Illustrative Math Alignment: Grade 8 Unit 2

Dilations, Similarity, and Introducing Slope

Lesson 5: More Dilations

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Math Clip Art--Geometry Concepts--Similarity with Transformations--09	Math Clip Art--Similarity with Transformations--09 Topic Geometry Concepts Description This image depicts a trapezoid reflected across a horizontal line. Reflection creates a mirror image of the original figure across a line of reflection. This transformation demonstrates how a figure can be flipped over a line to create its mirror image. The resulting trapezoid is congruent to the original but inverted vertically. Teacher's Script: "Observe how the trapezoid is reflected across the horizontal line. What properties of the trapezoid remain unchanged? How does the orientation of the trapezoid change after reflection? Can you identify the line of reflection?"	Proportions
Math Clip Art--Geometry Concepts--Similarity with Transformations--10	Math Clip Art--Similarity with Transformations--10 Topic Geometry Concepts Description This image shows a kite reflected across a horizontal line, followed by a dilation. Reflection creates a mirror image of the original figure across a line of reflection. Dilation changes the size of the original figure but keeps it proportional. This transformation demonstrates how a figure can be flipped over a line to create its mirror image. The resulting kite is similar to the original but inverted vertically.	Proportions
Math Clip Art--Geometry Concepts--Similarity with Transformations--11	Math Clip Art--Similarity with Transformations--11 Topic Geometry Concepts Description This image illustrates an irregular hexagon reflected across a vertical line. Reflection creates a mirror image of the original figure across a line of reflection. This transformation demonstrates how a complex figure can be flipped over a vertical line to create its mirror image. The resulting hexagon is congruent to the original but reversed horizontally.	Proportions
Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Labeled	Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Labeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Unlabeled	Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Unlabeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Labeled	Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Labeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Unlabeled	Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Unlabeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Labeled	Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Labeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Unlabeled	Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Unlabeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Labeled	Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Labeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Unlabeled	Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Unlabeled This is from a collection of triangular shapes. They come labeled and unlabeled.	Applications of Triangles
Math Clip Art: Comparing Right Triangles	Math Clip Art: Comparing Right Triangles Use these clip art images and the background grid to test if the right triangles are similar.	Right Triangles
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 1	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 1 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5, 4, and 6, while triangle DEF also has sides 5, 4, and 6. The SSS Postulate ensures that these triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 2	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 2 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5, 4, and 6, while triangle DEF has sides 5, 7, and 6. Because corresponding sides are not all congruent, then the triangles are not congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 3	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 3 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5 and 6, and triangle DEF has sides 5 and 6, but no information on the third side. As a result, there isn't enough information to know if they are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 4	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 4 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are congruent. Both triangles have two sides (5 and 6) and an included angle of 60°. As a results of the SAS Postulate, the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 5	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 5 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5 and 6 with an included angle of 60°, and triangle DEF has sides 5 and 6 with an included angle of 62°. Because corresponding angles are not congruent, then the triangles are not congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 6	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 6 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are congruent. Both triangles have angles of 63°, 60°, and 57°, but no information on side lengths. As a result, we can't conclude if the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 7	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 7 Topic Geometric Shapes Description Determine if two triangles on a grid are congruent. Both triangles appear identical in shape and orientation, positioned differently on the grid. Using the grid, you can see that corresponding sides are congruent. Therefore the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 8	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 8 Topic Geometric Shapes Description Determine if two shapes on a grid are congruent. The shapes appear as two congruent triangles within a diamond shape, with each triangle reflected across the center axis. Using the grid, you can see that corresonding sides are congruent. Therefore, the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 9	Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 9 Topic Geometric Shapes Description Determine if two triangles on a grid are congruent. The triangles have differing shapes and orientations on the grid, suggesting different side lengths or angles. Therefore, they are not congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 1	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 1 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10, 8, and 12, while triangle DEF has sides 5, 4, and 6. Triangles are similar if corresponding sides are proportional. Here, the ratio of corresponding sides is 2:1 (10:5, 8:4, 12:6). Thus, the triangles are similar. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 10	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 10 Topic Geometric Shapes Description Determine if two right triangles are similar. The triangle on the left has angle 40°, and the other triangle has a 60° angle. For similarity, all corresponding angles must be congruent. The angles in each triangle do not match, so the triangles aren't similar. Therefore, the answer is No, the triangles aren't similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 2	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 2 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10, 8, and 12, while triangle DEF has sides 5, 3, and 6. Triangles are similar if all corresponding sides are proportional. Only two pairs of sides here have a 2:1 ratio, so the triangles are not similar. Therefore, the answer is No, the triangles arenÕt similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 3	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 3 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Both triangles have angles 60°, 57°, and 63°. Triangles are similar if corresponding angles are equal. Since all corresponding angles match, the triangles are similar. Therefore, the answer is Yes, the triangles are similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 4	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 4 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has angles 75°, 42°, and an unknown angle, and triangle DEF has angles 75° and 63° and an unknown angles. Triangles are similar if all corresponding angles are congruent. Solving for the unknown angles confirms angle congruency, so the triangles are similar. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 5	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 5 Topic Geometric Shapes Description Determine if two right triangles, ABC and DEF, are similar. Triangle ABC has angles 90° and 40°, and triangle DEF has angles 90° and 50°. Triangles are similar if all corresponding angles are congruent. Solving reveals congruent corresponding angles. Therefore, the answer is Yes, the triangles are similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 6	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 6 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has a right angle and an angle of 40°, and triangle DEF has a right angle and 45°. Not all corresponding angles are congruent, so the triangles are not similar. Therefore, the answer is No, the triangles aren't similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 7	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 7 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10 and 12 with a 45° angle, and triangle DEF has sides 5 and 6 with a 45° angle. Triangles are similar if two pairs of sides are proportional, and the included angle is congruent. The side ratio is 2:1, and both included angles are 45°. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 8	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 8 Topic Geometric Shapes Description Determine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10 and 12 with a 45° angle, and triangle DEF has sides 5 and 6 with a 40° angle. Triangles are similar if two pairs of sides are proportional, and the included angles are congruent. Here, the included angles differ (45° vs. 40°), so the triangles are not similar. Therefore, the answer is No, the triangles aren't similar.	Definition of a Triangle
Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 9	Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 9 Topic Geometric Shapes Description Determine if two right triangles are similar. The triangle on the left has sides 10 and 15 and an included right angle, while the other triangle has sides 5 and 7.5 and an included right angle. Triangles are similar if two pairs of corresponding sides are proportional, and corresponding included angles are congruent. The side ratio is 2:1, and the included right angles are congruent. Therefore, the answer is Yes, the triangles are similar.	Definition of a Triangle
Math Example--Geometric Transformation--Transformations: Example 1	Math Example--Geometric Transformation--Transformations: Example 1 Topic Transformations Description This example demonstrates a simple translation in geometric transformations. Two "T" shapes are shown on a grid, labeled "Before" and "After". The "Before" shape on the left is translated to become the "After" shape on the right, illustrating how an object can move position without changing its size or orientation.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 10	Math Example--Geometric Transformation--Transformations: Example 10 Topic Transformations Description This example showcases a combination of reflection and translation in geometric transformations. Two circular "O" shapes are displayed on a grid, one labeled "Before" and the other "After". A red dashed line indicates the axis of reflection. The transformation involves first reflecting the shape across this axis and then translating it to a new position. This demonstrates how multiple transformations can be applied in sequence to create more complex movements.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 11	Math Example--Geometric Transformation--Transformations: Example 11 Topic Transformations Description This example demonstrates a single rotation in geometric transformations. Two circular "O" shapes are shown on a grid, one labeled "Before" and the other "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
Math Example--Geometric Transformation--Transformations: Example 12	Math Example--Geometric Transformation--Transformations: Example 12 Topic Transformations Description This example illustrates a combination of rotation and reflection in geometric transformations. Two circular "O" shapes are displayed on a grid, one labeled "Before" and the other "After". The transformation involves two steps: first, a rotation around a point indicated by a pushpin, which serves as the axis of rotation, followed by a reflection across 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
Math Example--Geometric Transformation--Transformations: Example 13	Math Example--Geometric Transformation--Transformations: Example 13 Topic Transformations Description This example demonstrates a single reflection in geometric transformations. Two grids are shown with the letter "O" before and after a transformation. The "Before" grid has the "O" on the left side, while the "After" grid has it on the right side. A red dashed line indicates the axis of reflection in the middle of the two "O"s. Below, there is a diagram showing the reflection transformation along this axis, illustrating how the shape is flipped across the line while maintaining its size and shape.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 14	Math Example--Geometric Transformation--Transformations: Example 14 Topic Transformations Description This example illustrates a complex combination of transformations: translation, rotation, and reflection. Two grids are shown with an "O" shape before and after transformations. The "Before" grid has an "O" with a red dot at its center, while the "After" grid shows the same "O" but rotated and reflected. Below, a step-by-step diagram illustrates three sequential transformations: first, a translation moves the shape, then a rotation turns it around a pushpin (indicating the axis of rotation), and finally, a reflection flips it across a red dashed line.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 15	Math Example--Geometric Transformation--Transformations: Example 15 Topic Transformations Description This example demonstrates a simple translation in geometric transformations. Two grids are displayed with the letter "F" before and after a transformation. The "Before" grid shows an "F" on the left side, and the "After" grid shows it on the right side. Below, there is an arrow indicating that only a translation occurred between the two positions of the letter, showcasing how an object can move position without changing its size or orientation.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 16	Math Example--Geometric Transformation--Transformations: Example 16 Topic Transformations Description This example demonstrates a combination of translation and rotation in geometric transformations. Two grids are shown with the letter "F" before and after transformations. The "Before" grid has an upright "F", while the "After" grid shows it rotated 90 degrees clockwise. Below, there is a diagram illustrating two steps: first, a translation moves the shape, then a rotation turns it around a pushpin axis.	Definition of Transformations
Math Example--Geometric Transformation--Transformations: Example 17	Math Example--Geometric Transformation--Transformations: Example 17 Topic Transformations Description This example illustrates a combination of reflection and translation in geometric transformations. The image shows a large letter "F" on a grid transforming into a letter "L". The transformation involves two steps: first, a reflection across an axis represented by a red dashed line, and then a translation to a new position.	Definition of Transformations

Illustrative Math Alignment: Grade 8 Unit 2

Dilations, Similarity, and Introducing Slope

Lesson 5: More Dilations

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