in 2011 Mathematics Frameworks Pre-Kindergarten Kindergarten Developing an understanding of whole numbers to 10, including concepts of one-to-one correspondence, counting, cardinality (the number of items in a set), and comparison. Representing, relating, and operating on whole numbers, initially with sets of objects. Describing shapes and space. Recognizing two-dimensional shapes, describing spatial relationships, and sorting and classifying objects by one or more attributes. Relatively: more learning time should be devoted to numbers than to other topics. Relatively: more learning time should be devoted to developing children s sense of number as quantity than to other mathematics topics. Comeau-Pittsfield 1
Grade 1 Grade 2 Developing understanding of addition, subtraction. Developing strategies for addition and subtraction within 20. Extending understanding of base-ten notation. Building fluency with addition and subtraction. Developing understanding of whole number relationships. Developing understanding place value, including grouping in tens and ones. Developing understanding of linear measurement. Developing understanding measuring lengths as iterating length units. Using standard units of measure. Describing and analyzing shapes. Reasoning about attributes of geometric shapes. Composing and decomposing geometric shapes. Comeau-Pittsfield 2
Grade 3 Grade 4 Developing understanding of multiplication and division Developing strategies for multiplication and division within 100. Developing understanding of fractions, especially unit fractions (fractions with the numerator 1). Developing understanding of the structure of rectangular arrays Developing understanding of area. Describing and analyzing twodimensional shapes. Developing understanding and fluency with multi-digit multiplication Developing understanding of dividing to find quotients involving multi-digit dividends. Developing an understanding of: fraction equivalence. addition and subtraction of fractions with like denominators. multiplication of fractions by whole numbers. 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. Comeau-Pittsfield 3
Grade 5 Grade 6 Developing fluency with addition and subtraction of fractions 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). Extending division to 2-digit divisors, integrating decimal fractions into the place value system. Developing understanding of operations with decimals to hundredths. Developing fluency with whole number and decimal operations. Develop understanding of volume Connecting ratio and rate to whole number multiplication and division Using concepts of ratio and rate to solve problems. Completing understanding of division of fractions. Extending the notion of number to the system of rational numbers, which includes negative numbers. Writing, interpreting, and using expressions and equations. Developing understanding of statistical thinking. Additional: Geometry in Grade 6 Comeau-Pittsfield 4
Grade 7 Grade 8 Developing understanding of and applying proportional relationships. Developing understanding of operations with rational numbers and working with expressions and linear equations. Solving problems involving scale drawings and informal geometric constructions. Working with two- and threedimensional shapes to solve problems involving area, surface area, and volume. Drawing inferences about populations based on samples. Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation. Solving linear equations and systems of linear equations. Grasping the concept of a function. Using functions to describe quantitative relationships. Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence. Understanding and applying the Pythagorean Theorem. Comeau-Pittsfield 5
Model Algebra 1 Model Geometry Areas Critical Deepen and extend understanding of linear and exponential relationships. Contrast linear and exponential relationships with each other. Engage in methods for analyzing, solving, and using quadratic functions. Extend the laws of exponents to square and cube roots. Apply linear models to data that exhibit a linear trend. Establish criteria for congruence of triangles based on rigid motions. Establish criteria for similarity of triangles based on dilations and proportional reasoning. Informally develop explanations of circumference, area, and volume formulas Apply the Pythagorean Theorem to the coordinate plane. 5. Prove basic geometric theorems. 6. Extend work with probability. Comeau-Pittsfield 6
Model Algebra II Pre-Calculus Relate arithmetic of rational expressions to arithmetic of rational numbers. Expand understandings of functions and graphing to include trigonometric functions. Synthesize and generalize functions. Extend understanding of exponential functions to logarithmic functions. Relate data display and summary statistics to probability. Explore a variety of data collection methods. Extend work with complex numbers. Expand understanding of logarithms and trigonometric functions. Use characteristics of polynomial and rational functions to sketch graphs of those functions. Perform operations with vectors. Comeau-Pittsfield 7