Mathematics Grade 8
Strand: GEOMETRY (8.G)
Understand congruence and similarity using physical models, transparencies, or geometry software (Standards 8.G.1-5)
. Understand and apply the Pythagorean Theorem and its converse (Standards 8.G.6-8)
. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres (Standard 8.G.9)
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
A Scaled Curve
The goal of this task is to motivate and prepare students for the formal definition of dilations and similarity transformations. While these notions are typically applied to triangles and quadrilaterals, having students engage with the concepts in a context where they don't have as much training (these more "random" curves) lead students to focus more on the properties of the transformations than the properties of the figure.
A Triangle's Interior Angles
The task gives students to demonstrate several Practice Standards. Practice Standards SMP2 (Reason abstractly and quantitatively), SMP7 (Look for and make use of structure), and SMP8 (look for and express regularity in repeated reasoning) are all illustrated by the process of taking an initial solved problem -- in this case, the argument for the single given triangle -- and looking for the key structures that allow them to repeat that reasoning for a more abstract general setting.
Calculating Distance Using the Pythagorean Theorem
In this interactive students must find the distance between two points on a plane by use the Pythagorean Theorum. They then use this skill to complete an activity involving an amusement park. They create a map of a park and then figure out the distance between attractions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Chapter 10 - Mathematical Foundation (UMSMP)
This is Chapter 10 of the Utah Middle School Math Grade 8 textbook. It provides a Mathematical Foundation for Angles, Triangles and Distance.
Chapter 10 - Student Workbook (UMSMP)
This is Chapter 10 of the Utah Middle School Math Grade 8 student workbook. It focuses Angles, Triangles and Distance.
Comparing Volumes of Cylinders, Spheres, and Cones
This interactive explains how to calculate the volumes of cylinders, cones and spheres.
Students then apply this understanding to an activity where cylinders, cones and spheres are filled with water so that their volumes can be compared. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
Congruence of Alternate Interior Angles via Rotations
This goal of this task is to experiment with rigid motions to help visualize why alternate interior angles (made by a transverse connecting two parallel lines) are congruent: this result can then be used to establish that the sum of the angles in a triangle is 180 degrees.
Find the Angle
The task is an example of a direct but non-trivial problem in which students have to reason with angles and angle measurements (and in particular, their knowledge of the sum of the angles in a triangle) to deduce information from a picture.
Find the Missing Angle
This task "Uses facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure (7.G.5)" except that it requires students to know, in addition, something about parallel lines, which students will not see until 8th grade.
Geometry (8.G) - 8th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 8 Geometry.
Grade 8 Math Module 2: The Concept of Congruence (EngageNY)
In this Grade 8 module, students learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence. Throughout Topic A, on the definitions and properties of the basic rigid motions, students verify experimentally their basic properties and, when feasible, deepen their understanding of these properties using reasoning. All the lessons of Topic B demonstrate to students the ability to sequence various combinations of rigid motions while maintaining the basic properties of individual rigid motions. Students learn that congruence is just a sequence of basic rigid motions in Topic C, and Topic D begins the learning of Pythagorean Theorem.
Grade 8 Math Module 3: Similarity (EngageNY)
In 8th grade Module 3, students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles. The module begins with the definition of dilation, properties of dilations, and compositions of dilations. One overarching goal of this module is to replace the common idea of same shape, different sizes with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.
Grade 8 Unit 1: Transformations, Congruence, and Similarity (Georgia Standards)
In this unit students will develop the concept of transformations and the effects that each type of transformation has on an object; explore the relationship between the original figure and its image in regards to their corresponding parts being moved an equal distance which leads to concept of congruence of figures; learn to describe transformations with both words and numbers; relate rigid motions to the concept of symmetry and to use them to prove congruence or similarity of two figures; physically manipulate figures to discover properties of similar and congruent figures; and focus on the sum of the angles of a triangle and use it to find the measures of angles formed by transversals (especially with parallel lines), find the measures of exterior angles of triangles, and to informally prove congruence.
Is this a rectangle?
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle.
Partitioning a Hexagon
The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.
Reflecting a rectangle over a diagonal
The goal of this task is to give students experience applying and reasoning about reflections of geometric figures using their growing understanding of the properties of rigid motions.
Rigid motions and congruent angles
The goal of this task is to use rigid motions to establish some fundamental results about angles made by intersecting lines. Both vertical angles and alternate interior angles are treated.
Same Size, Same Shape?
The purpose of the task is to help students transition from the informal notion of congruence as "same size, same shape" that they learn in elementary school and begin to develop a definition of congruence in terms of rigid transformations.
Similar Triangles I
The goal of this task is to prepare students for the angle-angle criterion for triangle similarity. Since the sum of the three angles in a triangle is always 180 degrees, having two pairs of congruent corresponding angles in two triangles tells us that the third pair of corresponding angles is also congruent.
Similar Triangles II
The goal of the task is to provide an informal argument for the AA criterion for triangle similarity, appropriate for an 8th grade audience.
The purpose of this task is to apply facts about angles (including congruence of vertical angles and alternate interior angles for parallel lines cut by a transverse) in order to calculate angle measures in the context of a map.
Tile Patterns I: octagons and squares
This task aims at explaining why four regular octagons can be placed around a central square, applying student knowledge of triangles and sums of angles in both triangles and more general polygons.
Tile Patterns II: hexagons
In this task one of the most important examples of a tiling, with regular hexagons, is studied in detail. This provides students an opportunity to use what they know about the sum of the angles in a triangle and also the sum of angles which make a line.
Triangle's Interior Angles
This problem has students argue that the interior angles of the given triangle sum to 180 degrees, and then generalize to an arbitrary triangle via an informal argument. The original argument requires students to make use the angle measure of a straight angle, and about alternate interior angles formed by a transversal cutting a pair of parallel lines.
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