# List of Benchmarks for Mathematics |

**Standard 2.** | Understands and applies basic and advanced properties of the concepts of numbers |

| ** Level Pre-K (Grade Pre-K)** |

| 1. | Understands that numbers represent the quantity of objects |

| 2. | Counts by ones to ten or higher |

| 3. | Counts objects |

| 4. | Understands one-to-one correspondence |

| 5. | Understands the concept of position in a sequence (e.g., first, last) |

| 6. | Knows the written numerals 0-9 |

| 7. | Knows the common language for comparing quantity of objects (e.g., "more than," "less than," "same as") |

| 8. | Understands that a whole object can be separated into parts |

| 9. | Knows that numbers are used in real-world situations |

| ** Level I (Grade K-2)** |

| 1. | Understands that numerals are symbols used to represent quantities or attributes of real-world objects |

| 2. | Counts whole numbers (i.e., both cardinal and ordinal numbers) |

| 3. | Understands symbolic, concrete, and pictorial representations of numbers (e.g., written numerals, objects in sets, number lines) A |

| 4. | Uses base ten concepts to compare whole number relationships (e.g., 4 is less than 10, 30 is 3 tens) and represent them in flexible ways (e.g., expanded form, decomposing numbers) |

| 5. | Understands the concept of a unit and its subdivision into equal parts (e.g., one object, such as a candy bar, and its division into equal parts to be shared among four people) |

| ** Level II (Grade 3-5)** |

| 1. | Understands basic number theory concepts (e.g., prime and composite numbers, factors, multiples, odd and even numbers, divisibility) A |

| 2. | Understands equivalent forms of basic percents, fractions, and decimals (e.g., 1/2 is equivalent to 50% is equivalent to .5) and when one form of a number might be more useful than another A |

| 3. | Understands the basic difference between odd and even numbers |

| 4. | Understands the basic meaning of place value A |

| 5. | Understands the concepts related to fractions (e.g., numerator and denominator, equivalence, relative magnitudes) and decimals (e.g.,relative magnitudes) A |

| 6. | Uses models (e.g., number lines, two-dimensional and three-dimensional regions) to identify, order, and compare numbers A |

| 7. | Uses base ten concepts to represent decimals and fractions in flexible ways (e.g., composing and decomposing numbers |

| ** Level III (Grade 6-8)** |

| 1. | Understands the relationships among equivalent number representations (e.g., whole numbers, positive and negative integers, fractions, ratios, decimals, percents, scientific notation, exponentials) and the advantages and disadvantages of each type of representation A |

| 2. | Understands the characteristics and properties (e.g., order relations, relative magnitude, base-ten place values) of the set of rational numbers and its subsets (e.g., whole numbers, fractions, decimals, integers) |

| 3. | Understands the role of positive and negative integers in the number system |

| 4. | Uses number theory concepts (e.g., divisibility and remainders, factors, multiples, prime, relatively prime) to solve problems A |

| 5. | Understands the characteristics and uses of exponents and scientific notation A |

| 6. | Understands the structure of numeration systems that are based on numbers other than 10 (e.g., base 60 for telling time and measuring angles, Roman numerals for dates and clock faces) |

| 7. | Understands the concepts of ratio, proportion, and percent and the relationships among them A |

| ** Level IV (Grade 9-12)** |

| 1. | Understands the properties (e.g., relative magnitude, density, absolute value) of the real number system, its subsystems (e.g., irrational numbers, natural numbers, integers, rational numbers), and complex numbers (e.g., imaginary numbers, conjugate numbers) |

| 2. | Understands the properties and basic theorems of roots, exponents (e.g., [bm][bn] = bm+n), and logarithms |

| 3. | Uses discrete structures (e.g., finite graphs, matrices, sequences) to represent and to solve problems |

| ** Level V (Grade (College Readiness))** |

| 1. | Understands the properties of rational exponents |

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A = Assessment items available |