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New Jersey Math Standards - 12th Grade
MathScore aligns to the New Jersey Math Standards for 12th Grade. The standards appear below along with the MathScore topics that match. If you click on a topic name, you will see sample problems at varying degrees of difficulty that MathScore generated. When students use our program, the difficulty of the problems will automatically adapt based on individual performance, resulting in not only true differentiated instruction, but a challenging game-like experience.
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Number and Numerical Operations
4.1.12 A. Number Sense1. Extend understanding of the number system to all real numbers.
2. Compare and order rational and irrational numbers.
3. Develop conjectures and informal proofs of properties of number systems and sets of numbers.
4.1.12 B. Numerical Operations
1. Extend understanding and use of operations to real numbers and algebraic procedures.
2. Develop, apply, and explain methods for solving problems involving rational and negative exponents. (Negative Exponents Of Fractional Bases , Roots Of Exponential Expressions )
3. Perform operations on matrices.
• Addition and subtraction
• Scalar multiplication
4. Understand and apply the laws of exponents to simplify expressions involving numbers raised to powers. (Simplifying Algebraic Expressions 2 , Exponents Of Fractional Bases , Negative Exponents Of Fractional Bases , Multiplying and Dividing Exponent Expressions , Exponent Rules For Fractions , Roots Of Exponential Expressions , Simplifying Radical Expressions , Adding and Subtracting Radical Expressions , Multiplying and Dividing Radical Expressions )
4.1.12 C. Estimation
1. Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.
Geometry and Measurement
4.2.12 A. Geometric Properties1. Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).
2. Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).
3. Apply the properties of geometric shapes.
• Parallel lines - transversal, alternate interior angles, corresponding angles (Identifying Angles , Angle Measurements 2 )
• Triangles
a. Conditions for congruence
b. Segment joining midpoints of two sides is parallel to and half the length of the third side
c. Triangle Inequality
d. Special right triangles (Triangle Angles 2 , Pythagorean Theorem , Solving For Angles )
• Minimal conditions for a shape to be a special quadrilateral
• Circles - arcs, central and inscribed angles, chords, tangents
• Self-similarity
4. Use reasoning and some form of proof to verify or refute conjectures and theorems.
• Verification or refutation of proposed proofs
• Simple proofs involving congruent triangles
• Counterexamples to incorrect conjectures
5. Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology).
• Perpendicular bisector of a line segment
• Bisector of an angle
• Perpendicular or parallel lines
4.2.12 B. Transforming Shapes
1. Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic representation, and, conversely, determine whether and how one representation can be transformed to another by a transformation or a sequence of transformations.
2. Recognize three-dimensional figures obtained through transformations of two- dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.
3. Determine whether two or more given shapes can be used to generate a tessellation.
4. Generate and analyze iterative geometric patterns.
• Fractals (e.g., Sierpinski's Triangle)
• Patterns in areas and perimeters of self-similar figures
• Outcome of extending iterative process indefinitely
4.2.12 C. Coordinate Geometry
1. Use coordinate geometry to represent and verify properties of lines and line segments.
• Distance between two points
• Midpoint and slope of a line segment (Determining Slope )
• Finding the intersection of two lines
• Lines with the same slope are parallel (Applied Linear Equations 2 )
• Lines that are perpendicular have slopes whose product is -1 (Applied Linear Equations 2 )
2. Show position and represent motion in the coordinate plane using vectors.
• Addition and subtraction of vectors
3. Find an equation of a circle given its center and radius and, given an equation of a circle in standard form, find its center and radius.
4.2.12 D. Units of Measurement
1. Understand and use the concept of significant digits.
2. Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation.
• Degree of accuracy of a given measurement tool
• Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements
4.2.12 E. Measuring Geometric Objects
1. Use techniques of indirect measurement to represent and solve problems.
• Similar triangles
• Pythagorean theorem
• Right triangle trigonometry (sine, cosine, tangent)
2. Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures.
• Approximation of area using grids of different sizes
• Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets
• Estimation of area, perimeter, volume, and surface area
Patterns and Algebra
4.3.12 A. Patterns1. 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.
4.3.12 B. Functions and Relationships
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 or relations, using algebraic and graphing techniques.
• Slope of a line (Determining Slope )
• Domain and range (Domain and Range )
• Intercepts (Quadratic X-Intercepts , Quadratic Zero Equations , Quadratic Formula )
• Continuity
• Maximum/minimum
• Estimating roots of equations
• Solutions of systems of equations (System of Equations Substitution , System of Equations Addition )
• Solutions of systems of linear inequalities using graphing techniques
• 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
4.3.12 C. Modeling
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 (Absolute Value 2 , Absolute Value Equations )
• Expressions, equations and inequalities
• Same function can model variety of phenomena
• Growth/decay and change in the natural world (Continuous Compound Interest )
• 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. (Independent and Dependent Variables )
3. Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).
4.3.12 D. Procedures
1. Evaluate and simplify expressions.
• Add and subtract polynomials (Simplifying Algebraic Expressions )
• Multiply a polynomial by a monomial or binomial (Foil Method )
• Divide a polynomial by a monomial
• Perform simple operations with rational expressions (Binomial Fraction Simplification , Polynomial Fraction Simplification )
• Evaluate polynomial and rational expressions (Variable Substitution 2 )
2. Select and use appropriate methods to solve equations and inequalities.
• Linear equations and inequalities - algebraically (Single Variable Equations 3 , Single Variable Inequalities )
• Quadratic equations - factoring (including trinomials when the coefficient of x2 is 1) and using the quadratic formula (Trinomial Factoring , Quadratic Zero Equations , Quadratic Formula , Quadratic X-Intercepts )
• Literal equations
• All types of equations and inequalities 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.
Data Analysis, Probability, and Discrete Mathematics
4.4.12 A. Data Analysis1. Use surveys and sampling techniques to generate data and draw conclusions about large groups.
• Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)
2. Evaluate the use of data in real-world contexts.
• Accuracy and reasonableness of conclusions drawn
• Correlation vs. causation
• Bias in conclusions drawn (e.g., influence of how data is displayed)
• Statistical claims based on sampling
3. Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome.
4. Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data.
5. Analyze data using technology, and use statistical terminology to describe conclusions.
• Measures of dispersion: variance, standard deviation, outliers
• Correlation coefficient
• Normal distribution (e.g., approximately 95% of the sample lies between two standard deviations on either side of the mean)
6. Distinguish between randomized experiments and observational studies.
4.4.12 B. Probability
1. Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair.
2. Use concepts and formulas of area to calculate geometric probabilities.
3. Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.
4. Determine probabilities in complex situations.
• Conditional events
• Complementary events
• Dependent and independent events (Object Picking Probability )
5. Estimate probabilities and make predictions based on experimental and theoretical probabilities.
6. Understand and use the "law of large numbers" (that experimental results tend to approach theoretical probabilities after a large number of trials).
4.4.12 C. Discrete Mathematics-Systematic Listing and Counting
1. Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).
2. Apply the multiplication rule of counting in complex situations, recognize the difference between situations with replacement and without replacement, and recognize the difference between ordered and unordered counting situations.
3. Justify solutions to counting problems.
4. Recognize and explain relationships involving combinations and Pascal's Triangle, and apply those methods to situations involving probability.
4.4.12 D. Discrete Mathematics-Vertex-Edge Graphs and Algorithms
1. Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems.
• Circuits that include every edge in a graph
• Circuits that include every vertex in a graph
• Scheduling problems (e.g., when project meetings should be scheduled to avoid conflicts) using graph coloring
• Applications to science (e.g., who-eats-whom graphs, genetic trees, molecular structures)
2. Explore strategies for making fair decisions.
• Combining individual preferences into a group decision (e.g., determining winner of an election or selection process)
• Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment)
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