• INTEGRATED MODERN ALGEBRA 

     

      

     

    PLEASE CHECK YOUR SCHOOL EMAIL FOR A GOOGLE CLASSROOM INVITE! 

     

    ALL WORK WILL BE POSTED ON GOOGLE CLASSROOM.

     

     

    Overview 

    The curriculum for Integrated Modern Algebra is based on the belief that mastery of learning takes place over an extended period of time. When a skill or concept is introduced and practiced, students develop familiarity with it. The intent of this course is to enable students to move toward independent learning within the context of review and extension of these skills with introduction to topics essential for further study of mathematics. Emphasis is placed on reinforcement of fundamental skills and concepts. As this course follows Algebra 1 and Plane Geometry, students who successfully complete this course will meet the NJDOE three­ year mathematics graduation requirement. Students who successfully complete and wish to continue to pursue mathematics at Wall High School can enroll in Algebra 2 CP as a senior. As this is a non­required precursor for Algebra 2 CP, students who have successfully completed Algebra 2 CP are not eligible to take this course.  

     

    Class Supplies

    - Large 3 Ring Binder

    - Lined Paper

    - Pencils or Pens

    - Graphing Calculator (recommended)

    - Charged Chromebook

     

     

    Syllabus

     

     

     

    Unit 1: Expressions, Equations, and Functions

     

    • Apply the order of operations to simplify expressions involving rational numbers. 
    • Interpret parts of an expression, such as terms, factors, and coefficients.
    • Simplify expressions using the commutative, associative and distributive properties.
    • Classify polynomials by degree and number of terms.
    • Add, subtract and multiply polynomials.
    • Factor monomials and trinomials. 
    • Identify special polynomials including difference of squares and perfect square trinomials.
    • Use a graphing calculator to compare equivalent expressions. 
    • Solve linear equations with one solution, no solution and infinitely many solutions. 
    • Use a graphing calculator to solve an equation in one variable by graphing both sides of the equation and finding the point of intersection. 
    • Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. (If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.)
    • Use the vertical line test to determine whether a graph represents a function.
    • Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
    • Identify families of functions considering linear, exponential, and quadratic functions.
    • Identify key features for linear, exponential and quadratic functions.
    • Create graphs for linear, exponential and quadratic functions by generating a table of values.
    • Create graphs for linear, exponential and quadratic functions using key features of the function type.
    • Introduce the square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

      

     

    Unit 2: Linear Functions

     

    • Prove that linear functions grow by equal differences over equal intervals.
    • Given tables of values determine which represent linear functions and explain reasoning.
    • Graph linear functions from a table, an equation or a described relationship.
    • Construct linear functions, including arithmetic sequences, given a graph, a description of a relationship, a pattern or two input-output pairs and include reading these from a table. (Find slope given two points, write equations given various types of information.)
    • ​Write a linear function in different but equivalent forms to reveal and explain different properties of the function. These forms include slope-intercept form, standard form and point-slope form each revealing different properties. .
    • Rearrange the equation of a line into different forms (translate between slope-intercept form, standard form, and point-slope form).
    • Use technology to explore the effects of the parameters m and b in the linear function f(x) = mx + b by holding first one parameter and then the other constant while allowing the other to vary. (Both in and out of context.)
    • Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval in order to identify linear functions.
    • Estimate the rate of change from a graph and compare rate of change associated with different intervals.
    • Find slopes of parallel and perpendicular lines and write equations for such.
    • Write both explicit and recursive formulas for arithmetic sequences and translate between the types. Graph the results.

     

     

    Unit 3:  Quadratics Functions

     

    • Operations with complex numbers
    • Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.
    • Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. (If appropriate, Derive the quadratic formula from this form.)
    • Investigate the graph of quadratic functions through the use of the graphing calculator.
    • Recognize transformations of the parent f(x) = x2 as vertical f(x) = x2 + k, horizontal f(x + k), stretch or reflections.
    • Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
    • Graph quadratic functions given in vertex form (f(x) = a(x - h)2 + k ) through the process of generating points in function notation and apply the meaning of symmetry to plot points.
    • Recognize that different forms of quadratic functions reveal different key features of its graph. (Standard Form: y-intercept, Vertex Form: Vertex & Max/Min Value, Factored Form: x-intercepts)
    • Relate the value of the discriminant to the type of root to expect for the graph of a quadratic function. (one real root: 1 x-intercept, two real roots: 2 x-intercepts, no real roots: no x-intercepts)
    • Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
    • Interpret models of quadratic functions given as equations or graphs.
    • Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes

     

     

    Unit 4: Systems

     

    • Given two data sets, write linear models to find the intersection of the two trend lines if it exists and explain the solution.
    • Recognize when linear systems have one solution, no solutions or infinitely many solutions. 
    • Translate algebraic verbal equations to represent linear systems and solve those systems.
    • Solve systems of linear equations through an algebraic method (elimination/linear combination, substitution) and check answer for correctness.
    • Solve systems of linear equations through a graphical approach both by hand and with a graphing calculator. Find approximate solutions when appropriate.
    • Explain why graphical approaches may only lead to approximate solutions while an algebraic approach produces precise solutions that can be represented graphically or numerically.
    • Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).
    • Solve a system involving a linear function and a quadratic function with the use of a graphing calculator.
    • Identify the number of solultions given a system of linear equations, a linear equation and quadrtic equation, or two quadratic equations. Use a graphical approach to support reasoning. 
    • Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality).
    • Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
    • Examine graphs created from Linear Programming concepts and answer questions regarding concept.  
     
    Unit 5: Exponents and Exponential Functions
    • Apply the laws of exponents to simplify expressions.
    • Interpret complicated exponential expressions by viewing one or more of their parts as a single entity. (Growth or Decay factor)
    • Use the properties of exponents to transform expressions for exponential functions.
    • Distinguish between situations that can be modeled with linear functions and with exponential functions.
    • Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    • Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
    • Recognize that geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
    • Write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
    • Graph geometric sequences.
    • Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table) and compare and contrast the results of each.

     

     

    Unit 6: Rational Functions

      

    • Identify Domain and Range from a graph using both interval notation and inequality notation. 
    • Sketch a possible graph of a function given the domain and range. 
    • Identify the Domain of a Rational Expression by inspection and with the use of the graphing calculator.
    • Identify the key characteristcs of a rational function from a graph, including the domain, range, vertical and horizontal asymptotes. 
    • Graph the function f(x) = 1/x, identifying the domain, range, vertical and horizontal asymptotes.
    • Identify the End Behavior of a Rational Function. 
    • Analyze graphs of rational functions identifying key characteristics including zeros, x- and y-intercepts, vertical and horizontal asymptotes, points of discontinuity and end behavior.

     

     

     

    Unit 7: Trig

      

    • Label a triangle in relation to the reference angle (opposite, adjacent & hypotenuse).
    • Determine the most appropriate trigonometric ratio (sine, cosine, tangent) to use for a given problem based on the information provided.
    • Solve for sides and angles of right triangles using trigonometry.
    • Explain why similar triangles have the same trigonometric ratio values.
    • Determine the exact value of the trigonometric ratios for 30, 45, and 60 degree angles.
    • Explain and use the relationship between the sine and cosine of complementary angles.
    • Define radian measure as the length of the arc on the unit circle subtended by the angle.
    • Write angles in both degrees and radians. 
    • Construct the unit circle from special right triangles.
    • Calculate exact values for sine, cosine, tangent trigonometric ratios for any radian or degree measure around the unit circle.
    • Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
    • Sketch an angle given in radians and degrees. Determine an angle measure in radians and degrees from a picture.
    • Use the relationship of the six trigonometric functions to a central angle of the unit circle to determine the exact trigonometric ratio of angles on the unit circle. (0º to 360º, 0 to 2pi)
    • Identify the angle that produce an exact value of a trigonometric function.

     

     

    Unit 8: Data and Trends

     

    • Represent data with plots on the real number line (dot plots, histograms, and box plots).
    • Calculate mean, median, mode and interquartile range.
    • Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. (Students should use spreadsheets, graphing calculators and statistical software for calculations.)
    • Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
    • Summarize categorical data for two categories in two-way frequency tables.
    • Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies).
    • Recognize possible associations and trends in the data.
    • Represent data on two quantitative variables on a scatter plot and describe how the variables are related.
    • Fit a linear function for scatter plots that suggest a linear association.
    • Interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data.
    • Use linear functions fitted to data to solve problems in the context of the data.
    • Informally assess the fit of a model function by plotting and analyzing residuals.
    • Compute (using technology) and interpret the correlation coefficient of a linear fit.
    • Distinguish between correlation and causation.