The student produces evidence that demonstrates
understanding of arithmetic and number concepts; that is, the student:
a
Adds, subtracts, multiplies, and divides whole
numbers, with and without calculators; that is: 
• 
adds, i.e., joins things together, increases; 
• 
subtracts, i.e., takes away, compares, finds the difference; 
• 
multiplies, i.e., uses repeated addition, counts by multiples, combines
things that come in groups, makes arrays, uses area models, computes
simple scales, uses simple rates; 
• 
divides, i.e., puts things into groups, shares equally; calculates
simple rates; 
• 
analyzes problem situations and contexts in order to figure out
when to add, subtract, multiply, or divide; 
• 
solves arithmetic problems by relating addition, subtraction, multiplication,
and division to one another; 
• 
computes answers mentally, e.g., 27 + 45, 30 x 4; 
• 
uses simple concepts of negative numbers, e.g., on a number line,
in counting, in temperature, “owing.” 


b
Demonstrates understanding of the base
ten place value system and uses this knowledge to solve arithmetic
tasks; that is: 
• 
counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less
than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than
9,000; 
• 
uses knowledge about ones, tens, hundreds, and thousands to figure
out answers to multiplication and division tasks, e.g., 36 x 10, 18
x 100, 7 x 1,000, 4,000 ÷ 4.

c
Estimates, approximates, rounds off,
uses landmark numbers, or uses exact numbers, as appropriate, in calculations.

d
Describes and compares quantities by
using concrete and real world models of simple fractions; that is:

• 
finds simple parts of wholes; 
• 
recognizes simple fractions as instructions to divide, e.g., 1/4
of something is the same as dividing something by 4; 
• 
recognizes the place of fractions on number lines, e.g., in measurement; 
• 
uses drawings, diagrams, or models to show what the numerator and
denominator mean, including when adding like fractions, e.g., 1/8
+ 5/8, or when showing
that 3/4 is more than
3/8; 
• 
uses beginning proportional reasoning and simple ratios, e.g., “about
half of the people.” 
e
Describes and compares quantities by using
simple decimals; that is: 
• 
adds, subtracts, multiplies, and divides money amounts; 
• 
recognizes relationships among simple fractions, decimals, and percents,
i.e., that 1¼2 is the same as 0.5, and 1/2
is the same as 50%, with concrete materials, diagrams, and in real
world situations, e.g., when discovering the chance of a coin landing
on heads or tails.

f
Describes and compares quantities by using whole numbers up to 10,000;
that is: 
• 
connects ideas of quantities to the real world, e.g., how many people
fit in the school’s cafeteria; how far away is a kilometer; 
• 
finds, identifies, and sorts numbers by their properties, e.g.,
odd, even, multiple, square. 
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 1/2 + 5/6 mentally or on paper but may opt to add 13/24 + 57/68
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 1/2 and 1 to conclude that the sum 1/2 + 5/6 must be between 1
and 11/2 (likewise, 13/24
+ 57/68)).
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.
