# Integrated Math 2

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In Integrated Math 2 students build on the probability, function, expressions and equations, and geometry work encountered previously. Integrated 2 also marks the introduction of triangle similarity and trigonometry, furthering students’ reasoning skills and language precision. Students will build on their work with probability from grade 7 to admit the notions of independence and conditional probability. Students encounter quadratic functions, comparing and contrasting them with already familiar linear and exponential functions. They develop their abilities to see structure in expressions to show that expressions involving several operations are equivalent (for example, grasping that “substitution” works at various levels of complexity), and they solve quadratic equations by writing a series of equivalent statements, justifying each step. Students continue to develop their picture of the complex number system by investigating how non-real solutions arise and how non-real numbers behave. Students use geometric transformations to define similarity, which leads to triangle similarity criteria and using these criteria to prove a wide variety of theorems and to solve problems. The definitions for sine, cosine, and tangent develop from the understanding that side ratios are predictable for triangles with given angle measures. Finally, students construct more sophisticated arguments for the circumference, area, and volume formulas that they learned in earlier grades.

## Units

#### Summary

• Describe events as subsets of a sample space (the set of outcomes) using characteristics of the outcomes or as unions, intersections, or complements of other subsets (“or,” “and” “not”). • Use the Addition Rule to compute probabilities of compound events in a uniform probability model, and interpret the result in terms of the model. • In a uniform probability model, understand the probability of A given B as the fraction of B's outcomes that also belong to A. • Understand the conditional probability of event A given event B as P(A and B)/P(B). • Understand that A and B are independent if P(A and B) = P(A) • P(B). • Interpret independence of A and B as saying that the probability of A given B is equal to the probability of A, and the probability of B given A is equal to the probability of B, i.e. P(A|B) = P(A) and P(B|A) = P(B). • Recognize independence in everyday situations and explain it in everyday language. • Determine whether events are independent. • Use data presented in two-way frequency tables to approximate conditional probabilities.

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#### Summary

• Construct quadratic functions and quadratic sequences. • Represent quadratic functions using recursive formulas, expressions, tables, and graphs. • Express quadratic functions in equivalent forms for different purposes; understand the relation between vertex form and the shape of the graph. • Find the average rate of change of a quadratic function over a unit interval and compare rates for successive intervals. • Describe properties that distinguish linear, exponential, and quadratic functions. • Model with quadratic functions.

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#### Summary

• Connect solving quadratic equations to finding zeros of quadratic functions. • Explore forms of quadratic equations that can be solved by seeing structure. • Understand and be able to use the method of factoring to solve factorable quadratic equations. • Understand and be able to use the method of completing the square to solve quadratic equations, and derive the quadratic formula. • Construct and solve quadratic equations by the most strategic method to solve problems in various contexts. • Express a quadratic function in the appropriate form for a given purpose, including vertex form. • Solve problems using systems consisting of a linear and a quadratic equation in two variables. • Derive the equation of a parabola given the focus and a directrix parallel to one of the axes.

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#### Summary

• Work with infinite decimal expansions of numbers on the number line. • Reason about operations with rational and irrational numbers. • Extend properties of integer exponents to rational exponents and write expressions with rational exponents as radicals. • Solve equations and real-world problems involving radicals and fractional exponents. • Note extraneous solutions and explain where they come from. • Discover a new type of number that is outside previously known number systems. • Perform operations with complex numbers. • Solve quadratic equations with complex solutions.

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• Verify experimentally properties of dilations, and use center and scale factor to describe them. • Use the definition of similarity to decide if two figures are similar. • Use the properties of similarity to establish AA criterion for two triangles to be similar. • Prove and use some theorems about triangles. • Prove and use slope criteria for parallel and perpendicular lines. • Construct points that partition a segment in a given ratio. • Explore why all circles are similar.

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#### Summary

• Using similarity, show that side ratios in right triangles are properties of the angles. • Define the trigonometric ratios for acute angles. • Explain and use the relationship between sine and cosine of complementary angles. • Use trigonometric ratios to solve a variety of modeling problems.

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#### Summary

• Use geometric shapes to describe objects and use measures of the shapes. • Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. • Give arguments that combine dissection and informal limits to yield the circumference and area formulas for a circle. • Give dissection arguments that yield the volume formula for prisms. • Use the volume formula for prisms and an informal limit argument to obtain the volume formula for cylinders. • Identify the shapes of two-dimensional cross-sections of three-dimensional objects. • Identify three-dimensional objects generated from rotations of two-dimensional shapes. • Obtain the formula for volume of a pyramid with square base via dissection. • Use Cavalieri’s Principle to obtain the formula for the volume of a pyramid from the formula for the volume of a pyramid with square base. • Use volume formulas to solve problems. • Solve volume problems involving the calculation of density.

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