Focus Areas for Integrated Math III
- Apply methods from probability and statistics to draw inferences and conclusions from data
- Expand understanding of functions to include polynomial, rational, and radical functions
- Expand right triangle trigonometry to include general triangles
- Consolidate functions and geometry to create models and solve contextual problems
Math Standards by Domain
Number and Quantity
What Students Learn:
Students extend polynomial identities to complex numbers and apply the Fundamental Theorem of Algebra to understand that polynomials of degree n have exactly n complex roots (counting multiplicity).
Key Skills:
- Extend polynomial identities to complex numbers
- Factor polynomials using complex numbers
- Apply Fundamental Theorem of Algebra to higher degree polynomials
Factor x⁴ + 4 using complex numbers:
x⁴ + 4 = (x² + 2x + 2)(x² - 2x + 2) = (x + 1 + i)(x + 1 - i)(x - 1 + i)(x - 1 - i)
Algebra
What Students Learn:
Students work with polynomials beyond quadratics, apply the Remainder Theorem, use polynomial identities, and rewrite rational expressions. They derive the geometric series formula.
Key Skills:
- Add, subtract, and multiply polynomials of any degree
- Apply Remainder Theorem: remainder when dividing p(x) by (x - a) is p(a)
- Identify zeros of polynomials and sketch graphs
- Prove polynomial identities (e.g., Pythagorean triples: (x² + y²)² = (x² - y²)² + (2xy)²)
- Apply Binomial Theorem using Pascal's Triangle
- Rewrite rational expressions a(x)/b(x) in form q(x) + r(x)/b(x)
- Add, subtract, multiply, divide rational expressions
- Derive geometric series formula: 1 + r + r² + ... + rⁿ⁻¹ = (1 - rⁿ)/(1 - r)
Find remainder when x³ - 4x² + 5x - 2 is divided by (x - 2):
p(2) = 8 - 16 + 10 - 2 = 0, so (x - 2) is a factor
Geometric series: 1 + 2 + 4 + 8 + 16 = (1 - 2⁵)/(1 - 2) = -31/(-1) = 31
What Students Learn:
Students solve radical and rational equations, understanding extraneous solutions. They work with exponential equations using logarithms and solve complex systems graphically.
Key Skills:
- Solve simple radical equations: √(2x + 3) = 5 → 2x + 3 = 25
- Solve rational equations and check for extraneous solutions
- Understand why squaring can introduce false solutions
- Solve equations graphically by finding intersections
- Work with polynomial, rational, radical, absolute value, exponential, and logarithmic functions
Solve √(x + 2) = x
Square both sides: x + 2 = x²
x² - x - 2 = 0 → (x - 2)(x + 1) = 0
x = 2 or x = -1. Check: √4 = 2 ✓, but √1 ≠ -1, so only x = 2 works
Functions
What Students Learn:
Students analyze polynomial, rational, and radical functions, identifying key features and comparing functions in different representations. They understand inverse functions deeply.
Key Skills:
- Interpret key features of any function type in context
- Relate domain to graphs and contexts
- Calculate average rate of change
- Graph square root, cube root, and piecewise functions
- Graph polynomials, identifying zeros and end behavior
- Graph exponential, logarithmic, and trigonometric functions
- Compare properties of functions in different representations
- Build functions by combining standard types
- Understand transformations across all function types
- Find inverses of simple functions
f(x) = x³ - 4x has zeros at x = 0, x = ±2
End behavior: as x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞
Inverse of f(x) = (x + 1)/(x - 1): swap and solve for y to get f⁻¹(x) = (x + 1)/(x - 1)
What Students Learn:
Students solve exponential equations using logarithms, prove logarithm laws, work with radian measure, and model periodic phenomena with trigonometric functions.
Key Skills:
- Express solutions to ab^(ct) = d as logarithms
- Prove logarithm laws: log(xy) = log(x) + log(y), log(x/y) = log(x) - log(y)
- Translate between logarithm bases using change of base formula
- Simplify logarithmic expressions using properties
- Understand radian measure as arc length on unit circle
- Extend trigonometric functions to all real numbers using unit circle
- Graph all 6 trigonometric functions
- Choose trig functions to model periodic phenomena with specific amplitude, frequency, midline
Solve 3(2^x) = 24: 2^x = 8, so x = log₂(8) = 3
Simplify log₃(27) + log₃(9) = log₃(243) = 5
Model: Temperature varies sinusoidally with high 75°F, low 55°F, period 24 hrs:
T(t) = 10sin(2π(t - 6)/24) + 65
Geometry
What Students Learn:
Students extend trigonometry beyond right triangles, deriving and applying the Laws of Sines and Cosines to solve any triangle. They understand ambiguous cases.
Key Skills:
- Derive triangle area formula A = (1/2)ab·sin(C)
- Prove Laws of Sines and Cosines
- Apply Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
- Apply Law of Cosines: c² = a² + b² - 2ab·cos(C)
- Determine if given measures define 0, 1, 2, or infinitely many triangles
- Solve surveying and real-world problems
Triangle with sides a = 7, b = 10, angle C = 30°
c² = 49 + 100 - 2(7)(10)cos(30°) = 149 - 140(0.866) ≈ 27.8
c ≈ 5.3
Area = (1/2)(7)(10)sin(30°) = 35(0.5) = 17.5 square units
What Students Learn:
Students work with conic sections, complete the square to identify circle and parabola equations, visualize 2D/3D relationships, and apply geometric concepts in modeling.
Key Skills:
- Given ax² + by² + cx + dy + e = 0, complete square to identify conic type
- Put circle and parabola equations in standard form
- Identify 2D cross-sections of 3D objects
- Identify 3D solids generated by rotating 2D shapes
- Use geometric shapes to model real objects
- Apply density concepts (persons/sq mi, BTUs/cu ft)
- Solve design problems with geometric methods
x² + y² + 6x - 4y - 3 = 0
Complete square: (x² + 6x + 9) + (y² - 4y + 4) = 3 + 9 + 4
(x + 3)² + (y - 2)² = 16
This is a circle with center (-3, 2) and radius 4
Statistics and Probability
What Students Learn:
Students understand statistical inference, distinguish between sample surveys, experiments, and observational studies, and make inferences about populations from sample data. They fit data to normal distributions.
Key Skills:
- Use mean and standard deviation to fit data to normal distribution
- Estimate population percentages using normal curves
- Use technology to estimate areas under normal curve
- Understand statistics as inference about population parameters
- Use simulation to decide if model is consistent with data
- Recognize purposes of surveys, experiments, observational studies
- Explain role of randomization in each type of study
- Estimate population mean/proportion from sample survey
- Develop margin of error through simulation
- Compare treatments using randomized experiments
- Use simulations to test significance of differences
- Evaluate reports based on data
- Use probability to make and analyze decisions
SAT scores have mean 1050, standard deviation 200, normally distributed
What percent score above 1250?
z = (1250 - 1050)/200 = 1.0
About 16% score above z = 1 (using standard normal table)
A sample of 100 students has mean 1070. Is this significantly different from 1050?
Standard error = 200/√100 = 20. Difference = 20, which is 1 standard error. Not highly significant.