9 th Grade Algebra Exploring Expressions, Equations, and Functions Translate verbal expressions into mathematical expressions containing variables. Solve problems by looking for a pattern. Use the order of operations to evaluate real number expressions. Evaluate algebraic expressions and formulas by substituting real numbers. Use, stem-and-leaf plots to interpret data. Solve open sentences performing arithmetic operations. Apply the associative, commutative, and distributive properties to simplify computations with real numbers (NAEP 12N5f). USE PROPERTIES OF NUMBERS TO DEMONSTRATE WHETHER ASSERTIONS ARE TRUE OR FALSE (NCTM 4N1d). Exploring Rational Numbers September State the coordinates of a point on a number line. Graph integers on a number line. Add integers using the number line. Interpret numerical data from a table. Display and interpret data on a line plot. October Use matrices, to interpret data Find the absolute value of a number. Add and subtract integers and rational numbers. Multiply and divide Rational numbers. Find square roots. Graph solutions of inequalities on a number line. Compare real numbers (NAEP 12N1c). Locate real numbers on a number line (including approximations of irrational numbers) (NAEP 8A2e). RECOGNIZE THAT THE SET OF REAL NUMBERS CONTAINS THE SET OF IRRATIONAL NUMBERS AND THE SET OF RATIONAL NUMBERS AND KNOW THE DIFFERENCE BETWEEN THEM (NCTM N4lb). Apply equivalent forms of real numbers to solve problems. DEVELOP AND use strategies to estimate the results of real number computations, determine the amount of error, AND JUDGE THE REASONABLENESS OF RESULTS (NAEP 12N2c) (NCTM 4N3b).
Solving Linear Equations Solve equations by using addition and subtraction. Solve equations by using multiplication and division. November Solve equations involving more than one operation. Solve problem that involve supplementary and complementary angles, and the sum of the angles of a triangle. DETERMINE THE PRECISION OF A GIVEN MEASURING TOOL (E.G., 1 DEGREE FOR A STANDARD PROTRACTOR) (NCTM 4M2a). Solving equations with the variable on both sides and equations with grouping symbols. Solving equations and formulas for a specified variable. Estimate from a graph or a set of data the mean and standard deviation of a normal distribution and DRAW CONCLUSIONS ABOUT THE DISTRIBUTION OF DATA USING MEASURES OF CENTER AND SPREAD (e.g., analyze a variety of summary statistics and graphical displays) (NAEP 12S2a) (NCTM 4D2a). Using Proportional Reasoning Solve proportions. Find the unknown measures of the sides of two similar triangles. Use trigonometric ratios to solve right triangles. Solve percent problems. Solve problems involving simple interest. Solve problems involving percent of change. Determine probabilities of dependent and independent events (e.g., use colored marbles with and without replacement) (NAEP 12S4f) (NCTM 4D4d) (NCTM 4D4d). December Solve mixture problems. Solve problems involving uniform motion. Solve problems involving direct and inverse variation Graphing Relations and Functions Graph ordered pairs on a coordinate plane. Solve problems by making a table. Identify the domain, range and inverse of a relation. Show relations as sets of ordered pairs, tables, mappings, and graphs. Identify independent and dependent variables and determine the domain and range of a function in a problem situation (NAEP 12A1h).
Approximate and interpret rates of change in graphical and numeric data (NCTM 4A4a). Graph linear equations. Evaluate and make a table for two-variable formulas and match a graph or table of values to its formula (NCTM 4A1f). Write equations to represent relations, given some of the solutions for the equation. Estimate from a graph or a set of data the mean and standard deviation of a normal distribution and DRAW CONCLUSIONS ABOUT THE DISTRIBUTION OF DATA USING MEASURES OF CENTER AND SPREAD (e.g., analyze a variety of summary statistics and graphical displays) (NAEP 12S2a) (NCTM 4D2a). January Analyzing Linear Equations Find the slope of a line, given the coordinates of two points on the line. Writing Equations in Point-Slope and Standard Form. ANALYZE BIVARIATE DATA AND IDENTIFY THE TYPE OF FUNCTION (LINEAR, QUADRATIC, EXPONENTIAL) THAT COULD BE USED TO MODEL THE DATA (NCTM 4D2e) Use scatter plots, to interpret data Determine the x- and y-intercepts of linear graphs from their equations. Write equations in slope - intercept form. Graph a line given any linear equation. Determine if two lines are parallel or perpendicular by their slopes. Write equations of lines passing through a given point, parallel or perpendicular to the graph of a given equation. Find the coordinates of the midpoint of a line segment in the coordinate plane. Solving Linear Inequalities February Solve linear inequalities by using addition and subtraction. Solve linear inequalities by using multiplication and division. Solve linear inequalities involving more than one operation. Solve open sentences involving absolute value and graph the solutions. GRAPH LINEAR INEQUALITIES IN TWO VARIABLES (NAEP 12A4c). Compute the probability of a compound event (e.g., toss a coin three times to find the probability of two heads) (NAEP 12S4a) (NCTM 4D4e) (NCTM 4D4e). Determine ALL POSSIBLE OUTCOMES OF A PARTICULAR EVENT OR ALL POSSIBLE arrangements of objects in a given set by applying counting strategies, combinations, and permutations (NAEP 12S4h) (NCTM 4N1c). Use box-and whisker-plots to interpret data.
Solving Systems of Linear Equations and Inequalities Represent and solve system of linear equations with 2 variables using simultaneous equations and by graphing (NAEP 12A2g) (MAP IVC9) (NCTM 4A2b). Solve systems of inequalities by graphing. Exploring Polynomials Multiply and divide monomials Order and compare numbers expressed in scientific notation to each other and to other forms of real numbers (NAEP 12N1c) March Compute with real numbers, including absolute value and numbers expressed in scientific notation (NAEP 12N3a) (NCTM 4N3a) Compute with integer exponents and whole number roots MENTALLY MULTIPLY AND DIVIDE BY POWERS OF 10 TO ESTIMATE RESULTS OF COMPUTATIONS INVOLVING NUMBERS EXPRESSED IN SCIENTIFIC NOTATION (NCTM 4Nla) Estimate the results of computations with integer powers and roots of real numbers Add and subtracting polynomials. Multiply a polynomial by a monomial. Multiply polynomials. Recognize AND GENERATE equivalent forms for algebraic expressions, including combining like terms and expanding binomials (NAEP 8A3b) (MAP IVA7) (NCTM 4A2a). Using Factoring April Find the prime factorization of integers. Find the greatest common factors for sets of monomials. Use the GCF, the distributive property, and grouping to factor polynomials. Factoring quadratic trinomials. Identify and factor binomials that are the difference of squares. Identify and factor perfect square trinomials. Use the zero product property to solve equations.
RECOGNIZE THAT TAKING THE NTH ROOT OF A NUMBER CORRESPONDS TO PRIME FACTORIZATION (NAEP 8N3f). Exploring Quadratic and Exponential Functions GRAPH QUADRATIC AND EXPONENTIAL EQUATIONS (NCTM 4A1b) Determine and interpret maxima or minima and zeros of quadratic functions, and linear functions where y = constant (NAEP 12A1f) (NCTM 4A1c). ANALYZE BIVARIATE DATA AND IDENTIFY THE TYPE OF FUNCTION (LINEAR, QUADRATIC, EXPONENTIAL) THAT COULD BE USED TO MODEL THE DATA (NCTM 4D2e). Use graphs to solve non-linear equations, including quadratics. Solving Quadratic Equations by Using the Quadratic Formula. Graph exponential functions. Determine if a set of data displays exponential behavior. Compute with integer exponents and whole number roots. May RECOGNIZE THAT TAKING THE NTH ROOT OF A NUMBER CORRESPONDS TO PRIME FACTORIZATION (NAEP 8N3f). Use the inverse operations of nth power and nth root to solve problems AND CHECK SOLUTIONS. Translate between and interpret quadratic and exponential relationships represented by words, symbols, tables, and graphs (NAEP 12A2a) (NCTM 4A1b). Determine when data represented in a table or graph represents a linear, quadratic, or exponential relationship (NAEP 12A2l) (NCTM 4A3a). Analyze how changing a parameter (i.e., k, b) in a quadratic or exponential function of the form y=k^x+b, y=kx^2+b, or y=k(x+b)^2 affects its graph (NAEP 12A1i, 12A1j) (NCTM 4A1d). REPRESENT and generalize sequences resulting from linear, quadratic, and exponential relationships using recursive or explicit formulas tables of values and graphs (NAEP 12A1a, 12A1b, 12A1l) (NCTM 4A1a). Algebraically represent situations and solve problems involving quadratic and exponential equations, including exponential growth and decay (NAEP 12A2m) (NCTM 4A2c). Model situations, make predictions and inferences, and solve problems using linear, quadratic, and exponential functions (NAEP 12A2l, 12A4i) (NCTM 4A3b). Analyze the nature of change of each variable in a non-linear relationship as suggested by a table of values, a graph or a formula (NCTM 4A4a).
10 th Grade Geometry September Discovering Points, Lines, Planes, and Angles Graph ordered pairs on a coordinate plane. Identify collinear points. Identify and model points, lines, and planes. Determine the appropriate units, scales, and tools for problem situations involving measurement (NAEP 12M12a) (NCTM 4M1a). Solve problems involving unit conversions (e.g., mi/hr to ft/sec) given the unit equivalencies (MAP IIB12) (NCTM 4M2d). Using the Pythagorean theorem find the distance between two points. Find the midpoint of a segment. Complete proofs involving segment theorems. Identify and classify angles. Use the angle addition postulate to find the measure of angles. Identify and use congruent angles and the bisector of an angle. Identify and use adjacent, vertical, complementary, supplementary, and linear pairs of angles, and perpendicular lines. Determine what information can and cannot be assumed from a diagram. Connecting Reasoning and Proof PRODUCE A VALID CONJECTURE USING INDUCTIVE REASONING BY GENERALIZING FROM A PATTERN OF OBSERVATIONS. Write an if-then statement and the converse, inverse, and contrapositive on an if-then statement. CONSTRUCT AND JUDGE THE VALIDITY OF A LOGICAL ARGUMENT AND GIVE COUNTEREXAMPLES TO DISPROVE A STATEMENT (NAEP 12G3g) (NCTM 4G1c). Complete proofs involving segment theorems. Complete proofs involving angle theorems. Using Perpendicular and Parallel Lines October Identify the relationship between two lines or two planes. Justify AND USE THEOREMS INVOLVING THE ANGLES FORMED BY PARALLEL LINES CUT BY A TRANSVERSAL (NAEP 12G3g).
DESCRIBE HOW TWO OR MORE OBJECTS ARE RELATED IN SPACE (E.G., SKEW LINES, THE POSSIBLE WAYS THREE PLANES MIGHT INTERSECT). Find the slope of lines and use the slope to identify parallel and perpendicular lines. Prove two lines are parallel based on given angle relationships. Recognize and use distance relationships among points, lines, and planes. Identify points, lines, and planes in spherical geometry. Identifying Congruent Triangles Identify the parts of triangles and classify triangles by their parts. Apply the Angle Sum Theorem and the Exterior Angle Theorem. Determine the measures of corresponding angles, sides, and corresponding parts of congruent and similar figures (NCTM 4G2b). PROVE CONGRUENCE OF TWO TRIANGLES OR THEIR CORRESPONDING COMPONENT PARTS (NCTM 4G1c). Use properties of isosceles and equilateral triangles. Applying Congruent Triangles Identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle. Recognize and use tests for congruence of right triangles. Use indirect reasoning and indirect proof to reach a conclusion. Use angle, side length and triangle inequality relationships to solve problems (NCTM 4G2b). Exploring Quadrilaterals November Recognize and apply the properties of a parallelogram. Recognize and apply the properties of rectangles. Recognize and apply the properties of squares and rhombi. Recognize and apply the properties of trapezoids. December Connecting Proportion and Similarity Recognize and use ratios and proportions.
JUSTIFY AND use theorems involving the properties of triangles, quadrilaterals, circles, and their component parts to verify congruence and similarity (NCTM 4G1c). MAKE AND use scale drawings and models to solve problems (NAEP 12M2f). Identify similar figures and triangles. Recognize and use proportional relationships of corresponding perimeters, altitudes, angle bisectors, and medians of similar triangles. Recognize and describe characteristics of fractals. January Applying Right Triangles and Trigonometry Find the Geometric Mean between two numbers. Use the Pythagorean Theorem and its converse. Use trigonometric functions, and angle and side relationships of special right triangles (30-60-right triangles and isosceles right triangles) to solve for an unknown length and determine distances and solve problems (NCTM 4G1b) (NCTM 4G1d). Find Trigonometric Ratios using right triangles. Use trigonometry to solve problems involving angles of elevation and depression. Use the Law of Sines to solve triangles. Use the Law of Cosines to solve triangles. Analyzing Circles Identify and use parts of circles. DEVELOP, UNDERSTAND, and use the formula for determining arc length (e.g., portion of a circle) (MAP IIIB6). Recognize and use relationships between arcs, chords and diameters. DEVELOP, UNDERSTAND, and apply properties of circles and of inscribed and circumscribed polygons (NAEP 12G3e) (NCTM 4G1a). INVESTIGATE RELATIONSHIPS AMONG CHORDS, SECANTS, TANGENTS, INSCRIBED ANGLES, AND INSCRIBED AND CIRCUMSCRIBED POLYGONS OF CIRCLES. Write and use the equations of a circle in the coordinate plane. Exploring Polygons and Area February Use measures of sides and of interior and exterior angles of polygons to classify figures and solve problems.
Create tessellations using translations and rotations. Find areas of parallelograms. Find areas of triangles, rhombi, and trapezoids. Determine perimeter and area of shapes of circles and polygons (annulus, etc.) in context (NAEP 12M1i) (NCTM 4M4b). Recognize nodes and edges as used in graph theory. March Geometric Probability Use area to solve problems involving geometric probability. Determine ALL POSSIBLE OUTCOMES OF A PARTICULAR EVENT OR ALL POSSIBLE arrangements of objects in a given set by applying counting strategies, combinations, and permutations (NAEP 12S4h) (NCTM 4N1c). Determine appropriate designs for simulations (surveys, observational studies, and experiments) and modeling to study a problem AND CONSTRUCT EMPIRICAL PROBABILITY DISTRIBUTIONS TO REPRESENT RESULTS. Identify examples of populations that are normally distributed (NAEP 12S2e). Make predictions about populations based on reported sample statistics (NAEP 12S3d). UNDERSTAND THAT INFERENCES ABOUT A POPULATION DRAWN FROM A SAMPLE INVOLVE UNCERTAINTY AND THAT THE ROLE OF STATISTICS IS TO MEASURE THAT UNCERTAINTY. Investigating Surface Area and Volume DETERMINE DEFINING PROPERTIES THAT CHARACTERIZE CLASSES OF THREE-DIMENSIONAL FIGURES AND THEIR COMPONENT PARTS (MAP IIIA13, IIIA14, IIIA15, IIIA16) (NCTM 3G1a). RECOGNIZE AND REPRESENT THREE-DIMENSIONAL FIGURES AND THEIR COMPONENT PARTS (NCTM 3G1a). MAKE A MODEL OF A THREE-DIMENSIONAL FIGURE FROM A TWO- DIMENSIONAL DRAWING AND MAKE A TWO-DIMENSIONAL REPRESENTATION OF A THREE-DIMENSIONAL OBJECT THROUGH SCALE DRAWINGS, PERSPECTIVE DRAWINGS, BLUEPRINTS OR COMPUTER SIMULATIONS (NCTM 4G4a).
Recognize representations of three-dimensional objects from different perspectives and identify cross-sections of three-dimensional objects (NAEP 12G4d) (MAP IIIA19) (NCTM 4G4b). Make two-dimensional nets for three-dimensional solids. Find the lateral area and surface area of prisms and cylinders. Find the lateral area and surface area of pyramids and cones. COMPARE AND CONTRAST THE FORMULAS FOR SURFACE AREA AND VOLUME OF CYLINDERS AND CONES. DEVELOP and use strategies and formulas for calculating surface area and volume of cones and spheres (NAEP 12M1k) (MAP IIIB10) (NCTM 4M2c). Use formulas to solve problems involving finding missing dimensions given perimeter, area, surface area and volume of polygons, circles, prisms, pyramids, cones, cylinders, and spheres (NCTM 4M2c). Determine the surface area and volume of a complex figure composed of a combination of two or more geometric figures or a figure derived from a regular solid (e.g., hemisphere, frustum of a cone) (NCTM 4M4b). DETERMINE A SHAPE THAT HAS MINIMUM OR MAXIMUM PERIMETER, AREA, SURFACE AREA, OR VOLUME UNDER SPECIFIED CONDITIONS. Identify congruent or similar solids. Coordinate Geometry April Determine the relative placement (e.g., intersecting, parallel, perpendicular) of two lines on a coordinate plane given the algebraic equations representing them. Calculate slope, distance and midpoint between points with an emphasis on practical applications (USE COORDINATE FORMULAS). Make inferences and predictions from data in histograms, scatter plots, and parallel box plots (NAEP 12S1a, 12S1d) (NCTM 4D1d). May Prove theorems using coordinate proof. Use coordinate geometry to determine whether a figure is symmetrical with respect to a line or a point. Find the magnitude and direction of a vector. Locate a point in space. Investigating Loci and Coordinate Transformations Locate, draw, and describe a locus in a plane or in space.
Find the locus of points that are solutions of a system of equations. MODEL, SKETCH, LABEL AND WHERE APPROPRIATE CONSTRUCT CONES AND SPHERES, AND BASIC ELEMENTS OF GEOMETRIC FIGURES (E.G., ALTITUDES, MIDPOINTS, MEDIANS, ANGLE BISECTORS, AND PERPENDICULAR BISECTORS) USING COMPASS AND STRAIGHTEDGE OR TECHNOLOGY (NCTM 4G4a). Determine whether a given pair of figures on a coordinate plane represent a translation, reflection, rotation and/or dilation (NCTM 4G3a). Determine the image of a figure on a coordinate graph under translations, reflections, and rotations (NCTM 4G3a). Given a figure and its image on a coordinate graph, determine the translation vector or locate the axis of reflection (NCTM 4G3a). Determine the coordinates of AND DRAW the dilation of a figure on a coordinate graph (NCTM 4G3a). Analyze the congruence, similarity, and line or rotational symmetry of figures using transformations (NCTM 3G3b).