Focus Areas for Algebra II
- Relate arithmetic of rational expressions to arithmetic of rational numbers
- Expand understandings of functions and graphing to include trigonometric functions
- Synthesize and generalize functions and extend understanding of exponential functions to logarithmic functions
- Relate data display and summary statistics to probability and explore a variety of data collection methods
Math Standards by Domain
Number and Quantity
What Students Learn:
Students extend the number system to include complex numbers. They learn to perform arithmetic with complex numbers, solve quadratic equations with complex solutions, and understand the Fundamental Theorem of Algebra.
Key Skills:
- Understand that i² = -1 and every complex number has the form a + bi
- Add, subtract, and multiply complex numbers using properties of operations
- Solve quadratic equations with real coefficients that have complex solutions
- Extend polynomial identities to complex numbers (e.g., x² + 4 = (x + 2i)(x - 2i))
- Know the Fundamental Theorem of Algebra: every polynomial has complex roots
Add complex numbers: (3 + 2i) + (5 - 4i) = 8 - 2i
Multiply: (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i
Solve x² + 4 = 0: x² = -4, so x = ±2i
Algebra
What Students Learn:
Students work with polynomials and rational expressions, learning to factor, apply the Remainder Theorem, use polynomial identities, and perform operations with rational expressions. They also work with geometric series.
Key Skills:
- Use structure to rewrite expressions (e.g., factor to reveal zeros)
- Derive and use the formula for the sum of a finite geometric series
- Apply the Remainder Theorem: the remainder when dividing by (x - a) is p(a)
- Identify zeros of polynomials from factorizations and sketch graphs
- Prove polynomial identities and use them for numerical relationships
- Apply the Binomial Theorem to expand (x + y)ⁿ
- Rewrite rational expressions in different forms using long division
- Add, subtract, multiply, and divide rational expressions
Sum of geometric series: 1 + 2 + 4 + 8 + ... + 512 = (2¹⁰ - 1)/(2 - 1) = 1023
Factor x³ - 8 to reveal zeros: x³ - 8 = (x - 2)(x² + 2x + 4), so x = 2 is a zero
Binomial expansion: (x + 2)³ = x³ + 3x²(2) + 3x(4) + 8 = x³ + 6x² + 12x + 8
What Students Learn:
Students solve radical and rational equations, being mindful of extraneous solutions. They continue working with absolute value equations and understand how graphs of equations represent solutions.
Key Skills:
- Solve radical equations: √(2x + 3) = 5, square both sides to get 2x + 3 = 25
- Solve rational equations and check for extraneous solutions
- Recognize when squaring both sides may introduce extraneous solutions
- Solve absolute value equations and inequalities
- Understand that graph intersections represent solutions
- Use technology to find approximate solutions
Solve √(x + 5) = x - 1: Square both sides: x + 5 = x² - 2x + 1, so x² - 3x - 4 = 0
Factor: (x - 4)(x + 1) = 0, giving x = 4 or x = -1
Check: x = 4 works (√9 = 3), but x = -1 doesn't (√4 ≠ -2). Only x = 4 is valid.
Functions
What Students Learn:
Students deepen their understanding of functions by analyzing key features, working with transformations, and graphing polynomial, rational, radical, and piecewise functions. They combine functions and find inverses.
Key Skills:
- Interpret key features: intercepts, intervals of increase/decrease, maxima/minima, end behavior
- Relate domain to graphs and contexts
- Calculate and interpret average rate of change
- Graph polynomial functions showing zeros and end behavior
- Graph square root, cube root, piecewise, absolute value functions
- Graph exponential and logarithmic functions
- Write functions in equivalent forms to reveal properties
- Compare functions represented in different ways
- Combine functions arithmetically
- Understand function transformations: shifts, stretches, reflections
- Find inverse functions algebraically
Graph f(x) = (x - 2)² + 3: This is a parabola shifted right 2 and up 3, with vertex (2, 3)
Find inverse of f(x) = 3x - 7: Switch x and y: x = 3y - 7, solve: y = (x + 7)/3, so f⁻¹(x) = (x + 7)/3
What Students Learn:
Students learn logarithms as the inverse of exponential functions. They prove and apply logarithm laws, change bases, and use logarithms to solve exponential equations.
Key Skills:
- Express solutions to exponential equations as logarithms: if 10ˣ = 50, then x = log₁₀(50)
- Evaluate logarithms using technology
- Prove logarithm laws: log(ab) = log(a) + log(b), log(aⁿ) = n·log(a)
- Use change of base formula: log_b(x) = log(x)/log(b)
- Simplify logarithmic expressions using properties
- Solve exponential equations using logarithms
Solve 5 · 2ˣ = 80: Divide by 5: 2ˣ = 16, take log: x·log(2) = log(16), so x = log(16)/log(2) = 4
Simplify log(8) + log(125) - log(1000) using base 10: log(8·125/1000) = log(1) = 0
What Students Learn:
Students extend trigonometry to the unit circle and all real numbers, learning radian measure, graphing trigonometric functions, modeling periodic phenomena, and using trigonometric identities.
Key Skills:
- Understand radian measure as arc length on the unit circle
- Use the unit circle to extend sine, cosine, tangent to all real numbers
- Graph all six trigonometric functions
- Model periodic phenomena using trig functions (amplitude, frequency, midline)
- Prove and use the Pythagorean identity: sin²(θ) + cos²(θ) = 1
- Find trig values given one value and the quadrant
Model temperature with T(t) = 15·sin(2π(t-3)/12) + 65, where t is hours after midnight. This has amplitude 15°, period 12 hours, midline at 65°, and peaks at 3 PM.
If sin(θ) = 3/5 in quadrant II, find cos(θ): Use sin²(θ) + cos²(θ) = 1: 9/25 + cos²(θ) = 1, so cos²(θ) = 16/25. In quadrant II, cos(θ) = -4/5.
Geometry
What Students Learn:
Students work with conic sections, learning to complete the square to put equations in standard form and identify whether an equation represents a circle or parabola (in Algebra II, ellipses and hyperbolas are typically introduced but not emphasized).
Key Skills:
- Complete the square to convert quadratic equations to standard form
- Identify whether an equation represents a circle or parabola
- Graph circles and parabolas from their equations
- Write equations of circles given center and radius
- Write equations of parabolas given key features
Given x² + y² + 6x - 4y - 3 = 0, complete the 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 learn about statistical inference, understanding how to make conclusions about populations from samples. They explore different data collection methods, use simulations to develop margins of error, and evaluate statistical reports critically.
Key Skills:
- Understand statistics as making inferences about population parameters from random samples
- Use simulation to test if a model is consistent with data
- Distinguish among sample surveys, experiments, and observational studies
- Explain the role of randomization in each study type
- Estimate population means or proportions from sample data
- Develop margins of error using simulation models
- Compare treatments using data from randomized experiments
- Use simulations to determine if differences are significant
- Evaluate reports based on data critically
A poll of 1000 voters finds 520 support a candidate. Estimate the population proportion: 520/1000 = 0.52 or 52%.
Using simulation, the margin of error might be ±3%, meaning we're confident the true population proportion is between 49% and 55%.
What Students Learn:
Students work with normal distributions, learning to fit data to normal curves, use mean and standard deviation to estimate percentages, and recognize when normal distribution is appropriate.
Key Skills:
- Use mean and standard deviation to fit data to a normal distribution
- Estimate population percentages using the normal curve
- Recognize data sets where normal distribution is inappropriate
- Use calculators, spreadsheets, and tables to estimate areas under the normal curve
- Interpret z-scores and percentiles
- Apply the 68-95-99.7 rule (empirical rule)
Test scores have mean 75 and standard deviation 8. Using the empirical rule:
• About 68% of scores are between 67 and 83 (within 1 standard deviation)
• About 95% are between 59 and 91 (within 2 standard deviations)
• About 99.7% are between 51 and 99 (within 3 standard deviations)