SOLVING CUBED POLYNOMIALS EQUATIONS

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UNDERSTAND VARIOUS TYPES OF PATTERNS ANDFUNCTIONAL RELATIONSHIPS

The student should be able to:

1. Compare and contrast number patterns.
2. Identify, describe, and extend geometric and numeric patterns including growing and shrinking patterns.
3. Investigate and describe situations involving inverse relationships (e.g., the more friends the fewer the cookies for each person, the larger the denominator in a unit fraction the smaller the quantity).

The student should be able to:

1. Create, recognize, extend, find, and write rules for number patterns,
2. Identify and describe relationships between two quantities that vary together (e.g., the length of a square and its area).
3. Identify, express, and verify generalizations and use them to make predictions (e.g., doubling a number then doubling again is the same as multiplying by four),

The student should be able to.

1. Recognize, create, and continue patterns (give an informal description for continuance of the pattern and/or generalize patterns through a verbal rule,
2. Represent, interpret, and describe function relationships through tables, graphs, and verbal rules.
3. Analyze, create, and generalize numeric and visual patterns paying particular attention to patterns that have a recursive nature.
4. Use patterns to solve mathematical and applied problems,
5. Represent a variety of relations and functions with tables, graphs, verbal rules, and, when possible, symbolic rules.

The student should be able to:

1. Recognize, create, and continue patterns and generalize the pattern by giving the rule for any term.
2. Use patterns to solve mathematical and applied problems,
3. Organize data into tables and plot points onto all four quadrants of a coordinate (Cartesian) system/grid and interpret resulting patterns or trends,

The student should be able to.

1. Analyze, create, and continue numeric and visual patterns.
2. Generalize a pattern by giving the rule, for the nth term and defend the generalization with logical arguments.
3. Solve, mathematical and real world problems using patterns,
4. Represent, interpret, and describe a variety of relations and functions with tables, graphs, verbal rules, and symbolic rules (input/output).

High School
The student should be able to:

1. See the patterns in arithmetic sequences and geometric sequences using recursion (formulas expressing each term as a function of one or more of the previous terms).
2. Relate the patterns in arithmetic sequences to linear equations.
3. Relate the patterns in geometric sequences to exponential equations (e.g., squared, cubed, nth power).
4. See patterns in other sequences (e.g., quadratic, cubic).
5. Recognize equivalent forms of an expression, equation, function, or relation.
6. Be familiar with classes of functions, including bear, quadractc, power polynomial, rational, absolute value, and exponential.
7. Select appropriate representations (numerical, graphical, verbal and symbolic) for the functions embedded in quantitative situations, and use them to interpret the situations represented.
8. Use a variety of symbolic representations to explore the behavior of functions and relations.

USESYMBOLIC FORMS TO REPRESENT AND ANALYZEMATHEMATICAL SITUATIONS AND STRUCTURES

The students should be able to:

1. Explore variables and solve equations using variables.
2. Formulate rules for number relationships,
3. Identify and use relationships between operations to solve problems (e.g. multiplication as the inverse of division.
4. Identify and use. algebraic properties of operations to solve problems, e.g., 28 x 7 is equivalent to (7 x 20) + (7 x 8) or (7 x 30) (7 x 2).

The students should be able to:

1. Generalize a rule for ordered pairs.
2. Develop the concept of variable as a useful tool for representing unknown quantities.
3. Use variables (boxes, letters, or other symbols) to solve problems of to describe general rules,

The students should be able to:

1. Write and solve equations with one variable, using concrete and/or informal methods that model everyday situations.
2. Explore the concept of variable, expression, and equation.
3. Solve problems involving simple formulas (i.e., A = I w, P = 21 + 2w)
4. Interpret relationships between tables and graphs.
5. Organize data into tables and plot points onto the first quadrant of a coordinate (Cartesian) system/grid.
6. Develop a sound conceptual understanding of equation and of variable.
7. Explore relationships between symbolic expressions and graphs,

The student should be able to:

1. Understand the concept of equations and inequalities using variables as they relate to everyday situations.
2. Simplify numeric and algebraic expressions.
3. Use a variety of methods and representations to create and solve single‑variable equations that may be applied to everyday situations.
4. Solve problems involving formulas,
5. Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
6. Explore relationships between symbolic expressions and graphs, paying particular attention to the horizontal and vertical intercepts, points of intersection, and slope (for linear relations).

The students should be able to:

1. Develop a sound conceptual understanding of equation and variable.
2. Translate verbal expressions into algebraic expressions.
3. Apply and simplify algebraic expressions in a wide variety of situations.
4. Use a variety of methods and representations to create and solve one and two variable linear equations that require one and two steps.
5. Solve problems that involve substitution and formulas.
6. Use symbolic algebra to represent situations and to solve real life problems, especially those involving) linear relationships.
7. Create and solve simple inequalities using a variety of methods and representations,

High School
The students will be able to:

USE MATHEMATICAL MODELS AND ANALYZE
CHANGE IN BOTH REAL AND ABSTRACT CONTEXTS

The student should be able to

1. Graph points on a number line.
2. Represent and describe relationships through the use of variables, ordered pairs. lists in tables, plots on graphs, and patterns,
3. Represent and investigate how a change in one variable relates to the change in a second variable. (e.g. the height of a plant over time).

The student should be able to:

1. Explore variables and solve equations using variables.
2. Identify mid describe situations with varying rates of change, (e. g. a fundraising effort brought its a small, steady amount of money in the beginning, but more each day as the deadline approached).

The student should be able to:

1. Model and solve contextualized problems using various representations such as graphs and tables to understand the purpose and utility of each representation.
2. Introduce different types of change related to discrete patterns.

The student should be able to:

1. Represent, interpret and describe functional relationships through tables, graphs, and verbal rules (input/output).
2. Interpret relationships between tables, graphs, verbal rules, and equations.
3. Develop an initial understanding of rate of change, with emphasis on the connections among slope of a line, constant rate of change, and their meaning in context.
4. Explore different types of change occurring in discrete patterns such as proportional and linear change.

The student should be able to:

1. Model and solve contextualized problems using various representations such graphs, tables, verbal rules, and equations to understand the purpose and use of each representation.
2. Model and solve simple inequalities using a variety of methods and representations.
3. Organize data into tables, plot points onto all four quadrants of a coordinate system/grid and interpret resulting patterns or trends.
4. Explore different types of change occurring in discrete patterns, such as proportional and linear change.
5. Graph linear functions in. a four quadrant (Cartesian) system/grid and interpret the results such as the slope and equation of a line by analyzing the line (e.g., y = mx + b, in is rise/run, b is y intercept).
6. Explain how the change in one variable affects the change in another variable (e.g., in distance equals rate times time, increasing time, increasing distance).

High School (Algebra 1)
The student should be able to:

1. Use characteristics of the graphs of linear functions, such as slope, intercepts, and transformations,
2. Collect, organize, and display two variable data, and use a line of best fit or curve of best fit as a model to make predictions,
3. Extend ideas of transformations of linear equations, such as vertical and horizontal shifts, to transformations of non-linear equations.
4. Recognize that a particular type of function can model many different situations,
5. Approximate and interpret accumulation and rates of change, both graphically and numerically, for functions representing a variety of situations.

April 23, 2000