
c Has facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications.
d Understands and uses number systems: natural, integer, rational, and real.
e Represents numbers in decimal or fraction form and in scientific notation, and graphs numbers on the number line and number pairs in the coordinate plane.
f Compares numbers using order relations, differences, ratios, proportions, percents, and proportional change.
g Carries out proportional reasoning in cases involving part-whole relationships and in cases involving expansions and contractions.
h Understands dimensionless numbers, such as proportions, percents, and multiplicative factors, as well as numbers with specific units of measure, such as numbers with length, time, and rate units.
i Carries out counting procedures such as those involving sets (unions and intersections) and arrangements (permutations and combinations).
j Uses concepts such as prime, relatively prime, factor, divisor, multiple, and divisibility in solving problems involving integers.
k Uses a scientific calculator effectively and efficiently in carrying out complex calculations.
l Recognizes and represents basic number patterns, such as patterns involving multiples, squares, or cubes.

 Geometry and Measurement Concepts

### Elementary School

The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:

a Gives and responds to directions about location, e.g., by using words such as “in front of,” “right,” and “above.”
b Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes.
c Uses simple two dimensional coordinate systems to find locations on a map and to represent points and simple figures.
d Uses many types of figures (angles, triangles, squares, rectangles, rhombi, parallelograms, quadrilaterals, polygons, prisms, pyramids, cubes, circles, and spheres) and identifies the figures by their properties, e.g., symmetry, number of faces, two- or three-dimensionality, no right angles.
e Solves problems by showing relationships between and among figures, e.g., using congruence and similarity, and using transformations including flips, slides, and rotations.
f Extends and creates geometric patterns using concrete and pictorial models.
g Uses basic ways of estimating and measuring the size of figures and objects in the real world, including length, width, perimeter, and area.
h Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations.
i Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time.
j Carries out simple unit conversions, such as between cm and m, and between hours and minutes.
k Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.

### Middle School

The student produces evidence that demonstrates understanding of geometry and measurement concepts in the following areas; that is, the student:

a Is familiar with assorted two- and three-dimensional objects, including squares, triangles, other polygons, circles, cubes, rectangular prisms, pyramids, spheres, and cylinders.
b Identifies similar and congruent shapes and uses transformations in the coordinate plane, i.e., translations, rotations, and reflections.
c Identifies three dimensional shapes from two dimensional perspectives; draws two dimensional sketches of three dimensional objects that preserve significant features.
d Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.
e Recognizes similarity and rotational and bilateral symmetry in two- and three-dimensional figures.
f Analyzes and generalizes geometric patterns, such as tessellations and sequences of shapes.
g Measures angles, weights, capacities, times, and temperatures using appropriate units.
h Chooses appropriate units of measure and converts with ease between like units, e.g., inches and miles, within a customary or metric system. (Conversions between customary and metric are not required.)
i Reasons proportionally in situations with similar figures.
j Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings.
k Models situations geometrically to formulate and solve problems.

### High School

The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:

a Models situations geometrically to formulate and solve problems.
b Works with two- and three-dimensional figures and their properties, including polygons and circles, cubes and pyramids, and cylinders, cones, and spheres.
c Uses congruence and similarity in describing relationships between figures.
d Visualizes objects, paths, and regions in space, including intersections and cross sections of three dimensional figures, and describes these using geometric language.
e Knows, uses, and derives formulas for perimeter, circumference, area, surface area, and volume of many types of figures.
f Uses the Pythagorean Theorem in many types of situations, and works through more than one proof of this theorem.
g Works with similar triangles, and extends the ideas to include simple uses of the three basic trigonometric functions.
h Analyzes figures in terms of their symmetries using, for example, concepts of reflection, rotation, and translation.
i Compares slope (rise over run) and angle of elevation as measures of steepness.
j Investigates geometric patterns, including sequences of growing shapes.
k Works with geometric measures of length, area, volume, and angle; and non-geometric measures such as weight and time.
l Uses quotient measures, such as speed and density, that give “per unit” amounts; and uses product measures, such as person-hours.
m Understands the structure of standard measurement systems, both SI and customary, including unit conversions and dimensional analysis.
n Solves problems involving scale, such as in maps and diagrams.
o Represents geometric curves and graphs of functions in standard coordinate systems.
p Analyzes geometric figures and proves simple things about them using deductive methods.
q Explores geometry using computer programs such as CAD software, Sketchpad programs, or LOGO.

 Function and Algebra Concepts

### Elementary School

The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

a Uses linear patterns to solve problems; that is:
• shows how one quantity determines another in a linear (“repeating”) pattern, i.e., describes, extends, and recognizes the linear pattern by its rule, such as, the total number of legs on a given number of horses can be calculated by counting by fours;
• shows how one quantity determines another quantity in a functional relationship based on a linear pattern, e.g., for the “number of people and total number of eyes,” figure out how many eyes 100 people have all together.
b Builds iterations of simple non-linear patterns, including multiplicative and squaring patterns (e.g., “growing” patterns) with concrete materials, and recognizes that these patterns are not linear.
c Uses the understanding that an equality relationship between two quantities remains the same as long as the same change is made to both quantities.
d Uses letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations with concrete materials, i.e., demonstrates understanding and use of a beginning concept of a variable.

### Middle School

The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

a Discovers, describes, and generalizes patterns, including linear, exponential, and simple quadratic relationships, i.e., those of the form f(n)=n² or f(n)=cn², for constant c, including A=, and represents them with variables and expressions.
b Represents relationships with tables, graphs in the coordinate plane, and verbal or symbolic rules.
c Analyzes tables, graphs, and rules to determine functional relationships.
d Finds solutions for unknown quantities in linear equations and in simple equations and inequalities.

### High School

The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

a Models given situations with formulas and functions, and interprets given formulas and functions in terms of situations.
b Describes, generalizes, and uses basic types of functions: linear, exponential, power, rational, square and square root, and cube and cube root.
c Utilizes the concepts of slope, evaluation, and inverse in working with functions.
d Works with rates of many kinds, expressed numerically, symbolically, and graphically.
e Represents constant rates as the slope of a straight line graph, and interprets slope as the amount of one quantity (y) per unit amount of another (x).
f Understands and uses linear functions as a mathematical representation of proportional relationships.
g Uses arithmetic sequences and geometric sequences and their sums, and sees these as the discrete forms of linear and exponential functions, respectively.
h Defines, uses, and manipulates expressions involving variables, parameters, constants, and unknowns in work with formulas, functions, equations, and inequalities.
i Represents functional relationships in formulas, tables, and graphs, and translates between pairs of these.
j Solves equations symbolically, graphically, and numerically, especially linear, quadratic, and exponential equations; and knows how to use the quadratic formula for solving quadratic equations.
k Makes predictions by interpolating or extrapolating from given data or a given graph.
l Understands the basic algebraic structure of number systems.
m Uses equations to represent curves such as lines, circles, and parabolas.
n Uses technology such as graphics calculators to represent and analyze functions and their graphs.
o Uses functions to analyze patterns and represent their structure.

 Statistics and Probability Concepts

### Elementary School

The student produces evidence that demonstrates understanding of statistics and probability concepts in the following areas; that is, the student:
a Collects and organizes data to answer a question or test a hypothesis by comparing sets of data.
b Displays data in line plots, graphs, tables, and charts.
c Makes statements and draws simple conclusions based on data; that is:
• reads data in line plots, graphs, tables, and charts;
• compares data in order to make true statements, e.g., “seven plants grew at least 5 cm”;
• identifies and uses the mode necessary for making true statements, e.g., “more people chose red”;
• makes true statements based on a simple concept of average (median and mean), for a small sample size and where the situation is made evident with concrete materials or clear representations;
• interprets data to determine the reasonableness of statements about the data, e.g., “twice as often,” “three times faster”;
• uses data, including statements about the data, to make a simple concluding statement about a situation, e.g., “This kind of plant grows better near sunlight because the seven plants that were near the window grew at least 5 cm.”
d Gathers data about an entire group or by sampling group members to understand the concept of sample, i.e., that a large sample leads to more reliable information, e.g., when flipping coins.
e Predicts results, analyzes data, and finds out why some results are more likely, less likely, or equally likely.
f Finds all possible combinations and arrangements within certain constraints involving a limited number of variables.

### Middle School

The student produces evidence that demonstrates understanding of statistics and probability concepts; that is, the student:
a Collects data, organizes data, and displays data with tables, charts, and graphs that are appropriate, i.e., consistent with the nature of the data.
b Analyzes data with respect to characteristics of frequency and distribution, including mode and range.
c Analyzes appropriately central tendencies of data by considering mean and median.
d Makes conclusions and recommendations based on data analysis.
e Critiques the conclusions and recommendations of others’ statistics.
f Considers the effects of missing or incorrect information.
g Formulates hypotheses to answer a question and uses data to test hypotheses.
h Represents and determines probability as a fraction of a set of equally likely outcomes; recognizes equally likely outcomes, and constructs sample spaces (including those described by numerical combinations and permutations).
i Makes predictions based on experimental or theoretical probabilities.
j Predicts the result of a series of trials once the probability for one trial is known.

### High School

The student demonstrates understanding of statistics and probability concepts; that is, the student:
a Organizes, analyzes, and displays single-variable data, choosing appropriate frequency distributions, circle graphs, line plots, histograms, and summary statistics.
b Organizes, analyzes, and displays two-variable data using scatter plots, estimated regression lines, and computer generated regression lines and correlation coefficients.
c Uses sampling techniques to draw inferences about large populations.
d Understands that making an inference about a population from a sample always involves uncertainty and that the role of statistics is to estimate the size of that uncertainty.
e Formulates hypotheses to answer a question and uses data to test hypotheses.
f Interprets representations of data, compares distributions of data, and critiques conclusions and the use of statistics, both in school materials and in public documents.
g Explores questions of experimental design, use of control groups, and reliability.
h Creates and uses models of probabilistic situations and understands the role of assumptions in this process.
i Uses concepts such as equally likely, sample space, outcome, and event in analyzing situations involving chance.
j Constructs appropriate sample spaces, and applies the addition and multiplication principles for probabilities.
k Uses the concept of a probability distribution to discuss whether an event is rare or reasonably likely.
l Chooses an appropriate probability model and uses it to arrive at a theoretical probability for a chance event.
m Uses relative frequencies based on empirical data to arrive at an experimental probability for a chance event.
n Designs simulations including Monte Carlo simulations to estimate probabilities.
o Works with the normal distribution in some of its basic applications.

 Problem Solving and Mathematical Reasoning

### Elementary School

The student demonstrates logical reasoning throughout work in mathematics, i.e., concepts and skills, problem solving, and projects; demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow; and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion.

Formulation
a Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps;
• uses previously learned strategies, skills, knowledge, and concepts to make decisions;
• uses strategies, such as using manipulatives or drawing sketches, to model problems.

Implementation
b The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• makes up and uses a variety of strategies and approaches to solving problems and uses or learns approaches that other people use, as appropriate;
• makes connections among concepts in order to solve problems;
• solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution.

Conclusion
c The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student:
• explains a pattern that can be used in similar situations;
• explains how the problem is similar to other problems he or she has solved;
• explains how the mathematics used in the problem is like other concepts in mathematics;
• explains how the problem solution can be applied to other school subjects and in real world situations;
• makes the solution into a general rule that applies to other circumstances.

### Middle School

The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion.

Formulation
a The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student:
• formulates and solves a variety of meaningful problems;
• extracts pertinent information from situations and figures out what additional information is needed.

Implementation
b The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• uses and invents a variety of approaches and understands and evaluates those of others;
• invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table;
• determines, where helpful, how to break a problem into simpler parts;
• solves for unknown or undecided quantities using algebra, graphing, sound reasoning, and other strategies;
• integrates concepts and techniques from different areas of mathematics;
• works effectively in teams when the nature of the task or the allotted time makes this an appropriate strategy.

Conclusion
c The student provides closure to the solution process through summary statements and general conclusions; that is, the student:
• verifies and interprets results with respect to the original problem situation;
• generalizes solutions and strategies to new problem situations.

Mathematical reasoning
d The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student:
• formulates conjectures and argues why they must be or seem true;
• makes sensible, reasonable estimates;
• makes justified, logical statements.

### High School

The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion.

Formulation
a The student participates in the formulation of problems; that is, given the statement of a problem situation, the student:
• fills out the formulation of a definite problem that is to be solved;
• extracts pertinent information from the situation as a basis for working on the problem;
• asks and answers a series of appropriate questions in pursuit of a solution and does so with minimal “scaffolding” in the form of detailed guiding questions.

Implementation
b The student makes the basic choices involved in planning and carrying out a solution; that is, the student:
• chooses and employs effective problem solving strategies in dealing with non-routine and multi-step problems;
• selects appropriate mathematical concepts and techniques from different areas of mathematics and applies them to the solution of the problem;
• applies mathematical concepts to new situations within mathematics and uses mathematics to model real world situations involving basic applications of mathematics in the physical and biological sciences, the social sciences, and business.

Conclusion
c The student provides closure to the solution process through summary statements and general conclusions; that is, the student:
• concludes a solution process with a useful summary of results;
• evaluates the degree to which the results obtained represent a good response to the initial problem;
• formulates generalizations of the results obtained;
• carries out extensions of the given problem to related problems.

Mathematical reasoning
d The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student:
• employs forms of mathematical reasoning and proof appropriate to the solution of the problem at hand, including deductive and inductive reasoning, making and testing conjectures, and using counterexamples and indirect proof;
• differentiates clearly between giving examples that support a conjecture and giving proof of the conjecture.

 Mathematical Skills and Tools

### Elementary School

The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task, perhaps referring to notes, books, or other students, perhaps working to reconstruct a method; that is, the student:

a Adds, subtracts, multiplies, and divides whole numbers correctly; that is:
• knows single digit addition, subtraction, multiplication, and division facts;
• adds and subtracts numbers with several digits;
• multiplies and divides numbers with one or two digits;
• multiplies and divides three digit numbers by one digit numbers.
b Estimates numerically and spatially.
c Measures length, area, perimeter, circumference, diameter, height, weight, and volume accurately in both the customary and metric systems.
d Computes time (in hours and minutes) and money (in dollars and cents).
e Refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle, square, rectangle, side, edge, face, cube, point, line, perimeter, area, and circle; and refers with assistance to rhombus, parallelogram, quadrilateral, polygon, polyhedron, angle, vertex, volume, diameter, circumference, sphere, prism, and pyramid.
f Uses +, -, x, ÷, /,, \$, ¢, %, and . (decimal point) correctly in number sentences and expressions.
g Reads, creates, and represents data on line plots, charts, tables, diagrams, bar graphs, simple circle graphs, and coordinate graphs.
h Uses recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, and advice from peers, as appropriate, to achieve solutions; that is, uses measuring devices, graded appropriately for given situations, such as rulers (customary to the 1/8 inch; metric to the millimeter), graph paper (customary to the inch or half-inch; metric to the centimeter), measuring cups (customary to the ounce; metric to the milliliter), and scales (customary to the pound or ounce; metric to the kilogram or gram).

### Middle School

The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task (perhaps referring to notes, or books, perhaps working to reconstruct a method); that is, the student:

a Computes accurately with arithmetic operations on rational numbers.
b Knows and uses the correct order of operations for arithmetic computations.
c Estimates numerically and spatially.
d Measures length, area, volume, weight, time, and temperature accurately.
e Refers to geometric shapes and terms correctly.
f Uses equations, formulas, and simple algebraic notation appropriately.
g Reads and organizes data on charts and graphs, including scatter plots, bar, line, and circle graphs, and Venn diagrams; calculates mean and median.
h Uses recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, and advice from peers, as appropriate, to achieve solutions.

### High School

The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task, perhaps referring to notes, or books, perhaps working to reconstruct a method; that is, the student:

a Carries out numerical calculations and symbol manipulations effectively, using mental computations, pencil and paper, or other technological aids, as appropriate.
b Uses a variety of methods to estimate the values, in appropriate units, of quantities met in applications, and rounds numbers used in applications to an appropriate degree of accuracy.
c Evaluates and analyzes formulas and functions of many kinds, using both pencil and paper and more advanced technology.
d Uses basic geometric terminology accurately, and deduces information about basic geometric figures in solving problems.
e Makes and uses rough sketches, schematic diagrams, or precise scale diagrams to enhance a solution.
f Uses the number line and Cartesian coordinates in the plane and in space.
g Creates and interprets graphs of many kinds, such as function graphs, circle graphs, scatter plots, regression lines, and histograms.
h Sets up and solves equations symbolically (when possible) and graphically.
i Knows how to use algorithms in mathematics, such as the Euclidean Algorithm.
j Uses technology to create graphs or spreadsheets that contribute to the understanding of a problem.
k Writes a simple computer program to carry out a computation or simulation to be repeated many times.
l Uses tools such as rulers, tapes, compasses, and protractors in solving problems.
m Knows standard methods to solve basic problems and uses these methods in approaching more complex problems.

 Mathematical Communication

### Elementary School

The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:

a Uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.
b Shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.
c Explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.
d Considers purpose and audience when communicating about mathematics.
e Comprehends mathematics from reading assignments and from other sources.

### Middle School

The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:

a Uses mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.
b Organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.
c Uses mathematical language to make complex situations easier to understand.
d Exhibits developing reasoning abilities by justifying statements and defending work.
e Shows understanding of concepts by explaining ideas not only to teachers and assessors but to fellow students or younger children.
f Comprehends mathematics from reading assignments and from other sources.

### High School

The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:

a Is familiar with basic mathematical terminology, standard notation and use of symbols, common conventions for graphing, and general features of effective mathematical communication styles.
b Uses mathematical representations with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
c Organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.
d Communicates logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
e Presents mathematical ideas effectively both orally and in writing.
f Explains mathematical concepts clearly enough to be of assistance to those who may be having difficulty with them.
g Writes narrative accounts of the history and process of work on a mathematical problem or extended project.
h Writes succinct accounts of the mathematical results obtained in a mathematical problem or extended project, with diagrams, graphs, tables, and formulas integrated into the text.
i Keeps narrative accounts of process separate from succinct accounts of results, and realizes that doing so can enhance the effectiveness of each.

 Putting Mathematics to Work

### Elementary School

The student conducts at least one large scale project each year, beginning in fourth grade, drawn from the following kinds and, over the course of elementary school, conducts projects drawn from at least two of the kinds.
A single project may draw on more than one kind.

a Data study, in which the student:
• develops a question and a hypothesis in a situation where data could help make a decision or recommendation;
• decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers;
• collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions;
• writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.
b Science study, in which the student:
• decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement;
• develops a prediction (a hypothesis) and develops procedures to test the hypothesis;
• collects and records data, represents and displays data, and compares results with predictions;
• writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers.
c Design of a physical structure, in which the student:
• decides on a structure to design, the size and budget constraints, and the scale of design;
• makes a first draft of the design, and revises and improves the design in response to input from peers and teachers;
• makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers.
d Management and planning, in which the student:
• decides on what to manage or plan, and the criteria to be used to see if the plan worked;
• identifies unexpected events that could disrupt the plan and further plans for such contingencies;
• identifies resources needed, e.g., materials, money, time, space, and other people;
• writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers;
• carries out the plan (optional);
• writes a report on the plan that includes resources, budget, and schedule, and acknowledges assistance received from parents, peers, and teachers.
• writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.
e Pure mathematics investigation, in which the student:
• decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns;
• describes a question or concept to investigate;
• decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models;
• carries out the investigation;
• writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.

### Middle School

The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds.
A single investigation or project may draw on more than one kind.

a Data study based on civic, economic, or social issues, in which the student:
• selects an issue to investigate;
• makes a hypothesis on an expected finding, if appropriate;
• gathers data;
• analyzes the data using concepts from Standard 4, e.g., considering mean and median, and the frequency and distribution of the data;
• shows how the study’s results compare with the hypothesis;
• uses pertinent statistics to summarize;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.
b Mathematical model of physical phenomena, often used in science studies, in which the student:
• carries out a study of a physical system using a mathematical representation of the structure;
• uses understanding from Standard 3, particularly with respect to the determination of the function governing behavior in the model;
• generalizes about the structure with a rule, i.e., a function, that clearly applies to the phenomenon and goes beyond statistical analysis of a pattern of numbers generated by the situation;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.
c Design of a physical structure, in which the student:
• generates a plan to build something of value, not necessarily monetary value;
• uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes;
• summarizes the important features of the structure;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.
d Management and planning, in which the student:
• determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling;
• notes any constraints that will affect the plan;
• determines a plan;
• uses concepts from any of Standards 1 to 4, depending on the nature of the project;
• considers the possibility of a more efficient solution;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.
e Pure mathematics investigation, in which the student:
• extends or “plays with,” as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers;
• uses concepts from any of Standards 1 to 4, e.g., an investigation of Pascal’s triangle would have roots in Standard 1 but could tie in concepts from geometry, algebra, and probability; investigations of derivations of geometric formulas would be rooted in Standard 2 but could require algebra;
• determines and expresses generalizations from patterns;
• makes conjectures on apparent properties and argues, short of formal proof, why they seem true;
• prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

### High School

The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of high school, conducts investigations or projects drawn from at least three of the kinds.
A single investigation or project may draw on more than one kind.

a Data study, in which the student:
• carries out a study of data relevant to current civic, economic, scientific, health, or social issues;
• uses methods of statistical inference to generalize from the data;
• prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.
b Mathematical model of a physical system or phenomenon, in which the student:
• carries out a study of a physical system or phenomenon by constructing a mathematical model based on functions to make generalizations about the structure of the system;
• uses structural analysis (a direct analysis of the structure of the system) rather than numerical or statistical analysis (an analysis of data about the system);
• prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.
c Design of a physical structure, in which the student:
• creates a design for a physical structure;
• uses general mathematical ideas and techniques to discuss specifications for building the structure;
• prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.
d Management and planning analysis, in which the student:
• carries out a study of a business or public policy situation involving issues such as optimization, cost-benefit projections, and risks;
• uses decision rules and strategies both to analyze options and balance trade-offs; and brings in mathematical ideas that serve to generalize the analysis across different conditions;
• prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.
e Pure mathematics investigation, in which the student:
• carries out a mathematical investigation of a phenomenon or concept in pure mathematics;
• uses methods of mathematical reasoning and justification to make generalizations about the phenomenon;
• prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.
f History of a mathematical idea, in which the student:
• carries out a historical study tracing the development of a mathematical concept and the people who contributed to it;
• includes a discussion of the actual mathematical content and its place in the curriculum of the present day;
• prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.