Strands |
8th
grade |
9th -12th
grade |
| A. Patterns and Relationships |
1. Recognize, describe, extend, and create patterns involving
whole numbers, rational numbers, and integers.
- Descriptions using tables, verbal and symbolic rules, graphs, simple equations
or expressions
- Finite and infinite sequences
- Arithmetic sequences (i.e., sequences generated by repeated addition of
a fixed number, positive or negative)
- Geometric sequences (i.e., sequences generated by repeated multiplication
by a fixed positive ratio, greater than 1 or less than 1)
- Generating sequences by using calculators to repeatedly apply a formula
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1. Use models and algebraic formulas to represent and analyze sequences and series
- Explicit formulas for nth terms
- Sums of finite arithmetic series
- Sums of finite and infinite geometric series
2. Develop an informal notion of limit
3.
Use inductive reasoning to form generalizations |
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| B. Functions and Relationships |
1. Graph functions, and understand and describe their general
behavior.
- Equations involving two variables
- Rates of change (informal notion of slope)
2. Recognize and describe the difference between linear and exponential growth,
using tables, graphs, and equations. |
1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.
2.
Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.
- Slope of a line or curve
- Domain and range
- Intercepts
- Continuity
- Maximum/minimum
- Estimating roots of equations
- Intersecting points as solutions of systems of equations
- Rates of change
3.
Understand and perform transformations on commonly-used functions.
- Translations, reflections, dilations
- Effects on linear and quadratic graphs of parameter changes in equations
- Using graphing calculators or computers for more complex functions
4. Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.
- Linear vs. non-linear
- Symmetry
- Increasing/decreasing on an interval
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| C. Modeling |
1. Analyze functional relationships to explain how a change in
one quantity can result in a change in another, using pictures, graphs,
charts, and equations.
2. Use patterns, relations, symbolic algebra, and linear functions to model
situations.
- Using concrete materials (manipulatives), tables, graphs, verbal rules,
algebraic expressions/equations/inequalities
- Growth situations, such as population growth and compound interest, using
recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social
studies standard 6.6)
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1. Use functions to model real-world phenomena and solve problems that involve varying quantities.
- Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)
- Direct and inverse variation
- Absolute value
- Expressions, equations and inequalities
- Same function can model variety of phenomena
- Growth/decay and change in the natural world
- Applications in mathematics, biology, and economics (including compound interest)
2. Analyze and describe how a change in an independent variable leads to change in a dependent one.
3. Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).
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| D. Procedures |
1. Use graphing techniques on a number line.
- Absolute value
- Arithmetic operations represented by vectors (arrows) (e.g., "-3 +
6" is "left 3, right 6")
2. Solve simple linear equations informally, graphically, and using formal
algebraic methods
- Multi-step, integer coefficients only (although answers may not be integers)
- Using paper-and-pencil, calculators, graphing calculators, spreadsheets,
and other technology
3. Solve simple linear inequalities.
4. Create, evaluate, and simplify algebraic expressions involving variables.
- Order of operations, including appropriate use of parentheses
- Distributive property
- Substitution of a number for a variable
- Translation of a verbal phrase or sentence into an algebraic expression,
equation, or inequality, and vice versa
3. Understand and apply the properties of operations, numbers, equations, and
inequalities.
- Additive inverse
- Multiplicative inverse
- Addition and multiplication properties of equality
- Addition and multiplication properties of inequalities
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1. Evaluate and simplify expressions.
- Add and subtract polynomials
- Multiply a polynomial by a monomial or binomial
- Divide a polynomial by a monomial
2. Select and use appropriate methods to solve equations and inequalities.
- Linear equations - algebraically
- Quadratic equations - factoring (when the coefficient of x2 is 1) and using the quadratic formula
- All types of equations using graphing, computer, and graphing calculator techniques
3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
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