Standards NCTM Curriculum Focal Points The National Council of Teachers of Mathematics (NCTM) created its Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006) to provide descriptions of the most significant mathematical concepts and skills at each grade level. These focal points were chosen according to three criteria: they are mathematically important, both for further study in mathematics and for use in applications in and outside of school; they conform to what is known about learning mathematics; they connect logically with the mathematics in earlier and later grade levels. According to the NCTM, “these curriculum focal points should be considered as major instructional goals and desirable learning expectations, not as a list of objectives for students to master.” It is important that none of the focal points be omitted; it is less important whether they are addressed a year earlier or later, or over a period of two years, compared with the suggested sequence in the document. On this page, each of the NCTM focal points is followed (in parentheses) by the corresponding New York State performance indicators. Focal Points Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or xcoordinate, changes by a specific amount, a, the output, or ycoordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and yintercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems. (7.A.78, 7.A.10) Geometry and Measurement: Analyzing two and threedimensional space and figures by using distance and angle Students use fundamental facts about distance and angles to describe and analyze figures and situations in two and threedimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra. (5.G.78, 7.G.59) Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose. (6.S.2, 6.S.48, 7.S.16)
Connections to the Focal Points Academic Language PreK8 Glossary of Mathematical Terms Development of Content Topics and Concepts 
Number of Content Performance Indicators by Grade Level  
Content Strand  PreK  K  TOTAL  
Number Sense and Operations Algebra 
9 1 
13

207 56 

Geometry Measurement 
2 2 
5 3 
87 80 

Statistics and Probability TOTAL 
4 18 
5 28 
67 497 
Mathematics Scope and Sequence – Middle School
Grade 8
SeptemberOctober  NovemberDecember  January March  PostMarch  
N U M B E R S E N S E A N D O P E R A T I O N S S T R A N D 
Operations 8.N.3 Read, write, and identify percents less than 1% and greater than 100% 8.N.4 Apply percents to: Tax, percent increase/decrease, simple interest, sale price, commission, interest rates, and gratuities Estimation 8.N.6 Justify the reasonableness of answers using estimation 
Operations 8.N.1 Develop and apply the laws of exponents for multiplication and division 8.N.2 Evaluate expressions with integral exponents 8.N. 3 Read, write, and identify percents less than 1% and greater than 100% 8.N.4 Apply percents to: Tax, percent increase/decrease, simple interest, sale price, commission, interest rates, and gratuities Estimation 8.N.6 Justify the reasonableness of answers using estimation 
Operations 8.N.4 Apply percents to: Tax, percent increase/decrease, simple interest, sale price, commission, interest rates, and gratuities Estimation 8.N.5 Estimate a percent of a quantity, given an application


A L G E B R A S T R A N D 
Variables and Expressions 8.A. 3 Describe a situation involving relationships that matches a given graph 8.A.4 Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship Patterns, Relations and Functions 8.A.15 Understand that numerical information can be represented in multiple ways: arithmetically, algebraically and graphically 8.A.16 Find sets of ordered pairs to satisfy an algebraic expression and then plot the ordered pairs and draw the line

Variables and Expressions 8.A.5 Use physical models to perform operations with polynomials (uses area of rectangles as a way to think about the distributive property) 8.A.6 Multiply and divide monomials 8.A.7 Add and subtract polynomials (integer coefficients) 8.A.8 Multiply a binomial by a monomial or binomial (integer coefficients) 8.A.9 Divide a polynomial by a monomial (integer coefficients) Equations and Inequalities 8.A.12 Apply algebra to determine the measure of angles formed by or contained in parallel lines cut by a transversal and by intersecting lines 
Variables and Expressions 8.A.1 Translate verbal sentences into algebraic inequalities 8.A. 2 Write verbal expressions that match given mathematical expressions 8.A.3 Describe a situation involving relationships that matches a given graph 8.A.5 Use physical models to perform operations with polynomials 8.A.10 Factor algebraic expressions using the GCF 8.A.11 Factor a trinomial in the form ax²+bx+c; a=1 and c having no more than 3 sets of factors Equations and Inequalities 8.A.13* Solve multistep inequalities and graph the solution set on a number line 8.A.14* Solve linear inequalities by combining like terms, using the distributive property, or moving variables to one side of the inequality (include multiplication or division of inequalities by a negative number) Patterns, Relations and Functions 8.A.15 Understand that numerical information can be represented in multiple ways: arithmetically, algebraically and graphically 
Patterns, Relations and Functions 8.A.17* Define and use correct terminology when referring to function (domain and range) 8.A.18* Determine if a relation is a function 8.A.19* Interpret multiple representations using equation, table of values and graph 
G E O M E T R Y S T R A N D 
Geometric Relationships 8.G.1 Identify pairs of vertical angles as congruent 8.G.2 Identify pairs of supplementary and complementary angles 8.G.3 Calculate the missing angle in a supplementary or complementary pair 8.G.4 Determine angle pair relationship when given two parallel lines cut by a transversal 8.G.5 Calculate the missing angle measurements when given two parallel lines cut by a transversal 8.G.6 Calculate the missing angle measurements when given two intersecting lines and an angle Coordinate Geometry 8.G.13* Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change 8.G.14* Determine the yintercept of a line from a graph and be able to explain the yintercept 8.G.15* Graph a line using a table of values 8.G. 16* Determine the equation of a line given the slope and the yintercept 8.G.17* Graph a line from an equation in slopeintercept form (y= mx + b) 
Transformational Geometry 8.G.7 Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations.) 8G.8 Draw the image of a figure under rotations of 90 and 180 degrees 8.G.9 Draw the image of a figure under a reflection over a given line 8.G.10 Draw the image of a figure under a translation 8.G.11 Draw the image of a figure under a dilation 8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation Coordinate Geometry 8.G.18* Solve systems of equations graphically (only linear, integral solutions, y=mx +b format, no vertical or horizontal lines) 8.G.19* Graph the solution set of an inequality on a number line 8.G.20* Distinguish between linear and nonlinear equations ax²+bx+c; a=1 (only graphically) 8.G.21* Recognize the characteristics of quadratics in tables, graphs, equations, and situations 
8.G.0*Construct the following using a straight edge and compass: Segment congruent to a segment; angle congruent to an angle; perpendicular bisector, and angle bisector Coordinate Geometry 8.G.20* Distinguish between linear and nonlinear equations ax²+bx+c; a=1 (only graphically) 8.G.21* Recognize the characteristics of quadratics in tables, graphs, equations, and situations 

M E A S U R E M E N T S T R A N D 
Units of Measurement 8.M.1 Solve equations/ proportions to convert to equivalent measurements within metric and customary measurement systems. 
