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Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 7 Unit 7

Expressions, Equations, and Inequalities

Lesson 7: Reasoning about Solving Equations (Part 1)

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 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

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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

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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

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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

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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).

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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).

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 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).

Related Resources

To see additional resources on this topic, click on the Related Resources tab.

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).

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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

To see additional resources on this topic, click on the Related Resources tab.

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).

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 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).

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 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

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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

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 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
Math Clip Art--Quadratics Concepts--Equations of Parabolas, Image 7 Math Clip Art--Quadratics Concepts--Equations of Parabolas 07 Math Clip Art--Quadratics Concepts--Equations of Parabolas 07

Topic

Quadratics

Description

This math clip art image advances the concept of deriving a parabola's equation. It shows a parabola with its focus at (0, f) and directrix, and introduces calculations for distances D1 and D2. These distances represent the key relationship that defines a parabola: the equidistance of any point on the parabola from the focus and the directrix.

Graphs of Quadratic Functions