#### 4.1: All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

4.1.A: Number Sense

4.1.A.2: Compare and order rational and irrational numbers.

4.1.B: Numerical Operations

4.1.B.4: Understand and apply the laws of exponents to simplify expressions involving numbers raised to powers.

#### 4.2: All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.

4.2.A: Geometric Properties

4.2.A.2: Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).

4.2.A.3: Apply the properties of geometric shapes.

4.2.A.3.1: Parallel lines – transversal, alternate interior angles, corresponding angles

4.2.A.3.2a: Conditions for congruence

4.2.A.3.2c: Triangle Inequality

4.2.A.3.3: Minimal conditions for a shape to be a special quadrilateral

4.2.A.3.4: Circles – arcs, central and inscribed angles, chords, tangents

4.2.A.4: Use reasoning and some form of proof to verify or refute conjectures and theorems.

4.2.A.4.1: Verification or refutation of proposed proofs

4.2.A.4.2: Simple proofs involving congruent triangles

4.2.B: Transforming Shapes

4.2.B.1: Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.

4.2.B.2: Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.

4.2.B.4: Generate and analyze iterative geometric patterns.

4.2.B.4.2: Patterns in areas and perimeters of self-similar figures

4.2.B.4.3: Outcome of extending iterative process indefinitely

4.2.C: Coordinate Geometry

4.2.C.1: Use coordinate geometry to represent and verify properties of lines.

4.2.C.1.1: Distance between two points

4.2.C.1.2: Midpoint and slope of a line segment

4.2.C.1.4: Lines with the same slope are parallel

4.2.C.1.5: Lines that are perpendicular have slopes whose product is –1

4.2.C.2: Show position and represent motion in the coordinate plane using vectors.

4.2.C.2.1: Addition and subtraction of vectors

4.2.E: Measuring Geometric Objects

4.2.E.1: Use techniques of indirect measurement to represent and solve problems.

4.2.E.1.1: Similar triangles

4.2.E.1.2: Pythagorean theorem

4.2.E.1.3: Right triangle trigonometry (sine, cosine, tangent)

4.2.E.2: Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures.

4.2.E.2.1: Approximation of area using grids of different sizes

4.2.E.2.2: Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets

4.2.E.2.3: Estimation of area, perimeter, volume, and surface area

#### 4.3: All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

4.3.A: Patterns

4.3.A.1: Use models and algebraic formulas to represent and analyze sequences and series.

4.3.A.1.1: Explicit formulas for nth terms

4.3.B: Functions and Relationships

4.3.B.1: Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.

4.3.B.2: Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

4.3.B.2.1: Slope of a line or curve

4.3.B.2.2: Domain and range

4.3.B.2.3: Intercepts

4.3.B.2.4: Continuity

4.3.B.2.5: Maximum/minimum

4.3.B.2.6: Estimating roots of equations

4.3.B.2.7: Intersecting points as solutions of systems of equations

4.3.B.2.8: Rates of change

4.3.B.3: Understand and perform transformations on commonly-used functions.

4.3.B.3.1: Translations, reflections, dilations

4.3.B.3.2: Effects on linear and quadratic graphs of parameter changes in equations

4.3.B.4: Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.

4.3.B.4.1: Linear vs. non-linear

4.3.C: Modeling

4.3.C.1: Use functions to model real-world phenomena and solve problems that involve varying quantities.

4.3.C.1.1: Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)

4.3.C.1.2: Direct and inverse variation

4.3.C.1.3: Absolute value

4.3.C.1.4: Expressions, equations and inequalities

4.3.C.1.5: Same function can model variety of phenomena

4.3.C.1.7: Applications in mathematics, biology, and economics (including compound interest)

4.3.C.2: Analyze and describe how a change in an independent variable leads to change in a dependent one.

4.3.C.3: Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).

4.3.D: Procedures

4.3.D.1: Evaluate and simplify expressions.

4.3.D.1.2: Multiply a polynomial by a monomial or binomial

4.3.D.1.3: Divide a polynomial by a monomial

4.3.D.2: Select and use appropriate methods to solve equations and inequalities.

4.3.D.2.1: Linear equations – algebraically

4.3.D.2.2: Quadratic equations – factoring (when the coefficient of x² is 1) and using the quadratic formula

4.3.D.2.3: All types of equations using graphing, computer, and graphing calculator techniques

#### 4.4: All students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.

4.4.A: Data Analysis

4.4.A.1: Use surveys and sampling techniques to generate data and draw conclusions about large groups.

4.4.A.1.1: Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)

4.4.A.3: Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome.

4.4.A.4: Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data.

4.4.A.5: Analyze data using technology, and use statistical terminology to describe conclusions.

4.4.A.5.2: Correlation coefficient

4.4.B: Probability

4.4.B.2: Use concepts and formulas of area to calculate geometric probabilities.

4.4.B.3: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.

4.4.B.4: Determine probabilities in complex situations.

4.4.B.4.3: Dependent and independent events

4.4.B.5: Estimate probabilities and make predictions based on experimental and theoretical probabilities.

4.4.B.6: Understand and use the "law of large numbers" (that experimental results tend to approach theoretical probabilities after a large number of trials).

4.4.C: Discrete Mathematics-Systematic Listing and Counting

4.4.C.1: Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).

4.4.C.4: Recognize and explain relationships involving combinations and Pascal’s Triangle, and apply those methods to situations involving probability.

Correlation last revised: 11/13/2008

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