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Number Sense

  • Rational Numbers:
    Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation.
  • Operations on Rational Numbers:
    Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
  • Fractions, Decimals and Percentages:
    Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
  • Irrational Numbers:
    Differentiate between rational and irrational numbers.
  • Terminating and Repeating Decimals:
    Know that every rational number is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions.
  • Word Problems:
    Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.
  • Negative Numbers:
    Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base.
  • Exponent Rules:
    Multiply, divide, and simplify rational numbers by using exponent rules.
  • Power of a Number:
    Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.
  • Absolute Value:
    Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.
Algebra and Functions
  • Use of Variables:
    Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
  • Order of Operations:
    Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5) 2.
  • Simplifying Equations:
    Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used.
  • Algebraic Terminology:
    Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly.
  • Graphical Representation:
    Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
  • Powers:
    Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
  • Roots:
    Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent.
  • Graph of Linear Functions:
    Graph linear functions, noting that the vertical change (change in y- value) per unit of horizontal change (change in x- value) is always the same and know that the ratio ("rise over run") is called the slope of a graph.
  • Linear Equations and Inequalities:
    Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.
Measurement and Geometry
  • Measurement System:
    Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters).
  • Reading Scales:
    Construct and read drawings and models made to scale.
  • Dimensional Analysis:
    Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
  • 2-D and 3-D Figures:
    Use of formulas for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
  • Conversion of Units:
    Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft2] = [144 in2], 1 cubic inch is approximately 16.38 cubic centimeters or [1 in3] = [16.38 cm3]).
  • Construction:
    Identify and construct basic elements of geometric figures (e.g., altitudes, mid-points, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge.
  • Plotting Graphs:
    Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
  • Pythagoras Theorem:
    Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
  • Congruency:
    Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.
  • Elements of 3-D Figures:
    Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect).
Statistics, Data Analysis, and Probability
  • Data Collection:
    Know various forms of display for data sets, including a stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data.
  • Data Representation:
    Represent two numerical variables on a scatter plot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level).
  • Data Interpretation:
    Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set.
Mathematical Reasoning
  • Analysis of a Problem:
    Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
  • Problem Formulation:
    Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
  • Breaking of Complex Problem in Simpler Parts:
    Determine when and how to break a problem into simpler parts. Estimation: Use estimation to verify the reasonableness of calculated results.
  • Use of Strategies:
    Apply strategies and results from simpler problems to more complex problems.
  • Estimation:
    Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
  • Inductive and Deductive:
    Reasoning: Make and test conjectures by using both inductive and deductive reasoning.
  • Use of Methods to Explain Mathematical Reasoning:
    Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
  • Accuracy:
    Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.
  • Evaluation of Result:
    Evaluate the reasonableness of the solution in the context of the original situation.
  • Problem Generalization:
    Develop generalizations of the results obtained and the strategies used and apply them to new problem situations.

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