Diocese of Covington - Education at PO Box 15550, Covington, KY 41015-0550 US - Mathematics Part 2

The content is directly aligned with Kentucky's academic expectations. These guidelines are designed to present the mathematical topics fundamental to mathematical literacy and mathematical power for all middle school students. Each content statement is interrelated with other statements and designed to be delivered in meaningful contexts, developing mathematical problem solving, communication, reasoning, and connections.

Features of middle school mathematics programs include exploration, communication, mathematical tools, manipulatives (concrete materials), calculators, hands-on activities, and group work. The students' interrelated mathematical explorations and experiences contribute to their confidence and ability to understand and address real quantitative, scientific, and technological issues.

Middle level problem solving, mathematical communications and mathematical reasoning should be a part of the mathematics curriculum.

Problem solving includes multiple strategies for modeling, interpreting, and formulating problems based in real-world situations, within and outside mathematics, and aids in investigating and understanding mathematical content.

Mathematical communication includes modeling problems including oral, written, concrete, visual, graphical, and algebraic methods to define, interpret, and argue mathematical ideas. Mathematical communication includes mathematical symbolic notation (letters and marks used in mathematics to name numbers, operations, sets, relations).

Mathematical connections include relating mathematical ideas within mathematics and with other disciplines using graphic, numerical, physical, algebraic, and verbal models.

Mathematical reasoning includes deductive and inductive reasoning necessary in developing conjectures and validating arguments. The guidelines included in this document for the middle levels are arranged sequentially by grade. However, it is the prerogative of school councils and local boards of education for schools exempt from school-based decision making to reorganize the content into a format that best meets the needs of their students, creating integrated, interdisciplinary, or multidisciplinary programs, or offer higher level coursework.

Each topic organizer is followed by the relevant academic expectations. Bulleted points denote the required content statements. Lists in parentheses (designated with an "e.g.") are suggestions for instruction and are not meant to be comprehensive. Schools or districts may arrange the content to meet the needs of their students. For example, they may offer the content in a grade-level arrangement or as integrated courses that focus on topics within units or alternate configurations. The mathematics content also provides connections to Kentucky's Learning Goal 5 (Think and Solve Problems) and Goal 6 (Connect and Integrate Knowledge). These connections provide a comprehensive link between essential content and the skills and abilities important to learning.

Students will:

• continue to develop number sense including percents with fractions and decimals (including percents greater than 100% and improper fractions).
• work fluently with operations (+, -, x, ÷) of fractions and decimals.
• develop meaning of ratio and proportion (describe and compare two sets of data using ratios and appropriate notations: 3:5, 3/5, 3 to 5).
• explore exponents (e.g., squares, cubes).
• simplify fractions with prime factorization (numbers that divide exactly into a given number).
• estimate with large and small quantities of objects.
• estimate and mentally compute using fractions and decimals.
• use prime numbers, composite numbers, factors, multiples, and divisibility to solve problems.
• compare, order, and convert between whole numbers, fractions, decimals, and percents using concrete materials, drawings or pictures, and mathematical symbols (<, >, =, order on a number line).
• explore how applications of properties (e.g., commutative, associative, inverse, identity) show relationships among numbers and operations.
• solve-real world problems using a combination of the four basic operations

Students will:

• find perimeter of regular and irregular polygons in metric and U.S. customary units.
• extend the use of measurement tools (e.g., rulers, scales, protractor, compass).
• find area of plane figures composed of squares and rectangles through subdividing and measuring and use square units appropriately.
• estimate, compare, and convert units of measures for length, weight/mass, and volume/capacity within the U.S. customary system and within the metric system:
  a) length (e.g., parts of an inch, inches, feet, yards, miles, millimeter, centimeter, kilometer;
  b) weight/mass (e.g., pounds, tons, grams, kilograms); and
  c) volume/capacity (e.g., cups, pints, quarts, gallons, milliliters, liters). (The intent of this standard is for students to make ballpark comparisons and not to memorize conversion factors between U.S. and metric units.)
• estimate and find angle measurement and segment measurements.
• formulate the rule that the sum of angle measurements is 180 degrees in a triangle and 360 degrees in a quadrilateral.
• identify properties and classify line segments, rays, planes, and points.
• recognize regular polygons; special quadrilaterals including squares, rectangles, rhombuses, trapezoids, and parallelograms; and special triangles including acute, obtuse, scalene, and isosceles.
• identify characteristics of lines (e.g., parallel, perpendicular).
• use lines of symmetry and sketch plane figures with multiple lines of symmetry.

Students will:

• collect, organize, analyze, and interpret data in a variety of graphical methods, including line plots, line graphs, bar graphs, and stem and leaf plots.
• make predictions, draw conclusions, and verify results from statistical data and probability experiments.
• select an appropriate graph to represent given data.
• compare data from various types of graphs.
• investigate solutions to probability problems, using counting techniques, tree diagrams, charts, and tables.
• recognize the role of probability in decision making.
• apply range and measures of central tendency (mean, median, mode).

Students will:

• recognize, create, and continue patterns (give an informal description for the continuance of the pattern and/or generalize patterns through a verbal rule).
• represent, interpret, and describe function relationships through tables, graphs, and verbal rules.
• write and solve equations with one variable, using concrete and/or informal methods that model everyday situations.
• explore the concept of variable, expression, and equation.
• solve problems involving simple formulas (i.e., A = 1w, P = 21 + 2w).
• interpret relationships between tables and graphs.
• organize data into tables and plot points onto the first quadrant of a coordinate (Cartesian) system/grid.

Students will:

• extend number sense for percents and integers.
• extend understanding of operations (=, -, x, ÷) to include integers.
• develop number sense for pi as one example of an irrational number.
• apply meaning of ratio and proportion to problems.
• use whole number exponents.
• extend and apply addition, subtraction, multiplication, and division of integers both concretely and symbolically (mental, pencil and paper, calculators).
• extend concepts and application of operations with fractions and decimals to include percents.
• compute percentages of numbers and use percentages in proportional reasoning.
• estimate and mentally compute using integers and percents.
• solve proportions.
• compare, order, and determine equivalent relationships among fractions, decimals, and percents.
• explain and apply properties (e.g., commutative, associative, distributive, inverse, identity).
• develop proportional thinking, rates, scaling, and similarity.
• extend positive and negative number operations
• use estimation to check reasonableness of results
• solve real-world problems using a combination of the four basic operations
• perform calculations involving conversions between units within a system (length, capacity, mass)

Students will:

• find circle measurements (radius, diameter, circumference, area) and the relationships among them.
• develop and use the formulas for area of triangles, parallelograms, and trapezoid; relate to the formula for area of rectangles (1 x w).
• investigate fixed area with changing perimeter and fixed perimeter with changing area.
• investigate area of polygons and other two-dimensional shapes.
• identify and classify characteristics of two-dimensional shapes, such as regular and irregular quadrilaterals, special triangles, and regular polygons.
• identify characteristics of angles (e.g., adjacent, vertical, corresponding, interior, exterior).
• represent three-dimensional geometric figures with special attention to developing spatial sense (e.g., top view, side view, three-dimensional shapes drawn on isometric dot paper).
• move shapes in a plane: (e.g., translate (slide), rotate (turn), reflect (flip)
• construct, analyze, and compare 2-dimensional figures (e.g. perpendicular bisector, angle bisector)
• develop and apply formulas for volume and surface area of prisms, pyramids, cylinders, etc. and investigate relationships between and among them
• extend conversion units (i.e. length, capacity, and mass) within the U.S. customary system and within the metric system
• identify temperatures of boiling water, normal body temperature, or water freezing in Fahrenheit and Celsius

Students will:

• collect, organize, analyze, and interpret data in a variety of graphical methods, including circle graphs, multiple line graphs, bar graphs, and stem and leaf plots.
• make predictions, draw conclusions, and verify results from statistical data and probability experiments.
• select an appropriate graph to represent given data and justify its use.
• compare data from various types of graphs.
• determine appropriate techniques to use when investigating solutions to probability problems (using counting techniques; tree diagrams; area models; and exhaustive, organized lists, charts, and tables).
• investigate and explain the role of probability in decision making.
• determine and apply the most appropriate measures of central tendency (e.g, mean, median, mode) and/or dispersion (e.g., range).
• design and conduct probability experiments.
• determine theoretical (mathematical) probabilities, compare to experimental results, and explain reasons why there might be differences, (e.g., express probability as a ratio, decimal, or a percent as appropriate for a given situation).
• explore concepts of randomness and independent events.
• determine and interpret clusters, quartiles, gaps, and outliers in data

Students will:

• recognize, create, and continue patterns and generalize the pattern by giving the rule for any term.
• represent, interpret, and describe functional relationships through tables, graphs, and verbal rules (input/output).
• understand the concept of equations and inequalities using variables as they relate to everyday situations.
• simplify numeric and algebraic expressions.
• use a variety of methods and representations to create and solve single-variable equations that may be applied to everyday situations.
• solve problems involving formulas.
• organize data into tables and plot points onto all four quadrants of a coordinate (Cartesian) system/grid and interpret resulting patterns or trends.
• interpret relationships between tables, graphs, verbal rules, and equations.

Students will:

• use percents, decimals, integers, and fractions (include percents less than 1).
• use percentages and proportions in consumer applications (e.g., simple interest, percentages of increase or decrease, discounts, unit pricing, sale prices).
• use irrational numbers (e.g., square roots).
• relate irrational and rational numbers (e.g., magnitude, order on a number line).
• determine the inverse relationship between addition and subtraction, multiplication and division, or raising to an exponent and taking the root of a number.
• extend positive and negative number operations

Students will:

• discover and apply the Pythagorean theorem.
• derive and use formulas for various rates (e.g., distance/time, miles per hour).
• develop and apply formulas for volume and surface area of cubes, cylinders, and rectangular prisms; and investigate relationships between and among them.
• develop and apply proportionality and relationships between scale models and actual figures.
• investigate transformations' congruence, proportionality, and similarity (e.g., enlargements, reductions, proportional triangles) in a coordinate plane.
• investigate counting techniques through shortest paths (e.g., networks).

Students will:

• collect, organize, analyze, and interpret data in a variety of graphical methods (e.g., circle graphs, scatter plots, box and whisker plots, histograms).
• make predictions, draw conclusions, and verify results from statistical data and probability experiments.
• select an appropriate graph to represent given data and justify its use.
• compare data from various types of graphs.
• recognize that statistics can be interpreted in many ways.
• analyze situations, such as games of chance, board games, or grading scales, and make predictions using knowledge of probability.
• identify and describe the number of possible arrangements of several objects, using a tree diagram or the basic counting principle, and make a sample space represented in the form of a list, picture, chart, or a tree diagram.
• investigate and explain the role of probability in everyday decision making.
• design and conduct probability experiments and interpret the results.
• explore concepts of randomness and independent events.
• determine theoretical (mathematical) probabilities, compare that to experimental results, and explain reasons why there might be differences (e.g., express probability as a ratio, decimal, percent as appropriate for a given situation).
• determine and interpret clusters, quartiles, gaps, and outliers in data.

Students will:

• recognize, create, and continue patterns (generalize the pattern by giving the rule for the nth term and defend the generalization).
• represent, interpret, and describe functional relationships through tables, graphs, and symbolic rules (input/output).
• explain how change in one variable affects change in another variable (e.g., in distance equals rate times time, increasing time, increases distance).
• use a variety of methods and representations to create and solve one- and two- variable linear equations that require two steps.
• simplify algebraic expressions.
• investigate inequalities using a variety of methods and representations.
• solve problems involving substitutions and formulas.
• organize data into tables, plot points onto all four quadrants of a coordinate (Cartesian) system/grid, interpret resulting patterns or trends.
• interpret and explain relationships between tables, graphs, verbal rules, and equations.
• graph linear functions in a four quadrant (Cartesian) system/grid and interpret the results.
• determine the slope and equation of a line by analyzing the line (e.g., Y = mx + b; m is rise/run, b is y - intercept).

  ---

  (Back)