#### 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.1: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well):

4.1.A.1.b: Percents

4.1.A.1.d: Roots

4.1.A.1.e: Absolute values

4.1.A.3: Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.

4.1.A.4: Compare and order numbers of all named types.

4.1.A.5: Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.

4.1.A.7: Construct meanings for common irrational numbers, such as p (pi) and the square root of 2.

4.1.B: Numerical Operations

4.1.B.1: Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

4.1.B.1.a: Pencil-and-paper

4.1.B.1.b: Mental math

4.1.B.1.c: Calculator

4.1.B.3: Find square and cube roots of numbers and understand the inverse nature of powers and roots.

4.1.B.4: Solve problems involving proportions and percents.

4.1.B.5: Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.

4.1.C: Estimation

4.1.C.1: Estimate square and cube roots of numbers.

4.1.C.2: Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.

#### 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.1: Understand and apply concepts involving lines, angles, and planes.

4.2.A.1.a: Complementary and supplementary angles

4.2.A.1.b: Vertical angles

4.2.A.1.c: Bisectors and perpendicular bisectors

4.2.A.1.d: Parallel, perpendicular, and intersecting planes

4.2.A.1.e: Intersection of plane with cube, cylinder, cone, and sphere

4.2.A.2: Understand and apply the Pythagorean theorem.

4.2.A.3: Understand and apply properties of polygons.

4.2.A.3.a: Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

4.2.A.3.c: Sum of measures of interior angles of a polygon

4.2.A.4: Understand and apply the concept of similarity.

4.2.A.4.a: Using proportions to find missing measures

4.2.A.4.c: Models of 3D objects

4.2.A.5: Use logic and reasoning to make and support conjectures about geometric objects.

4.2.B: Transforming Shapes

4.2.B.1: Understand and apply transformations.

4.2.B.1.a: Finding the image, given the pre-image, and vice-versa

4.2.B.1.b: Sequence of transformations needed to map one figure onto another

4.2.B.1.c: Reflections, rotations, and translations result in images congruent to the pre-image

4.2.B.1.d: Dilations (stretching/shrinking) result in images similar to the pre-image

4.2.B.2: Use iterative procedures to generate geometric patterns.

4.2.B.2.c: Construction of initial stages

4.2.B.2.d: Patterns in successive stages (e.g., number of triangles in each stage of Sierpinskiâ€™s Triangle)

4.2.C: Coordinate Geometry

4.2.C.1: Use coordinates in four quadrants to represent geometric concepts.

4.2.C.2: Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).

4.2.D: Units of Measurement

4.2.D.3: Recognize that the degree of precision needed in calculations depends on how the results will be used and the instruments used to generate the measurements.

4.2.D.4: Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.

4.2.D.6: Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile).

4.2.E: Measuring Geometric Objects

4.2.E.1: Develop and apply strategies for finding perimeter and area.

4.2.E.1.a: Geometric figures made by combining triangles, rectangles and circles or parts of circles

4.2.E.1.b: Estimation of area using grids of various sizes

4.2.E.1.c: Impact of a dilation on the perimeter and area of a 2-dimensional figure

4.2.E.2: Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).

4.2.E.3: Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure.

4.2.E.3.a: Volume - prism, cone, pyramid

4.2.E.3.b: Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base)

4.2.E.3.c: Impact of a dilation on the surface area and volume of a three-dimensional figure

#### 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: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

4.3.A.1.a: Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

4.3.A.1.b: Finite and infinite sequences

4.3.A.1.c: Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

4.3.A.1.d: Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)

4.3.A.1.e: Generating sequences by using calculators to repeatedly apply a formula

4.3.B: Functions and Relationships

4.3.B.1: Graph functions, and understand and describe their general behavior.

4.3.B.1.a: Equations involving two variables

4.3.B.1.b: Rates of change (informal notion of slope)

4.3.B.2: Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.

4.3.C: Modeling

4.3.C.1: Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

4.3.C.2: Use patterns, relations, symbolic algebra, and linear functions to model situations.

4.3.C.2.a: Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities

4.3.C.2.b: Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

4.3.D: Procedures

4.3.D.1: Use graphing techniques on a number line.

4.3.D.1.a: Absolute value

4.3.D.2: Solve simple linear equations informally, graphically, and using formal algebraic methods.

4.3.D.2.a: Multi-step, integer coefficients only (although answers may not be integers)

4.3.D.3: Solve simple linear inequalities.

4.3.D.4: Create, evaluate, and simplify algebraic expressions involving variables.

4.3.D.4.a: Order of operations, including appropriate use of parentheses

4.3.D.4.d: Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa

4.3.D.5: Understand and apply the properties of operations, numbers, equations, and inequalities.

4.3.D.5.c: Addition and multiplication properties of equality

#### 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: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).

4.4.A.1.a: Type of display most appropriate for given data

4.4.A.1.b: Box-and-whisker plot, upper quartile, lower quartile

4.4.A.1.c: Scatter plot

4.4.A.1.e: Finding the median and mean (weighted average) using frequency data.

4.4.A.1.f: Effect of additional data on measures of central tendency

4.4.A.3: Estimate lines of best fit and use them to interpolate within the range of the data.

4.4.A.4: Use surveys and sampling techniques to generate data and draw conclusions about large groups.

4.4.B: Probability

4.4.B.1: Interpret probabilities as ratios, percents, and decimals.

4.4.B.2: Determine probabilities of compound events.

4.4.B.3: Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag, without replacement, are both red).

4.4.B.4: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.

4.4.B.4.a: Frequency, relative frequency

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

4.4.C: Discrete Mathematics-Systematic Listing and Counting

4.4.C.1: Apply the multiplication principle of counting.

4.4.C.1.a: Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

4.4.C.1.c: Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)

Correlation last revised: 11/13/2008

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