# Standards Detail

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Subject Area - CC.2:
Mathematics
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.PREK: GRADE Pre-Kindergarten
Standard - CC.2.1.PREK.A.1

Know number names and the count sequence.

Standard - CC.2.1.PREK.A.2

Count to tell the number of objects.

Standard - CC.2.1.PREK.A.3

Compare numbers.

• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.K: GRADE K
Standard - CC.2.1.K.A.1

Know number names and write and recite the count sequence.

Standard - CC.2.1.K.A.2

Apply one-to-one correspondence to count the number of objects.

Standard - CC.2.1.K.A.3

Apply the concept of magnitude to compare numbers and quantities.

Standard - CC.2.1.K.B.1

Use place value to compose and decompose numbers within 19.

• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.1: GRADE 1
Standard - CC.2.1.1.B.1

Extend the counting sequence to read and write numerals to represent objects.

Standard - CC.2.1.1.B.2

Use place value concepts to represent amounts of tens and ones and to compare two digit numbers.

Standard - CC.2.1.1.B.3

Use place value concepts and properties of operations to add and subtract within 100.

• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.2: GRADE 2
Standard - CC.2.1.2.B.1

Use place value concepts to represent amounts of tens and ones and to compare three digit numbers.

Standard - CC.2.1.2.B.2

Use place value concepts to read, write, and skip count to 1000.

Standard - CC.2.1.2.B.3

Use place value understanding and properties of operations to add and subtract within 1000.

• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.3: GRADE 3
Standard - CC.2.1.3.B.1

Apply place value understanding and properties of operations to perform multi-digit arithmetic.

• Assessment Anchor - M03.A-T.1 Use place-value understanding and properties of operations to perform multi-digit arithmetic.
• Anchor Descriptor - M03.A-T.1.1

Apply place-value strategies to solve problems.

• Eligible Content - M03.A-T.1.1.1 Round two- and three-digit whole numbers to the nearest ten or hundred, respectively.
• Eligible Content - M03.A-T.1.1.2 Add two- and three-digit whole numbers (limit sums from 100 through 1,000) and/or subtract two- and three-digit numbers from three-digit whole numbers.
• Eligible Content - M03.A-T.1.1.3 Multiply one-digit whole numbers by two-digit multiples of 10 (from 10 through 90).
• Eligible Content - M03.A-T.1.1.4 Order a set of whole numbers from least to greatest or greatest to least (up through 9,999, and limit sets to no more than four numbers).
Standard - CC.2.1.3.C.1

Explore and develop an understanding of fractions as numbers.

• Assessment Anchor - M03.A-F.1 Develop an understanding of fractions as numbers.
• Anchor Descriptor - M03.A-F.1.1 Develop and apply number theory concepts to compare quantities and magnitudes of fractions and whole numbers.
• Eligible Content - M03.A-F.1.1.1 Demonstrate that when a whole or set is partitioned into y equal parts, the fraction 1/y represents 1 part of the whole and/or the fraction x/y represents x equal parts of the whole (limit denominators to 2, 3, 4, 6, and 8; limit numerators to whole numbers less than the denominator; and no simplification necessary).
• Eligible Content - M03.A-F.1.1.2 Represent fractions on a number line (limit denominators to 2, 3, 4, 6, and 8; limit numerators to whole numbers less than the denominator; and no simplification necessary).
• Eligible Content - M03.A-F.1.1.3 Recognize and generate simple equivalent fractions (limit the denominators to 1, 2, 3, 4, 6, and 8 and limit numerators to whole numbers less than the denominator). Example 1: 1/2 = 2/4 Example 2: 4/6 = 2/3
• Eligible Content - M03.A-F.1.1.4 Express whole numbers as fractions, and/or generate fractions that are equivalent to whole numbers (limit denominators to 1, 2, 3, 4, 6, and 8). Example 1: Express 3 in the form 3 = 3/1. Example 2: Recognize that 6/1 = 6.
• Eligible Content - M03.A-F.1.1.5 Compare two fractions with the same denominator (limit denominators to 1, 2, 3, 4, 6, and 8), using the symbols >, =, or <, and/or justify the conclusions.
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.4: GRADE 4
Standard - CC.2.1.4.B.1

Apply place value concepts to show an understanding of multi-digit whole numbers.

• Assessment Anchor - M04.A-T.1 Generalize place-value understanding for multi-digit whole numbers.
• Anchor Descriptor - M04.A-T.1.1 Apply place-value and numeration concepts to compare, find equivalencies, and round.
• Eligible Content - M04.A-T.1.1.1 Demonstrate an understanding that in a multi-digit whole number (through 1,000,000), a digit in one place represents ten times what it represents in the place to its right. Example: Recognize that in the number 770, the 7 in the hundreds place is ten times the 7 in the tens place.
• Eligible Content - M04.A-T.1.1.2 Read and write whole numbers in expanded, standard, and word form through 1,000,000.
• Eligible Content - M04.A-T.1.1.3 Compare two multi-digit numbers through 1,000,000 based on meanings of the digits in each place, using >, =, and < symbols.
Standard - CC.2.1.4.B.2

Use place value understanding and properties of operations to perform multi-digit arithmetic.

• Assessment Anchor - M04.A-T.2 Use place-value understanding and properties of operations to perform multi-digit arithmetic.
• Anchor Descriptor - M04.A-T.2.1 Use operations to solve problems.
• Eligible Content - M04.A-T.2.1.1 Add and subtract multi-digit whole numbers (limit sums and subtrahends up to and including 1,000,000).
• Eligible Content - M04.A-T.2.1.2 Multiply a whole number of up to four digits by a one-digit whole number and multiply 2 two-digit numbers.
• Eligible Content - M04.A-T.2.1.3 Divide up to four-digit dividends by one-digit divisors with answers written as whole-number quotients and remainders.
• Eligible Content - M04.A-T.2.1.4 Estimate the answer to addition, subtraction, and multiplication problems using whole numbers through six digits (for multiplication, no more than 2 digits × 1 digit, excluding powers of 10).
Standard - CC.2.1.4.C.1

Extend the understanding of fractions to show equivalence and ordering.

• Assessment Anchor - M04.A-F.1 Extend understanding of fraction equivalence and ordering.
• Anchor Descriptor - M04.A-F.1.1 Find equivalencies and compare fractions.
• Eligible Content - M04.A-F.1.1.1 Recognize and generate equivalent fractions.
• Eligible Content - M04.A-F.1.1.2 Compare two fractions with different numerators and different denominators (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100) using the symbols >, =, or < and justify the conclusions.
Standard - CC.2.1.4.C.2

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

• Assessment Anchor - M04.A-F.2 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
• Anchor Descriptor - M04.A-F.2.1 Solve problems involving fractions and whole numbers (straight computation or word problems).
• Eligible Content - M04.A-F.2.1.1 Add and subtract fractions with a common denominator (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100; answers do not need to be simplified; and no improper fractions as the final answer).
• Eligible Content - M04.A-F.2.1.2 Decompose a fraction or a mixed number into a sum of fractions with the same denominator (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100), recording the decomposition by an equation. Justify decompositions (e.g., by using a visual fraction model). Example 1: 3/8 = 1/8 + 1/8 + 1/8 OR 3/8 = 1/8 + 2/8 Example 2: 2 1/12 = 1 + 1 + 1/12 = 12/12 + 12/12 + 1/12
• Eligible Content - M04.A-F.2.1.3 Add and subtract mixed numbers with a common denominator (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100; no regrouping with subtraction; fractions do not need to be simplified; and no improper fractions as the final answers).
• Eligible Content - M04.A-F.2.1.4 Solve word problems involving addition and subtraction of fractions referring to the same whole or set and having like denominators (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100).
• Eligible Content - M04.A-F.2.1.5 Multiply a whole number by a unit fraction (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100 and final answers do not need to be simplified or written as a mixed number). Example: 5 × (1/4) = 5/4
• Eligible Content - M04.A-F.2.1.6 Multiply a whole number by a non-unit fraction (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100 and final answers do not need to be simplified or written as a mixed number). Example: 3 × (5/6) = 15/6
• Eligible Content - M04.A-F.2.1.7 Solve word problems involving multiplication of a whole number by a fraction (denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100).
Standard - CC.2.1.4.C.3

Connect decimal notation to fractions, and compare decimal fractions (base 10 denominator, e.g, 19/100).

• Assessment Anchor - M04.A-F.3 Understand decimal notation for fractions and compare decimal fractions.
• Anchor Descriptor - M04.A-F.3.1 Use operations to solve problems involving decimals, including converting between fractions and decimals (may include word problems).
• Eligible Content - M04.A-F.3.1.1 Add two fractions with respective denominators 10 and 100. Example: Express 3/10 as 30/100, and add 3/10 + 4/100 = 30/100 + 4/100 = 34/100.
• Eligible Content - M04.A-F.3.1.2 Use decimal notation for fractions with denominators 10 or 100. Example: Rewrite 0.62 as 62/100 and vice versa.
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.5: GRADE 5
Standard - CC.2.1.5.B.1

Apply place value to show an understanding of operations and rounding as they pertain to whole numbers and decimals.

• Assessment Anchor - M05.A-T.1 Understand the place-value system.
• Anchor Descriptor - M05.A-T.1.1 Demonstrate understanding of place-value of whole numbers and decimals, and compare quantities or magnitudes of numbers.
• Eligible Content - M05.A-T.1.1.1 Demonstrate an understanding that in a multi-digit number, a digit in one place represents 1/10 of what it represents in the place to its left. Example: Recognize that in the number 770, the 7 in the tens place is 1/10 the 7 in the hundreds place.
• Eligible Content - M05.A-T.1.1.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10 and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Example 1: 4 × 10^2 = 400 Example 2: 0.05 ÷ 10^3 = 0.00005
• Eligible Content - M05.A-T.1.1.3 Read and write decimals to thousandths using base-ten numerals, word form, and expanded form. Example: 347.392 = 300 + 40 + 7 + 0.3 + 0.09 + 0.002 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (0.1) + 9 × (0.01) + 2 × (0.001)
• Eligible Content - M05.A-T.1.1.4 Compare two decimals to thousandths based on meanings of the digits in each place using >, =, and < symbols.
• Eligible Content - M05.A-T.1.1.5 Round decimals to any place (limit rounding to ones, tenths, hundredths, or thousandths place).
Standard - CC.2.1.5.B.2

Extend an understanding of operations with whole numbers to perform operations including decimals.

• Assessment Anchor - M05.A-T.2 Perform operations with multi-digit whole numbers and with decimals to hundredths.
• Anchor Descriptor - M05.A-T.2.1 Use whole numbers and decimals to compute accurately (straight computation or word problems).
• Eligible Content - M05.A-T.2.1.1 Multiply multi-digit whole numbers (not to exceed three-digit by three-digit).
• Eligible Content - M05.A-T.2.1.2 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors.
• Eligible Content - M05.A-T.2.1.3 Add, subtract, multiply, and divide decimals to hundredths (no divisors with decimals).
Standard - CC.2.1.5.C.1

Use the understanding of equivalency to add and subtract fractions.

• Assessment Anchor - M05.A-F.1 Use equivalent fractions as a strategy to add and subtract fractions.
• Anchor Descriptor - M05.A-F.1.1 Solve addition and subtraction problems involving fractions (straight computation or word problems).
• Eligible Content - M05.A-F.1.1.1 Add and subtract fractions (including mixed numbers) with unlike denominators. (May include multiple methods and representations.) Example: 2/3 + 5/4 = 8/12 + 15/12 = 23/12
Standard - CC.2.1.5.C.2

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

• Assessment Anchor - M05.A-F.2 Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
• Anchor Descriptor - M05.A-F.2.1 Solve multiplication and division problems involving fractions and whole numbers (straight computation or word problems).
• Eligible Content - M05.A-F.2.1.1 Solve word problems involving division of whole numbers leading to answers in the form of fractions (including mixed numbers).
• Eligible Content - M05.A-F.2.1.2 Multiply a fraction (including mixed numbers) by a fraction.
• Eligible Content - M05.A-F.2.1.3 Demonstrate an understanding of multiplication as scaling (resizing). Example 1: Comparing the size of a product to the size of one factor on the basis of the size of the other factor without performing the indicated multiplication. Example 2: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.
• Eligible Content - M05.A-F.2.1.4 Divide unit fractions by whole numbers and whole numbers by unit fractions.
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.6: GRADE 6
Standard - CC.2.1.6.D.1

Understand ratio concepts and use ratio reasoning to solve problems.

• Assessment Anchor - M06.A-R.1 Understand ratio concepts and use ratio reasoning to solve problems.
• Anchor Descriptor - M06.A-R.1.1 Represent and/or solve realworld and mathematical problems using rates, ratios, and/or percents.
• Eligible Content - M06.A-R.1.1.1 Use ratio language and notation (such as 3 to 4, 3:4, 3/4) to describe a ratio relationship between two quantities. Example 1: “The ratio of girls to boys in a math class is 2:3 because for every 2 girls there are 3 boys.” Example 2: “For every five votes candidate A received, candidate B received four votes.”
• Eligible Content - M06.A-R.1.1.2 Find the unit rate a/b associated with a ratio a:b (with b ≠ 0) and use rate language in the context of a ratio relationship. Example 1: “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” Example 2: “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.”
• Eligible Content - M06.A-R.1.1.3 Construct tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and/or plot the pairs of values on the coordinate plane. Use tables to compare ratios.
• Eligible Content - M06.A-R.1.1.4 Solve unit rate problems including those involving unit pricing and constant speed. Example: If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
• Eligible Content - M06.A-R.1.1.5 Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percentage.
Standard - CC.2.1.6.E.1

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• Assessment Anchor - M06.A-N.1 Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
• Anchor Descriptor - M06.A-N.1.1 Solve real-world and mathematical problems involving division of fractions.
• Eligible Content - M06.A-N.1.1.1 Interpret and compute quotients of fractions (including mixed numbers), and solve word problems involving division of fractions by fractions. Example 1: Given a story context for (2/3) ÷ (3/4), explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.) Example 2: How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Example 3: How many 2 1/4-foot pieces can be cut from a 15 1/2-foot board?
Standard - CC.2.1.6.E.2

Identify and choose appropriate processes to compute fluently with multi-digit numbers.

• Assessment Anchor - M06.A-N.2 Compute with multi-digit numbers and find common factors and multiples.
• Anchor Descriptor - M06.A-N.2.1 Compute with multi-digit numbers using the four arithmetic operations with or without a calculator.
• Eligible Content - M06.A-N.2.1.1 Solve problems involving operations (+, –, ×, and ÷) with whole numbers, decimals (through thousandths), straight computation, or word problems.
• Anchor Descriptor - M06.A-N.2.2 Apply number theory concepts (specifically, factors and multiples).
• Eligible Content - M06.A-N.2.2.1 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12.
• Eligible Content - M06.A-N.2.2.2 Apply the distributive property to express a sum of two whole numbers, 1 through 100, with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: Express 36 + 8 as 4(9 + 2).
Standard - CC.2.1.6.E.3

Develop and/or apply number theory concepts to find common factors and multiples.

• Assessment Anchor - A1.1.1

Operations with Real Numbers and Expressions

• Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
• Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
• Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
• Eligible Content - A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10).
• Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
• Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
• Eligible Content - A1.1.1.5.1 Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).
• Eligible Content - A1.1.1.5.2

Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is equal to 1 after factoring out all monomial factors).

• Eligible Content - A1.1.1.5.3 Simplify/reduce a rational algebraic expression.
• Assessment Anchor - M06.A-N.2 Compute with multi-digit numbers and find common factors and multiples.
• Anchor Descriptor - M06.A-N.2.1 Compute with multi-digit numbers using the four arithmetic operations with or without a calculator.
• Eligible Content - M06.A-N.2.1.1 Solve problems involving operations (+, –, ×, and ÷) with whole numbers, decimals (through thousandths), straight computation, or word problems.
• Anchor Descriptor - M06.A-N.2.2 Apply number theory concepts (specifically, factors and multiples).
• Eligible Content - M06.A-N.2.2.1 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12.
• Eligible Content - M06.A-N.2.2.2 Apply the distributive property to express a sum of two whole numbers, 1 through 100, with a common factor as a multiple of a sum of two whole numbers with no common factor. Example: Express 36 + 8 as 4(9 + 2).
Standard - CC.2.1.6.E.4

Apply and extend previous understandings of numbers to the system of rational numbers.

• Assessment Anchor - M06.A-N.3 Apply and extend previous understandings of numbers to the system of rational numbers.
• Anchor Descriptor - M06.A-N.3.1 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values and locations on the number line and coordinate plane.
• Eligible Content - M06.A-N.3.1.1 Represent quantities in real-world contexts using positive and negative numbers, explaining the meaning of 0 in each situation (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge).
• Eligible Content - M06.A-N.3.1.2 Determine the opposite of a number and recognize that the opposite of the opposite of a number is the number itself (e.g., –(–3) = 3; 0 is its own opposite).
• Eligible Content - M06.A-N.3.1.3 Locate and plot integers and other rational numbers on a horizontal or vertical number line; locate and plot pairs of integers and other rational numbers on a coordinate plane.
• Anchor Descriptor - M06.A-N.3.2 Understand ordering and absolute value of rational numbers.
• Eligible Content - M06.A-N.3.2.1 Write, interpret, and explain statements of order for rational numbers in real-world contexts. Example: Write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.
• Eligible Content - M06.A-N.3.2.2 Interpret the absolute value of a rational number as its distance from 0 on the number line and as a magnitude for a positive or negative quantity in a real-world situation. Example: For an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars, and recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
• Eligible Content - M06.A-N.3.2.3 Solve real-world and mathematical problems by plotting points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.7: GRADE 7
Standard - CC.2.1.7.D.1

Analyze proportional relationships and use them to model and solve real-world and mathematical problems.

• Assessment Anchor - M07.A-R.1 Demonstrate an understanding of proportional relationships.
• Anchor Descriptor - M07.A-R.1.1 Analyze, recognize, and represent proportional relationships and use them to solve real-world and mathematical problems.
• Eligible Content - M07.A-R.1.1.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. Example: If a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2 / 1/4 miles per hour, equivalently 2 miles per hour.
• Eligible Content - M07.A-R.1.1.2 Determine whether two quantities are proportionally related (e.g., by testing for equivalent ratios in a table, graphing on a coordinate plane and observing whether the graph is a straight line through the origin).
• Eligible Content - M07.A-R.1.1.3 Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• Eligible Content - M07.A-R.1.1.4 Represent proportional relationships by equations. Example: If total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
• Eligible Content - M07.A-R.1.1.5 Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r), where r is the unit rate.
• Eligible Content - M07.A-R.1.1.6 Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease.
Standard - CC.2.1.7.E.1

Apply and extend previous understandings of operations with fractions to operations with rational numbers.

• Assessment Anchor - M07.A-N.1 Apply and extend previous understandings of operations to add, subtract, multiply, and divide rational numbers.
• Anchor Descriptor - M07.A-N.1.1 Solve real-world and mathematical problems involving the four operations with rational numbers.
• Eligible Content - M07.A-N.1.1.1 Apply properties of operations to add and subtract rational numbers, including real-world contexts.
• Eligible Content - M07.A-N.1.1.2 Represent addition and subtraction on a horizontal or vertical number line.
• Eligible Content - M07.A-N.1.1.3 Apply properties of operations to multiply and divide rational numbers, including real-world contexts; demonstrate that the decimal form of a rational number terminates or eventually repeats.
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.8: GRADE 8
Standard - CC.2.1.8.E.1

Distinguish between rational and irrational numbers using their properties.

• Assessment Anchor - A1.1.1

Operations with Real Numbers and Expressions

• Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
• Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
• Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
• Eligible Content - A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10).
• Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
• Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
• Eligible Content - A1.1.1.5.1 Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).
• Eligible Content - A1.1.1.5.2

Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is equal to 1 after factoring out all monomial factors).

• Eligible Content - A1.1.1.5.3 Simplify/reduce a rational algebraic expression.
• Assessment Anchor - M08.A-N.1 Demonstrate an understanding of rational and irrational numbers.
• Anchor Descriptor - M08.A-N.1.1 Apply concepts of rational and irrational numbers.
• Eligible Content - M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths).
• Eligible Content - M08.A-N.1.1.2 Convert a terminating or repeating decimal to a rational number (limit repeating decimals to thousandths).
• Eligible Content - M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). Example: √5 is between 2 and 3 but closer to 2.
• Eligible Content - M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational numbers.
• Eligible Content - M08.A-N.1.1.5 Locate/identify rational and irrational numbers at their approximate locations on a number line.
Standard - CC.2.1.8.E.4

Estimate irrational numbers by comparing them to rational numbers.

• Assessment Anchor - A1.1.1

Operations with Real Numbers and Expressions

• Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
• Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
• Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
• Eligible Content - A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10).
• Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
• Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
• Eligible Content - A1.1.1.5.1 Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).
• Eligible Content - A1.1.1.5.2

Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is equal to 1 after factoring out all monomial factors).

• Eligible Content - A1.1.1.5.3 Simplify/reduce a rational algebraic expression.
• Assessment Anchor - M08.A-N.1 Demonstrate an understanding of rational and irrational numbers.
• Anchor Descriptor - M08.A-N.1.1 Apply concepts of rational and irrational numbers.
• Eligible Content - M08.A-N.1.1.1 Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths).
• Eligible Content - M08.A-N.1.1.2 Convert a terminating or repeating decimal to a rational number (limit repeating decimals to thousandths).
• Eligible Content - M08.A-N.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). Example: √5 is between 2 and 3 but closer to 2.
• Eligible Content - M08.A-N.1.1.4 Use rational approximations of irrational numbers to compare and order irrational numbers.
• Eligible Content - M08.A-N.1.1.5 Locate/identify rational and irrational numbers at their approximate locations on a number line.
• Standard Area - CC.2.1: Numbers and Operations
• Grade Level - CC.2.1.HS: GRADE High School
Standard - CC.2.1.HS.F.1

Apply and extend the properties of exponents to solve problems with rational exponents

• Assessment Anchor - A1.1.1

Operations with Real Numbers and Expressions

• Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
• Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
• Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
• Eligible Content - A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10).
• Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
• Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
• Eligible Content - A1.1.1.5.1 Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).
• Eligible Content - A1.1.1.5.2

Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is equal to 1 after factoring out all monomial factors).

• Eligible Content - A1.1.1.5.3 Simplify/reduce a rational algebraic expression.
• Assessment Anchor - A2.1.2

Non-Linear Expressions

• Anchor Descriptor - A2.1.2.1 Use exponents, roots and/or absolute values to represent equivalent forms or to solve problems.
• Eligible Content - A2.1.2.1.1 Use exponential expressions to represent rational numbers.
• Eligible Content - A2.1.2.1.2 Simplify/evaluate expressions involving positive and negative exponents and/or roots (may contain all types of real numbers - exponents should not exceed power of 10).
• Eligible Content - A2.1.2.1.3

Simplify/evaluate expressions involving multiplying with exponents (e.g. x6 * x7 = x13), powers of powers (e.g., (x6)7=x42) and powers of products (2x2)3=8x6 (limit to rational exponents).

• Eligible Content - A2.1.2.1.4

Simplify or evaluate expressions involving logarithms and exponents (e.g. log28 = 3 or log42 = ½).

• Anchor Descriptor - A2.1.2.2 Simplify expressions involving polynomials.
• Eligible Content - A2.1.2.2.1

Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is not equal to 0).

• Eligible Content - A2.1.2.2.2 Simplify rational algebraic expressions.
Standard - CC.2.1.HS.F.2

Apply properties of rational and irrational numbers to solve real world or mathematical problems.

• Assessment Anchor - A1.1.1

Operations with Real Numbers and Expressions

• Anchor Descriptor - A1.1.1.1 Represent and/or use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, square roots, and exponents).
• Anchor Descriptor - A1.1.1.2 Apply number theory concepts to show relationships between real numbers in problem solving settings.
• Anchor Descriptor - A1.1.1.3 Use exponents, roots and/or absolute value to solve problems.
• Eligible Content - A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots and/or absolute value to solve problems (exponents should be integers from -10 to 10).
• Anchor Descriptor - A1.1.1.4 Use estimation strategies in problem-solving situations.
• Anchor Descriptor - A1.1.1.5 Simplify expressions involving polynomials.
• Eligible Content - A1.1.1.5.1 Add, subtract and/or multiply polynomial expressions (express answers in simplest form – nothing larger than a binomial multiplied by a trinomial).
• Eligible Content - A1.1.1.5.2

Factor algebraic expressions, including difference of squares and trinomials (trinomials limited to the form ax2+bx+c where a is equal to 1 after factoring out all monomial factors).

• Eligible Content - A1.1.1.5.3 Simplify/reduce a rational algebraic expression.
• Alternate Eligible Content - CC.2.1.HS.F.2.a Convert between fractions and decimals in a real-world problem.
Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
Standard - CC.2.1.HS.F.3

Apply quantitative reasoning to choose and Interpret units and scales in formulas, graphs and data displays.

• Assessment Anchor - A1.1.2

Linear Equations

• Anchor Descriptor - A1.1.2.1 Write, solve and/or graph linear equations and inequalities using various methods.
• Anchor Descriptor - A1.1.2.2 Write, solve and/or graph systems of linear equations using various methods.
• Eligible Content - A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
• Eligible Content - A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation (systems of 2 linear equations only).
• Assessment Anchor - A1.2.1

Functions

• Anchor Descriptor - A1.2.1.1 Analyze and/or use patterns or relations.
• Anchor Descriptor - A1.2.1.2 Interpret and/or use linear functions and their equations, graphs or tables.
• Assessment Anchor - A2.2.2

Applications of Functions

• Anchor Descriptor - A2.2.2.1 Create, interpret and/or use polynomial, exponential and/or logarithmic functions and their equations, graphs or tables.
• Eligible Content - A2.2.2.1.1 Create, interpret and/or use the equation, graph or table of a polynomial function (including quadratics).
• Eligible Content - A2.2.2.1.2 Create, interpret and/or use the equation, graph or table of an exponential or logarithmic function (including common and natural logarithms).
• Eligible Content - A2.2.2.1.3

Determine, use and/or interpret minimum and maximum values over a specified interval of a graph of a polynomial, exponential or logarithmic function.

• Eligible Content - A2.2.2.1.4

Translate a polynomial, exponential or logarithmic function from one representation to another (graph, table and equation).

• Anchor Descriptor - A2.2.2.2 Describe and/or determine families of functions.
• Eligible Content - A2.2.2.2.1

Identify or describe the effect of changing parameters within a family of functions (e.g., y = x2 and y = x2 + 3, or y = x2 and y = 3x2).

• Assessment Anchor - A2.2.3

Data Analysis

• Anchor Descriptor - A2.2.3.1 Analyze and/or interpret data on a scatter plot and/or use a scatter plot to make predictions.
• Eligible Content - A2.2.3.1.1 Draw, identify, find and/or write an equation for a regression model (lines and curves of best fit) for a scatter plot.
• Eligible Content - A2.2.3.1.2 Make predictions using the equations or graphs of regression models (lines and curves of best fit) of scatter plots.
• Anchor Descriptor - A2.2.3.2 Apply probability to practical situations.
• Eligible Content - A2.2.3.2.1 Use combinations, permutations, and the fundamental counting principle to solve problems.
• Eligible Content - A2.2.3.2.2 Use odds to find probability and/or use probability to find odds.
• Eligible Content - A2.2.3.2.3 Use probability for independent, dependent or compound events to predict outcomes.
• Alternate Eligible Content - CC.2.1.HS.F.3.a Identify and interpret scale in a real-world problem.
Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
Standard - CC.2.1.HS.F.4

Use units as a way to understand problems and to guide the solution of multi-step problems.

• Assessment Anchor - A1.1.2

Linear Equations

• Anchor Descriptor - A1.1.2.1 Write, solve and/or graph linear equations and inequalities using various methods.
• Anchor Descriptor - A1.1.2.2 Write, solve and/or graph systems of linear equations using various methods.
• Eligible Content - A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
• Eligible Content - A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation (systems of 2 linear equations only).
• Assessment Anchor - A1.2.1

Functions

• Anchor Descriptor - A1.2.1.1 Analyze and/or use patterns or relations.
• Anchor Descriptor - A1.2.1.2 Interpret and/or use linear functions and their equations, graphs or tables.
• Assessment Anchor - A2.2.2

Applications of Functions

• Anchor Descriptor - A2.2.2.1 Create, interpret and/or use polynomial, exponential and/or logarithmic functions and their equations, graphs or tables.
• Eligible Content - A2.2.2.1.1 Create, interpret and/or use the equation, graph or table of a polynomial function (including quadratics).
• Eligible Content - A2.2.2.1.2 Create, interpret and/or use the equation, graph or table of an exponential or logarithmic function (including common and natural logarithms).
• Eligible Content - A2.2.2.1.3

Determine, use and/or interpret minimum and maximum values over a specified interval of a graph of a polynomial, exponential or logarithmic function.

• Eligible Content - A2.2.2.1.4

Translate a polynomial, exponential or logarithmic function from one representation to another (graph, table and equation).

• Anchor Descriptor - A2.2.2.2 Describe and/or determine families of functions.
• Eligible Content - A2.2.2.2.1

Identify or describe the effect of changing parameters within a family of functions (e.g., y = x2 and y = x2 + 3, or y = x2 and y = 3x2).

• Alternate Eligible Content - CC.2.1.HS.F.4.a Determine the necessary units and solve a real-world problem.
Alternate Eligible Content is designed for students assessed using the PA Alternate System of Assessment (PASA). Essentialized Example resources assist teachers in designing instruction that reduces content complexity while maintaining alignment to academic standards.
Standard - CC.2.1.HS.F.5

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

• Assessment Anchor - A1.1.2

Linear Equations

• Anchor Descriptor - A1.1.2.1 Write, solve and/or graph linear equations and inequalities using various methods.
• Anchor Descriptor - A1.1.2.2 Write, solve and/or graph systems of linear equations using various methods.
• Eligible Content - A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution and/or elimination (limit systems to 2 linear equations).
• Eligible Content - A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation (systems of 2 linear equations only).
• Assessment Anchor - A1.1.3

Linear Inequalities

• Anchor Descriptor - A1.1.3.1 Write, solve and/or graph linear inequalities using various methods.
• Anchor Descriptor - A1.1.3.2 Write, solve and/or graph systems of linear inequalities using various methods.
• Assessment Anchor - A2.2.3

Data Analysis

• Anchor Descriptor - A2.2.3.1 Analyze and/or interpret data on a scatter plot and/or use a scatter plot to make predictions.
• Eligible Content - A2.2.3.1.1 Draw, identify, find and/or write an equation for a regression model (lines and curves of best fit) for a scatter plot.
• Eligible Content - A2.2.3.1.2 Make predictions using the equations or graphs of regression models (lines and curves of best fit) of scatter plots.
• Anchor Descriptor - A2.2.3.2 Apply probability to practical situations.
• Eligible Content - A2.2.3.2.1 Use combinations, permutations, and the fundamental counting principle to solve problems.
• Eligible Content - A2.2.3.2.2 Use odds to find probability and/or use probability to find odds.
• Eligible Content - A2.2.3.2.3 Use probability for independent, dependent or compound events to predict outcomes.
Standard - CC.2.1.HS.F.6

Extend the knowledge of arithmetic operations and apply to complex numbers.

• Assessment Anchor - A2.1.1

Operations with Complex Numbers

• Anchor Descriptor - A2.1.1.1 Represent and/or use imaginary numbers in equivalent forms (e.g., square roots and exponents).
• Eligible Content - A2.1.1.1.1

Simplify square roots in terms of i. (e.g., √–24 = 2i√6).

• Eligible Content - A2.1.1.1.2

Simplify/evaluate expressions involving imaginary numbers powers of i (e.g., i6 + i3 = –1 + i).

• Anchor Descriptor - A2.1.1.2 Apply the order of operations in computation and in problem-solving situations.
• Eligible Content - A2.1.1.2.1 Add and subtract complex numbers(e.g., (7 – 3i) – (2 + i) = 5 – 4i).
• Eligible Content - A2.1.1.2.2 Multiply and divide complex numbers (e.g., (7 – 3i)(2 + i) = 17 + i).
Standard - CC.2.1.HS.F.7

Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems.

• Assessment Anchor - A2.2.1

Patterns, Relations, and Functions

• Anchor Descriptor - A2.2.1.1 Analyze and/or use patterns or relations.
• Eligible Content - A2.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern with a rule algebraically and/or graphically.
• Eligible Content - A2.2.1.1.2

Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, find the 20th term).

• Eligible Content - A2.2.1.1.3 Determine the domain, range or inverse of a relation.
• Eligible Content - A2.2.1.1.4 Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increasing/decreasing, intercepts, zeros, and asymptotes).