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Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 7 Unit 7

Expressions, Equations, and Inequalities

Lesson 8: Reasoning about Solving Equations (Part 2)

Use the following Media4Math resources with this Illustrative Math lesson.

Thumbnail Image Title Body Curriculum Topic
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 03, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 04, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 05, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

Related Resources

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 06, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

Related Resources

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 07, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

Related Resources

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 08, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 09, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

To see the complete collection of Quizzes, click on this link.

 

Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 1 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 1

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 2 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 2

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Quiz Library

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Solving Two-Step Equations
Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 3 Interactive Quiz: Solving Two-Step Subtraction and Multiplication Equations, Quiz 10, Level 3

This is part of a collection of math quizzes on the topic of solving two-step equations. Some quizzes are interactive and some are in PDF format.

To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key).

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Solving Two-Step Equations
Equations Word Search Puzzle Interactive Word Search Puzzle: Equations Interactive Word Search Puzzle: Equations

Review key vocabulary on the topic of equations with this interactive and printable word search puzzle.

This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide.

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Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 1 Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 1 Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 1

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 2 Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 2 Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 2

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 3 Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 3 Math Clip Art--Equations and Inequalities--Equation Models--Balance Scale 3

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 1 Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 1 Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 1

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 2 Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 2 Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 2

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 3 Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 3 Math Clip Art--Equations and Inequalities--Equation Models--Balanced Equation 3

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 1 Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 1 Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 1 Left

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 1 Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 1 Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 1 Right

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 2 Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 2 Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 2 Left

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 2 Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 2 Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 2 Right

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 3 Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 3 Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 3 Left

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 3 Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 3 Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation 3 Right

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation Left Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation Left

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation Right Math Clip Art--Equations and Inequalities--Equation Models--Unbalanced Equation Right

This is part of a collection of math clip art images about equations and inequalities.

Applications of Equations and Inequalities
Math Clip Art--Equations and Inequalities--Inequalities Math Clip Art--Equations and Inequalities--Inequalities Math Clip Art--Equations and Inequalities--Inequalities

This is part of a collection of math clip art images about equations and inequalities.

Inequalities
Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 1 Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 1 Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 1

This is part of a collection of math clip art images about equations and inequalities.

Inequalities
Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 2 Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 2 Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 2

This is part of a collection of math clip art images about equations and inequalities.

Inequalities
Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 3 Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 3 Math Clip Art--Equations and Inequalities--Inequalities--Compound Inequalities Set 3

This is part of a collection of math clip art images about equations and inequalities.

Inequalities
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 1 Math Clip Art--Quadratics Concepts--Equations of Parabolas 01 Math Clip Art--Quadratics Concepts--Equations of Parabolas 01

Topic

Quadratics

Description

This math clip art image introduces the concept of equations of parabolas, a fundamental topic in quadratic functions. The image showcases a blue parabola drawn on graph paper, with clearly labeled x and y axes. Points are marked on the graph, providing a visual representation of key points on a parabola.

Graphs of Quadratic Functions
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 2 Math Clip Art--Quadratics Concepts--Equations of Parabolas 02 Math Clip Art--Quadratics Concepts--Equations of Parabolas 02

Topic

Quadratics

Description

This math clip art image illustrates the geometric definition of a parabola. It shows a red parabola on a grid, with clearly labeled focus and directrix. The parabola is symmetric around the vertical axis, demonstrating a key property of parabolas.

Graphs of Quadratic Functions
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 3 Math Clip Art--Quadratics Concepts--Equations of Parabolas 03 Math Clip Art--Quadratics Concepts--Equations of Parabolas 03

Topic

Quadratics

Description

This math clip art image builds upon the previous concept, showing a red parabola opening upwards on a grid. The focus is clearly marked above the vertex, and the directrix is represented as a horizontal line below the vertex. This visual representation emphasizes a crucial property of parabolas: the focus and directrix are equidistant from the vertex.

Graphs of Quadratic Functions
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 4 Math Clip Art--Quadratics Concepts--Equations of Parabolas 04 Math Clip Art--Quadratics Concepts--Equations of Parabolas 04

Topic

Quadratics

Description

This math clip art image presents a red parabola on a grid with its equation f(x) = 1/4x2. The focus and directrix are clearly labeled, demonstrating the symmetry of the parabola. A specific point (2, 1) is highlighted to show that its distance from the focus is equal to its distance from the directrix.

Graphs of Quadratic Functions
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 5 Math Clip Art--Quadratics Concepts--Equations of Parabolas 05 Math Clip Art--Quadratics Concepts--Equations of Parabolas 05

Topic

Quadratics

Description

This math clip art image introduces a more general approach to understanding parabolas. It depicts a parabola with its focus at (0, f) and includes a directrix. The key feature of this image is the marking of a general point with coordinates (x, y) on the parabola.

Graphs of Quadratic Functions
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 6 Math Clip Art--Quadratics Concepts--Equations of Parabolas 06 Math Clip Art--Quadratics Concepts--Equations of Parabolas 06

Topic

Quadratics

Description

This math clip art image builds upon the previous concept, showing a parabola with its focus at (0, f) and a directrix line below it. The image specifically labels the point on the directrix as (x, -f), corresponding to the x-coordinate of a point on the parabola.

Graphs of Quadratic Functions