Number and Operation Concepts

The student demonstrates understanding of a mathematical concept by using it to solve problems, by representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and by explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

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.

Examples of activities through which students might demonstrate understanding of number and operation concepts include:

 Show how to enlarge a picture by a factor of 2 using repeated enlargements at a fixed setting on a photocopy machine that can only enlarge up to 155%. Do the same for enlargements by a factor of 3, 4, and 5. 1a, 1c, 1g, 1h Discuss the relationship between the “Order of Operations” conventions of arithmetic and the order in which numbers and operation symbols are entered in a calculator. Do all calculators use the same order? 1a, 1c, 1k Give a reasoned estimate of the volume of gasoline your car uses in a year. How does this compare to the volume of liquid you drink in a year? (Balanced Assessment) 1a, 1c, 2k Show that there must have been at least one misprint in a newspaper report on an election that says: Yes votes - 13,657 (42%); No votes - 186,491 (58%). Suggest two different specific places a misprint might have occurred. (Balanced Assessment) 1a, 1f, 1g, 1h Make and prove a conjecture about the sum of any sequence of consecutive odd numbers beginning with the number 1. 1a, 1l It is sometimes convenient to represent physical phenomena using logarithmic scales. Discuss why this is so, and illustrate with a description of pH scales (acidity), decibel scales (sound intensity), and Richter scales (earthquake intensity). 1b, 1c, 1d, 1e What proportion of two digit numbers contain the digit 7? What about three digit numbers? 1d, 1e, 1i Figure out how many pages it would take to write out all the numbers from 1 to 1,000,000. (Balanced Assessment) 1d, 1e, 1l If 10% of U.S. citizens have a certain trait, and four out of five people with the trait are men, what proportion of men have the trait and what proportion of women have the trait? Explain whether the answer depends on the proportion of U.S. citizens who are women and, if so, how. (Balanced Assessment) 1f, 1g, 1h Simpson’s Paradox is this: X may have a better record than Y in each of two possible categories but Y’s overall record for the combined categories may be better than X’s. Explain how this can happen. 1g Find a simple relationship between the least common multiple of two numbers, the greatest common divisor of the two numbers, and the product of the two numbers. Prove that the relationship is true for all pairs of positive integers. 1j

 Geometry and Measurement Concepts High School

The student demonstrates understanding of a mathematical concept by using it to solve problems, by representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and by explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

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.

Examples of activities through which students might demonstrate understanding of geometry and measurement concepts include:
 A model tower is made of small cubes of the same size. There are four types of cubes used in the tower: vertex, edge, face, and interior, having respectively 3, 2, 1, and 0 faces exposed. If a new tower, of the same shape but three times as tall, is to be built using the same sort of cubes, show how the numbers of each of the four types of cubes need to be increased. Generalize to a tower n times as tall as the original. 2a, 2b, 2c, 2d, 2j, 2n Figure out which of two ways of rolling an 8.5" by 11" piece of paper into a cylinder gives the greater volume. Is there a way to get even greater volume using a sheet of paper with the same area but different shape? (Balanced Assessment) 2a, 2b, 2d, 2e Explain which is a better fit, a round peg in a square hole or a square peg in a round hole. Go on to the case of a cube in a sphere vs. a sphere in a cube. (Balanced Assessment) 2a, 2b, 2e, 2f Suppose that you are on a cliff looking out to sea on a clear day. Show that the distance to the horizon in miles is about equal to 1.2, where h is the height in feet of the cliff above sea level. Derive a similar expression in terms of meters and kilometers. (Balanced Assessment) 2a, 2d, 2f Can a cube be dissected into four or fewer congruent square-base pyramids? What about triangle-base pyramids? In each case, show how it can be done or why it cannot be done. 2a, 2b, 2d, 2p Given three cities on a map, find a place that is the same distance from all of them. Determine if there is always such a place. Are there ever many such places? (Balanced Assessment) 2a, 2b, 2d, 2p A circular glass table top has broken, and all you have is one piece. The piece contains a section of the circular edge, but not the center. Describe and apply two different methods for finding the radius of the original top (so that you can order a new top). (Balanced Assessment) 2a, 2b, 2p An isoscles trapezoid has height h and bases of lengths b and c. What must be the relationship among the lengths h, b, and c if we are to be able to inscribe a circle in the trapezoid? 2a, 2b, 2p Explore the relation between the length of a person’s shadow (made by a streetlight) and the person’s height and distance from the light. Extend the analysis to include the rate of change of shadow length when the person is moving. (Balanced Assessment) 2a, 2g, 2l

 Function and Algebra Concepts High School

The student demonstrates understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

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.

Examples of activities through which students might demonstrate understanding of function and algebra concepts include:
 A used car is bought for \$9,500. If the car depreciates at 5% per year, how much will the car be worth after one year? Five years? Twelve years? n years? (College Preparatory Mathematics) 3a, 3b, 3c Express the diameter of a circle as a function of its area and sketch a graph of this function. 3a, 3b, 3c, 3h If a half gallon carton of milk is left out on the counter, its temperature T in degrees Fahrenheit can be approximated by the formula T = 70 - (), where t is the time in minutes it has been out of the refrigerator. (This formula works as long as t is greater than about 20 minutes.) Find a formula that will let you figure out how long the milk has been there from its temperature T. Graph this formula. (College Preparatory Mathematics) 3a, 3b, 3c, 3h Use measurements from shopping carts that are nested together to find a formula for the number of carts that will fit in a space of any given length, and a formula for the amount of space needed for any given number of carts. (Balanced Assessment) 3a, 3b, 3c, 3f, 3h Express the concentration of bleach as a function of the amount of water added to three liters of a 12% solution of bleach. 3a, 3b, 3c, 1h The quantity 1 + x is sometimes used as an approximation for the quantity if x is positive and small (much less than 1). Use graphs to show why this makes sense. Over what range of values of x does this approximation yield less than a 5% error? Find the sum of the infinite geometric series 1 + x + x² + x³ + ... (assuming 0 < x < 1) and show how it sheds light on why the approximation works. 3b, 3c, 3g, 3h, 3i Design a staircase that rises a total of 11 feet, given that the slope must be between .55 and .85, and that the rise plus the run on each step must be between 17 and 18 inches. (Balanced Assessment) 3c, 3h, A1a You have a green candle 12.4 cm tall that cost \$0.45; after burning for four minutes it is 11.2 cm tall. You also have a red candle 8.9 cm tall that cost \$0.40; after burning for ten minutes it is 7.5 cm tall. Analyze the burning rates with functions and graphs. If they are both lit at the same time, predict when (if ever) they will be the same height, and when each will burn down completely. Which costs less per minute to use? (College Preparatory Mathematics) 3d, 3e, 3f

 Statistics and Probability Concepts High School

The student demonstrates understanding of a mathematical concept by using it to solve problems, representing it in multiple ways (through numbers, graphs, symbols, diagrams, or words, as appropriate), and explaining it to someone else. All three ways of demonstrating understanding—use, represent, and explain—are required to meet this standard.

The student demonstrates understanding of statistics and probability concepts; that is, the student:
 a Organizes, analyzes, and displays single-variable data, choosing appropriate frequency distribution, 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.

Examples of activities through which students might demonstrate understanding of statistics and probability concepts include:
 Compare a frequency distribution of salaries of women in a company with a frequency distribution of salaries of men. Describe and quantify similarities and differences in the distributions, and interpret these. 4a, 4f Analyze and interpret prominent features of a scatter plot of several hundred data points, each giving the age of death of a person and the average number of cigarettes smoked per day by that person. 4b, 4f Make an estimate of the number of beads in a large container using the following method. Select a sample of beads, mark these beads, return them to the container, and mix them in thoroughly. Then re-sample and count the proportion of marked beads. Compare your result with another method of estimating the number, for example, one based on weighing the beads. 4c Two integers, each between 1 and 9 are selected at random, and then added. Determine the possible sums and the probability of each. Generalize to two integers between 1 and n. Generalize to three integers between 1 and 9. (Balanced Assessment) 4h, 4i, 4j Suppose it is known that 1% of \$100 bills in circulation are counterfeit. Suppose also that there is a quick test for counterfeit bills, but that the test is imperfect: 5% of the time the test gives a false negative (pronouncing a counterfeit bill as genuine) and 15% of the time the test gives a false positive (pronouncing a genuine bill as counterfeit). Find the probability that a bill that tests negative is actually counterfeit. Find the probability that a bill that tests positive is actually genuine. 4h, 4j, 4l Player A has a one out of six chance of hitting the target on any throw, while player B has a two out of ten chance. They alternate turns, with A going first. The first one to hit the target wins. Who is favored? 4i, 4j In a game, you toss a quarter (diameter 24 mm) onto a large grid of squares formed by vertical and horizontal lines 24 mm apart. You win if the quarter covers an intersection of two lines. What are the odds of winning? Express your answer in terms of . 4l, 4m

 Problem Solving and Mathematical Reasoning 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 a proof of the conjecture.

Examples of activities through which students might demonstrate facility with problem solving and reasoning include:
 A regular hexagon “rolls” around a stationary regular octagon of the same side length until it returns to its starting position. Figure out how many times the hexagon (i) rotates about the octagon and (ii) revolves on its axis. Generalize to an m-gon rolling around an n-gon. (Balanced Assessment) 5a, 5b, 5c, 5d, 1j Create a mathematical model that will give an estimate for the volume of a bottle, given a front view and top view of the bottle drawn to scale. Repeat for bottles of different shapes. (New Standards Released Task) 5a, 5b, 2a, 2b, 2d, 2e Classify quadrilaterals according to two criteria: the number of right angles, and the number of pairs of parallel sides. For every possible combination of number of right angles and number of pairs of parallel sides, either give an example of such a quadrilateral, or show why such an example is impossible. (New Standards Released Task) 5b, 5d, 2b, 2p An earthquake generates two types of “waves” that travel through the Earth: “P-waves,” which travel at 5.6 km/sec, and “S-waves,” which travel at 3.4 km/sec. After an earthquake, the P-waves arrive at one recording station 15 seconds before the S-waves. Use functions, graphs, and equations to explain how far the recording station was from the epicenter of the earthquake. Show the flaw in this attempted solution: “The epicenter is 33 km away because the difference in velocities is 2.2 km/sec, and in 15 seconds that’s 33 km.” 5a, 5b, 5c, 3a Analyze the relationship between the number of pairs of eyelet holes in a shoe and the length of the shoelace. (New Standards Released Task) 5a, 5b, 5c, 3a, 3f In a game for many players in which each player rolls three dice and adds the three numbers, show how to assign scores to each possible sum so that sums with the same probability get the same score, sums with twice the probability get half the score, and so on. 5a, 5b, 5c, 5d, 4h, 4l Investigate different ways of running a wire from the floor at one corner of a room to the ceiling at the opposite corner. Find the shortest wire under each of the following restrictions: (i) you can only run the wire along the edges of walls; (ii) you can also run the wire across the face of a wall; (iii) you can even run the wire through the air. (Balanced Assessment) 5b, 5c, 2d, 2f, 3b, 3h Explore rectangular spaces enclosed by line segments laid out on a square lattice of dots. Check that the numbers of line segments, dots, and spaces enclosed seem to be related by the formula L + 1 = D + S. Justify this formula by reasoning as follows: the formula holds for the simplest arrangement of line segments and dots, and it is not changed through any of the possible ways of adding to an arrangement. (Balanced Assessment) 5d

 Mathematical Skills and Tools 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.

Examples of activities through which students might demonstrate facility with mathematical skills and tools include:
 Given that Celsius temperature C can be computed from the Fahrenheit temperature F by the formula C = ()(F-32), find a formula for computing F from C. 6a If the temperature of an aluminum bar is increased from 0 to T degrees Celsius, its length is increased by a factor of aT, where a = 23.8 x 10 is the coefficient of thermal expansion for aluminum. By how many millimeters would a 1 meter bar increase if raised from 0 to 40 degrees Celsius? 6a Use the local phone book to find the approximate relative frequency of last names beginning with each of the 26 letters of the alphabet. Make a histogram and a circle graph of this information. Decide how you would divide the names into four roughly equal groups. 6a, 6b, 6g The braking distance in feet for a car is given by the formula 0.026 s + st, where s is the speed of the car in feet per second, and t is the reaction time in seconds of the driver. What is the braking distance at a speed of 60 miles per hour if the reaction time is second? 6a, 6c Write the general equation for a straight line that uses as parameters the x-intercept A and the y-intercept B. 6a, 6g Make a one-tenth size scale diagram of an archery target with these specifications: There are five target regions, bounded by concentric circles with radii equal to 10 cm, 15 cm…, 35 cm. Compute the area of each region. 6d, 6e Given the riser height and tread width of the steps on stairs of many kinds, make a scatter plot of the data. Find a line that seems to fit the data in two ways, by eye and using a calculator that can compute a regression line. Compare the result with the rule of thumb that riser height plus tread width should range from about 40 to 45 cm. 6g The function V = x (40 - 2x) (30 - 2x) gives the volume in cubic centimeters of a tray of depth x formed from a rectangle of dimensions 30 cm by 40 cm. Graph this function. What is the volume if the depth is 10 cm? What is the largest volume such a tray can have? What depth gives this largest volume? 6g, 6h, 6j Describe an algorithm for converting any distance given in miles and feet to decimal miles, and another algorithm for converting the other way. Do the same for converting decimal hours to hours, minutes, and seconds. 6i

 Mathematical Communication 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. j Reads mathematics texts and other writing about mathematics with understanding.

Examples of activities through which students might demonstrate facility with mathematical communication include:
 Discuss the mathematics underlying a sign along a highway that says “7% Grade Next 3 Miles.” Use representations such as tables, formulas, graphs, and diagrams. Explain carefully concepts such as slope, steepness, grade, and gradient. (Balanced Assessment) 7b, 7e Suppose in a certain country every adult gets married, and every married couple keeps having children until they have a daughter, then stops. Describe the effect on the population and the ratio of males to females over time. Assume a probability of one-half that a birth is a girl. 7c, 7d, 7e Design a unit of instruction for middle school about proportional relationships. Show the relevance and interconnection of concepts such as percent, ratio, similarity, and linear functions. 7f Prepare review materials that summarize the basic skills and tools used in an instructional unit from a mathematics text (assuming the unit does not already have such a summary). 7f Read a book written for the general public that discusses different advanced fields of mathematics and report on one of these fields. 7j

 Putting Mathematics to Work 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.

Examples of data study projects include:
 Carry out a study of the circulation of books in a library based on type of book and number of users, and showing the progression over a period of years. 3k, 4a, 4f, 4g, 5 Carry out a study of the students in a district in terms of their proficiency in using writing in mathematics, and how that proficiency changed over a period of years. 3k, 4a, 4g, 5 Carry out a study of several kinds of data about auto races and trends in these data over a number of years. 3k, 4a, 4g, 5 Carry out a study of the circulation of books in a library over a period of time. Represent the relative number of borrowers for each type of book and analyze any change over time. Represent the number of borrowers for the most popular book titles and look for a correlation with the number of copies of each title the library has. 4a, 4b, 4g, 5 Analyze selected newspapers and magazines for accuracy and clarity of graphical presentations of data, discussing the most common and effective types of presentation used, and identifying misleading graphical practices. 4f, 5, 7a, 7b

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.

Examples of mathematical modeling projects include:
 Analyze the change in shape undergone under thermal expansion of a long bridge. 2a, 2b, 3a, 3b, 3e, 3f, 3i, S1b, S1e Analyze the characteristics of an irrigation system for large fields that has a central water feed and rotating spray arms that sweep out a circle. 2a, 2b, 2e, 2l, 3a, 3d, 5 Construct pendulums with various lengths of rods and masses of bobs. Measure their periods when released from various heights. Determine which of these parameters the period depends on. Create a formula for the period in terms of these parameters, and compare these results with the analysis of a pendulum in a physics book. 3a, 3b, 3h, 3i, 3n, 5, S1d, S1e

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.

Examples of projects to design a physical structure include:
 Make a plan for the layout of a housing development to be created on a large tract of land, according to given specifications such as lot size, house setbacks, and street widths. Take into consideration given information on the relation between development cost and possible sale prices. 2a, 2b, 2e, 2k, 2n, 3a, 3i, 5 Design and make a model for a wheelchair access ramp to an 11' high platform, given that the ramp must fit in a 30' by 30' space and must conform to the provisions of the Americans with Disabilities Act. 2a, 2g, 2i, 3a, 3b, 3c, 5, A1a Design seating plans for a large theater given specifications on the size and shape of the space, the allowable width of aisles, the required spacing between rows, and the allowable sizes and spacing of seats. Find the plan that allows for the maximum number of seats. Suggest how that plan might have to be modified to take other features into consideration, such as staggering seats in successive rows for better viewing. 2a, 2b, 3a, 3e, 5

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.

Examples of management and planning projects include:

 Create a schedule for practices and events at the school gymnasium and swimming pool, taking into account home and away games, junior varsity and varsity, and boys’ and girls’ teams. 1i, 3a, 3i, 5, A1c Make a business plan for publication of a magazine, taking into account different requirements in the production of the magazine, such as quality of paper, use of color, cover stock, and the relationship between selling price and circulation. 3a, 3i, 5, A1a

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.

Examples of projects include:
 Carry out an investigation of the many properties of Pascal’s triangle. 1b, 1i, 1l, 3a, 3b, 3i, 3o, 5 Create a schedule for a ping-pong tournament among ten players in which each player plays each other player exactly once. Arrange the schedule so that no players have to sit out while others are playing. Try to do the same for a tournament with sixteen players. Then (this is much harder) say what you can about the general case of a tournament with 2n players. Create effective and revealing representations for the schedules. (Balanced Assessment) 1i, 5, A1c Make a study of different mathematical types of spirals, the properties they share, and the ways in which they are different. 2o, 3m, 5 Make an inquiry into what distributions of objects of two colors result in a probability of roughly ½ that the objects are the same color when two of the objects are selected at random. (For example, three of one color and six of another color is such a distribution.) 4h, 4i, 4j, 4k, 4l, 5

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.

Examples of projects include:
 Read and report on the history of the Pythagorean Theorem, including a discussion of some of the basic ways of proving the theorem and of its uses within and outside mathematics. 2f, 2p, 5, 7e, 7j Carry out a historical study of the concept of “function” in mathematics, including a report on the most important function concepts and types currently in use. Base part of the work on interviews with people from other fields who use mathematics in their work. 3, 5, 7e, 7j