# Math

##### Rigor requires that conceptual understanding, procedural skill and fluency, and application be approached with equal intensity. Teachers help students make steady progress toward procedural skills and computational fluency. Teachers also support students' ability to access concepts from a number of different perspectives and to apply mathematics to solve real-world problems.

Kindergarten students develop an understanding of the relationship between numbers, quantities, and counting. Instructional time focuses on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects; and (2) describing shapes and space. Kindergarten students also work toward fluency with addition and subtraction of whole numbers within 5.

In grade one, students develop the concept of place value by viewing 10 ones as a unit called a ten, an essential concept in the base-ten number system. In grade one, instructional time should focuses on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole-number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of and composing and decomposing geometric shapes. Students also work toward fluency in addition and subtraction with whole numbers within 10.

In grade two, students continue to build upon their mathematical foundation as they focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes. Students also work toward fluency with addition and subtraction within 20 using mental strategies and within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction. They know from memory all sums of two one-digit numbers.

In grade three, students continue to build upon their mathematical foundation as they focus on four critical areas: (1) developing understanding of multiplication and division, as well as strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with a numerator of 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes. Students also work toward fluency with addition and subtraction within 1000 and multiplication and division within 100. By the end of grade three, students know all products of two one-digit numbers from memory.

In grade four, students continue to build a strong foundation for higher mathematics and focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; and (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. Students also work toward fluency in addition and subtraction within 1,000,000 using the standard algorithm.

In grade five, students continue to build a strong foundation for higher mathematics and focus on three critical areas: (1) developing fluency with addition and subtraction of fractions and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to two-digit divisors, integrating decimal fractions into the place-value system, developing understanding of operations with decimals to hundredths, and developing fluency with whole-number and decimal operations; and (3) developing understanding of volume. Students also fluently multiply multi-digit whole numbers using the standard algorithm.

Grade six is an especially important year for bridging the concrete concepts of arithmetic and the abstract thinking of algebra. Students complete developing their skills with operations on rational numbers and delve into the multiplicative thinking demanded by proportional reasoning. Instructional time focuses on four critical areas: (1) connecting ratio, rate, and percentage to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. Students also work toward fluency with multi-digit division and multi-digit decimal operations.

In grade seven, instructional time focuses on four critical areas: (1) developing understanding of and applying proportional relationships, including percentages; (2) developing understanding of operations with rational numbers (i.e., positive and negative numbers) and working with expressions and linear equations; (3) solving problems that involve scale drawings and informal geometric constructions and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples. Students also work toward fluently solving equations, such as 7x + 2.5 = 13 and ½(x + 5) = 34.

In grade eight, instructional time focuses on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling the relationship between two quantities (e.g., absences and math scores) with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence and understanding and applying the Pythagorean Theorem. Students also work towards fluency with solving sets of two equations with two unknowns.