Mathematics Curriculum: Algebra 2
School District of South Orange and Maplewood
Mathematics Department
Appendix C: Level Distinctions in Algebra 2
Leveling is not a function of intelligence or mathematical talent or the ability to learn. Leveling in math begins with a consideration of the mathematical content that has to be developed, takes a measure of students’ prior learning, and enacts a plan to maximize learning across a spectrum of student achievement.
This appendix described the parameters of content by level for each learning objective as delineated in the content outline. At each level, students will develop every learning objective prescribed by State Standards (adopted in 2008). Variations and modifications in the content outline are based on evidence of foundational knowledge, the instructional time required for students to obtain mastery of essential aspects of content, and the opportunity to make mathematical decisions as the content is developed.
The mathematics identified here establishes the expectations for mastery in each level.
1. Graph real world phenomena and solve problems that involve variation. / These objectives identify deficiencies and gaps in 11-2 students' prerequisite knowledge and skills that typically lead to problems over the course of the school year. Most notably, these are fractions, negative numbers, graph interpretation, and algebraic computation. In obj 1-3, these crucial topics are developed/clarified at the beginning of the course to avoid sticking points during the year. / The connections regarding this objective are made among students in relatively shorter times than for level 2. / Prior mastery of topic expected. / Prior mastery of topic expected.
2. Apply and explain methods for solving problems involving integers and rational numbers. / The connections regarding this objective are made among students in relatively shorter times than for level 2. / Prior mastery of topic expected. / Prior mastery of topic expected.
3. Evaluate and simplify polynomial expressions / The connections regarding this objective are made among students in relatively shorter times than for level 2. / Prior mastery of topic expected. / Prior mastery of topic expected.
4. Use the properties of real numbers to evaluate expressions and formulas, and solve equations / -Overt discussion and activities to distinguish understanding between expressions and equations.
-Extended practice on the types of solutions linear equations can have (no solution, infinite solutions, and 1 solution) with most examples being teacher-generated.
-Remediate approach to making mathematical decisions to solve equations
-Integer coefficients and common fractions in linear equations
-Stress on students describing their problem-solving approach (minimum formalization) / -Brief practice to review the types of solutions linear equations can have (no solution, infinite solutions, and 1 solution)
-More formal use of vocabulary is required for student-generated examples.
-Rational coefficients in linear equations are now required.
-Student are required to generate examples on assessment. / Prior mastery of topic expected. / Prior mastery of topic expected.
Objective / Level 2 / Level 3 / Level 4 / Level 5
5. Solve absolute value equations / -Frame absolute value as the distance away from zero.
-Beginning example:
=constant.
-Students will be responsible for integer coefficient equations embedded in absolute value equations. / -Students will be responsible for integer coefficient equations embedded in absolute value equations.
-Build upon prior knowledge of absolute value as a scenario usually requiring two cases. / Prior familiarity expected
-Solving multistep absolute value equations
-Extend to more rigorous algebraic processes such as
3|2x – 5|= 15and |3x – 7| = x+ 2 / Prior mastery of topic expected.
6. Solve and graph inequalities / -Primary focus on linear inequalities with integer coefficients and checking validity of solutions
-Expressing solutions to inequalities as intervals on a number line.
-Open and closed interval notations. / -Solving absolute value inequalities with integer and simple rational solutions.
-Write the absolute value inequality from a number line.
-Solve absolute value inequalities using defs of intersection/union. / Prior familiarity expected
Solving absolute value inequalities with non standard solutions
-Write the absolute value inequality from a number line / Note: Prior familiarity expected. Diagnostic exam administered to determine need for review.
Topic extensions:
-Multiple representations both on a number line and a coordinate plane
7. Recognize, represent, and use linear functions to represent real world phenomena and solve problems. / -Discuss differences between discrete and continuous function
-Responsible for scenarios involving continuous data sets only
-Distinguishing between and finding x and y intercepts (limited to integer values)
-Describing domain and range with inequality notation or by a verbal expression / -Discuss differences between discrete and continuous function
-Responsible for scenarios involving continuous and discrete data sets.
-Distinguishing between and finding x and y intercepts (rational values possible)
-Describing domain and range with inequality notation or by a verbal expression / Prior familiarity expected
-Describing domain and range with inequality notation
-Introduction to interval notation
-No limit to types of values used in or found in problems / Note: Prior familiarity expected. Diagnostic exam administered to determine need for review.
Topic extensions:
-Describing domain and range with interval notation
-Writing equation from abstract point sets
8. Analyze and determine the rate of change using appropriate graphing technologies. / -Re-teach graphical and algebraic representation of slope
-Using computer software (green globs, equation grapher) to reinforce understanding
-Scatterplot would only include integer coordinates and a limited number of points. (n <10) / -Student-driven discussion to recall slope definition.
-Student exposure to best-fit lines found via graphing utility vs. finding by hand(student choice on assessment)
-Scatterplot would only include integer coordinates and a larger set of points (n >10) / -Discussion of what a correlation coefficient is and interpreting correlation coefficient as a measure of goodness of fit.
Finding the value numerically on the calculator.
-Assessment includes finding the line of best fit on a calculator / Note: Prior familiarity expected. Diagnostic exam administered to determine need for review.
Topic extensions:
-Interpreting the meaning of slope in real life situations
-Interpreting correlation coefficient as a measure of goodness of fit
-Finding the line of best fit on a calculator and its use in extrapolation/interpolation.
Objective / Level 2 / Level 3 / Level 4 / Level 5
9. Select and use appropriate methods for solving linear equations. / -Students are given a list of linear formats from which to choose.
-Solutions limited to integers and common fractions / -Students required to know linear formats for use on assessments
-Information embedded in problems are easily identifiable
-Solutions may be rational. / -Heavier use of information embedded in verbal passages (not very easily identified)
-Generate parallel/perpendicular line sets to satisfy a given condition. / Note: Prior familiarity expected. Diagnostic exam administered to determine need for review.
Topic extensions:
-Multi-variable representations of linear equations and the concept of “partial slope”.
10. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.
Identify and compare the properties of classes of functions. / -Limited to identity function, absolute value function and quadratic function.
-Use of computer software and graphing calculators to reinforce parent function identification
-Basic dilations of graphs / -Extend discussion to include piecewise functions where students generate a graph from a given piecewise function.
-Comparisons between non-common dilations (rational coefficients) / -Calculating dilation constant from a given point on a function.
-Piecewise functions will also be written given a graphical representation of the function only
-Generate piecewise functions from given scenarios involving real world examples(i.e. shipping fees) / Note: Prior familiarity expected. Diagnostic exam administered to determine need for review.
Topic extensions:
-Representing absolute value functions in piecewise form
-Real world examples of piecewise functions (i.e. shipping fees, tax brackets)
11. Perform transformations on commonly-used functions. / -Limited to identity function, absolute value function and quadratic function.
-Describing changes in domain and range as verbal expressions only / -Function list extended to include square root functions
-Describing changes in domain and range as verbal expressions or inequality notation / Function list extended to include square root functions and cubic functions
Topic extensions:
-Describing changes in domain and range in interval notation / Topic extensions:
-Describing changes in domain and range in interval notation
12. Graph linear and absolute value inequalities / -Teacher-led discussion to explore the idea of a half-plane
-Reinforcement of open/closed sets and their respective symbols / -Student-generated ideas of definition of a half-plane
-Extend to graphs of absolute value inequalities / -Linear Inequalities can be represented in standard form.
-Extend topic to 3 or more graphs on a coordinate plane.
-Can include absolute value and linear inequalities on a single coordinate plane / Brief diagnostic review of topic
-Extend topic to 3 or more graphs on a coordinate plane.
-Can include absolute value and linear inequalities on a single coordinate plane
Objective / Level 2 / Level 3 / Level 4 / Level 5
13. Solve systems of linear equations / -Limited to 2x2 systems of equations.
-Integer coefficients for all equations
-Coefficients are integers and solutions limited to integers and common fractions (halves, quarters, etc)
-Systems given are in same linear format / -No limit to solutions for 2x2 systems.
-Extend topic to include 3x3 systems.
-All solutions to 3x3 systems kept as integers.
-Systems given are in mixed linear formats / -No limit to coefficients and solutions.
-Graphical representation of 2x2 systems / Topic extensions:
-Solving 3 dimensional systems of linear equations.
-Graphical representation of the case of no solution and infinite solutions for 3 dimensional systems.
-Specifying the vector solutions of an infinite solution 3x3 system.
14. Solve real world problems using systems of inequalities. / -feasible regions limited to 3 lines with only one being oblique
-distinguish between points on boundary lines and those inside the feasible region / -Feasible regions limited to 3 oblique lines or 4 lines (at most 2 being oblique)
-uniform formats presented / Brief diagnostic review of topic
-mixed formats presented / Topic extensions:
-Modeling real-world scenarios as linear inequalities (e.g. animal dietary requirements).
15. Use linear programming to solve real-world problems** / Not Taught / -Constraints given
-No more than 3 constraints
-Only one line as oblique
-All integer solutions / -No constraints given
-All integer solutions
-no limit to formats of lines
-compound inequalities often
-feasible regions extended to uncommon quadrilaterals and pentagons
-extend to discrete math with lattice points / -No constraints given
-Non integer solutions likely
16. Describe and perform operations on matrices. Solve systems through matrix multiplication, using inverses. / Not Taught / -Operations limited to 2x2 matrices by hand
-Systems of equations limited to a 2x2 systems using inverses / - Writing and solving systems of equations of 3x3 system using matrices on graphing utility
- Use and application of Cramer’s Rule
-See the advantage of using matrices to solve systems / - Writing and solving systems of equations of 3x3 systems using matrices
- Use and application of Cramer’s Rule
-See the advantages of using matrices to solve systems
Topic extensions:
- Solving nxn and nxm systems.
-Introduction to other matrix-oriented techniques such as Gaussian elimination
Objective / Level 2 / Level 3 / Level 4 / Level 5
17. Recognize and use connections among significant values of a quadratic function, points on the graph of the function, and its symbolic representation. / -Graphing quadratics from tables
- The axis of symmetry will only be an integer value
-Intercepts of parabolas limited to integers
- Quadratics will have integer coefficients / -Graphing quadratics from tables and using the axis of symmetry
- The axis of symmetry will only be an integer value or simple fraction
-find standard form of a quadratic given integer (distinct) roots only.
-Sum and difference of cubes formulas for factoring given
-Use of zero product property for quadratics given in factored form / -Graphing quadratics using the axis of symmetry
-The axis of symmetry unlimited
-find standard form of a quadratic given any type of roots (rational, irrational, and imaginary)
- Ability to interchangeably express quadratic functions in factored form, vertex form, and standard form
-Leading coefficient can be composite. / -Graphing quadratic functions in standard form, vertex form and/or intercept/factored form.
-Ability to interchangeably express quadratic functions in standard, vertex or factored form.
-Finding the standard form of a quadratic given any roots (rational, irrational, and imaginary)
18. Identify properties of imaginary and complex numbers and operate on them.
Solve simple quadratic equations with imaginary solutions. / -Limited to complex outcomes with integer coefficients
-Complex conjugates limited to monomials
-Manipulation of complex numbers in a+bi (standard form) via FOIL and distribution. / -Extend to binomial complex numbers to powers greater than or equal to 3.
-Complex conjugates extended to binomials with integer coefficients. / -Unlimited exposure to operations complex numbers (and conjugates) with rational and irrational coefficients
- application of binomial theorem introduced / -Complex number operations.
-Graphical representation of imaginary numbers
19. Solve quadratic equations by completing the square. / -Solving limited to simple quadratics where b=0 / -Extend topic to include solving perfect square binomials
-Extend exposure to quadratic trinomials where b is not a factor of the leading coefficient / -Distinguish between methods of solving quadratics needed when leading coefficient is not equal to one, but still an integer / -Solving general quadratics (when leading coefficient not necessarily equal to one) using completing the square.
Topic extensions:
-Discovering the completing the square algorithm as a natural extension of how to form a perfect square trinomial.
-Geometric interpretation of completing the square.
Objective / Level 2 / Level 3 / Level 4 / Level 5
20. Solving quadratic equations using the Quadratic Formula / -Reinforce standard form (quadratics must be set equal to zero)
-Distinguish between possible types of solution sets (one real, two real, or two complex)
-Use discriminant to generalize outcomes for solutions