Algebra 1
Algebra 1
Year at a Glance
Algebra 1 focuses on four critical areas: (1) using units and relationships between quantities; (2) reasoning with equations and expressions; (3) analyzing and using linear, exponential, and quadratic functions; and (4) interpreting and displaying data using descriptive statistics. Algebra 1 is a foundational course for high school mathematics, focusing on the relationship between verbal descriptions, tables, equations, and graphs of linear, exponential, and quadratic functions. Algebraic modeling and applications are also a key focus, balancing conceptual understanding with procedural fluency.
Prerequisite(s):
8th Grade Mathematics
Credit:
1.0 (MA1110A, MA1110B)
Note: The Year at a Glance reflects the order of the Unit Plans and does not necessarily reflect the precise instructional order of evidence outcomes.
The highlighted evidence outcomes are the priority for all students, serving as the essential concepts and skills. It is recommended that the remaining evidence outcomes listed be addressed as time allows, representing the full breadth of the curriculum.
Algebra 1 Families of Functions
The following evidence outcomes apply throughout Algebra 1, focusing on the families of linear, exponential, and quadratic functions.
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (CCSS: HS.N-Q.A.1)
Define appropriate quantities for the purpose of descriptive modeling. (CCSS: HS.N-Q.A.2)
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (CCSS: HS.N-Q.A.3)
Interpret expressions that represent a quantity in terms of its context. (CCSS: HS.A-SSE.A.1)
Interpret parts of an expression, such as terms, factors, and coefficients. (CCSS: HS.A-SSE.A.1.a)
Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. (CCSS: HS.A-SSE.A.1.b)
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). (CCSS: HS.A-SSE.A.2)
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and exponential functions. (CCSS: HS.A-CED.A.1)
Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales. (CCSS: HS.A-CED.A.2)
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (CCSS: HS.A-CED.A.3)
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. (CCSS: HS.A-REI.A.1)
Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (CCSS: HS.A-REI.D.10)
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, and exponential functions. (CCSS: HS.A-REI.D.11)
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (CCSS: HS.F-IF.A.2)
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (CCSS: HS.F-IF.B.4)
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (CCSS: HS.F-IF.B.5)
Calculate and interpret the average rate of change, presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph. (CCSS: HS.F-IF.B.6)
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (CCSS: HS.F-IF.C.9)
Write a function that describes a relationship between two quantities. (CCSS: HS.F-BF.A.1)
Determine an explicit expression, a recursive process, or steps for calculation from a context. (CCSS: HS.F-BF.A.1.a)
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant to a decaying exponential, and relate these functions to the model. (CCSS: HS.F-BF.A.1.b)
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k both positive and negative; find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. (CCSS: HS.F-BF.B.3)
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (CCSS: HS.S-ID.B.6)
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. (CCSS: HS.S-ID.B.6.a)
1st Semester
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. (CCSS: HS.A-CED.A.4)
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (CCSS: HS.A-REI.B.3)
Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (CCSS: HS.F-IF.A.1)
Write arithmetic sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (CCSS: HS.F-BF.A.2)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (CCSS: HS.F-IF.C.7)
Graph linear functions and show intercepts. (CCSS: HS.F-IF.C.7.a)
Graph piecewise-defined functions, including step functions and absolute value functions. (CCSS: HS.F-IF.C.7.b)
Distinguish between situations that can be modeled with linear functions and with exponential functions. (CCSS: HS.F-LE.A.1)
Prove that linear functions grow by equal differences over equal intervals. (CCSS: HS.F-LE.A.1.a)
Identify situations in which one quantity changes at a constant rate per unit interval relative to another. (CCSS: HS.F-LE.A.1.b)
Construct linear functions, including arithmetic sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (CCSS: HS.F-LE.A.2)
Interpret the parameters in a linear function in terms of a context. (CCSS: HS.F-LE.B.5)
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (CCSS: HS.S-ID.B.6)
Informally assess the fit of a function by plotting and analyzing residuals. (CCSS: HS.S-ID.B.6.b)
Fit a linear function for a scatter plot that suggests a linear association. (CCSS: HS.S-ID.B.6.c)
Distinguish between correlation and causation. (CCSS: HS.S-ID.C.9)
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. (CCSS: HS.S-ID.C.7)
Using technology, compute and interpret the correlation coefficient of a linear fit. (CCSS: HS.S-ID.C.8)
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. (CCSS: HS.A-REI.C.5)
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (CCSS: HS.A-REI.C.6)
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (CCSS: HS.A-REI.D.12)
2nd Semester
Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1. (CCSS: HS.F-IF.A.3)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (CCSS: HS.F-IF.C.7)
Graph exponential functions, showing intercepts and end behavior. (CCSS: HS.F-IF.C.7.e)
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (CCSS: HS.F-IF.C.8)
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. (CCSS: HS.F-IF.C.8.b)
Write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (CCSS: HS.F-BF.A.2)
Distinguish between situations that can be modeled with linear functions and with exponential functions. (CCSS: HS.F-LE.A.1)
Prove that exponential functions grow by equal factors over equal intervals. (CCSS: HS.F-LE.A.1.a)
Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (CCSS: HS.F-LE.A.1.c)
Construct exponential functions, including geometric sequences, given a description of a relationship. (CCSS: HS.F-LE.A.2)
Interpret the parameters in an exponential function in terms of a context. (CCSS: HS.F-LE.B.5)
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (CCSS: HS.A-SSE.B.3)
Factor a quadratic expression to reveal the zeros of the function it defines. (CCSS: HS.A-SSE.B.3.a)
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (CCSS: HS.A-SSE.B.3.b)
Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (CCSS: HS.A-APR.A.1)
Solve quadratic equations in one variable. (CCSS:HS.A-REI.B.4)
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x − p)2 = q that has the same solutions. Derive the quadratic formula from this form. (CCSS: HS.A-REI.B.4.a)
Solve quadratic equations (e.g., for x2 = 49) by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. (CCSS: HS.A-REI.B.4.b)
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (CCSS: HS.F-IF.C.7)
Graph quadratic functions and show intercepts, maxima, and minima. (CCSS: HS.F-IF.C.7.a)
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (CCSS: HS.F-IF.C.8)
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (CCSS: HS.F-IF.C.8.a)
Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (CCSS: HS.F-LE.A.3)
Model data in context with plots on the real number line (dot plots, histograms, and box plots). (CCSS: HS.S-ID.A.1)
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. (CCSS: HS.S-ID.A.2)
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (CCSS: HS.S-ID.A.3)
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. (CCSS: HS.S-ID.B.5)
Accelerated Algebra 1
Year at a Glance
Accelerated Algebra 1 focuses on four critical areas in greater depth: (1) using units and relationships between quantities; (2) reasoning with equations and expressions; (3) analyzing and using linear, exponential, and quadratic functions; and (4) interpreting and displaying data using descriptive statistics. Accelerated Algebra 1 focuses on the relationship between verbal descriptions, tables, equations, and graphs of linear, exponential, and quadratic functions and is intended for students who took Algebra 1 in middle school and need further concept reinforcement through modeling and application contexts. Students in Accelerated Algebra 1 may engage in additional topics outside of a typical Algebra 1 course.
Prerequisite(s):
8th Grade Mathematics
Credit:
1.0 (MA1113A, MA1113B)
Accelerated Algebra 1 students can extend their learning beyond the Colorado Academic Standards with the following concepts and lessons:
Lesson 2.5: Solving Compound Inequalities
Lesson 2.6: Solving Absolute Value Inequalities
Lesson 6.5: Solving Exponential Equations
Lesson 9.1: Properties of Radicals
Lesson 10.1: Graphing Square Root Functions
Lesson 10.2: Graphing Cube Root Functions
Lesson 10.3: Solving Radical Equations