Illustrative Math Alignment: Grade 8 Unit 2
Dilations, Similarity, and Introducing Slope
Lesson 12: Using Equations for Lines
Use the following Media4Math resources with this Illustrative Math lesson.
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Math Example--Geometric Transformation--Transformations: Example 23 | Math Example--Geometric Transformation--Transformations: Example 23TopicTransformations DescriptionThis example illustrates a combination of translation and rotation in geometric transformations. The image shows an uppercase letter "Z" on a grid labeled "Before" and an uppercase letter "N" on another grid labeled "After." Below, there are illustrations showing the two steps of the transformation: first a translation (moving the figure) followed by a rotation (turning the figure). A pushpin indicates the axis of rotation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 24 | Math Example--Geometric Transformation--Transformations: Example 24TopicTransformations DescriptionThis example demonstrates a combination of reflection and translation in geometric transformations. The image shows an uppercase letter "Z" on a grid labeled "Before" and a rotated and translated version of the same letter in another position on another grid labeled "After." Below, there are illustrations showing the two steps of the transformation: first, a reflection (flipping the figure across an axis shown by the red dashed line) and then a translation (moving the figure). |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 25 | Math Example--Geometric Transformation--Transformations: Example 25TopicTransformations DescriptionThis example demonstrates a single rotation in geometric transformations. The image shows a transformation of the letter "Z" from an upright position to a rotated position resembling "N". The grid background helps visualize the transformation, and a pushpin icon indicates the axis of rotation, illustrating how an object can change its orientation without altering its size or shape. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 26 | Math Example--Geometric Transformation--Transformations: Example 26TopicTransformations DescriptionThis example illustrates a combination of rotation and reflection in geometric transformations. The image shows a transformation of the letter "Z" into an upside-down "N". A pushpin indicates the axis of rotation, and a red dashed line represents the axis of reflection. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in orientation and position. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 27 | Math Example--Geometric Transformation--Transformations: Example 27TopicTransformations DescriptionThis example demonstrates a single reflection in geometric transformations. The image shows a reflection of the letter "Z" across a horizontal red dashed line, resulting in a backward "Z". The grid background helps visualize the reflection, illustrating how an object can be flipped over a line while maintaining its size and shape but changing its orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 28 | Math Example--Geometric Transformation--Transformations: Example 28TopicTransformations DescriptionThis example demonstrates a complex combination of transformations: translation, rotation, and reflection. The image shows a transformation of the letter "Z" into an upside-down "N". This transformation involves three steps: first, a translation moves the shape, then a rotation turns it (indicated by a pushpin showing the axis of rotation), and finally, a reflection flips it across a red dashed line representing the axis of reflection. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 29 | Math Example--Geometric Transformation--Transformations: Example 29TopicTransformations DescriptionThis example demonstrates a simple translation in geometric transformations. The image shows a "Before" and "After" transformation of a black square on a grid. The square moves horizontally to the right, and a dotted arrow indicates the direction of movement. This showcases how an object can move position without changing its size or orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 3 | Math Example--Geometric Transformation--Transformations: Example 3TopicTransformations DescriptionThis example showcases a combination of reflection and translation in geometric transformations. Two "T" shapes are displayed on a grid, labeled "Before" and "After". The shape is first reflected across a red dashed line (the axis of reflection) and then translated to a new position. This demonstrates how multiple transformations can be applied in sequence to create more complex movements. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 30 | Math Example--Geometric Transformation--Transformations: Example 30TopicTransformations DescriptionThis example illustrates a combination of translation and rotation in geometric transformations. The image depicts a "Before" and "After" transformation of a black square on a grid. The square moves down and rotates 90 degrees clockwise. A pushpin marks the axis of rotation, and dotted arrows indicate the movement and rotation. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 31 | Math Example--Geometric Transformation--Transformations: Example 31TopicTransformations DescriptionThis example demonstrates a combination of reflection and translation in geometric transformations. The image shows a "Before" and "After" transformation of a black rectangle on a grid. The rectangle first reflects across a horizontal axis, marked by a red dashed line, and then translates horizontally to the right. This illustrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 32 | Math Example--Geometric Transformation--Transformations: Example 32TopicTransformations DescriptionThis example demonstrates a single rotation in geometric transformations. The image shows a "Before" and "After" transformation of a black rectangle on a grid. The rectangle rotates 90 degrees clockwise around an axis marked by a pushpin. This illustrates how an object can change its orientation without altering its size or shape. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 33 | Math Example--Geometric Transformation--Transformations: Example 33TopicTransformations DescriptionThis example illustrates a combination of rotation and reflection in geometric transformations. The image shows a transformation of a black L-shaped figure on a grid. The "Before" figure is at the top left, and the "After" figure is at the top right. Below, there is a sequence of two transformations: first, a rotation (indicated by a pushpin showing the axis of rotation), followed by a reflection (shown by a red dashed line representing the axis of reflection). |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 34 | Math Example--Geometric Transformation--Transformations: Example 34TopicTransformations DescriptionThis example demonstrates a single reflection in geometric transformations. The image shows a black L-shaped figure on a grid. The "Before" figure is at the top left, and the "After" figure is at the top right. Below, there is a single transformation: a reflection across a vertical axis shown with a red dashed line. This showcases how an object can be flipped over a line while maintaining its size and shape but changing its orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 35 | Math Example--Geometric Transformation--Transformations: Example 35TopicTransformations DescriptionThis example demonstrates a complex combination of transformations: translation, rotation, and reflection. The image shows a black L-shaped figure with a red dot on a grid. The "Before" figure is at the top left, and the "After" figure is at the top right. Below, there are three transformations: first, a translation moves the shape, then a rotation turns it around an axis shown by a pushpin, and finally, a reflection flips it across an axis shown by a red dashed line. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 36 | Math Example--Geometric Transformation--Transformations: Example 36TopicTransformations DescriptionThis example demonstrates a simple translation in geometric transformations. The image shows two hexagons on grids. The "Before" figure is at the top left, and the "After" figure is at the top right. Below, there is a single transformation: translation, as indicated by an arrow showing movement from one position to another. This showcases how an object can move position without changing its size or orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 37 | Math Example--Geometric Transformation--Transformations: Example 37TopicTransformations DescriptionThis example illustrates a combination of translation and rotation in geometric transformations. The image shows a hexagon on a grid, with a "Before" and "After" transformation. The pushpin indicates the axis of rotation, and the hexagon is translated and rotated. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 38 | Math Example--Geometric Transformation--Transformations: Example 38TopicTransformations DescriptionThis example demonstrates a combination of reflection and translation in geometric transformations. The image shows a hexagon on a grid, with a "Before" and "After" transformation. A red dashed line shows the axis of reflection, followed by a translation. This illustrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 39 | Math Example--Geometric Transformation--Transformations: Example 39TopicTransformations DescriptionThis example demonstrates a single rotation in geometric transformations. The image shows a hexagon on a grid, with a "Before" and "After" transformation. A pushpin indicates the axis of rotation, and only one transformation occurs. This illustrates how an object can change its orientation without altering its size or shape. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 4 | Math Example--Geometric Transformation--Transformations: Example 4TopicTransformations DescriptionThis example demonstrates a single rotation in geometric transformations. Two "T" shapes are shown on a grid, labeled "Before" and "After". The shape undergoes a rotation around a point indicated by a pushpin, which serves as the axis of rotation. This illustrates how an object can change its orientation without altering its size or position relative to the rotation point. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 40 | Math Example--Geometric Transformation--Transformations: Example 40TopicTransformations DescriptionThis example illustrates a combination of rotation and reflection in geometric transformations. The image shows a hexagon on a grid, with a "Before" and "After" transformation. A pushpin indicates the axis of rotation, and a red dashed line shows the axis of reflection. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in orientation and position. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 41 | Math Example--Geometric Transformation--Transformations: Example 41TopicTransformations DescriptionThis example demonstrates a single reflection in geometric transformations. The image shows two hexagons on a grid, labeled "Before" and "After." The hexagon is reflected over a vertical axis, with a red dashed line indicating the reflection axis. This illustrates how an object can be flipped over a line while maintaining its size and shape but changing its orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 42 | Math Example--Geometric Transformation--Transformations: Example 42TopicTransformations DescriptionThis example demonstrates a complex combination of transformations: translation, rotation, and reflection. The image shows two hexagons on a grid, labeled "Before" and "After." There are three transformations: first, a translation moves the shape, then a rotation turns it (indicated by a pushpin showing the axis of rotation), and finally, a reflection flips it across a red dashed line representing the axis of reflection. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 43 | Math Example--Geometric Transformation--Transformations: Example 43TopicTransformations DescriptionThis example demonstrates a simple translation in geometric transformations. The image shows two house icons on a grid, labeled "Before" and "After." The house is translated horizontally to the right, with an arrow indicating the movement direction. This showcases how an object can move position without changing its size or orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 44 | Math Example--Geometric Transformation--Transformations: Example 44TopicTransformations DescriptionThis example illustrates a combination of translation and rotation in geometric transformations. The image shows two house icons on a grid, labeled "Before" and "After." Two transformations occur: translation (shown by an arrow) and rotation (shown by a pushpin). This demonstrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 45 | Math Example--Geometric Transformation--Transformations: Example 45TopicTransformations DescriptionThis example demonstrates a combination of reflection and translation in geometric transformations. The image shows two figures labeled as "Before" and "After." The left figure illustrates a shape on a grid, while the right figure shows the same shape after a transformation involving reflection and translation, marked with a dashed red line. This illustrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 46 | Math Example--Geometric Transformation--Transformations: Example 46TopicTransformations DescriptionThis example demonstrates a single rotation in geometric transformations. The image presents two figures labeled "Before" and "After." The left figure shows a shape on a grid, while the right one displays the same shape after a rotation around a specified point marked with a pushpin. This illustrates how an object can change its orientation without altering its size or shape. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 47 | Math Example--Geometric Transformation--Transformations: Example 47TopicTransformations DescriptionThis example illustrates a combination of rotation and reflection in geometric transformations. Two figures are presented, labeled "Before" and "After." The left figure shows the original shape on a grid, while the right figure depicts the shape after undergoing a combination of transformations: first, a rotation around a point indicated by a pushpin, followed by a reflection across a dashed red line. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 48 | Math Example--Geometric Transformation--Transformations: Example 48TopicTransformations DescriptionThis example demonstrates a single reflection in geometric transformations. The image shows two figures labeled as "Before" and "After." The left figure includes a shape positioned within a grid, and the right figure illustrates the transformed shape after reflecting across a dashed red line. This showcases how an object can be flipped over a line while maintaining its size and shape but changing its orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 49 | Math Example--Geometric Transformation--Transformations: Example 49TopicTransformations DescriptionThis example demonstrates a complex combination of transformations: translation, rotation, and reflection. The image shows two house-shaped figures on a grid. The "Before" figure is on the left, and the "After" figure is on the right. The transformation involves three steps: first, a translation moves the shape, then a rotation turns it around an axis marked by a pushpin, and finally, a reflection flips it across a dashed red line representing the axis of reflection. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 5 | Math Example--Geometric Transformation--Transformations: Example 5TopicTransformations DescriptionThis example illustrates a combination of rotation and reflection in geometric transformations. Two grids labeled "Before" and "After" show a black "T" shape in different orientations. The shape first undergoes a rotation, indicated by a pushpin showing the axis of rotation, and then a reflection, shown by a red dashed line representing the axis of reflection. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in orientation and position. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 50 | Math Example--Geometric Transformation--Transformations: Example 50TopicTransformations DescriptionThis example demonstrates a simple translation in geometric transformations. The image shows two star-shaped figures on a grid. The "Before" figure is on the left, and the "After" figure is on the right. The transformation consists of a simple translation from left to right, showcasing how an object can move position without changing its size or orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 51 | Math Example--Geometric Transformation--Transformations: Example 51TopicTransformations DescriptionThis example illustrates a combination of translation and rotation in geometric transformations. The image shows two star-shaped figures on a grid. The "Before" figure is on the left, and the "After" figure is on the right. There are two transformations: first, a translation moves the shape, then a rotation turns it around an axis marked by a pushpin. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 52 | Math Example--Geometric Transformation--Transformations: Example 52TopicTransformations DescriptionThis example demonstrates a combination of reflection and translation in geometric transformations. The image shows two star-shaped figures on a grid. The "Before" figure is on the left, and the "After" figure is on the right, located lower and to the right. There are two transformations: first, a reflection across an axis shown by a red dashed line, followed by a translation to move the shape to its final position. This illustrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 53 | Math Example--Geometric Transformation--Transformations: Example 53TopicTransformations DescriptionThis example demonstrates a single rotation in geometric transformations. The image shows a star on a grid labeled "Before" and "After." The star has been rotated to a new position on the grid, as indicated by a pushpin that marks the axis of rotation. This illustrates how an object can change its orientation without altering its size or shape. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 54 | Math Example--Geometric Transformation--Transformations: Example 54TopicTransformations DescriptionThis example illustrates a combination of rotation and reflection in geometric transformations. The image displays a star on a grid with "Before" and "After" labels. The transformation involves two steps: first, a rotation around an axis marked by a pushpin, and second, a reflection across a vertical axis shown by a red dashed line. This demonstrates how multiple transformations can be applied sequentially to create more complex changes in orientation and position. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 55 | Math Example--Geometric Transformation--Transformations: Example 55TopicTransformations DescriptionThis example demonstrates a single reflection in geometric transformations. In this image, a star is shown before and after undergoing a reflection. The red dashed line represents the axis of reflection, which is horizontal in this case. This showcases how an object can be flipped over a line while maintaining its size and shape but changing its orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 56 | Math Example--Geometric Transformation--Transformations: Example 56TopicTransformations DescriptionThis example demonstrates a complex combination of transformations: translation, rotation, and reflection. The image shows three transformations applied to the star: first, a translation moves the star to another position, then a rotation turns it around an axis indicated by a pushpin, and finally a reflection flips it across a vertical axis shown by a red dashed line. This illustrates how multiple transformations can be applied sequentially to create more complex changes in position and orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 6 | Math Example--Geometric Transformation--Transformations: Example 6TopicTransformations DescriptionThis example demonstrates a single reflection in geometric transformations. Two grids labeled "Before" and "After" show a black "T" shape in different orientations. The solution diagram below illustrates a reflection of the shape across a red dashed line, which represents the axis of reflection. This transformation showcases how an object can be flipped over a line while maintaining its size and shape. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 7 | Math Example--Geometric Transformation--Transformations: Example 7TopicTransformations DescriptionThis example illustrates a complex combination of transformations: translation, rotation, and reflection. Two grids labeled "Before" and "After" show a black "T" shape in different orientations. The solution diagram below demonstrates three sequential transformations: first, a translation moves the shape, then a rotation turns it (indicated by a pushpin showing the axis of rotation), and finally, a reflection flips it across a red dashed line representing the axis of reflection. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 8 | Math Example--Geometric Transformation--Transformations: Example 8TopicTransformations DescriptionThis example demonstrates a simple translation in geometric transformations. Two grids labeled "Before" and "After" show an "O" shape slightly shifted between the grids. The solution diagram below illustrates a translation of the shape from one position to another, showcasing how an object can move position without changing its size or orientation. |
Definition of Transformations |
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Math Example--Geometric Transformation--Transformations: Example 9 | Math Example--Geometric Transformation--Transformations: Example 9TopicTransformations DescriptionThis example illustrates a combination of transformations: translation and rotation. Two circular "O" shapes are shown on a grid, one labeled "Before" and the other "After". The solution involves a translation (movement) and a rotation, with a pushpin indicating the axis of rotation. This demonstrates how multiple transformations can be applied sequentially to a shape. |
Definition of Transformations |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 1 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 1TopicLinear Functions DescriptionThis example demonstrates how to graph a linear function with a slope of 2 and a y-intercept of 3. The process involves three key steps: first, plotting the y-intercept at (0, 3); second, using the slope to find another point on the line; and finally, connecting these points to form the line. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 10 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 10TopicLinear Functions DescriptionThis example illustrates the process of graphing a linear function with a slope of -4 and a y-intercept of 0. The method involves three main steps: plotting the y-intercept at the origin (0, 0), using the slope to determine a second point on the line, and connecting these points to create the linear graph. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 11 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 11TopicLinear Functions DescriptionThis example demonstrates the process of graphing a linear function with a slope of 0 and a y-intercept of 5. The procedure involves three key steps: plotting the y-intercept at (0, 5), recognizing that a slope of 0 results in a horizontal line, and drawing the line parallel to the x-axis through the y-intercept. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 12 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 12TopicLinear Functions DescriptionThis example illustrates the process of graphing a linear function with a slope of 0 and a y-intercept of -5. The method involves three main steps: plotting the y-intercept at (0, -5), recognizing that a slope of 0 results in a horizontal line, and drawing the line parallel to the x-axis through the y-intercept. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 13 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 13TopicLinear Functions DescriptionThis example demonstrates how to graph a linear function with a slope of 0 and a y-intercept of 0. The process involves recognizing that this special case results in a horizontal line coinciding with the x-axis. The line passes through the origin (0, 0) and extends infinitely in both directions along the x-axis. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 2 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 2TopicLinear Functions DescriptionThis example illustrates the process of graphing a linear function with a slope of 0.5 and a y-intercept of 3. The method involves three main steps: plotting the y-intercept at (0, 3), using the slope to determine a second point on the line, and connecting these points to create the linear graph. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 3 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 3TopicLinear Functions DescriptionThis example demonstrates the process of graphing a linear function with a slope of 5 and a y-intercept of -4. The procedure involves three key steps: plotting the y-intercept at (0, -4), using the slope to determine a second point on the line, and connecting these points to form the linear graph. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 4 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 4TopicLinear Functions DescriptionThis example illustrates the process of graphing a linear function with a slope of 0.1 and a y-intercept of -4. The method involves three main steps: plotting the y-intercept at (0, -4), using the slope to determine a second point on the line, and connecting these points to create the linear graph. |
Slope-Intercept Form |
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Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 5 | Math Example--Linear Function Concepts--Graphs of Linear Functions in Slope-Intercept Form: Example 5TopicLinear Functions DescriptionThis example demonstrates how to graph a linear function with a slope of -4 and a y-intercept of 5. The process involves three key steps: first, plotting the y-intercept at (0, 5); second, using the slope to find another point on the line; and finally, connecting these points to form the line. |
Slope-Intercept Form |
