# Algebra 1

In Algebra 1, students build on the descriptive statistics, expressions and equations, and functions work first encountered in the middle grades while using more formal reasoning and precise language as they think deeper about the mathematics. Students add to the statistical work from the middle grades by working with standard deviation, describing statistical distributions more precisely, and measuring goodness-of-fit with residuals and the correlation coefficient. Students further their work with linear equations and inequalities as they transition from representations tied to tangible objects to working with abstract expressions. Students develop their abilities to see structure in expressions to show that expressions involving several operations are equivalent (for example, grasping that “substitution” works at various levels of complexity), and they solve linear and quadratic equations by writing a series of equivalent statements, justifying each step. Students formalize their concept of function and encounter exponential and quadratic functions as well as other examples of non-linear functions. A function that arises from a real context requires students to attend to an appropriate domain and to the meaning of various features of the function in the context. As they explore various functions, students should also leverage the power of making connections between graphical, tabular, symbolic, and contextual representations.

## Units

A1.1 One Variable Statistics

#### Summary

• Create dot plots, histograms, and box plots (S-ID.A.1).
• Use available classroom technology to create histograms and box plots and calculate measures of center and spread (S-ID.A.1).
• Use terms such as “flat,” “skewed,” “bell-shaped,” and “symmetric” to describe data distributions (S-ID.A.2).
• Analyze and compare data sets (S-ID.A.3).
• Understand relationships between mean and median for symmetrical and skewed data distributions (S-ID.A.2).
• Recognize outliers when they exist, and know to investigate their source (S-ID.A.3).
• Know that outliers affect the mean, but not the median of a data set (S-ID.A.3).
• Describe variability by calculating deviations from the mean (S-ID.A.2).
• Compare two data sets with the same means but different variabilities, and contrast them by calculating the deviation of each data point from the mean (S-ID.A.2).
• Understand that IQR is a description of variability better suited to a skewed distribution (S-ID.A.3).
• Work with two-way tables (S-ID.B.5).

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A1.2 Linear Equations, Inequalities and Systems

#### Summary

• Explain each step in solving a simple equation in one variable (A-REI.A.1).
• Create and solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (A-CED.A.1$^\star$, A-REI.B.3).
• Model constraints and relationships between quantities by equations and inequalities, and by systems of equations and inequalities, and interpret solutions (A-CED.A.2$^\star$, A-CED.A.3$^\star$).
• Solve systems of linear equations approximately by graphing and exactly by algebraic methods (A-REI.C.6).
• Understand the principles behind the method of elimination (A-REI.C.5).
• Graph the solution set to a linear inequality in two variables as a half-plane (A-REI.D.12).
• Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes (A-REI.D.12).

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A1.3 Bivariate Statistics

#### Summary

• Represent data on two quantitative variables on a scatter plot (S-ID.B.6).
• Describe how two quantitative variables on a scatter plot are related (S-ID.B.6).
• Interpret the slope and the intercept of a linear model in the context of the data (S-ID.B.7).
• Use available technology to find lines of best fit (S-ID.B.6a).
• Assess the goodness of fit of a line to a small data set by plotting and analyzing residuals (S-ID.B.6b).
• Fit a linear function for a scatter plot that suggests a linear association (S-ID.B.6c).
• Use available technology to compute correlation coefficients (S-ID.B.8).
• Understand that the correlation coefficient measures the “tightness” of a line fitted to data (S-ID.B.8).
• Understand that correlation does not necessarily imply causality (S-ID.B.9).

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A1.4 Functions

#### Summary

• Interpret key features of graphs in terms of the quantities represented (F-IF.B.4$^\star$).
• Sketch graphs showing key features of the graph by hand and using technology (F-IF.C.7$^\star$).
• Understand that a function from one set (the domain) to another set (the range) assigns to each element of the domain exactly one element of the range (F-IF.A.1).
• Use function notation (F-IF.A.2).
• Interpret statements that use function notation in various contexts (F-IF.A.2).
• Work with graphs of piecewise-defined functions, including step functions (F-IF.C.7b$^\star$).
• Relate the domain of a function to its graph (F-IF.B.5$^\star$).
• Relate the domain of a function to the quantitative relationship it describes (F-IF.B.5$^\star$).
• Calculate and interpret the average rate of change of a function over a specified interval (F-IF.B.6$^\star$).
• Estimate the average rate of change of a function from its graph (F-IF.B.6$^\star$).
• Solve for x such that f(x) = c, when f is a linear function (F-BF.B.4a).
• Write an expression for the inverse of a linear function (F-BF.B.4a).

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A1.5 Exponential Functions 1

#### Summary

• Distinguish between the growth laws of linear and exponential functions and recognize when a situation can be modeled by a linear function versus an exponential function (F-LE.A.1$^\star$).
• Graph exponential functions and understand how changing by a constant factor over equal intervals affects the graph (F-IF.C.7e$^\star$).
• Model situations of growth and decay with exponential functions expressed in various different forms given a graph, a description of the situation, or two input-output pairs (including reading these from a table) (F-LE.A.2$^\star$).
• Understand that over time a quantity increasing exponentially will eventually exceed a quantity increasing linearly (F-LE.A.3$^\star$).
• Understand the form of different expressions for exponential functions in terms of change by a constant factor over equal intervals (F-LE.B.5$^\star$).

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#### Summary

• Construct quadratic functions and quadratic sequences (A-CED.A.2$^\star$, F-IF.A.3, F-BF.A.1a$^\star$).
• Represent quadratic functions using recursive formulas, expressions, tables, and graphs (A-SSE.A.1$^\star$, A-CED.A.2$^\star$, F-IF.A.7a$^\star$, F-BF.A.1a$^\star$).
• Express quadratic functions in equivalent forms for different purposes; understand the relation between vertex form and the shape of the graph (A-SSE.A.1, A-SSE.B.3$^\star$, F-IF.A.7a$^\star$, F-IF.C.8, F-BF.B.3).
• Find the average rate of change of a quadratic function over a unit interval and compare rates for successive intervals (F-IF.B.6$^\star$).
• Describe properties that distinguish linear, exponential, and quadratic functions (F-LE.A.3$^\star$).
• Model with quadratic functions (A-CED.A.2$^\star$, F-IF.B.4$^\star$, F-IF.C.7a$^\star$, F-BF.A.1a$^\star$).

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#### Summary

• Connect solving quadratic equations to finding zeros of quadratic functions (A-SSE.B.3a$^\star$, F-IF.C.8a$^\star$).
• Explore forms of quadratic equations that can be solved by seeing structure (A-REI.B.4b).
• Understand and be able to use the method of factoring to solve factorable quadratic equations (A-REI.A.1, A-REI.B.4b).
• Understand and be able to use the method of completing the square to solve quadratic equations, and derive the quadratic formula (A-SSE.B.3b$^\star$, A-REI.A.1, A-REI.B.4).
• Construct and solve quadratic equations by the most strategic method to solve problems in various contexts (A-CED.A.1$^\star$, A-REI.B.4b).
• Express a quadratic function in the appropriate form for a given purpose, including vertex form (A-SSE.B.3b$^\star$, F-IF.C.8a$^\star$).
• Solve problems using systems consisting of a linear and a quadratic equation in two variables (A-REI.C.7).
• Derive the equation of a parabola given the focus and a directrix parallel to one of the axes (G-GPE.A.2).

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#### Sherri Martinie says:

This offers a great opportunity for schools and school districts to compare what they are currently doing with this model. Perhaps what they are doing is better. Perhaps there are things here that would be more effective. Maybe they have felt like something is missing or not quite right and the answer is here. The point is they can actually think critically about what is best for their own population of students. For some time now, there have been a number of sources to go to for course and unit models. This provides yet another good example. I also like that it coordinates well with other projects on this sight. It is helpful to go to one place to find such rich resources and examples.

#### Michael Fell says:

Disappointing that 5 years into common core that units for algebra 1, which is the key tested course, are finally under construction. Most districts, including mine, have already created the units from scratch and have moved forward from such work.

#### Bill says:

The goal of the Common Core initiative was to produce voluntary common standards, not to produce curriculum. That has always been the prerogative of states and districts, and remains so. This effort is not in any way a continuation of the standards writing work. It is simply intended to provide a model for those who want one.

#### Michael Fell says:

Thanks Bill, but that is not what has happened