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 the conceptual understanding
standards.
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 rootextraction (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 “parttopart”
and “wholetowhole” 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
baseten place values (e.g., recognizes relationships to “benchmark”
numbers
and 1 to conclude that the sum
+
must be between 1 and 1
(likewise,
+ )).
Examples of activities through which students might
demonstrate understanding of number and operation concepts include:

Sara placed one number on each
side of two disks. She said that if she flipped the disks and added
the two numbers facing up, the sum was always one of the following
numbers: 1, .23, .87, .1. What numbers could Sara have placed on each
face of the two disks? (Balanced Assessment) 1a,
5a, 5b 

How can you compute a 15%,
10%, or 20% tip, other than by multiplying an amount by 0.15, 0.1,
or 0.2 on paper? 1a, 1c, 1e 

Mrs. Brown wrote a number on
the chalkboard and said, “I know that to be sure I find all the
factor pairs for this number I must check each number from 1 to 29.”
What numbers could Mrs. Brown have written? What is the smallest number
Mrs. Brown could have written? What is the largest number Mrs. Brown
could have written? (Middle Grades Mathematics Project: Factors and
Multiples) 1b, 1d 

Burger Jack’s ads claim
that their pound
burger contains 50% more beef than Winnie’s pound
burger. Is their claim true? 1c, 1e 

The members of the school marching
band wanted to arrange themselves into rows with exactly the same
number of band members in each row. They tried rows of two, three,
and four, but there was always one band member left over. Finally,
they were able to arrange themselves into rows with exactly five in
each row. What is the least number of members in the marching band?
(Creative Problem Solving in Mathematics) 1d 

Is the sum of two consecutive
integers odd or even? Always? How about the product of two consecutive
numbers? Why? What can you say about three consecutive numbers? 1d 

Find the last two digits of
.
(NCTM, Mathematics Teaching in the Middle School) 1d 

If, in a school of 1,000 lockers,
one student opens every locker, another student closes every other
locker (2nd, 4th, 6th, etc.), a third student changes every third
locker (opens closed lockers and closes open lockers), and so on,
until the thousandth student changes the thousandth locker, which
lockers are open? Why? 1d, 5a, 5b,
5c 

A certain clock gains one minute
of time every hour. If the clock shows the correct time now, when
will it show the correct time again? (Creative Problem Solving in
Mathematics) 1f, 2h 

How much space would a million
shoe boxes fill? How large would a pile of a million pennies be? How
about a million sheets of notebook paper? 1g 

How small would a millionth
of a sheet of notebook paper be? 1g 

Middle
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 in the following areas;
that is, the student:
a
Is familiar with assorted two and threedimensional
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
threedimensional 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.
Examples of activities through which students
might demonstrate understanding of geometry and measurement concepts include:

Jan says rectangles and rhombi are completely different.
Joan says they are almost exactly the same. Explain and illustrate
why both students are partially correct. Compare and contrast the
two figures. (College Preparatory Mathematics, “Rhombus and Rectangle”)
Can you think of a special shape that is both a rectangle and a rhombus?
2a 

Jamaal installed two sprinkler heads to water a square
patch of lawn twelve feet on a side. He placed the sprinkler heads
in the middle of two opposite edges of the lawn. If the spray pattern
of each sprinkler is semicircular and just reaches the two nearest
corners, what percentage of the lawn is not watered by the two sprinkler
heads? (College Preparatory Mathematics, “Watering the Lawn”)
2a, 2d 

Examine logos of businesses in the yellow pages for
rotational and bilateral symmetry. 2b, 2e 

From square grid paper, cut “jackets” for
“space food packages,” which are actually blocks of cubes
(same size as the squares of the grid); each cube contains one day’s
supply of food pellets, and each square of “material” costs
$1. For different space armor jackets cut from the grid paper, find
the number of days the food supply will last (the volume of the package).
Also find the cost of the jacket (the surface area) and the dimensions
of the package. (Middle Grades Mathematics Project: Mouse and Elephant:
Measuring Growth) 2c 

A rectangular garden has two semicircular flower beds
on opposite ends. Find the area of the entire figure. (College Preparatory
Mathematics, “The Garden”) 2d


Investigate the area around your school and neighborhood
to describe the size of an acre and a square mile. 2d 

Domino’s Pizza is offering a free 16" pizza
to the first student who can tell them the greatest number of 1"
x 2" x "
dominoes it will take to fill their takeout shop. The dimensions
of the takeout shop are 10' x 11' x 8'. How many is it? (College
Preparatory Mathematics, “The Domino Effect”) 2d,
2h, 1f 

Display data from a survey with an accurately drawn
and divided pie chart: the angles should be accurately computed and
measured so that the regions of the circle are proportional to the
percentages of the survey results. 2g, 1f,
6g 

Stand nine feet from a friend and hold a onefoot ruler
vertically in front of yourself. Line up the top of the ruler with
the top of your friend and the bottom of the ruler with the feet of
your friend. If the ruler is two feet from your eyes, how tall is
your friend? (Middle Grades Mathematics Project: Similarity and Equivalent
Fractions) Measure the actual distance from your eyes to the top of
the ruler. According to the relationships of your similar triangles,
how tall is your friend? Measure your friend’s height. Was your
calculation accurate? 2i 

Construct an enlargement of some simple geometric shape
by a scale factor of 3 by utilizing grid paper and also by measuring
from a projection point. 2j 

Make a twodimensional, cardboard or paper replica
of yourself using measurements of lengths and widths of body parts
that are half those of your own body. (UCSMP, Transition Mathematics)
2j 

Create a poster by enlarging, to scale, a favorite cartoon
panel. 2j 

Make a scale drawing of your bedroom.
2j 

Middle
School 
This standard describes the foundation expected of middle school students
in preparation for high school mathematics. Some students will take a
course in algebra before high school, and their understanding of functions
and algebra should surpass what is described below.
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 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=r²,
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.
Examples of activities through which students
might demonstrate understanding of function and algebra concepts include:

Graph and explain the growth of population over time
of a colony of organisms that doubles once a day. 3a 

Use diagrams, tables, graphs, words, and formulas to
show the relationships between the length of the sides of a square
and its perimeter and area. 3a, 3b 

A ball is dropped from a height of sixteen feet. At
its first bounce, the ball reaches a peak of eight feet. Each successive
time the ball bounces, it reaches a peak height that is half that
of the previous bounce. How many times will the ball bounce until
it bounces to a peak height of one foot? Make a table that shows the
peak bounce height of the ball for the number of bounces. Make a graph
and write an algebraic expression that show the relationship between
the number of bounces and the peak height of the ball. (Balanced Assessment)
3b 

Examine areas that can be enclosed by 24 feet of fencing
and figure out the maximum area. 3b 

Your principal wants to hire you to work for her for
ten days. She will pay you either $6.00 each day for all ten days;
or $1.00 the first day, $2.00 the second day, $3.00 the third day,
and so on; or $0.10 the first day and each day thereafter twice the
amount of the day before. Under which arrangement would you earn the
most money? Under which arrangement would you earn the least money?
(NCTM, Mathematics Teaching in the Middle School) 3c 

Investigate the following situation: Bricklayers use
the rule N=7·L·H to determine the number N of bricks
needed to build a wall L feet long and H feet high. Examine a brick
wall or portion of a brick wall to see whether this seems to be true.
If it works, why? If not, what would be a better formula? (UCSMP,
Transition Mathematics) 3c 

When inquiring about the dimensions of a lot, you are
told, “I can’t remember. It’s shaped like a rectangle,
and I know that they needed ninety meters of fencing to enclose it.
Oh, yes! The crew putting up the fence remarked that the lot is exactly
twice as long as it is wide.” What are the dimensions of the
lot? (Creative Problem Solving in Mathematics) 3d 

Middle
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 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.
Examples of activities through which students
might demonstrate understanding of statistics and probability concepts
include:

From a sample news headline,
an article, and a table of data, select and construct appropriate
graphs or other visual representations of the data. Decide whether
or not the headline seems appropriate. Write a letter to the editor.
(Balanced Assessment) 4a, 4d, 4e,
7e 

In a game to see who could
best guess when 30 seconds had passed, the actual times (measured
by a stopwatch) of Gilligan’s guesses were 31, 25, 32, 27, and
28 seconds; Skipper’s guesses were 37, 19, 40, 36, and 22 seconds;
Ginger’s guesses were 32, 38, 24, 32, and 32 seconds. Who do
you think is best at estimating 30 seconds? Why? (Balanced Assessment)
4b, 4c 

When tossing two fair numbered
cubes and finding the sum of the two numbers turned up, if the sum
is seven, then Keisha gets seven points, but if the sum is not seven,
then Shawna gets one point. Is the game fair or not? Explain your
reasoning to Keisha and Shawna. (Balanced Assessment)
4h 

Decide whether it is most advantageous
to use three tetrahedra, two cubes, or one dodecahedron to arrive
at a specific number (like 10) when rolling polyhedral dice. 4h,
1g 

Lottery players often say,
“Well, my numbers have to come up sometime!” Analyze a state
lottery game to see how many number combinations there are and how
many weeks, months, or years it will take for all of them to be drawn.
4h, 4j 

A poll was taken of 40 students
on their favorite school lunch. The results show hamburgers and fries,
14; pizza and salad, 13; spaghetti and salad, 8; hot dogs and beans,
5; liver and spinach, 0. Assuming this is an accurate sample, if a
student is chosen at random, what is the probability that he or she
favors each lunch offering? If there are 400 students in the school,
how many prefer hamburgers? Hot dogs? Liver? (Middle Grades Mathematics
Project: Probability) 4i 

Middle 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 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 sensemaking 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.
Examples of activities through which
students might demonstrate problem solving and mathematical reasoning
include:

Aaron’s Pizzeria
will pay you $75 if you and a partner can eat his 26" diameter
pizza in one hour. To practice for this eating event, you and
your friend go to Round Label Pizza. How many of Round Label’s
12" diameter pizzas would the two of you need to eat to
equal the amount of pizza you hope to eat at Aaron’s? (College
Preparatory Mathematics, “Pizza Problem”) 5a,
5b, 2d 

Design a carnival game
in which paper “parachutes” are dropped onto a target
of five concentric rings, for which a player scores five points
for landing in the center, four points in the next ring, three
in the next, two in the next, and one point for landing in the
outer ring. Experiment and create rules for the game so that
it is fun, makes a profit, and people feel they have a chance
of winning. (Balanced Assessment) 5a,
5b, 4a, 8d 

Your rich uncle has just
died and has left you $1 billion. If you accept the money, you
must count it for eight hours a day at the rate of $1 per second.
When you are finished counting, the $1 billion is yours, and
then you may start to spend it. Should you accept your uncle’s
offer? Why or why not? How long will it take to count the money?
(College Preparatory Mathematics, “Big Bucks”) 5a,
5b, 5c, 1f, 1g 

Starting with the numbers
from 1 to 36, find out all you can about writing them as sums
of consecutive whole numbers. What kinds of numbers can be written
as the sum of two consecutive whole numbers; of three consecutive
whole numbers; of four consecutive whole numbers? Which numbers
cannot be written as the sum of consecutive whole numbers? What
patterns do you notice? Why do you think they occur? (Balanced
Assessment) 5b, 5c, 5d, 1d 

A diagonal of the
3 x 5 grid rectangle passes through seven of the squares.
In the 4 x 4 grid rectangle, the diagonal passes through only
four squares. Come up with at least two conjectures about
grid rectangles, diagonals, and the squares they pass through.
(College Preparatory Mathematics, “On Grids”) 5d,
1d



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.
Examples of activities through which
students might demonstrate facility with mathematical skills and
tools include:

How (many ways) can you
use four 4s to create an expression that has a value equal to
1? (e.g., =
…) (Creative Problem Solving in Mathematics) 6b,
6h 

Figure out how long it
would take to say your name a million times; how long it would
take to count to a million. 6c,
1f, 1g, 5a, 5b, 5c 

Accurately describe a
geometric design on a 10 x 10 grid to a friend by telephone.
(Balanced Assessment) 6e, 7c 

Use the formula A=
bh for areas of triangles measured with customary and metric
rulers. 6f 

Analyze advertisements
for different music clubs and decide which offers best value
for money. 6g 


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.
Examples of activities through which students
might demonstrate facility with mathematical communication include:

Use diagrams, tables, graphs,
words, and formulas to show the relationship of the length of the
sides of a square to its perimeter and area. 7a 

Use boxandwhiskers plots,
stemandleaf plots, and bar graphs to compare characteristics of
the boys and girls in the class; compare the kinds of information
provided by the different displays. 7a,
7b 

Use symbols and a Cartesian
map to explain to another student how to get from your home to school.
7c 

Make the following conjectures:
What happens to the area of a square when you double its perimeter?
What happens to the area when you triple its perimeter? Investigate
to see if this is true and, if so, explain why. What does doubling
the circumference of a circle do to its area? Explain. 7d,
2d, 3b 

Your fifth grade cousin is
convinced that the probability of rolling a 12 on two numbered cubes
is .
Explain to your cousin why this is incorrect, and convince your cousin
of the actual probability of getting 12. 7e,
4h 

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.
Examples of investigations or projects include:

Gather and analyze data from
the neighborhood and compare the data with published statistics for
the city, state, or nation. 8a, 4a,
4b, 4c, 4d 

Compare the growth of a set
of plants under a variety of conditions, e.g., amount of water, fertilizer,
duration and exposure to sunlight.
8b, 3b, S2a, S3d 

Design and equip a recreational
area on one acre with a limited budget. 8c,
1a, 2a, 2d, 2h, 2j, A1a 

Analyze and concoct games of
chance for a school carnival. 8d,
4h, 4i, A1c 

Discover relationships among,
and properties of, the numbers in Pascal’s triangle. Read to
find more relationships and properties. 8e,
1e, 4h, 7f 
