Mathematics I

MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.
a. Represent functions using function notation.
b. Graph the basic functions f(x) = xn where n = 1 to 3, f(x) = , f(x) = , and f(x) = 1/x.
c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.
d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.
e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior.
f. Recognize sequences as functions with domains that are whole numbers.
g. Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families.
h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd or neither.
i. Understand that any equation in x can be interpreted as the equation f(x) = g(x), and interpret the solutions of the equation as the x-value(s) of the intersection point(s) of the graphs of y = f(x) and y = g(x).

MM1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions.
a. Simplify algebraic and numeric expressions involving square root.
b. Perform operations with square roots.
c. Add, subtract, multiply, and divide polynomials.
d. Add, subtract, multiply, and divide rational expressions.
e. Factor expressions by greatest common factor, grouping, trial and error, and special products limited to the formulas below.
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2
(x + y) (x – y) = x2 – y2
(x + a) (x + b) = x2 + (a + b) x + ab
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x – y)3 = x3 – 3x2y + 3xy2 – y3
f. Use the binomial theorem to expand polynomials.
g. Use area and volume models for polynomial arithmetic.

MM1A3. Students will solve simple equations.
a. Solve quadratic equations in the form ax2 + bx + c = 0 where a = 1, by using factorization and finding square roots where applicable.
b. Solve equations involving radicals such as , using algebraic techniques.
c. Use a variety of techniques, including technology, tables, and graphs to solve equations resulting from the investigation of .
d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1.

 In this unit, use of the binomial theorem limits n to two or three, in future units, the value of n will be increased.
(a±b)n = an ± ncn-1 an-1b + ncn-2 an-2b2 ± ……+ nc1 abn-1 ± bn

MM1G2. Students will understand and use the language of mathematical argument and justification.
a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate.
b. Understand and use the relationships among a statement and its converse, inverse, and contrapositive.

MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons.
a. Determine the sum of interior and exterior angles in a polygon.
b. Understand and use the triangle inequality, the side-angle inequality, and the exterior-angle inequality.
c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).
d. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite.
e. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.
MM1D1 Students will determine the number of outcomes related to a given event.
a. Apply the addition and multiplication principles of counting
b. Calculate and use simple permutations and combinations
MM1D2. Students will use the basic laws of probabilities.
a. Find the probabilities of mutually exclusive events
b. Find probabilities of dependent events
c. Calculate conditional probabilities
d. Use expected value to predict outcomes
MM1D3. Students will relate samples to a population.
a. Compare summary statistics (mean, median, quartiles, and interquartile range) from one sample data distribution to another sample data distribution in describing center and variability of the data distributions.
b. Compare the averages of summary statistics from a large number of samples to the corresponding population parameters
c. Understand that a random sample is used to improve the chance of selecting a representative sample.
MM1D4. Students will explore variability of data by determining the mean absolute deviation (the averages of the absolute values of the deviations).