Focus Areas for Integrated Math II

Math Standards by Domain

Number and Quantity

What Students Learn:

Students extend exponent properties to rational exponents and explore relationships between rational and irrational numbers.

Key Skills:
  • Explain how rational exponents follow from integer exponent properties
  • Understand 5^(1/3) is the cube root of 5
  • Rewrite expressions with radicals and rational exponents
  • Explain why sum/product of rationals is rational
  • Explain why rational + irrational = irrational
  • Explain why (nonzero rational) × irrational = irrational
Example:

Rewrite ∛(x⁵) = x^(5/3)

Why is 3 + √2 irrational? Since √2 is irrational and 3 is rational, their sum must be irrational.

What Students Learn:

Students learn about complex numbers, including i = √(-1), and perform arithmetic operations with complex numbers. They solve quadratic equations with complex solutions.

Key Skills:
  • Know i² = -1 and every complex number has form a + bi
  • Add, subtract, and multiply complex numbers using i² = -1
  • Solve quadratic equations with complex solutions
  • Extend polynomial identities to complex numbers (e.g., x² + 4 = (x + 2i)(x - 2i))
  • Understand the Fundamental Theorem of Algebra for quadratics
Example:

Solve x² + 9 = 0: x² = -9, so x = ±3i

Multiply (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i

Algebra

What Students Learn:

Students interpret quadratic and exponential expressions, factor to reveal zeros, complete the square to find vertex form, and transform exponential expressions.

Key Skills:
  • Interpret expressions in context
  • Use structure to rewrite expressions (e.g., x⁴ - y⁴ = (x² - y²)(x² + y²))
  • Factor quadratics to reveal zeros
  • Complete the square to reveal maximum/minimum
  • Transform exponential expressions (e.g., 1.15^t = (1.15^(1/12))^(12t) ≈ 1.012^(12t))
Example:

Factor x² + 6x + 8 = (x + 2)(x + 4) to find zeros at x = -2 and x = -4

Complete the square: x² + 6x + 5 = (x + 3)² - 4, showing minimum at x = -3

What Students Learn:

Students understand polynomials form a system like integers, closed under addition, subtraction, and multiplication. They add, subtract, and multiply polynomials.

Key Skills:
  • Add polynomials by combining like terms
  • Subtract polynomials by distributing negatives
  • Multiply polynomials using distributive property
  • Recognize closure property of polynomials
Example:

Multiply (x² + 2x - 1)(x - 3):

= x³ - 3x² + 2x² - 6x - x + 3

= x³ - x² - 7x + 3

What Students Learn:

Students solve quadratic equations using multiple methods and solve linear-quadratic systems. They recognize when quadratics have complex solutions.

Key Skills:
  • Complete the square to transform any quadratic
  • Derive the quadratic formula from completing the square
  • Solve by inspection, square roots, completing square, formula, and factoring
  • Recognize complex solutions and write as a ± bi
  • Solve linear-quadratic systems algebraically and graphically
Example:

Solve x² + 4x - 1 = 0 using quadratic formula:

x = (-4 ± √(16 + 4))/2 = (-4 ± √20)/2 = -2 ± √5

Functions

What Students Learn:

Students interpret quadratic functions, analyze their key features, and learn to graph various function types including absolute value, step, and piecewise functions.

Key Skills:
  • Interpret key features of quadratic functions
  • Relate domain to context
  • Calculate average rate of change
  • Graph quadratics showing intercepts, maxima/minima
  • Graph square root, cube root, piecewise, step, and absolute value functions
  • Factor and complete square to reveal properties
  • Compare functions in different representations
Example:

f(x) = -(x - 3)² + 5 has vertex (3, 5) which is a maximum

The negative coefficient means parabola opens downward

What Students Learn:

Students build quadratic and exponential functions from context, understand transformations, and find inverse functions.

Key Skills:
  • Write functions from context
  • Combine functions arithmetically
  • Identify transformation effects: f(x) + k, kf(x), f(kx), f(x + k)
  • Recognize even and odd functions
  • Find inverses by solving f(x) = c
Example:

If f(x) = x², then f(x + 2) shifts left 2 units

f(x) + 2 shifts up 2 units

What Students Learn:

Students apply quadratic functions to physical problems like projectile motion and understand exponential growth eventually exceeds polynomial growth.

Key Skills:
  • Apply quadratics to motion under gravity
  • Model situations with appropriate function types
  • Understand exponential eventually exceeds linear and quadratic
Example:

A ball thrown upward: h(t) = -16t² + 64t + 5

Maximum height at t = -64/(2×-16) = 2 seconds

h(2) = -64 + 128 + 5 = 69 feet

Geometry

What Students Learn:

Students understand similarity through dilations, prove similarity theorems, and apply trigonometric ratios to solve right triangle problems.

Key Skills:
  • Verify dilation properties experimentally
  • Use similarity transformations to determine if figures are similar
  • Establish AA similarity criterion
  • Prove theorems: parallel line divides sides proportionally
  • Use Pythagorean Theorem proved via similarity
  • Understand trigonometric ratios arise from similarity
  • Explain sine and cosine of complementary angles
  • Use trig ratios to solve right triangles
  • Derive trig ratios for 30-60-90 and 45-45-90 triangles
Example:

In a 30-60-90 triangle, if hypotenuse = 10:

Short leg = 5, long leg = 5√3

sin(30°) = 1/2, cos(30°) = √3/2

What Students Learn:

Students prove circle theorems, work with inscribed and circumscribed circles, and understand radian measure and arc length.

Key Skills:
  • Prove all circles are similar
  • Identify relationships among inscribed angles, radii, chords
  • Understand central, inscribed, circumscribed angles
  • Construct inscribed and circumscribed circles of triangles
  • Derive that arc length is proportional to radius
  • Define radian measure
  • Derive area of sector formula
  • Convert between degrees and radians
  • Derive circle and parabola equations
Example:

Circle with center (2, -3) and radius 5:

(x - 2)² + (y + 3)² = 25

Arc length with θ = π/3 radians, r = 6: s = rθ = 6(π/3) = 2π

What Students Learn:

Students derive and apply volume formulas, understand scale factor effects on length/area/volume, and apply triangle inequality relationships.

Key Skills:
  • Derive formulas for circumference, area of circle
  • Derive volume formulas for cylinder, pyramid, cone
  • Use volume formulas to solve problems
  • Understand scale factor k affects: length by k, area by k², volume by k³
  • Verify triangle inequalities experimentally
Example:

If a cube has side length 3, its volume is 27 cubic units

If scaled by factor 2, new side = 6, new volume = 216

Volume multiplied by 2³ = 8

Statistics and Probability

What Students Learn:

Students understand independence and conditional probability, use set theory language, and compute probabilities of compound events using rules.

Key Skills:
  • Describe events as subsets using unions, intersections, complements
  • Understand independence: P(A and B) = P(A) × P(B)
  • Understand conditional probability: P(A|B) = P(A and B)/P(B)
  • Construct and interpret two-way frequency tables
  • Use tables to determine independence and approximate conditional probabilities
  • Recognize conditional probability in everyday situations
  • Find P(A|B) from outcomes
  • Apply Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
  • Apply Multiplication Rule and use permutations/combinations
  • Use probability to make fair decisions
Example:

In a school: 60% play sports (S), 40% play instrument (I), 25% do both

P(S or I) = 0.60 + 0.40 - 0.25 = 0.75

P(I|S) = P(I and S)/P(S) = 0.25/0.60 ≈ 0.42

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