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8.4.1
Number and Numerical Operations
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8.4.1.8 A
Number Sense
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8.4.1.8 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):
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8.4.1.8 A.1.a
Rational numbers
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8.4.1.8 A.1.b
Percents
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8.4.1.8 A.1.c
Exponents
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8.4.1.8 A.1.d
Roots
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8.4.1.8 A.1.e
Absolute values
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8.4.1.8 A.1.f
Numbers represented in scientific notation
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8.4.1.8 A.2
Demonstrate a sense of the relative magnitudes of numbers.
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8.4.1.8 A.3
Understand and use ratios, rates, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.
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8.4.1.8 A.4
Compare and order numbers of all named types.
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8.4.1.8 A.5
Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.
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8.4.1.8 A.6
Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.
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8.4.1.8 A.6.a
5/7 = 0. 714285714285... = 0.714285 repeating
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8.4.1.8 A.7
Construct meanings for common irrational numbers, such as pi and the square root of 2.
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8.4.1.8 B
Numerical Operations
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8.4.1.8 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:
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8.4.1.8 B.1.a
Pencil-and-paper
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8.4.1.8 B.1.b
Mental math
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8.4.1.8 B.1.c
Calculator
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8.4.1.8 B.2
Use exponentiation to find whole number powers of numbers.
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8.4.1.8 B.3
Find square and cube roots of numbers and understand the inverse nature of powers and roots.
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8.4.1.8 B.4
Solve problems involving proportions and percents.
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8.4.1.8 B.5
Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.
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8.4.1.8 C
Estimation
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8.4.1.8 C.1
Estimate square and cube roots of numbers.
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8.4.1.8 C.2
Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.
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8.4.1.8 C.3
Recognize the limitations of estimation and assess the amount of error resulting from estimation.
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8.4.2
Geometry and Measurement
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8.4.2.8 A
Geometric Properties
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8.4.2.8 A.1
Understand and apply concepts involving lines, angles, and planes.
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8.4.2.8 A.1.a
Complementary and supplementary angles
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8.4.2.8 A.1.b
Vertical angles
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8.4.2.8 A.1.c
Bisectors and perpendicular bisectors
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8.4.2.8 A.1.d
Parallel, perpendicular, and intersecting planes
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8.4.2.8 A.1.e
Intersection of plane with cube, cylinder, cone, and sphere
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8.4.2.8 A.2
Understand and apply the Pythagorean theorem.
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8.4.2.8 A.3
Understand and apply properties of polygons.
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8.4.2.8 A.3.a
Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi
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8.4.2.8 A.3.b
Regular polygons
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8.4.2.8 A.3.c
Sum of measures of interior angles of a polygon
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8.4.2.8 A.3.d
Which polygons can be used alone to generate a tessellation and why
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8.4.2.8 A.4
Understand and apply the concept of similarity.
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8.4.2.8 A.4.a
Using proportions to find missing measures
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8.4.2.8 A.4.b
Scale drawings
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8.4.2.8 A.4.c
Models of 3D objects
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8.4.2.8 A.5
Use logic and reasoning to make and support conjectures about geometric objects.
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8.4.2.8 A.6
Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology).
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8.4.2.8 A.6.a
Congruent angles or line segments
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8.4.2.8 A.6.b
Midpoint of a line segment
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8.4.2.8 A.7
Create two-dimensional representations (e.g., nets or projective views) for the surfaces of three-dimensional objects.
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8.4.2.8 B
Transforming Shapes
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8.4.2.8 B.1
Understand and apply transformations.
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8.4.2.8 B.1.a
Finding the image, given the pre-image, and vice-versa
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8.4.2.8 B.1.b
Sequence of transformations needed to map one figure onto another
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8.4.2.8 B.1.c
Reflections, rotations, and translations result in images congruent to the pre-image
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8.4.2.8 B.1.d
Dilations (stretching/shrinking) result in images similar to the pre-image
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8.4.2.8 B.2
Use iterative procedures to generate geometric patterns.
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8.4.2.8 B.2.a
Fractals (e.g., the Koch Snowflake)
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8.4.2.8 B.2.b
Self-similarity
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8.4.2.8 B.2.c
Construction of initial stages
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8.4.2.8 B.2.d
Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski's Triangle)
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8.4.2.8 C
Coordinate Geometry
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8.4.2.8 C.1
Use coordinates in four quadrants to represent geometric concepts.
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8.4.2.8 C.2
Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).
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8.4.2.8 D
Units of Measurement
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8.4.2.8 D.1
Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4'3" plus 7'10" equals 12'1").
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8.4.2.8 D.2
Use approximate equivalents between standard and metric systems to estimate measurements (e.g., 5 kilometers is about 3 miles).
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8.4.2.8 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.
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8.4.2.8 D.4
Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.
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8.4.2.8 D.5
Recognize that all measurements of continuous quantities are approximations.
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8.4.2.8 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).
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8.4.2.8 E
Measuring Geometric Objects
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8.4.2.8 E.1
Develop and apply strategies for finding perimeter and area.
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8.4.2.8 E.1.a
Geometric figures made by combining triangles, rectangles and circles or parts of circles
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8.4.2.8 E.1.b
Estimation of area using grids of various sizes
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8.4.2.8 E.1.c
Impact of a dilation on the perimeter and area of a 2-dimensional figure
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8.4.2.8 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).
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8.4.2.8 E.3
Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure.
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8.4.2.8 E.3.a
Volume - prism, cone, pyramid
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8.4.2.8 E.3.b
Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base)
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8.4.2.8 E.3.c
Impact of a dilation on the surface area and volume of a three-dimensional figure
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8.4.2.8 E.4
Use formulas to find the volume and surface area of a sphere.
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8.4.3
Patterns and Algebra
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8.4.3.8 A
Patterns
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8.4.3.8 A.1
Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.
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8.4.3.8 A.1.a
Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions
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8.4.3.8 A.1.b
Finite and infinite sequences
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8.4.3.8 A.1.c
Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)
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8.4.3.8 A.1.d
Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)
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8.4.3.8 A.1.e
Generating sequences by using calculators to repeatedly apply a formula
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8.4.3.8 B
Functions and Relationships
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8.4.3.8 B.1
Graph functions, and understand and describe their general behavior.
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8.4.3.8 B.1.a
Equations involving two variables
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8.4.3.8 B.1.b
Rates of change (informal notion of slope)
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8.4.3.8 B.2
Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.
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8.4.3.8 C
Modeling
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8.4.3.8 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.
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8.4.3.8 C.2
Use patterns, relations, symbolic algebra, and linear functions to model situations.
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8.4.3.8 C.2.a
Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities
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8.4.3.8 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)
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8.4.3.8 D
Procedures
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8.4.3.8 D.1
Use graphing techniques on a number line.
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8.4.3.8 D.1.a
Absolute value
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8.4.3.8 D.1.b
Arithmetic operations represented by vectors (arrows) (e.g., "-3 + 6" is "left 3, right 6")
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8.4.3.8 D.2
Solve simple linear equations informally, graphically, and using formal algebraic methods.
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8.4.3.8 D.2.a
Multi-step, integer coefficients only (although answers may not be integers)
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8.4.3.8 D.2.b
Simple literal equations (e.g., A = lw)
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8.4.3.8 D.2.c
Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology
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8.4.3.8 D.3
Solve simple linear inequalities.
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8.4.3.8 D.4
Create, evaluate, and simplify algebraic expressions involving variables.
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8.4.3.8 D.4.a
Order of operations, including appropriate use of parentheses
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8.4.3.8 D.4.b
Distributive property
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8.4.3.8 D.4.c
Substitution of a number for a variable
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8.4.3.8 D.4.d
Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa
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8.4.3.8 D.5
Understand and apply the properties of operations, numbers, equations, and inequalities.
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8.4.3.8 D.5.a
Additive inverse
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8.4.3.8 D.5.b
Multiplicative inverse
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8.4.3.8 D.5.c
Addition and multiplication properties of equality
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8.4.3.8 D.5.d
Addition and multiplication properties of inequalities
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8.4.4
Data Analysis, Probability, and Discrete Mathematics
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8.4.4.8 A
Data Analysis
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8.4.4.8 A.1
Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).
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8.4.4.8 A.1.a
Type of display most appropriate for given data
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8.4.4.8 A.1.b
Box-and-whisker plot, upper quartile, lower quartile
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8.4.4.8 A.1.c
Scatter plot
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8.4.4.8 A.1.d
Calculators and computer used to record and process information
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8.4.4.8 A.1.e
Finding the median and mean (weighted average) using frequency data.
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8.4.4.8 A.1.f
Effect of additional data on measures of central tendency
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8.4.4.8 A.2
Make inferences and formulate and evaluate arguments based on displays and analysis of data sets.
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8.4.4.8 A.3
Estimate lines of best fit and use them to interpolate within the range of the data.
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8.4.4.8 A.4
Use surveys and sampling techniques to generate data and draw conclusions about large groups.
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8.4.4.8 B
Probability
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8.4.4.8 B.1
Interpret probabilities as ratios, percents, and decimals.
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8.4.4.8 B.2
Determine probabilities of compound events.
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8.4.4.8 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).
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8.4.4.8 B.4
Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.
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8.4.4.8 B.4.a
Frequency, relative frequency
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8.4.4.8 B.5
Estimate probabilities and make predictions based on experimental and theoretical probabilities.
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8.4.4.8 B.6
Play and analyze probability-based games, and discuss the concepts of fairness and expected value.
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8.4.4.8 C
Discrete Mathematics-Systematic Listing and Counting
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8.4.4.8 C.1
Apply the multiplication principle of counting.
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8.4.4.8 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)
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8.4.4.8 C.1.b
Factorial notation
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8.4.4.8 C.1.c
Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)
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8.4.4.8 C.2
Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).
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8.4.4.8 C.3
Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.
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8.4.4.8 D
Discrete Mathematics-Vertex-Edge Graphs and Algorithms
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8.4.4.8 D.1
Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems.
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8.4.4.8 D.1.a
Finding the shortest network connecting specified sites
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8.4.4.8 D.1.b
Finding a minimal route that includes every street (e.g., for trash pick-up)
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8.4.4.8 D.1.c
Finding the shortest route on a map from one site to another
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8.4.4.8 D.1.d
Finding the shortest circuit on a map that makes a tour of specified sites
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8.4.4.8 D.1.e
Limitations of computers (e.g., the number of routes for a delivery truck visiting n sites is n!, so finding the shortest circuit by examining all circuits would overwhelm the capacity of any computer, now or in the future, even if n is less than 100)
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8.4.5
Mathematical Processes
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8.4.5 A
Problem Solving
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8.4.5 A.1
Learn mathematics through problem solving, inquiry, and discovery.
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8.4.5 A.2
Solve problems that arise in mathematics and in other contexts.
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8.4.5 A.2.a
Open-ended problems
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8.4.5 A.2.b
Non-routine problems
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8.4.5 A.2.c
Problems with multiple solutions
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8.4.5 A.2.d
Problems that can be solved in several ways
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8.4.5 A.3
Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.
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8.4.5 A.4
Pose problems of various types and levels of difficulty.
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8.4.5 A.5
Monitor their progress and reflect on the process of their problem solving activity.
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8.4.5 A.6
Distinguish relevant from irrelevant information, and identify missing information.
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8.4.5 B
Communication
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8.4.5 B.1
Use communication to organize and clarify their mathematical thinking.
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8.4.5 B.1.a
Reading and writing
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8.4.5 B.1.b
Discussion, listening, and questioning
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8.4.5 B.2
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.
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8.4.5 B.3
Analyze and evaluate the mathematical thinking and strategies of others.
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8.4.5 B.4
Use the language of mathematics to express mathematical ideas precisely.
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8.4.5 C
Connections
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8.4.5 C.1
Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).
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8.4.5 C.2
Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).
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8.4.5 C.3
Recognize that mathematics is used in a variety of contexts outside of mathematics.
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8.4.5 C.4
Apply mathematics in practical situations and in other disciplines.
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8.4.5 C.5
Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).
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8.4.5 C.6
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
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8.4.5 D
Reasoning
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8.4.5 D.1
Recognize that mathematical facts, procedures, and claims must be justified.
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8.4.5 D.2
Use reasoning to support their mathematical conclusions and problem solutions.
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8.4.5 D.3
Select and use various types of reasoning and methods of proof.
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8.4.5 D.4
Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.
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8.4.5 D.5
Make and investigate mathematical conjectures.
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8.4.5 D.5.a
Counterexamples as a means of disproving conjectures
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8.4.5 D.5.b
Verifying conjectures using informal reasoning or proofs.
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8.4.5 D.6
Evaluate examples of mathematical reasoning and determine whether they are valid.
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8.4.5 E
Representations
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8.4.5 E.1
Create and use representations to organize, record, and communicate mathematical ideas.
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8.4.5 E.1.a
Concrete representations (e.g., base-ten blocks or algebra tiles)
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8.4.5 E.1.b
Pictorial representations (e.g., diagrams, charts, or tables)
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8.4.5 E.1.c
Symbolic representations (e.g., a formula)
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8.4.5 E.1.d
Graphical representations (e.g., a line graph)
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8.4.5 E.2
Select, apply, and translate among mathematical representations to solve problems.
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8.4.5 E.3
Use representations to model and interpret physical, social, and mathematical phenomena.
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8.4.5 F
Technology
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8.4.5 F.1
Use technology to gather, analyze, and communicate mathematical information.
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8.4.5 F.2
Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.
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8.4.5 F.3
Use graphing calculators and computer software to investigate properties of functions and their graphs.
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8.4.5 F.4
Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).
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8.4.5 F.5
Use computer software to make and verify conjectures about geometric objects.
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8.4.5 F.6
Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).
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