Focus Areas for Algebra I
- Deepen and extend understanding of linear relationships
- Contrast linear relationships with exponential relationships
- Apply linear models to data
- Develop understanding of functions and function notation
- Solve quadratic equations and compare linear, quadratic, and exponential models
Math Standards by Domain
Number and Quantity
What Students Learn:
Students extend their understanding of exponents to include rational exponents and learn to work with radicals. They explore the properties of rational and irrational numbers, understanding how these number systems behave under different operations.
Key Skills:
- Explain how rational exponents extend from integer exponents (e.g., 5^(1/3) is the cube root of 5)
- Rewrite expressions with radicals using rational exponents and vice versa
- Understand that the sum or product of two rational numbers is rational
- Explain why adding or multiplying a rational and irrational number gives an irrational result
Rewrite √(x³) using a rational exponent: x^(3/2)
Explain why 3 + √2 is irrational: Since √2 is irrational and 3 is rational, their sum must be irrational.
What Students Learn:
Students learn to use units strategically in problem-solving, choose appropriate levels of accuracy, and define quantities for modeling real-world situations. This develops mathematical precision and practical application skills.
Key Skills:
- Use units to understand and solve multi-step problems
- Choose and interpret units in formulas (e.g., d = rt where d is in miles, r in mph, t in hours)
- Choose appropriate scales and origins for graphs
- Define quantities for descriptive modeling
- Choose accuracy levels appropriate to measurement limitations
If a car travels at 60 mph for 2.5 hours, how far does it go? Use the formula d = rt: d = 60 × 2.5 = 150 miles. The units help us verify our answer makes sense.
Algebra
What Students Learn:
Students learn to interpret and analyze algebraic expressions by identifying their structure. They see how different forms of the same expression can reveal different properties, such as factoring to find zeros or completing the square to find vertex form.
Key Skills:
- Interpret parts of expressions (terms, factors, coefficients) in context
- View complex expressions as single entities (e.g., P(1+r)ⁿ as P times a growth factor)
- Use structure to rewrite expressions in useful forms
- Factor quadratics to reveal zeros: x² - 5x + 6 = (x-2)(x-3)
- Complete the square to find maximum/minimum values
- Transform exponential expressions using exponent properties
For the expression 1000(1.05)ᵗ representing money in a savings account, interpret: 1000 is the initial deposit, 1.05 represents 5% annual growth, and t is the number of years.
What Students Learn:
Students understand that polynomials form a system similar to integers, closed under addition, subtraction, and multiplication. They perform operations with polynomials and understand the connection between polynomial arithmetic and integer arithmetic.
Key Skills:
- Add polynomials: (2x² + 3x) + (x² - 5x) = 3x² - 2x
- Subtract polynomials by distributing the negative sign
- Multiply polynomials using distributive property
- Multiply binomials: (x + 3)(x - 2) = x² + x - 6
- Recognize closure: polynomial operations always produce polynomials
Multiply (2x + 5)(x - 3):
= 2x² - 6x + 5x - 15 = 2x² - x - 15
What Students Learn:
Students create equations and inequalities to model real-world situations. They learn to represent constraints, create multi-variable equations, and rearrange formulas to solve for specific quantities of interest.
Key Skills:
- Create equations with absolute value, linear, quadratic, and exponential functions
- Create equations in two or more variables and graph them
- Represent constraints using systems of equations or inequalities
- Interpret solutions as viable or non-viable in context
- Rearrange formulas to solve for different variables (e.g., solve A = πr² for r)
A phone plan costs $30/month plus $0.10 per text. Write an equation for total cost C based on t texts: C = 30 + 0.10t
Rearrange the area formula A = lw to solve for width: w = A/l
What Students Learn:
Students solve various types of equations and systems, including linear equations, quadratic equations, and systems of equations. They learn multiple solution methods and understand when solutions from graphs represent the intersection of equations.
Key Skills:
- Solve linear equations and inequalities, including with literal coefficients
- Solve absolute value equations and inequalities
- Solve quadratic equations by factoring, completing the square, and using the quadratic formula
- Solve systems of linear equations using substitution, elimination, and graphing
- Solve systems with one linear and one quadratic equation
- Understand that graph intersections represent equation solutions
- Graph linear inequalities and systems of inequalities
Solve x² - 7x + 12 = 0 by factoring:
(x - 3)(x - 4) = 0, so x = 3 or x = 4
Solve the system: y = 2x + 1 and y = -x + 7
Set equal: 2x + 1 = -x + 7, so 3x = 6, x = 2, y = 5. Solution: (2, 5)
Functions
What Students Learn:
Students develop a deep understanding of functions, including function notation, domain and range, and how to interpret key features of graphs. They learn to analyze functions represented in multiple ways and calculate rates of change.
Key Skills:
- Understand functions assign each input exactly one output
- Use function notation: evaluate f(3) when f(x) = 2x + 5
- Recognize sequences as functions with integer domains
- Interpret key features: intercepts, intervals of increase/decrease, maxima/minima
- Relate domain to graphs and real-world contexts
- Calculate average rate of change over an interval
- Graph linear, quadratic, exponential, and absolute value functions
- Compare functions represented in different forms
If f(x) = x² - 4x + 3, find f(5): f(5) = 25 - 20 + 3 = 8
Find the average rate of change from x = 1 to x = 3: [f(3) - f(1)] / (3 - 1) = (0 - 0) / 2 = 0
What Students Learn:
Students learn to construct functions from contexts, combine functions arithmetically, write arithmetic and geometric sequences, and understand function transformations and inverses.
Key Skills:
- Write functions from descriptions or patterns
- Determine explicit and recursive formulas from context
- Combine functions: (f + g)(x), (f · g)(x)
- Write arithmetic sequences: aₙ = a₁ + (n-1)d
- Write geometric sequences: aₙ = a₁ · rⁿ⁻¹
- Translate between recursive and explicit forms
- Understand transformations: f(x) + k shifts up, f(x + k) shifts left
- Find inverse functions by switching x and y and solving
Write a function for a sequence: 3, 7, 11, 15, ...
This is arithmetic with a₁ = 3 and d = 4, so aₙ = 3 + 4(n-1) = 4n - 1
Find the inverse of f(x) = 2x - 6: Switch to get x = 2y - 6, solve: y = (x + 6)/2, so f⁻¹(x) = (x + 6)/2
What Students Learn:
Students distinguish between linear and exponential growth, construct and compare these models, and apply them to real-world situations including quadratic applications. They learn to identify which type of function best models a given situation.
Key Skills:
- Distinguish linear (constant rate of change) from exponential (constant growth factor)
- Prove linear functions change by equal differences over equal intervals
- Prove exponential functions change by equal factors over equal intervals
- Construct linear and exponential functions from graphs, descriptions, or data points
- Recognize that exponential growth eventually exceeds linear or polynomial growth
- Interpret parameters: in f(x) = a · bˣ, a is the initial value, b is the growth factor
- Apply quadratic functions to projectile motion and other physical problems
Linear vs. Exponential: A population that grows by 100 people per year is linear. A population that grows by 5% per year is exponential.
Model: A ball is thrown upward from 5 feet with initial velocity 40 ft/s. Its height is h(t) = -16t² + 40t + 5
Statistics and Probability
What Students Learn:
Students learn to represent, summarize, and interpret categorical and quantitative data. They create various data displays, calculate and interpret statistics, fit models to data, and understand the difference between correlation and causation.
Key Skills:
- Represent data with dot plots, histograms, and box plots
- Use statistics appropriate to data shape (mean/median, standard deviation/IQR)
- Compare center and spread of multiple data sets
- Interpret differences in shape, center, and spread; identify outliers
- Summarize categorical data in two-way frequency tables
- Create and interpret scatter plots showing relationships between variables
- Fit functions to data and use them to make predictions
- Interpret slope and intercept of linear models in context
- Compute and interpret correlation coefficients
- Distinguish between correlation and causation
A scatter plot shows hours studied vs. test scores with a positive correlation. The line of best fit is y = 5x + 60. The slope 5 means each additional hour of studying is associated with a 5-point increase in score. The y-intercept 60 represents the predicted score with 0 hours of study.
Note: Correlation doesn't prove studying causes higher scores (though it likely does) - there could be other factors.