Cluster 1: Interpret the structure of expressions. (Algebra 1 - Major Cluster) (Algebra 2 - Major Cluster)Archived

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Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.A-SSE.1
Title: Interpret the structure of expressions. (Algebra 1 - Major Cluster) (Algebra 2 - Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Grade: 912
Domain-Subdomain: Algebra: Seeing Structure in Expressions

Related Standards

This cluster includes the following benchmarks.

Related Access Points

This cluster includes the following access points.

Access Points

MAFS.912.A-SSE.1.AP.1a
Identify the different parts of the expression and explain their meaning within the context of a problem.
MAFS.912.A-SSE.1.AP.2a
Rewrite algebraic expressions in different equivalent forms, such as factoring or combining like terms.
MAFS.912.A-SSE.1.AP.2b
Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to factor completely.
MAFS.912.A-SSE.1.AP.1b
Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts.
MAFS.912.A-SSE.1.AP.2c
Simplify expressions including combining like terms, using the distributive property, and other operations with polynomials.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Formative Assessments

Interpreting Basic Tax:

Students are asked to interpret the parts of an equation used to calculate the total purchase price including tax of a set of items.

Type: Formative Assessment

Rewriting Numerical Expressions:

Students are asked to rewrite numerical expressions to find efficient ways to calculate.

Type: Formative Assessment

Determine the Width:

Students are asked to find the width of a rectangle whose area and length are given as polynomials.

Type: Formative Assessment

Quadratic Expressions:

Students are asked to identify equivalent quadratic expressions and to name the form in which each expression is written.

Type: Formative Assessment

Finding Missing Values:

Students are asked to rewrite quadratic expressions and identify parts of the expressions.

Type: Formative Assessment

Dot Expressions:

Students are asked to explain how parts of an algebraic expression relate to the number and type of symbols in a sequence of diagrams.

Type: Formative Assessment

What Happens?:

Students are asked to determine how the volume of a cone will change when its dimensions are changed.

Type: Formative Assessment

Lesson Plans

Free Fall Clock and Reaction Time!:

This will be a lesson designed to introduce students to the concept of 9.81 m/s2 as a sort of clock that can be used for solving all kinematics equations where a = g.

Type: Lesson Plan

Sorting Equations and Identities:

This lesson is intended to help you assess how well students are able to:

  • Recognize the differences between equations and identities.
  • Substitute numbers into algebraic statements in order to test their validity in special cases.
  • Resist common errors when manipulating expressions such as 2(x – 3) = 2x – 3; (x + 3)2 = x2 + 32.
  • Carry out correct algebraic manipulations.
It also aims to encourage discussion on some common misconceptions about algebra.

Type: Lesson Plan

Manipulating Polynomials:

This lesson unit is intended to help you assess how well students are able to manipulate and calculate with polynomials. In particular, it aims to identify and help students who have difficulties in switching between visual and algebraic representations of polynomial expressions, performing arithmetic operations on algebraic representations of polynomials, factorizing and expanding appropriately when it helps to make the operations easier.

Type: Lesson Plan

Math Is Exponentially Fun!:

The students will informally learn the rules for exponents: product of powers, powers of powers, zero and negative exponents. The activities provide the teacher with a progression of steps that help lead students to determine results without knowing the rules formally. The closing activity is hands-on to help reinforce all rules.

Type: Lesson Plan

Original Student Tutorials

Multistep Factoring: Quadratics:

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Polynomials when "a" Does Not Equal 1, Snowflake Method:

Learn to factor quadratic trinomials when the coefficient a does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method:

Learn how to factor quadratic polynomials when the leading coefficient (a) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

The Diamond Game: Factoring Quadratics when a = 1:

Learn how to factor quadratics when the coefficient a = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Identifying Parts of Linear Expressions:

Learn to identify and interpret parts of linear expressions in terms of mathematical or real-world contexts in this original tutorial.

Type: Original Student Tutorial

Solving Rational Equations: Using Common Denominators:

Learn how to solve rational functions by getting common denominators in this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials Using Special Cases:

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Perspectives Video: Professional/Enthusiast

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Tasks

Quadrupling Leads to Halving:

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

Type: Problem-Solving Task

A Cubic Identity:

Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.

Type: Problem-Solving Task

The Physics Professor:

Students write explanations of the structure and function of a mathematical expression.

Type: Problem-Solving Task

Equivalent Expressions:

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term.

Type: Problem-Solving Task

Throwing Horseshoes:

Students evaluate equivalent constructions of the same expression to determine which is the most useful for determining a maximum value.

Type: Problem-Solving Task

The Bank Account:

Students explore an expression that calculates the balance of a bank account with compounding interest.

Type: Problem-Solving Task

Radius of a Cylinder:

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

Increasing or Decreasing? Variation 1:

Students examine variable expression that is a complex fraction with two distinct unit fractions in the denominator. Students are asked to consider how increasing one variable will affect the value of the entire expression. The variable expression is used in physics and describes the combined resistance of two resistors in parallel.

Type: Problem-Solving Task

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

Animal Populations:

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Type: Problem-Solving Task

Seeing Dots:

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task

Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

Using this CMAP

To view an introduction on the CMAP tool, please .

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Type: Unit/Lesson Sequence

Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

Original Student Tutorials

Multistep Factoring: Quadratics:

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Polynomials when "a" Does Not Equal 1, Snowflake Method:

Learn to factor quadratic trinomials when the coefficient a does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method:

Learn how to factor quadratic polynomials when the leading coefficient (a) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

The Diamond Game: Factoring Quadratics when a = 1:

Learn how to factor quadratics when the coefficient a = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Identifying Parts of Linear Expressions:

Learn to identify and interpret parts of linear expressions in terms of mathematical or real-world contexts in this original tutorial.

Type: Original Student Tutorial

Solving Rational Equations: Using Common Denominators:

Learn how to solve rational functions by getting common denominators in this interactive tutorial.

Type: Original Student Tutorial

Factoring Polynomials Using Special Cases:

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

Type: Original Student Tutorial

Perspectives Video: Professional/Enthusiast

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Tasks

Quadrupling Leads to Halving:

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

Type: Problem-Solving Task

A Cubic Identity:

Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.

Type: Problem-Solving Task

Equivalent Expressions:

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term.

Type: Problem-Solving Task

Radius of a Cylinder:

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

Animal Populations:

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Type: Problem-Solving Task

Seeing Dots:

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Perspectives Video: Professional/Enthusiast

Base 16 Notation in Computing:

Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Tasks

Quadrupling Leads to Halving:

Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.

Type: Problem-Solving Task

A Cubic Identity:

Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.

Type: Problem-Solving Task

Equivalent Expressions:

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term.

Type: Problem-Solving Task

Radius of a Cylinder:

Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.

Type: Problem-Solving Task

Mixing Fertilizer:

Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.

Type: Problem-Solving Task

Mixing Candies:

Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.

Type: Problem-Solving Task

Kitchen Floor Tiles:

This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

Type: Problem-Solving Task

Delivery Trucks:

This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

Type: Problem-Solving Task

Animal Populations:

In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.

Type: Problem-Solving Task

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Type: Problem-Solving Task

Seeing Dots:

The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.

Type: Problem-Solving Task