Numbers And The Number System: Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
Ratio: Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ).
Proportions: Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/ 21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.
Percentages: Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.
Number Operations: Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.
LCM and HCF: Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).
Algebra and Functions
Linear Equations in One Variable.
Solving Algebraic Expressions: Write and evaluate an algebraic expression for a given situation, using up to three variables.
The Commutative, Associative, and Distributive Properties: Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
Use of Calculator: Solve problems manually by using the correct order of operations or by using a scientific calculator.
Conversion of Units: Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
Rate: Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.
Word Problems: Solve problems involving rates, average speed, distance, and time.
Symbolic Form: Express in symbolic form simple relationships arising from geometry.
Measurement and Geometry
Circle: Understand the concept of a constant such as ; know the formulas for the circumference and area of a circle.
Pi: Know common estimates of (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
Volume of figures: Know and use the formulas for the volume of triangular prisms and cylinders (area of base x height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
Angles: Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
Properties of Angles: Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
Drawing Quadrilaterals: Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).
Statistics, Data Analysis, and Probability
Mean, Median, Mode: Compute the range, mean, median, and mode of data sets.
Comparison Among Different Methods of Central Tendencies: Know why a specific measure of central tendency (mean, median) provides the most useful information in a given context.
Data Collection: Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population.
Data Interpretation: Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might have influenced the conclusions reached.
Sampling Errors: Identify data that represent sampling errors and explain why the sample (and the display) might be biased.
Outcomes of An Event: Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
Probability: Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven).
Representation of Probability: Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1- P is the probability of an event not occurring.
Independent and Dependent Events: Understand the difference between independent and dependent events.
Mathematical Reasoning
Problem Analysis: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
Problem Formulation and Justification: Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
Breaking Complex Problem into Simpler Parts: Determine when and how to break a problem into simpler parts.
Estimation: Use of estimation to verify the reasonableness of calculated results.
Using Strategies: Apply strategies and results from simpler problems to more complex problems.
Graphical Solution: Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques.
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.
Calculation of Results and Verification: Make precise calculations and check the validity of the results from the context of the problem.
Generalization: Develop generalizations of the results obtained and the strategies used and apply them in new problem situations.
