unit 1 algebra basics homework 7 answer key

Free Printable Math Worksheets for Algebra 1

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• Writing variable expressions
• Order of operations
• Evaluating expressions
• Number sets
• Adding rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• The distributive property
• Combining like terms
• Percent of change
• One-step equations
• Two-step equations
• Multi-step equations
• Absolute value equations
• Solving proportions
• Percent problems
• Distance-rate-time word problems
• Mixture word problems
• Work word problems
• Literal Equations
• Graphing one-variable inequalities
• One-step inequalities
• Two-step inequalities
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Finding slope from a graph
• Finding slope from two points
• Finding slope from an equation
• Graphing lines using slope-intercept form
• Graphing lines using standard form
• Writing linear equations
• Graphing linear inequalities
• Graphing absolute value equations
• Direct variation
• Solving systems of equations by graphing
• Solving systems of equations by elimination
• Solving systems of equations by substitution
• Systems of equations word problems
• Graphing systems of inequalities
• Discrete exponential growth and decay word problems
• Exponential functions and graphs
• Writing numbers in scientific notation
• Operations with scientific notation
• Addition and subtraction with scientific notation
• Naming polynomials
• Adding and subtracting polynomials
• Multiplying polynomials
• Multiplying special case polynomials
• Factoring special case polynomials
• Factoring by grouping
• Dividing polynomials
• Graphing quadratic inequalities
• Completing the square
• By taking square roots
• By factoring
• With the quadratic formula
• By completing the square
• Simplifying radicals
• Adding and subtracting radical expressions
• Multiplying radicals
• Dividing radicals
• Using the distance formula
• Using the midpoint formula
• Simplifying rational expressions
• Finding excluded values / restricted values
• Multiplying rational expressions
• Dividing rational expressions
• Adding and subtracting rational expressions
• Finding trig. ratios
• Finding angles of triangles
• Finding side lengths of triangles
• Visualizing data
• Center and spread of data
• Scatter plots
• Using statistical models

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Review articles, unit 1: proportional relationships, unit 2: rates and percentages, unit 3: integers: addition and subtraction, unit 4: rational numbers: addition and subtraction, unit 5: negative numbers: multiplication and division, unit 6: expressions, equations, & inequalities, unit 7: statistics and probability, unit 8: scale copies, unit 9: geometry.

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Unit 1 – The Building Blocks of Algebra

This unit revolves around the concept of equivalency.  Within this larger framework, we review and develop the real number properties and use them to justify equivalency amongst algebraic expressions.  Students get work in mindful manipulation of algebraic expressions and actively seek structure within expressions to understand equivalency.

Rates, Patterns, and Problem Solving

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Variables and Expressions

The Commutative and Associative Properties

The Distributive Property

Equivalent Expressions

Seeing Structure in Expressions

Exponents as Repeated Multiplication

More Complex Equivalency

More Structure Work

Translating English to Algebra

Algebraic Puzzles

Unit Review

Unit #1 Review – Building Blocks

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit #1 Assessment Form A

EDITABLE ASSESSMENT

Unit #1 Assessment.Form B

Unit #1 Assessment.Form C

Unit #1 Assessment.Form D

Unit #1 Exit Tickets

Unit #1 Mid-Unit Quiz (through Lesson #6) – Form A

Unit #1 Mid-Unit Quiz (through Lesson #6) – Form B

Unit #1 Mid-Unit Quiz (through Lesson #6) – Form C

U01.AO.01 – Skill Building Set #1

EDITABLE RESOURCE

U01.AO.02 – Skill Building Set #2

U01.AO.03 – Mindful Manipulation Practice (Enrichment)

U01.AO.04 – Additional Practice Translating English to Algebra

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Geometry Basics (Geometry Curriculum - Unit 1) | All Things Algebra®

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Description

This Geometry Basics Unit Bundle contains guided notes, homework assignments, three quizzes, dictionary, study guide and a unit test that cover the following topics:

• Points, Lines, and Planes

• Segment Addition Postulate

• The Distance Formula

• The Midpoint Formula

• Partitioning a Segment

• Naming and Classifying Angles

• Angle Addition Postulate

• Angle Relationships (Adjacent, Vertical, Complementary, Supplementary, Linear Pair)

• Solving for Missing Measures using Algebra

• Special Relationships: Perpendicular and Angle Bisectors

• Constructions (Perpendicular bisectors, perpendicular line through a point on the line, perpendicular line through a point not on the line, line parallel to a given line through a given point, angle bisector, congruent angles)

Note: This unit was updated on 8/25/19 to include partitioning a segment. If you do not teach this topic, I included the old unit without this topic in the download as well.

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Geometry First Semester Notes Bundle

Geometry Curriculum

Geometry Curriculum (with Activities)

More Geometry Units:

Unit 2 – Logic and Proof

Unit 3 – Parallel and Perpendicular Lines

Unit 4 – Congruent Triangles

Unit 5 – Relationships in Triangles

Unit 6 – Similar Triangles

Unit 7 – Right Triangles and Trigonometry Unit 8 – Polygons and Quadrilaterals

Unit 9 – Transformations

Unit 10 – Circles

Unit 11 – Volume and Surface Area Unit 12 – Probability

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

© All Things Algebra (Gina Wilson), 2012-present

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Applying the Quadratic Formula (Part 1)

Select all  the equations that have 2 solutions.

\((x + 3)^2 = 9\)

\((x - 5)^2 = \text- 5\)

\((x + 2)^2-6 = 0\)

\((x - 9)^2+25 = 0\)

\((x + 10)^2 = 1\)

\((x - 8)^2 = 0\)

\(5=(x+1)(x+1)\)

For access, consult one of our IM Certified Partners .

A frog jumps in the air. The height, in inches, of the frog is modeled by the function \(h(t) = 60t-75t^2\) , where \(t\) is the time after it jumped, measured in seconds.

Solve  \(60t - 75t^2 = 0\) . What do the solutions tell us about the jumping frog?

A tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeled by the equation \(f(t) = 4 + 12t - 16t^2\) , where \(t\) is measured in seconds since the ball was thrown.

• Find the solutions to the equation \(0 = 4 + 12t - 16t^2\) .
• What do the solutions tell us about the tennis ball?

Rewrite each quadratic expression in standard form.

• \((x+1)(7x+2)\)
• \((8x+1)(x-5)\)
• \((2x+1)(2x-1)\)
• \((4+x)(3x-2)\)

Find the missing expression in parentheses so that each pair of quadratic expressions is equivalent. Show that your expression meets this requirement.

• \((4x-1)(\underline{\hspace{1in}})\) and \(16x^2 -8x +1\)
• \((9x + 2)(\underline{\hspace{1in}})\)  and  \(9x^2 -16x -4\)
• \((\underline{\hspace{1in}})(\text-x + 5)\)  and  \(\text-7x^2 +36x-5\)

The number of downloads of a song during a week is a function, \(f\) , of the number of weeks, \(w\) , since the song was released. The equation  \(f(w) = 100,\!000 \boldcdot \left(\frac{9}{10}\right)^w\)  defines this function.

• What does the number 100,000 tell you about the downloads? What about the  \(\frac{9}{10}\) ?
• Is \(f(\text-1)\) meaningful in this situation? Explain your reasoning.

Consider the equation \(4x^2 - 4x -15 = 0\) .

• Identify the values of \(a\) , \(b\) , and \(c\) that you would substitute into the quadratic formula to solve the equation.

Evaluate each expression using the values of \(a\) , \(b\) , and \(c\) .

​​​​ \(\text- b\)

​​​​​ \(b^2 - 4ac\)

\(\sqrt{b^2 - 4ac}\)

\(\text- b \pm \sqrt{b^2 - 4ac}\)

\(\dfrac{\text- b \pm \sqrt{b^2 - 4ac}}{2a}\)

• The solutions to the equation are  \(x=\text-\frac 32\) and \(x=\frac52\) . Do these match the values of the last expression you evaluated in the previous question?
• Describe the graph of \(y=\text-x^2\) . (Does it open upward or downward? Where is its \(y\) -intercept? What about its \(x\) -intercepts?)

Without graphing, describe how adding \(16x\) to \(\text-x^2\) would change each feature of the graph of \(y = \text-x^2\) . (If you get stuck, consider writing the expression in factored form.)

• the \(x\) -intercepts
• the \(y\) -intercept
• the direction of opening of the U-shape graph

Answer: 0.0816

Step 1: Calculate the number of students who play soccer: 17% of the total number of students. Step 2: Calculate the number of students who play football: 23% of the total number of students. Step 3: Calculate the number of football players who also play soccer: 48% of the number of football players. Step 4: Calculate the probability of a randomly selected student playing both sports by dividing the number of students who play both sports by the total number of students.

7 Best AI for Math Tools (July 2024)

Unite.AI is committed to rigorous editorial standards. We may receive compensation when you click on links to products we review. Please view our affiliate disclosure .

Table Of Contents

The way we approach education, particularly in mathematics, has changed a lot over the past few years. As students face increasingly complex math problems, AI-powered tools have emerged as invaluable resources for learning, problem-solving, and skill development. These innovative applications offer features ranging from instant problem recognition to step-by-step explanations, catering to learners at all levels.

In this article, we explore some of the best AI math tools that are transforming the landscape of mathematical education and empowering students and educators to tackle challenges.

1. Socratic

Socratic by Google is a free mobile app designed to help high school and university students with their studies by providing relevant educational resources and detailed explanations to their questions. Powered by Google's artificial intelligence, Socratic allows students to ask questions using voice, text, or images, and delivers high-quality learning materials tailored to their needs.

Acquired by Google in 2018, Socratic has become a go-to study companion for students looking for quick, reliable answers and in-depth explanations across a wide range of subjects, including math, science, literature, and social studies. By leveraging advanced AI algorithms, the app identifies the core concepts behind each question and curates the most relevant content from trusted sources across the web.

Key features of Socratic by Google:

• Versatile input methods: Students can ask questions by typing, speaking, or simply snapping a picture of their handwritten notes or textbook pages.
• AI-powered content curation: Socratic's algorithms analyze each question to identify key concepts and find the most relevant explanations, videos, and step-by-step solutions.
• Comprehensive subject coverage: The app supports a wide range of academic subjects, including Algebra, Geometry, Biology, Chemistry, Physics, History, and Literature.
• Expert-created study guides: Socratic partners with educators to provide visually engaging explanations and study guides for each subject, ensuring the content aligns with educational standards.
• Personalized learning experience: By understanding each student's question and learning style, Socratic tailors its content recommendations to their individual

Visit Socratic →

2. Photomath

Photomath is a popular mobile app that uses advanced computer vision and artificial intelligence to provide instant solutions to math problems. By simply pointing your smartphone camera at a math equation, Photomath can recognize and solve it, providing step-by-step explanations to help users understand the problem-solving process. The app covers a wide range of math topics, from basic arithmetic to advanced calculus, making it a valuable tool for students of all levels.

One of the key strengths of Photomath is its ability to recognize handwritten equations, not just printed text. This feature uses a neural network model that has been trained on over 100,000 images of handwritten math expressions, achieving an impressive 98% accuracy rate.

Key features of Photomath:

• Instant problem solving: Simply point your camera at a math problem, and Photomath will provide the solution in real-time, along with a step-by-step explanation of how to arrive at the answer.
• Handwriting recognition: Photomath can recognize and solve handwritten math problems with high accuracy, thanks to its advanced neural network model.
• Step-by-step explanations: The app breaks down each problem into clear, easy-to-follow steps, helping users understand the problem-solving process and learn at their own pace.
• Interactive graphs: Photomath generates interactive graphs to help visualize problems, allowing users to explore key features like roots, minimum and maximum values, and more.
• Wide range of topics: The app covers math topics from basic arithmetic to advanced calculus, including algebra, trigonometry, statistics, and more, making it suitable for students at all levels.

Visit Photomath →

3. MathGPTPro

MathGPTPro is a cutting-edge AI-powered math learning app for homework help and math education. By leveraging advanced Math GPT (Generative Pre-trained Transformer) AI technology, MathGPTPro provides real-time, interactive solutions to math problems across a wide range of subjects, including algebra, calculus, physics, chemistry, engineering, biology, and statistics.

The app's Beta model boasts 25% higher accuracy than GPT-4 in math and STEM-related problem-solving. MathGPTPro goes beyond traditional math solvers by offering an immersive learning experience. The AI delves into the logic behind each question, providing not just answers but a comprehensive understanding of methods and principles. With its photomath recognition technology, students can simply snap a picture of a problem, and MathGPTPro instantly analyzes and solves any equation or problem.

Key features of MathGPTPro:

• AI-enhanced understanding: MathGPTPro provides step-by-step solutions and detailed explanations, ensuring students grasp the underlying concepts.
• Photomath simplicity: Users can easily input problems by taking a photo, and the app's advanced recognition technology does the rest.
• Wide range of subjects: MathGPTPro covers various STEM subjects, including math, physics, chemistry, engineering, biology, and statistics.
• Interactive learning: The app offers animated tutorials and interactive problem-solving, making learning engaging and enjoyable.
• Always-ready homework help: With MathGPTPro, students have access to a personal tutor 24/7, providing assistance whenever needed.

Visit MathGPTPro →

Mathway is a powerful math problem solver app that provides instant solutions to a wide range of math problems, from basic arithmetic to advanced calculus. Developed by Chegg, a leading education technology company, Mathway has become a go-to resource for students seeking quick, reliable answers to their math questions .

Mathway can recognize and solve problems entered in various formats, including typed equations, handwritten expressions, and even photos of textbook pages. The app's advanced AI algorithms analyze the input and provide accurate, step-by-step solutions, making it easy for students to follow along and learn from the problem-solving process. This versatility makes Mathway an indispensable tool for students who may struggle with typing out complex equations or prefer to work with handwritten notes.

Key features of Mathway:

• Comprehensive math coverage: Mathway supports a wide range of math subjects, including basic math, pre-algebra, algebra, geometry, trigonometry, precalculus, calculus, and statistics.
• Step-by-step explanations: The app provides detailed, step-by-step solutions to help students understand the problem-solving process and reinforce their learning.
• Multiple input methods: Students can enter problems by typing, handwriting, or simply taking a picture of their textbook or worksheet.
• Graphing capabilities: Mathway includes a powerful graphing calculator that allows users to plot and analyze functions, making it a valuable tool for advanced math courses.
• Customizable experience: Users can select their preferred solving method and customize the app's settings to suit their learning style and needs.

Visit Mathway →

5. Symbolab

Symbolab is an advanced mathematics education platform that leverages artificial intelligence to provide step-by-step solutions to a wide range of math problems. Developed by Israeli startup EqsQuest Ltd. and released in 2011, Symbolab has become a go-to resource for students seeking to understand complex mathematical concepts and improve their problem-solving skills.

Symbolab offers comprehensive coverage of math subjects, ranging from pre-algebra and geometry to calculus, trigonometry, and beyond. The platform's AI algorithms can interpret and solve problems entered in various formats, including typed equations, handwritten expressions, and even photos of textbook pages. This versatility makes Symbolab accessible to students with different learning styles and preferences.

Key features of Symbolab:

• Step-by-step solutions: Symbolab provides detailed, step-by-step explanations for each problem, helping students understand the problem-solving process and reinforce their learning.
• Handwriting recognition: The platform's advanced AI can recognize and solve handwritten math problems with high accuracy, making it easy for students to get help with their handwritten homework or notes.
• Comprehensive subject coverage: Symbolab supports a wide array of math topics, including algebra, geometry, trigonometry, calculus, statistics, and more.
• Interactive graphing: The platform offers a powerful graphing calculator that allows users to plot and analyze functions, making it a valuable tool for visualizing complex mathematical concepts.
• Practice and quizzes: With a paid subscription, students can access personalized practice problems and quizzes to hone their skills and track their progress.

Visit Symbolab →

6. GeoGebra

GeoGebra is an interactive mathematics software suite for learning and teaching mathematics and science from primary school to university level. Developed by Markus Hohenwarter in 2001 as part of his master's thesis at the University of Salzburg, GeoGebra has evolved into a leading provider of dynamic mathematics software, supporting STEM education and innovations in teaching and learning worldwide.

GeoGebra connects geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package. This dynamic linkage between different mathematical representations allows students to explore and understand mathematical concepts in a visual and interactive way. GeoGebra's user-friendly interface and wide range of tools make it accessible to students of all levels.

Key features of GeoGebra:

• Dynamic geometry: GeoGebra allows users to create and manipulate geometric constructions with points, vectors, segments, lines, polygons, and conic sections. These constructions can be modified dynamically, enabling students to explore geometric relationships and properties.
• Algebraic input: Equations and coordinates can be entered directly using the keyboard. GeoGebra's algebra view displays the symbolic representation of constructed objects, facilitating the connection between geometry and algebra.
• Spreadsheet integration: GeoGebra includes a spreadsheet view that allows users to explore mathematical concepts using a familiar interface. Data from the spreadsheet can be easily plotted in the graphics view.
• Interactive graphing: Functions can be defined algebraically and then dynamically modified using sliders. Students can investigate the behavior of functions and the effects of parameters on their graphs.
• Scripting and tools : GeoGebra offers a range of scripting options and tools for creating interactive learning materials, simulations, and animations. These can be easily shared online through the GeoGebra Materials platform.

Visit GeoGebra →

7. MathPapa

MathPapa is an online algebra calculator and AI educational tool designed to help students learn and practice algebra concepts. The platform offers step-by-step solutions to a wide range of algebra problems, making it an invaluable resource for students seeking to improve their understanding of the subject.

One of the key strengths of MathPapa is its comprehensive coverage of algebra topics, including linear equations, quadratic equations, inequalities, graphing, factoring, and systems of equations. The platform's AI-powered calculator can solve problems entered in various formats, such as typed equations or expressions, providing detailed, step-by-step explanations to help students follow the problem-solving process.

Key features of MathPapa:

• Step-by-step solutions : MathPapa provides detailed, step-by-step explanations for each problem, breaking down the solution process to help students understand the underlying concepts and techniques.
• Algebra lessons: The platform offers a collection of algebra lessons, covering key topics and providing clear explanations and examples to support student learning.
• Practice problems: MathPapa includes a practice section where students can test their understanding by solving algebra problems and receiving immediate feedback on their answers.
• Mobile app: The MathPapa mobile app, available for Android devices, allows students to access the calculator and learning resources on the go, making it convenient to practice and learn algebra anytime, anywhere.
• Free and premium options: While MathPapa offers a free version with basic features, students can upgrade to a premium subscription for access to additional resources, such as unlimited step-by-step solutions and ad-free experience.

Visit MathPapa →

Enhancing Mathematical Understanding with AI

The integration of AI in education has opened up new possibilities for students and educators alike. These advanced tools not only solve math problems but also explain mathematical concepts in ways that traditional methods often struggle to achieve. By providing step-by-step solutions, AI math tools help students grasp the underlying principles behind complex equations and formulas.

For advanced math students, these AI-powered applications offer an opportunity to explore challenging topics and push their boundaries. Whether it's tackling calculus, linear algebra, or statistics, these tools can provide instant feedback and guidance, allowing students to progress at their own pace. Additionally, the visual representations and interactive elements offered by many of these platforms make abstract mathematical concepts more tangible and easier to understand.

While various AI-based applications have gained popularity, it's important to remember that they should complement, not replace, traditional learning methods. Educators can leverage these AI tools to create more engaging lesson plans, address individual student needs, and provide personalized support. By combining human expertise with AI capabilities, we can create a more effective and inclusive math learning environment that helps students overcome math challenges and develop strong math skills for the future.

As technology continues to evolve, we can expect even more sophisticated AI-powered educational tools to emerge, further enhancing our ability to teach and learn mathematics effectively. The future of math education looks bright, with AI serving as a powerful ally in nurturing the next generation of mathematical thinkers and problem-solvers.

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Alex McFarland is an AI journalist and writer exploring the latest developments in artificial intelligence. He has collaborated with numerous AI startups and publications worldwide.

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