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 rootextraction 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 partwhole
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 

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 nongeometric 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 personhours. 
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
squarebase pyramids? What about trianglebase 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 

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 

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 singlevariable data,
choosing appropriate frequency distribution, circle graphs, line plots,
histograms, and summary statistics. 
b 
Organizes, analyzes, and displays twovariable 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 resample
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 

High
School 
The student demonstrates problem solving by using
mathematical concepts and skills to solve nonroutine 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 nonroutine and multistep 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 mgon
rolling around an ngon. (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: “Pwaves,” which travel
at 5.6 km/sec, and “Swaves,” which travel at 3.4 km/sec.
After an earthquake, the Pwaves arrive at one recording station 15
seconds before the Swaves. 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 

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 = ()(F32),
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 xintercept A and the yintercept B. 6a,
6g 

Make a onetenth 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 

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 onehalf 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 

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, costbenefit projections, and
risks; 
• 
uses decision rules and strategies both to analyze options
and balance tradeoffs; 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 pingpong 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 
