0016  Understand the number system and the concept of place value. 

For example: 

• Analyze the structures and properties of the base-10 and other numeral systems (e.g., expanded form of a number, visual representations of place value, numeration systems of ancient cultures). 
  
• Recognize decimal expansions. 
  
• Use scientific notation in the real world. 
  
• Analyze procedures (e.g., rounding, regrouping) for estimation. 
  
• Determine the reasonableness of estimates. 
  
• Identify subsets of the real numbers (e.g., integer, rational, irrational) and their characteristics.
  

0017  Understand integers, fractions, decimals, percents, and mixed numbers. 

For example: 

• Understand the meanings and models of integers, fractions, decimals, percents, and mixed numbers and apply them to the solution of word problems. 
  
• Analyze and convert among various representations of numbers (e.g., graphic, numeric, symbolic, verbal). 
  
• Use number lines. 
  
• Compare, sort, order, and round numbers. 
  
• Recognize equivalent representations of numbers (e.g., fractions, decimals, percents). 

0018 Understand and apply principles of number theory.
For example:

• Identify prime and composite numbers and their characteristics. 
  
• Find the prime factorization of a number and recognize its uses. 
  
• Demonstrate knowledge of the divisibility rules and why they work. 
  
• Find the least common multiple (LCM) and greatest common factor (GCF) of a set of numbers. 
  
• Apply the LCM and GCF in real-world situations. 
  

0019 Understand operations on numbers.
For example:

• Understand the meaning and models of operations on numbers (e.g., integers, fractions, decimals). 
  
• Analyze and justify standard and nonstandard computational algorithms and mental math techniques (e.g., by application of the arithmetic properties, such as commutative, associative, distributive). 
  
• Evaluate the validity of nonstandard or unfamiliar computational strategies. 
  
• Recognize and analyze various representations (e.g., graphic, pictorial, verbal) of number operations. 
  
• Recognize relationships among operations (e.g., addition and subtraction, addition and multiplication, multiplication and exponentiation). 
  
• Identify and apply the arithmetic properties and the transitive properties of equality and inequality. 
  
• Apply the order of operations. 
  
• Apply the laws of exponents. 
  
• Demonstrate fluency in arithmetic computation, including operations on fractions. 
  
• Interpret the concept of absolute value. 
  
• Apply appropriate strategies (e.g., proportional thinking, ratios) to estimate quantities in real-world situations. 
  
• Solve problems using arithmetic operations with various representations of numbers. 

0020  Understand algebra as generalized arithmetic. 

For example: 

• Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically. 
  
• Manipulate simple algebraic expressions and solve linear equations and inequalities. 
  
• Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties. 
  
• Use algebra to solve word problems involving fractions, ratios, proportions, and percents. 
  
• Identify variables and derive algebraic expressions that represent real-world situations. 
  

0021  Understand the concept of function.

For example: 

• Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). 
  
• Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). 
  
• Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations. 
  
• Use qualitative graphs to represent functional relationships in the real world. 
  
• Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships. 
  

0022  Understand linear functions and linear equations. 

For example: 

• Recognize the formula and graph of a linear function. 
  
• Distinguish between linear and nonlinear functions. 
  
• Find a linear equation that represents a graph. 
  
• Analyze the relationships among proportions, constant rates, and linear functions. 
  
• Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation. 
  
• Select the linear equation that best models a real-world situation. 
  

0027 Apply mathematical knowledge and reasoning to communicate multiple solutions in detail to a problem involving two or more of the following subareas: Numbers and Operations, Functions and Algebra, Geometry and Measurement, and Statistics and Probability.

(Refer to objectives 0016 through 0026 and associated descriptive statements.)

0023 Understand and apply concepts of measurement.
For example:

• Estimate and calculate measurements using customary, metric, and nonstandard units of measurement. 
  
• Use unit conversions and dimensional analysis to solve measurement problems. 
  
• Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures. 
  
• Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions. 
  
• Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations. 
  

0024 Understand and apply concepts of geometry.
For example:

• Classify and analyze polygons using attributes of sides and angles, including real-world applications. 
  
• Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices. 
  
• Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems. 
  
• Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings). 
  
• Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem). 


0025 Understand descriptive statistics.
For example:

• Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data. 
  
• Select appropriate ways to present data and communicate statistical information (e.g., tables, graphs, line plots, Venn diagrams). 
  
• Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles). 
  
• Compare different data sets. 
  

0026 Understand and apply basic concepts of probability.
For example:

• Calculate the probabilities of simple and compound events and of independent and dependent events. 
  
• Recognize and apply the concept of conditional probability. 
  
• Recognize the difference between experimentally and theoretically determined probabilities in real-world situations. 
  
• Apply knowledge of combinations and permutations to the computation of probabilities.