Rounding and Estimation: Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers.
Percentages: Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.
Powers: Understand and compute positive integer powers of non-negative integers; compute examples as repeated multiplication.
Factors: Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3).
Number Lime: Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.
Mathematical Operations: Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results.
Division: Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors.
Word Problems: Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form.
Fractions: Understand the concept of multiplication and division of fractions.
Algebra and Functions
Graphs: Use information taken from a graph or equation to answer questions about a problem situation.
Solving Expression: Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.
Ordered Pairs: Identify and graph ordered pairs in the four quadrants of the coordinate plane.
Problems on Ordered Pairs: Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.
Measurement and Geometry
Area of Polygons: Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram).
Surface Areas: Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.
Volume: Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to compute the volume of rectangular solids.
Units of Measure: Differentiate between, and use appropriate units of measures for, two-and three-dimensional objects (i.e., find the perimeter, area, volume).
Angles: Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).
Triangles: Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.
2-D and 3-D Figures: Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.
Statistics, Data Analysis, and Probability
Mean, Median and Mode: Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.
Representing Data: Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.
Fractions and Percentages: Use fractions and percentages to compare data sets of different sizes.
Ordered Pairs: Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.
Analysis of a Problem: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.
Estimation: Use estimation to verify the reasonableness of calculated results.
Use of Strategies: Apply strategies and results from simpler problems to more complex problems.
Methods to explain Reasoning: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
Approximation: relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.
Calculating Results and Verification: Checking the validity of the results from the context of the problem.
Generalization of result: Develop generalizations of the results obtained and apply them in other circumstances.