# Mathematics

## What Do Students Learn in Algebra 1?

Algebra 1 formalizes and extends students’ understanding and application of functions. Students primarily explore linear functions (as well as linear piecewise, absolute value, and step functions), quadratic functions, and exponential functions. Within these parent functions, students develop a deep understanding of the features of each function—graphically and algebraically—and use these to guide creation of models and analysis of situations.

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## Units

Unit 1

11 Lessons

### Functions, Graphs and Features

Students are introduced to the main features of functions that they will learn throughout the year, providing students with a conceptual understanding of how functions are used to model various situations.

Unit 2

22 Lessons

### Descriptive Statistics

Students analyze contextual situations, focusing on single variable data and bivariate data, and are introduced to the concept of using data to make predictions and judgments about a situation.

Unit 3

12 Lessons

### Linear Expressions & Single-Variable Equations/Inequalities

Students become proficient at manipulating and solving single-variable linear equations and inequalities, and using them to model and interpret contextual situations.

Unit 4

14 Lessons

### Linear Equations, Inequalities and Systems

Students manipulate, graph, and model with two-variable linear equations and inequalities, are introduced to inverse functions, and continue studying linear systems of equations and inequalities.

Unit 5

16 Lessons

### Functions and Transformations

Students take a deeper look at piecewise functions and absolute value functions, and study how transformations of functions can be identified graphically and represented algebraically.

Unit 6

22 Lessons

### Exponents and Exponential Functions

Students extend their understanding of properties of exponents to include rational exponents and radicals, and investigate rates of change in linear and exponential sequences and functions.

Unit 7

13 Lessons

Students investigate and understand the features that are unique to quadratic functions, and they learn to factor quadratic equations in order to reveal the roots of the equation.

Unit 8

15 Lessons

Students continue their study of quadratic equations, learning new strategies to determine the vertex and roots of quadratic equations and applying these in various real-world contexts.

## How Do Students Build Algebra 1 Skills Across Units?

In Unit 1, Functions, Graphs, and Features, students are introduced to all of the main features of functions they will learn throughout the year through basic graphical modeling of contextual situations. Students will learn function notation and use this to analyze and express features of functions represented in graphs and contextually. Students will use the tools of domain and range, rates of change, intercepts, and where a function is changing to describe contextual situations.

In Unit 2, Statistics, students continue to analyze contextual situations, but in this unit, they focus on single-variable data and then bivariate data. This is the first unit where students are introduced to the concept of using data to make predictions and judgments about a situation. Univariate data is described through shape, center, and spread by using mathematical calculations to support their reasoning. Students begin to make judgments about whether data is consistent (analysis of spread) and whether mean or median is a better representation of a situation (center). Bivariate data is analyzed for whether the variables are related (correlation) and whether a linear model is the best function to fit a set of data (analysis of residuals); students also develop a linear model that can be used to predict future events. In Unit 2, students are introduced to the modeling cycle and complete a project on univariate data analysis and another on bivariate data analysis.

In Unit 3, Linear Expressions & Single-Variable Equations/Inequalities, students become proficient at manipulating and solving single-variable linear equations and inequalities, as well as using linear expressions to model contextual situations. Domain and range are introduced again through the lens of a “constraint” with inequalities. The understanding students develop in this unit builds the foundation for Unit 4, Unit 5, and Unit 6, as well as provides an algebraic outlet for modeling contextual situations started in Unit 1 and continued in Unit 2.

In Unit 4, Linear Equations, Inequalities, and Systems, students become proficient at manipulating, identifying features, graphing, and modeling with two-variable linear equations and inequalities. Students are introduced to inverse functions and formalize their understanding on linear systems of equations and inequalities to model and analyze contextual situations. Proficiency of algebraic manipulation and solving, graphing skills, and identification of features of functions are essential groundwork to build future concepts studied in Unit 5, Unit 6, Unit 7, and Unit 8.

In Unit 5, Piecewise Functions and Transformations, students revisit work in Unit 1, Unit 3, and Unit 4 to formalize their understanding of domain and range to model linear piecewise functions. The algebraic work done in Unit 3 and Unit 4 build to working with absolute value functions. Students are introduced to the concept of a function transformation—a key concept in identifying functions that model situations but are shifted, reflected, or dilated to represent the characteristics of the particular situation. Absolute value functions are used to bridge piecewise functions and provide a low-entry parent function to understand how different function transformations are represented algebraically and how domain and range are affected by transformations.

In Unit 6, Exponents and Exponential Functions, students review exponent rules studied in middle school and extend their understanding to include rational exponents. In this unit, students will be operating with polynomials as an extension of work done in Unit 3 with expressions, and utilizing exponent rules reviewed in this unit. Students formalize the conceptual understanding of the power of exponents to increase or decrease values at increasing or decreasing rates, respectively, to model with exponential functions. Students use the understanding about linear functions developed in Unit 2, Unit 3, Unit 4, and Unit 5 to make comparisons to exponential situations in terms of algebraic modeling, use of the function in contextual situations, and graphical analysis. The understanding from this unit carries through to quadratics as well as into Algebra 2 with exponential modeling and logarithms.

In Unit 7, Quadratic Functions and Solutions, students begin a deep study of quadratic functions—a key function for students to develop a deep understanding of in Algebra 1. Students pull together their understanding of graphical analysis from Unit 1, algebraic manipulation from Unit 3, and linear equations and inequalities from Unit 4 to develop an understanding of what a solution means in context, graphically, and algebraically within quadratics. The concepts learned in this unit will be directly applied in the next unit and throughout Algebra 2, where students will be expected to be fluent in analyzing and solving quadratic functions and equations.

In Unit 8, Quadratic Equations and Applications, students wrap up their study of quadratic functions in Algebra 1, diving deep into all forms of quadratic equations, methods to solve quadratic equations, and methods to identify features from equations. Students apply their understanding of how to graphically and algebraically analyze, manipulate, and solve quadratic functions to model contextual situations. Students will be expected to be fluent in analyzing and solving quadratic functions and equations in Algebra 2.

This course follows the 2017 Massachusetts Curriculum Frameworks and incorporates foundational material from middle school where it is supportive of the current standards.

## Algebra 1 FAQs

### What math skills do students learn in Algebra 1?

In Algebra 1, students primarily explore linear functions, quadratic functions, and exponential functions. Within these parent functions, students develop a deep understanding of the features of each function—graphically and algebraically—and use these to guide creation of models and analysis of situations.

### What Common Core Math Standards do students focus on in Algebra 1?

The major work of the course focuses on the following Common Core State Standards:

• A.APR.A Perform arithmetic operations on polynomials
• A.CED.A Create equations that describe numbers or relationships
• A.REI.A Understand solving equations as a process of reasoning and explain the reasoning
• A.REI.B Solve equations and inequalities in one variable
• A.REI.D Represent and solve equations and inequalities graphically
• A.SSE.A Interpret the structure of expressions
• F.IF.A Understand the concept of a function and use function notation
• F.IF.B Interpret functions that arise in application in terms of context
• S.ID.C Interpret linear models

### How do students build upon their middle school math skills in Algebra 1?

In Algebra 1, students continue their work with exponents and extend their understanding to include exponent functions, and extend their work with equations to include quadratic equations.

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