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Common Core Math

High School Algebra Teaching and Learning Resources

Domain: REI: Reasoning with Equations and Inequalities

Common Core Math

Select the cluster for resources on this page:

  • REI-A: Understand solving equations as a process of reasoning and explain the reasoning

  • REI-B: Solve equations and inequalities in one variable

  • REI-C: Solve systems of equations

  • REI-D: Represent and solve equations and inequalities graphically

  • REI Unit: Reasoning with Equations and Inequalities Performance Task identified by New York City School System.

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REI-A: Understand solving equations as a process of reasoning and explain the reasoning

Standards:

 

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with REI-A, REI-B, REI-C and REI-D:

absolute value rules linear inequality polynomial rational function
Cramer's rule linear system of equations quadratic equation solve
exponential function logarithm rules quadratic formula solve graphically
extraneous solution matrix radical rules system of equations
linear equation matrix inverse rational equation system of inequalities

 

LearnZillion:

  • Lesson set: Solve and explain simple algebraic equations: This is a series of five video lessons that also include guided practice on this topic.  Videos demonstrate how to recognize properties of unknown values by using real number examples, and addition and subtraction with real number examples; using additive and multiplicative inverses to solve algebraic equations; and using multiple operations to solve linear equations. Aligns with REI-A.1.
  • Lesson set: Justify solutions to equations in terms of equation properties: Six lessons delve into solving equations using a balance scale, distributive property, using properties, and working backwards.  Aligns with REI-A.1.
  • Lesson set: Solve simple rational and radical equations in one variable: This is a series of four video lessons that also include guided practice on this topic.  Videos demonstrate how to solve rational expressions by using additive and multiplicative inverses, identify extraneous solutions in rational equations by checking solutions, solve radical expressions by using rules of exponents, and determine if a radical equation has an extraneous solution by solving.  Align with REI-A.2.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled."  Additional simulations are included.  Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.

PurpleMath:

Just Math Tutorials, videos by Patrick JMT: Solving an Equation with a Single Radical: After viewing the first example, view the related math videos with examples 2 and 3.

Shodor Interactivate: Solving Equations: Discussions and activities are designed to help students understand the concepts behind and methods of solving equations. This lesson is best implemented with students working in groups of 2-4.  Also see Equation Solver, which includes the virtual manipulative.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 6 Rational Exponents and Radical Functions and  Ch. 8 Rational Functions for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook.  See multiple choice below for related sections to use within the eWorkbook.  Use with REI-A.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives: US Common Core State Standard Math HSA-REI.A.1: 5 manipulatives: Solving quadratic equations with integer coefficients, solution of quadratic equations, two-step equations, solving multiplication equations, solving simple equations.

Multiple Choice:

Khan Academy: Practice questions with videos.

Glencoe Online Learning Center, Algebra 2 (2010) self-check quizzes:

McDougal Littell ClassZone, Algebra 2, 2011:

OpenEd:

  • Explain solving equations: 5 questions (4 multiple choice, 1 free response).  Includes related review resources.  Aligns with REI-A.1.

Constructed-response:

Khan Academy: Practice questions with videos.

OpenEd:

Performance tasks:

Illustrative Mathematics: Algebra:

Inside Mathematics: MARS Task: Magic Squares: Students find cell values in magic squares and simple algebraic notation.  This task aligns with standards REI-A.1, REI-A.2.

Mathematics Vision Project, Secondary 1 Student Edition:

  • Getting Ready Module:  This module contains 10 classroom tasks.  Task 5: Cafeteria Actions and Reactions; Task 7: Solving Equations, Literally; Task 8: Cafeteria Conundrums; Task 9: Greater Than; and Task 10: Taking Sides align to algebra standard REI-A.1.

Mathematics Vision Project, Secondary 3 Student Edition:

  • Module 4: Rational Functions: This module contains 7 classroom tasks.  Multiple standards are addressed. Standards REI-A.2, REI-D.11 are among those found in task 7.

 

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REI-B: Solve equations and inequalities in one variable

Standards:

Technology-enhanced investigations:

Art of Problem Solving: Introduction to Algebra Videos: These are the videos that go with each of the sections within the 13-chapter textbook, Introduction to Algebra.  View Ch. 3 One-Variable Linear Equations, Ch. 4, section 5: Introduction to Multi-Variable Equations, Ch. 9 Introduction to Inequalities, Ch. 10 Quadratic Equations-Part 1 (includes factoring); and Ch. 13 Quadratic Equations-Part 2 ( includes completing the square, proving the quadratic formula, using the quadratic formula, the discriminant, and quadratic problem solving).

Study.com: Algebra: Each of the following algebra chapters includes video lessons with sample problems and a short quiz with each video.  There is a multiple choice chapter test following the last video in each chapter.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  The Quadratic Equation: It’s Hip to Be Squared: "The lesson begins with three quadratics and their graphs (three parabolas):  y = x2 - 2x + (0 or 1 or 2). The prerequisite or co-requisite is some working experience with algebra, like factoring x2 -2x into x(x-2). The objective is to connect four things: the formula for y, the graph of y (a parabola), the roots of y and the minimum or maximum of y."  Aligns with Algebra standards SSE-B.3.a, CED-A.2, and REI-B.4.b and Function standards IF-B.4, IF-C.7.a, IF-C.8.a, BF-B.3.

LearnZillion:

  • Lesson set: Solve linear equations and inequalities in one variable: This is a series of eight video lessons that also include guided practice on this topic.  Videos demonstrate how to solve equations and word problems involving one variable, including when variables are on both sides in a equation or when some coefficients are letters; how to solve inequalities and inequality word problems using additive and multiplicative inverses; and how to solve compound inequalities.  Aligns with REI-B.3.
  • Lesson set: Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula, and factoring: This is a series of ten video lessons that also include guided practice on this topic.  Lessons address each method to solve a quadratic and also include how to determine when the quadratic has no real solutions and how to choose the best method to solve a quadratic. Aligns with REI-B.4.b.

WisWeb: Virtual Manipulatives:

  • Solving Equations with Balance Strategy: game.  Learners solve linear equations of increasing difficulty with this game, showing their steps indicating operations to be performed on each side of the given equation, or you can enter your own equation.  Use this applet with standard REI-B.3.
  • Linear Equations, Test: Learners solve linear equations, showing their step by step solutions with this Java virtual manipulative.  There are five levels of difficulty for problems in this test. Use this applet with standard REI-B.3.

Thinking Mathematics:

Wisc-Online, Learning Objects:

Purple Math:

Algebrahelp.com: Calculator Index: Lists the online calculators provided at the site for solving and working with algebra equations, simplifying expressions, graphing functions, prime factoring numbers, operations with fractions, solving proportions, and so on.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 1 Equations and Inequalities and Ch. 4 Quadratic Functions and Factoring for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook.  See multiple choice below for related sections to use within the eWorkbook.  Use with REI-B.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

  • US Common Core State Standard Math HSA-REI.B.3: 5 manipulatives: Graphs of inequalities, two-step equations, solving multiplication equations, solving simple equations, algebra quiz (Choose the type of problem for your quiz. You can also select the number of problems and different forms).
  • US Common Core State Standard Math HSA-REI.B.4: 6 manipulatives: Complex zeroes of quadratic functions, solving quadratic equations with integer coefficients, identifying trinomials that are perfect squares, solution of quadratic equations, polynomial roots in the complex plane, algebra quiz (Choose the type of problem for your quiz. You can also select the number of problems and different forms).

Multiple-choice:

CK-12: Algebra: Video examples, practice problems online, and real world applications.  The following align with REI-B.3:

Glencoe Online Learning Center, Algebra 2 (2010) self-check quizzes:

Khan Academy: See Algebra I: Quadratic Equations and Algebra 2: Polynomial Expressions, Equations and Functions: Practice questions with videos.  Use with REI-B.4:

McDougal Littell ClassZone, Algebra 2, 2011:

OpenEd:

  • Solving quadratic equations: 5 questions (4 multiple choice, 1 free response).  Includes related review resources.  Aligns with REI-B.4.

Constructed-response:

Achieve the Core: Quadratic Equations Mini-Assessment (pdf or Word docx) can be used to evaluate students' understanding of algebra standards REI-B.4 and CED-A.1 in order to prepare to teach this material or to check for student ability to demonstrate understanding and apply these concepts.

CK-12: Algebra: Video examples, short-answer practice problems online, and real world applications.  The following align with REI-B.3:

Khan Academy: See Algebra I: One-variable Linear Equations; Algebra 1: One-variable Linear Inequalities; Algebra I: Quadratic Equations; Algebra 2: Rational Equations and Functions; Algebra 2: Polynomial Expressions, Equations and Functions: Practice questions with videos.  Use with REI-B.3 and REI-B.4.  For example:

Louisiana Department of Education: Multiple choice and constructed response Sample Test Items: Algebra 1 (from 2016-2017): Constructed Response samples: Falling Rock Function aligns with IF-A.2; Water Spray aligns with IF-B.4; and Solving Quadratics aligns with algebra standard REI-B.4.

L. Spector: The Math Page: Skill in Algebra.  Students see worked examples of concepts and following the presentation within each section, students are presented problems in constructed response form.  They would solve using paper-pencil and then by using their mouse to roll over a box, they see the solution to problems presented.  Use with standards REI-B.3 and REI-B.4.

OpenEd:

Shodor Interactivate: Algebra Quiz: Learners solve algebraic linear and quadratic equations of one variable. Choose difficulty level, question types, and time limit.

S.O.S. Math: Solving Equations: Linear and Quadratic: Detailed solutions are provided as feedback to equations presented.

Performance tasks:

Illustrative Mathematics: Algebra:

Inside Mathematics: MARS Task: Quadratic (2009): Students work with a quadratic function in various forms.  This task aligns with function standards IF-B.4, IF-C.7, IF-C.8 and algebra standard REI-B.4.

Mathematics Assessment Project: Standards: High School: Algebra: http://map.mathshell.org/stds.php?standardid=1190

Mathematics Vision Project, Secondary 1 Student Edition:

  • Getting Ready Module: This module contains 10 classroom tasks.  Task 6: Elvira’s Equations; Task 7: Solving Equations, Literally; Task 8: Cafeteria Conundrums; Task 9: Greater Than; and Task 10: Taking Sides align to algebra standard REI-B.3.
  • Module 3: Arithmetic and Geometric Sequences: This module contains 10 classroom tasks.  Task 8: What Does It Mean?; and Task 9: Geometric Meanies align to standard REI-B.3.
  • Module 4: Linear and Exponential Functions: This module contains 10 classroom tasks.  Task 10: X Marks the Spot aligns to standard REI-B.3.

 

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REI-C: Solve systems of equations

Standards:

 

Technology-enhanced investigations:

Art of Problem Solving: Introduction to Algebra Videos: These are the videos that go with each of the sections within the 13-chapter textbook, Introduction to Algebra.  View Ch. 5 Multi-Variable Linear Equations (solving systems, including word problems.)

Concord Consortium STEM resources: Mathematics:  Each of the following includes an applet, student activity, teacher lesson plan, and student assessment.

  • Solving Systems of Equations: Two linear equations, with two variables, are presented and students use graphs to solve the system of equations. Students graph the lines and solve the problem using the graphs.  Aligns with CED-A.2 and REI-C.6.
  • Systems of Equations: Word Problems 1: Students explore two real-life scenarios that involve solving systems of equations. The first situation involves mixing chemicals to create a new solution while the second situation involves cost and revenue for a bake sale.  Aligns with CED-A.2 and REI-C.6.
  • Systems of Equations: Word Problems 2: Students explore two new scenarios that involve solving systems of equations. The first situation involves the travel time for a round trip on an airplane, while the second situation involves calculating the numbers of students and adults attending a high school musical.  Aligns with CED-A.2 and REI-C.6.

LearnZillion:

Math Planet, Algebra 1: Following align with REI-C.6.  Each includes an example and video instruction.

Math Planet, Algebra 2: Following align with REI-C.8 and REI-C.9.  Each includes an example and video instruction.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

  • US Common Core State Standard Math HSA-REI.C.6: 11 manipulatives: Lines-2 points, When two runners meet, Two-by-two linear systems, Intersecting lines using slope-intercept form, Finding a pattern (linear equations), Two points determine a line, Lines: point-slope, Lines: slope-intercept, a linear programming scenario, the Fundamental Theorem of linear programming, and algebra quiz.
  • US Common Core State Standard Math HSA-REI.C.7: 4 manipulatives: the Wire Problem, Maximizing the area of some geometric figures of fixed perimeter, the Intersection of a line and a circle, and Finding a tangent line to a parabola.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 3 Linear Systems and Matrices for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. Use with REI-C.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  Selfish Drivers: The Braess Paradox and Traffic Planning: "The idea of this lesson is to introduce, in a simplified manner, the so-called Braess Paradox by providing simple examples to clarify that the addition of some new roads to a network does not always lead to an improvement in the liquidity of the traffic; in some cases it might even increase the time required to get from one point to another if all drivers are selfish. The educational objectives for this lesson can be summarized as follows: providing a practical application of mathematics; indicating the importance of scientific research; training the students on methods of counting and surveying all the possibilities in a given problem; and introducing Nash Equilibrium and Braess Paradox in a simplified manner by visual examples."  Aligns with CED-A.3 and REI-C.6.

NCTM Illuminations:

  • Supply and Demand: This activity focuses on having students create and solve a system of linear equations in a real-world setting. By solving a system of two equations in two unknowns, students will find the equilibrium point for supply and demand.
  • Movement with Functions: In this lesson, students use remote-controlled cars to create a system of equations.
  • Everything Balances Out in the End: Students use the Balance Pans - Expressions Tool to explore algebraic expressions. They determine if algebraic expressions are equal. They balance pans to solve a system of equations and use graphing to find the solutions to a system of equations.

National Security Agency: Making Connections: Matrices, Graphs, and Linear Systems: Students use graphing calculators to solve systems of linear equations in two ways. They first solve the systems by graphing the equations and finding the point of intersection.  Next, they will solve systems of equations by writing related matrices and finding the solution by using inverse matrices.

PBS Mathline: Meadows or Malls?: Students explore the use of matrices and graphing calculators to solve a complex linear programming problem with six variables.

Purple Math:

YouTube.com video lessons:

Following YouTube video lessons align with  REI-C.8 and REI-C.9:

Multiple-choice:

Glencoe Online Learning Center, Algebra 1 (2010) self-check quiz:

Glencoe Online Learning Center, Algebra 2 (2010) self-check quizzes:

McDougal Littell ClassZone, Algebra 2, 2011:

OpenEd:

Constructed-response:

L. Spector: The Math Page: Skill in Algebra.  Students see worked examples of concepts and following the presentation within each section, students are presented problems in constructed response form.  They would solve using paper-pencil and then by using their mouse to roll over a box, they see the solution to problems presented.  Use with standards REI-C.5 and REI-C.6 and REI-C.8.

OpenEd:

Wisc­Online, Learning Objects: Solving Systems of Linear Equations Using the Addition and Subtraction Method and Solving Systems of Linear Equations using the Substitution Method: Students review each method for solving simultaneous equations and complete a series of practice problems using each method.

Algebra Lab: Solving Systems of Linear Equations: Students are provided explanation for review and then complete problems.  Steps in solutions are provided as feedback.

MATHguide.com: Solve systems of linear equations: See examples for solving systems of linear equations with addition, multiplication/addition, and substitution.  Follow each lesson with a quizmaster.  Fill in and check answers.

Khan Academy: see Precalculus: Matrices: Practice questions with videos.  Use the following with REI-C.8 and REI-C.9:

Performance tasks:

Georgia Standards of Excellence Framework:

  • Pre-Calculus Unit 5: Matrices.  This unit contains five tasks: Central High Booster Club, Walk Like a Mathematician, Candy? What Candy?, An Okefenoke Food Web, and a Culminating Task: Vacationing in Georgia.  Tasks align with VM-C.6, VM-C.7, VM-C.8, VM-C.9, VM-C.10, VM-C.12 and algebra standards REI-C.8, REI-C.9.
  • Pre-Calculus Unit 6: Conics.  This unit contains eight tasks: Our Only Focus: Circles and Parabolas Review, The Focus is the Foci: Ellipses and Hyperbolas, Deriving the General Equation of a Parabola, Parabolas in Other Directions, Writing the Equations of Parabolas, The Intersection of a Line and a Quadratic, A Conic Application, and Culminating Task: Dr. Cone's New House.  Tasks align with GPE-A.1, GPE-A.2, and GPE-A.3, and algebra standard REI-C.7.

Illustrative Mathematics Project: Algebra:

Inside Mathematics: MARS Tasks:

  • Coffee: Students use a chart to solve simultaneous equations.  This task aligns with function standard BF-A.1 and algebra standard REI-C.6.
  • Graphs (2006): Students use knowledge of graphs and their features to identify the linear equations of graphs shown and write equations of lines.  This task aligns with algebra standards REI-C.6, REI-D.10, and REI-D.11.

Mathematics Assessment Project: Standards: High School: Algebra: http://map.mathshell.org/stds.php?standardid=1190

Mathematics Vision Project, Secondary 1 Student Edition: Module 2: Systems of Equations and Inequalities: This module contains 10 classroom tasks.  Standards addressed in Module 1 include REI-C.5, REI-C.6, REI-D.12, and CED-A.2, CED-A.3, and CED-A.4.

  • Task 7: Shopping for Cats and Dogs; and Task 8: Can You Get to the Point, Too? align to both standards REI-C.5 and REI-C.6.
  • Task 6: Get to the Point; and Task 10: Taken Out of Context align to standard REI-C.6.

 

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REI-D: Represent and solve equations and inequalities graphically

Standards:

 

Technology-enhanced investigations:

Art of Problem Solving: Introduction to Algebra Videos: These are the videos that go with each of the sections within the 13-chapter textbook, Introduction to Algebra.  View Ch. 8 Graphing Lines, including systems of linear equations; and Ch. 9, section 4 Graphing Linear Inequalities.

Concord Consortium STEM resources: Mathematics:  Each of the following includes an applet, student activity, teacher lesson plan, and student assessment.

  • y= mx + b: Learn to graph a line using its slope and y-intercept, or to identify the slope and y-intercept from a linear equation written in slope-intercept form.  Aligns with standard CED-A.2, and REI-D.10.
  • Linear Equations: Points, Intercepts, and Slopes, Oh My!: Students learn to graph lines using the x- and y-intercepts of equations, as well as how to graph a line when given an equation in point-slope form.  Aligns with standard CED-A.2, and REI-D.10.
  • Linear Equations: Word Problems:  Learn to apply linear equations and their graphs to real-world problems. Write an equation based on a word problem, and graph the line described by your equation. Relate the equation and the line to the situation described in words.  Aligns with CED-A.2 and REI-D.10.

LearnZillion:

Southeast Comprehensive Center of SEDL: Video: Represent and solve equations and inequalities graphically: This video aligns with standard REI-D.11.  View online or download.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 2 Linear Equations and Functions, Ch. 3 Linear Systems and Matrices and Ch. 4 Quadratic Functions and Factoring for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. Use with REI-D.

NCTM Illuminations: Supply and Demand: This activity focuses on having students create and solve a system of linear equations in a real-world setting. By solving a system of two equations in two unknowns, students will find the equilibrium point for supply and demand.

Purple Math: Graphing Linear Inequalities

Shodor Interactivate:

  • Graphit: Graph linear equations and inequalities.
  • Linear inequalities: Graph up to four linear inequalities in two variables on the same set of axes to model examples of intersections of inequalities.  Activity worksheets and discussion are also provided.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

Multiple-choice:

Glencoe Online Learning Center, Algebra 1 (2010) self-check quiz: Ch. 6, Lesson 8: Systems of Inequalities

Glencoe Online Learning Center, Algebra 2 (2010) self-check quiz: Ch. 3, Lesson 3: Solving Systems of Inequalities by Graphing

McDougal Littell ClassZone, Algebra 2, 2011:

Constructed-response:

L. Spector: The Math Page: Students see worked examples of concepts and following the presentation within each section, students are presented problems in constructed response form.  They would solve using paper-pencil and then by using their mouse to roll over a box, they see the solution to problems presented.  Use the following with REI-D.10 and REI-D.11:

OpenEd:

Khan Academy: see Algebra 1: Functions and Algebra 2: Advanced Equations and Inequalities: Practice questions with videos.  Use the following with REI-D.12:

Performance tasks:

Illustrative Mathematics: Algebra:

  • Taxi!: Aligns with REI-D.10 and function standard LE-B.5.
  • Two Squares are Equal: Solve the quadratic using as many methods as possible.  This aligns with standards REI-B.4 and REI-D.11.
  • Population and Food Supply: This involves linear and exponential functions.  Aligns with REI-D.11 and function standards LE-A.2 and LE-A.3.
  • Solution Sets: This task involves determining linear inequalities and begins with a given set of equations and shaded regions on a graph. Aligns with REI-D.12.
  • Fishing Adventures 3: Students are given a real world problem in which they write equations and graph inequalities to solve it.  Aligns with REI-D.12 and CED-A.3.

Inside Mathematics: MARS Task: Graphs (2006): Students use knowledge of graphs and their features to identify the linear equations of graphs shown and write equations of lines.  This task aligns with algebra standards REI-C.6, REI-D.10, and REI-D.11.

Mathematics Assessment Project: Standards: High School: Algebra: http://map.mathshell.org/stds.php?standardid=1190

Mathematics Vision Project, Secondary 1 Student Edition:

  • Module 2: Systems of Equations and Inequalities: This module contains 10 classroom tasks.  Standards addressed in Module 1 include REI-C.5, REI-C.6, REI-D.10, REI-D.12, and CED-A.2, CED-A.3, and CED-A.4.  Task 6 aligns with REI-D.10; Task 2: Too Big or Not Too Big, That is the Question; Task 3: Some of One, None of the Other; Task 4: Pampering and Feeding; and Task 5: All for One, One for All, parts 1 & 2, align with REI-D.12.
  • Module 5: Features of Functions: This module contains nine classroom tasks.  Task 4: Water Park; Task 5: Pooling it Together; Task 6: Interpreting Functions; and Task 9: Match That Function align with REI-D.11.

Mathematics Vision Project, Secondary 3 Student Edition:

  • Module 4: Rational Functions: This module contains 7 classroom tasks.  Multiple standards are addressed. Standards REI-A.2, REI-D.11 are among those found in task 7.

 

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REI Unit: Reasoning with Equations and Inequalities Performance Task

Performance task:

New York City School System: High School Algebra: The Cycle Shop: Unit Overview: “This packet contains a curriculum-embedded CCLS aligned task and instructional supports. The task is embedded in a 4-5 week unit on reasoning with Equations and Inequalities.” Algebra standards addressed include REI-A.1, REI-B.3, REI-C.6, and CED-A.2, CED-A.3.  Mathematical Practices standards include MP.1, MP.3, MP.4, and MP.7.

 

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