# Integrated Math II. IM2.1.2 Interpret given situations as functions in graphs, formulas, and words.

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1 Standard 1: Algebra and Functions Students graph linear inequalities in two variables and quadratics. They model data with linear equations. IM2.1.1 Graph a linear inequality in two variables. IM2.1.2 Interpret given situations as functions in graphs, formulas, and words. IM2.1.3 Find a linear equation that models a data set using the median fit method and use the model to make predictions. IM2.1.4 Graph quadratic functions. Apply transformations to quadratic functions. Find and interpret the zeros and maximum or minimum value of quadratic functions. Standard 2: Geometry and Measurement Students identify and describe types of triangles. They define and apply the trigonometric relations. Students apply theorems to triangles and circles. IM2.2.1 Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems. IM2.2.2 Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used. IM2.2.3 Find the measures of interior and exterior angles of polygons, justifying the method used. IM2.2.4 Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. IM2.2.5 Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors. IM2.2.6 Use properties of congruent and similar triangles to solve problems involving lengths and areas. IM2.2.7 Find and use measures of sides, perimeters, and areas of triangles. Relate these measures to each other using formulas. 1

2 IM2.2.8 Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and the hinge theorem. IM2.2.9 State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. IM Use special right triangles (30º - 60º and 45º - 45º) to solve problems. IM Define and use the trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. IM Know and use the relationship sin 2 x + cos 2 x = 1. IM Solve word problems involving right triangles. IM Find the center of a given circle. Construct the circle that passes through three given points not on a line. IM Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent. IM Prove theorems related to circles. IM Construct tangents to circles and circumscribe and inscribe circles. IM Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). IM Define and identify congruent and concentric circles. IM Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems. 2

3 IM Describe sets of points on spheres: chords, tangents, and great circles. Standard 3: Data Analysis and Statistics Students interpret scatterplots and analyze correlation. IM2.3.1 Describe the association between two variables by interpreting a scatterplot. IM2.3.2 Interpret correlation coefficients. IM2.3.3 Make predictions from the least squares regression line or its equation. IM2.3.4 Understand that a correlation between two variables does not necessarily imply one directly causes the other. IM2.3.5 Understand the effects of outliers on correlation coefficients, on the least squares regression line, and on the interpretations of correlation coefficients and regression lines in real-life contexts. Standard 4: Probability Students construct probability distributions, understand fundamental probability concepts, and use counting principles. IM2.4.1 Construct a probability distribution by simulation and use it to understand and analyze the probabilistic situation. IM2.4.2 Explore the geometric, or waiting-time, distribution. IM2.4.3 Understand fundamental concepts of probability (i.e., independent events, multiplication rule, and expected value). IM2.4.4 Understand and apply counting principles to compute combinations and permutations. IM2.4.5 Use the basic counting principle, combinations, and permutations to compute probabilities. 3

4 Standard 5: Discrete Mathematics Students use graphs and networks as mathematical models and use matrices to solve problems. IM2.5.1 Experience in mathematical modeling by building and using vertex-edge graph models to solve problems in a variety of real-world settings. IM2.5.2 Develop the skill of algorithmic problem solving: designing, using, and analyzing systematic procedures for problem solving. IM2.5.3 Optimize networks in different ways and in different contexts by finding minimal spanning trees, shortest paths, and Hamiltonian paths. IM2.5.4 Use matrices to organize and displaying data in a variety of real-world settings. IM2.5.5 Develop mathematical modeling skills by building matrix models and then apply the models to solve problems. IM2.5.6 Apply matrix operations to solve problems (i.e., row sums, scalar multiplication, addition, subtraction, and matrix multiplication). IM2.5.7 Use matrices and inverse matrices to answer questions that involve systems of linear equations. IM2.5.8 Build and use matrix representations to model polygons, transformations, and computer animations. Standard 6: Trigonometry Students apply trigonometric ratios to right triangles. IM2.6.1 Explore properties and applications of the sine, cosine, and tangent ratios for the lengths of sides of right triangles. Standard 7: Mathematical Reasoning and Problem Solving 4

5 Students use a variety of strategies to solve problems and develop and evaluate mathematical arguments and proofs. IM2.7.1 Use the properties of the real number system and the order of operations to justify the steps of simplifying functions and solving equations. IM2.7.2 Make conjectures about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. IM2.7.3 Write and interpret statements of the form if then and if and only if. IM2.7.4 State, use, and examine the validity of the converse, inverse, and contrapositive of if then statements. IM2.7.5 Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs. IM2.7.6 Perform basic constructions, describing and justifying the procedures used. Distinguish between constructing and drawing geometric figures. IM2.7.7 Decide if a given algebraic statement is true always, sometimes, or never (statements involving quadratic expressions). IM2.7.8 Understand that the logic of equation solving begins with the assumption that the variable is a number that satisfies the equation, and that the steps taken when solving equations create new equations that have, in most cases, the same solution as the original. Understand that similar logic applies to solving systems of equations simultaneously. IM2.7.9 Use counterexamples to show that statements are false. 5

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