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Arithmetic and Number Concepts/Number and Operation Concepts
Elementary School
The student produces evidence that demonstrates understanding of arithmetic and number concepts; that is, the student:
a Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is:
adds, i.e., joins things together, increases;
subtracts, i.e., takes away, compares, finds the difference;
multiplies, i.e., uses repeated addition, counts by multiples, combines things that come in groups, makes arrays, uses area models, computes simple scales, uses simple rates;
divides, i.e., puts things into groups, shares equally; calculates simple rates;
analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide;
solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another;
computes answers mentally, e.g., 27 + 45, 30 x 4;
uses simple concepts of negative numbers, e.g., on a number line, in counting, in temperature, “owing.”
b Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is:
counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000;
uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.
c Estimates, approximates, rounds off, uses landmark numbers, or uses exact numbers, as appropriate, in calculations.
d Describes and compares quantities by using concrete and real world models of simple fractions; that is:
finds simple parts of wholes;
recognizes simple fractions as instructions to divide, e.g., 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.


Middle School
The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:
a Consistently and accurately adds, subtracts, multiplies, and divides rational numbers using appropriate methods (e.g., the student can add 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 root-extraction (e.g., squares and square roots, cubes and cube roots); uses the inverse operation to determine unknown quantities in equations.

c Consistently and accurately applies and converts the different kinds and forms of rational numbers.

d Is familiar with characteristics of numbers (e.g., divisibility, prime factorization) and with properties of operations (e.g., commutativity and associativity), short of formal statements.

e Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.

f Uses ratios and rates to express “part-to-part” and “whole-to-whole” relationships, and reasons proportionally to solve problems involving equivalent fractions, equal ratios, or constant rates, recognizing the multiplicative nature of these problems in the constant factor of change.

g Orders numbers with the > and < relationships and by location on a number line; estimates and compares rational numbers using sense of the magnitudes and relative magnitudes of numbers and of base-ten place values (e.g., recognizes relationships to “benchmark” numbers 1/2 and 1 to conclude that the sum 1/2 + 5/6 must be between 1 and 11/2 (likewise, 13/24 + 57/68)).


High School
The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:
a Uses addition, subtraction, multiplication, division, exponentiation, and root-extraction in forming and working with numerical and algebraic expressions.

b Understands and uses operations such as opposite, reciprocal, raising to a power, taking a root, and taking a logarithm.

c Has facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications.

d Understands and uses number systems: natural, integer, rational, and real.

e Represents numbers in decimal or fraction form and in scientific notation, and graphs numbers on the number line and number pairs in the coordinate plane.

f Compares numbers using order relations, differences, ratios, proportions, percents, and proportional change.

g Carries out proportional reasoning in cases involving part-whole relationships and in cases involving expansions and contractions.

h Understands dimensionless numbers, such as proportions, percents, and multiplicative factors, as well as numbers with specific units of measure, such as numbers with length, time, and rate units.

i Carries out counting procedures such as those involving sets (unions and intersections) and arrangements (permutations and combinations).

j Uses concepts such as prime, relatively prime, factor, divisor, multiple, and divisibility in solving problems involving integers.

k Uses a scientific calculator effectively and efficiently in carrying out complex calculations.

l Recognizes and represents basic number patterns, such as patterns involving multiples, squares, or cubes.