1. Understand and Use Numbers

Compare magnitudes and relative magnitudes of rational numbers, including integers, fractions, decimals, percents, and absolute values.

Write, use, and talk about rational numbers, including percents, ratios, positive real numbers, and π (pi) in regards to order, absolute value, position on a number line, and to solve problems.

Locate the position of rational number and positive real numbers on a number line.

Convert between standard form, scientific notation, and exponential form.

Apply number theory concepts (primes, composites, prime factorization, LCM, GCF, least common denominator) for two or more fractions.

Recognize pertinent information for problem solving.

Understand the concept of ratio using concrete and pictorial models; write in a variety of ways.

Apply integers in one- and two-step common real-world situations.

Use appropriate vocabulary.

2. Perform computations accurately

Write and recall the common equivalent forms of numbers using exponents, scientific notation (including negative exponents), decimals, fractions, percent, and percent of halves, thirds, fourths, fifths, and tenths.

Add, subtract, multiply, divide, and order rational numbers.

Add, subtract, multiply and divide monomials

Add, subtract and multiply polynomials

Evaluate numerical expressions (add, subtract, multiply, divide) with whole number exponents and exponential numbers with the same base.

Evaluate numerical expressions with rational numbers using the order of operations.

Identify equivalent roots & radicals.

Estimate square roots using perfect squares.

Select and use an appropriate method of computation from mental math, paper and pencil, calculator, or a combination of the three.

 Use a variety of strategies including common mathematical formulas to compute problems drawn from real-life situations.

Solve story problems requiring the addition, subtraction, and multiplication of fractions.

Solve story problems requiring the addition, subtraction, multiplication, and division of integers.

Solve story problems with inequalities.

Write a number sentence to solve one-step word problems.

Interpret real-world scenarios involving integers presented on number lines.

Solve story problems which include irrelevant information.

Use different strategies to solve problems such as working backwards, guess and check, patterns pictures, tables, charts and graphs, inverse operation and breaking it up into an easier problem.

Calculate numbers expressed in scientific notation (to include negative exponents)

Calculate percents and percent of change.

Use appropriate vocabulary and notations.

3.Estimate and judge reasonableness of results

Estimate to predict computation results.

Identify when estimation is appropriate and apply to problem-solving situations.

Identify whether a given estimate is an overestimate or underestimate.

Use a four-function calculator to solve complex grade-level problems. 

Formulate conjectures and justify (short of formal proof) why they must be or seem to be true.

Recognize deductive and inductive reasoning.

Introduce the math strategy of compatible numbers in estimating for all four operations.

Use estimation to determine if solutions to word problems are reasonable.

Use appropriate vocabulary and notations.

1.  Understand and use U.S. customary and metric measurements

Select and use appropriate units and tools to make formal measurements in both systems. 

Apply estimation of measurement to real-world and content problems using standard measuring devices in standard and metric systems. 

Compare the differences and relationships among measures of perimeter, area, and volume (capacity) within both systems.

Calculate circumference, perimeter, or area of triangles, circles, and quadrilaterals, and the volume and surface area of rectangular prisms, or find a missing dimension given the area and other dimensions in multiple converted units, i.e., meters and centimeters.  (know the formulas)

Solve problems which require converting units of measurement within a system and between systems.

Solve problems using involving area of circles and the perimeter and area of rectangles and triangles.

Understand what will happen to the perimeter if the sides are lengthened or shortened.

Recognize which unit is the most appropriate to use in a given situation.

Know and use appropriate vocabulary and notations.

2. Apply concepts of rates, ratios, proportions and other derived or indirect measurements.

Recognize rates as a derived measurement

Use rates, proportions, ratios, and map scales in problem-solving situations (i.e. velocity, hourly wages, etc.).

Write ratios in simplest form and determine unit rates in real-world situations.

Set up and solve simple proportions.

Understand the concept of ratio using concrete and pictorial models and write in a variety of ways.

3. Apply dimensional analysis    

Illustrate the interrelationship of measurement units through dimensional analysis conversions.

Apply estimation of measurement to real-world and content problems using actual measuring devices.

Discover and discuss the need for standard units of measure and the need for different units of measure.

1. Use algebraic symbolism as a tool to represent mathematical relationships

Use and interpret expressions, equations, inequalities, and polynomials using models and manipulatives.

Translate simple word statements and story problems into algebraic expressions and equations.

Recognize like terms

Create and evaluate algebraic expressions and equations from word problems.

Compare integers with ordering symbols (<, >, =) and with phrases

Use symbols “<,” “>,” “=,” “≠,” “≤, and ”≥” to express relationships.

2. Evaluate algebraic expressions.

Use, identify, and apply the following properties in evaluating algebraic expressions: commutative, associative, identity, zero (multiplication by 0), inverse, distributive, and substitution.

Use and apply the rules for order of operations, and evaluate expressions with one or two unknown variables.

Solve for the missing element in a given equation.

Recognize like terms.

Multiply and divide exponential numbers with the same base & integral exponents.

Multiply & divide monomials. 

Add, subtract, & multiply polynomials, multiply a polynomial by a monomial.

Simplify algebraic expressions.

3. Solve algebraic equations and inequalities.

Solve one- and two-step equations and inequalities. 

Solve one-variable equations and inequalities using inverse operations and simplifying terms to solve problems involving fractions, decimals and integers.

Solve for the value of a variable in two variable equations.

Solve word problems with equations and inequalities.

Match graphical representations with simple linear equations.

Solve and graph linear equations.

Find the solution to two equations.

4.  Understand the concept of functions.

Extend patterns and identify a rule (function) that generates the pattern using real numbers [including finding any term in a sequence given an expression or rule for the nth term, identifying the equivalent form of a whole number to the nth power, recognizing and continuing a number pattern and/or geometric representation (e.g. Fibonacci sequence, triangular numbers), and complete a function table using whole numbers based on a given rule].

Use relationships to explain how a change in one quantity may result in a change in another, and identify the relationship as positive, negative, or neither [direct variation in terms of graphs, rates (slopes) and constants (proportional)].

Use appropriate vocabulary and notations.

5.  Represent equations, inequalities and functions in a variety of formats.

Represent a set of data in a table, as a graph, and as a mathematical relationship. 

Find the solution to two equations.

6.  Apply functions to a variety of problems.

Use patterns and linear functions to represent and solve problems.

1.  Apply concepts of size, shape and spatial relationships

Describe and classify relationships among types of  one-, two- and three-dimensional geometric figures using their defined properties, and based on similarities and differences.

Draw and measure various angles and shapes using appropriate tools. 

Understand and apply the fundamental concepts, properties, and relationships among points, lines, rays, planes, and angles (including complementary and supplementary) and 2- and 3-dimensional shapes [including edges, vertices, faces, bases, acute, right, obtuse, etc.; recognizing circle terminology: center, chord, arc, diameter, semicircle, and circumference; horizontal, vertical, congruence, symmetry, curve, perpendicular, parallel, and intersection; and identify congruent triangles according to corresponding parts (SSS,SAS,ASA)]

Identify, discuss, and model the effects of geometric transformations of reflections, translations, rotations, and scaling (dilation) on various shapes.

Identify congruence, similarities, and line symmetry of shapes.

Explain the concept of surface area and volume (capacity).

Discuss geometric relationships such as similarity and congruency using manipulatives.

Understand the effects of changing dimensions on perimeter, area, and volume.

Use appropriate vocabulary and symbols.

2. Apply the geometry of right triangles

Use the Pythagorean theorem to calculate the measure of one side of a right triangle when the other two sides are known.

Use triangle sum property to calculate the value of a missing angle.

3. Apply graphing in two dimensions

Identify and plot (graph) points, ordered pairs and analyze points in all quadrants of the coordinate plane.

Apply graphs to problem solving.

Locate the vertex on a coordinate grid.

1. Understand data analysis       

Graph, analyze and interpret data, tables, charts, and graphs, including frequency tables, scatter plots, broken line graphs, line plots, vertical and horizontal bar graphs, histograms, circle graphs, stem & leaf plots, box-and-whisker plots, Venn diagrams, and pictographs.

Explain and justify conclusions drawn from tables, charts and graphs. 

Use appropriate vocabulary and notations.

2. Collect, organize and display data      

Collect, organize and display discrete and continuous data with appropriate notation in tables, charts and graphs, including scatter plots, broken line graphs, line plots, bar graphs, histograms, and stem-and-leaf plots.

3. Apply simple statistical measurements

Choose and calculate the appropriate measure of central tendency (mean, median, mode and range) for a given group of numbers.

Explore and explain the significance of distribution of data, including range, frequency, gaps, clusters, and informal distribution.

4. Understand basic concepts of probability

Explain that probability ranges from 0% to 100% and identify a situation as having high or low probability.

Recognize equally likely outcomes

Model situations of probability using simulations.

Find the probability of independent and dependent events.

Find the odds of a simple event using permutations and combinations  

Evaluate expressions with factorials.

Use the language of probability.

5. Make predictions or decisions based on data and discuss the difference between theoretical and real world probability.

Make predictions based on experimental and theoretical probabilities.

Conduct statistical experiments and interpret results using tables, charts or graphs.

Use a tree diagram to determine the number of possible outcomes of an event.

Use appropriate vocabulary and notations.

6.  Apply appropriate technology and models to find solutions to problems

Be able to make a sketch, use manipulatives, or geometric area to aid in organizing & visualizing data.