 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).
• 3D Transmographer
This lesson contains an applet that allows students to explore translations, reflections, and rotations.
• A rectangle in the coordinate plane
This task provides an opportunity to apply the Pythagorean theorem to multiple triangles in order to determine the length of the hypotenuse; the converse of the Pythagorean theorem is also required in order to conclude that certain angles are right angles.
• 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.
• Applying the Pythagorean Theorem in a mathematical context
This task reads "Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale. Is the shaded triangle a right triangle? Provide a proof for your answer."
• Are These Shapes Congruent?
This cool interactive will allow students to conceptualize whether two shapes are congruent my twisting and turning them. The student then applies an understanding of congruency by diagramming and building shapes on a graph in the accompanying classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Are They Similar?
This goal of this task is to provide experience applying transformations to show that two polygons are similar.
• Bird and Dog Race
The purpose of this task is for students to use the Pythagorean Theorem as a problem-solving tool to calculate the distance between two points on a grid.
• 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.
• Chapter 8 - Mathematical Foundation (UMSMP)
This is Chapter 8 of the Utah Middle School Math Grade 8 textbook. It provides a Mathematical Foundation for Integer Exponents, Scientific Notation and Volume.
• Chapter 8 - Student Workbook (UMSMP)
This is Chapter 8 of the Utah Middle School Math Grade 8 student workbook. It focuses on Integer Exponents, Scientific Notation and Volume.
• Chapter 9 - Mathematical Foundation (UMSMP)
This is Chapter 9 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Transformations, Congruence and Similarity.
• Chapter 9 - Student Workbook (UMSMP)
This is Chapter 9 of the Utah Middle School Math: Grade 8 student workbook. It focuses on these topics: Transformations, Congruence and Similarity.
• Circle Sandwich
The purpose of this task is to apply knowledge about triangles, circles, and squares in order to calculate and compare two different areas.
• Comparing Snow Cones
This task asks students to use formulas for the volumes of cones, cylinders, and spheres to solve a real-world problem.
• 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.
• Congruent Rectangles
This task is designed to give students insight into the effects of translations, rotations, and reflections on geometric figures in the context of showing that two figures are congruent.
• Congruent Segments
When given two line segments with the same length this task asks students to describe a sequence of reflections that exhibits a congruence between them.
• Congruent Triangles
This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation.
• Converse of the Pythagorean Theorem
This task is for instruction purposes. Part (b) is subtle and the solution presented here uses a "dynamic" view of triangles with two side lengths fixed.
• Creating Similar Triangles
The purpose of this task is to apply rigid motions and dilations to show that triangles are similar.
• Cutting a rectangle into two congruent triangles
This task shows the congruence of two triangles in a particular geometric context arising by cutting a rectangle in half along the diagonal.
• Different Areas?
The goal of this task is to motivate a discussion of similarity and slope via a counterintuitive geometric construction where it appears as if area is not conserved by cutting and reassembling a simple shape.
• Effects of Dilations on Length, Area, and Angles
The purpose of this task is for students to study the impact of dilations on different measurements: segment lengths, area, and angle measure.
• Escaramuza: 2D Drawing
The real-life equestrian event known as Escaramuza is used to help student make 2D drawings to make triangles. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Escaramuza: Coordinates, Reflection, Rotation
A real-life equestrian event known as Escaramuza is used to demonstrate how to draw a two-dimensional diagram and then represent it on a coordinate plane. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Escaramuza: Symmetry, Reflection, Rotation
The real-life equestrian event known as Escaramuza is used to teach students how to diagram 2D representations on an x-y graph and then reflect and rotate the figure. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• 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.
• Finding isosceles triangles
This task looks at some triangles in the coordinate plane and how to reason that these triangles are isosceles.
• Finding the distance between points
The goal of this task is to establish the distance formula between two points in the plane and its relationship with the Pythagorean Theorem.
• Flower Vases
The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.
• 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.
• Geometry in Tessellations
In this lesson students will learn about lines, angles, planes, and experiment with the area and perimeter of polygons.
• Glasses
This task gives students an opportunity to work with volumes of cylinders, spheres and cones.
• 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.
• Grade 8 Unit 3: Geometric Applications of Exponents (Georgia Standards)
In this unit students will distinguish between rational and irrational numbers; find or estimate the square and cubed root of non-negative numbers, including 0; interpret square and cubed roots as both points of a line segment and lengths on a number line; use the properties of real numbers (commutative, associative, distributive, inverse, and identity) and the order of operations to simplify and evaluate numeric and algebraic expressions involving integer exponents, square and cubed roots; work with radical expressions and approximate them as rational numbers; solve problems involving the volume of a cylinder, cone, and sphere; determine the relationship between the hypotenuse and legs of a right triangle; use deductive reasoning to prove the Pythagorean Theorem and its converse; apply the Pythagorean Theorem to determine unknown side lengths in right triangles; determine if a triangle is a right triangle, Pythagorean triple; apply the Pythagorean Theorem to find the distance between two points in a coordinate system; and solve problems involving the Pythagorean Theorem.
• Identifying Unknown Transformations
This applet allows the student to drag a shape and then observe the changes to its behavior. They then determine whether the alteration is due to reflection, a rotation, or a translation/slide transformation.
• Introduction to Fractals: Infinity, Self-Similarity and Recursion
This lesson is designed to help students understand aspects of fractals, specifically self-similarity and recursion.
• 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.
• IXL Game: Pythagorean theorem
This game will help eighth graders understand the pythagorean theorem via word problems. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Modeling: Making Matchsticks
This lesson unit is intended to help educators assess how well students are able to interpret a situation and represent the variables mathematically, select appropriate mathematical methods, interpret and evaluate the data generated, and communicate their reasoning clearly.
• Origami Silver Rectangle
The purpose of this task is to apply geometry in order analyze the shape of a rectangle obtained by folding paper. The central geometric ideas involved are reflections (used to model the paper folds), analysis of angles in triangles, and the Pythagorean Theorem.
• 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.
• Point Reflection
The purpose of this task is for students to apply a reflection to a single point.
• Points from Directions
This task provides a slightly more involved use of similarity, requiring students to translate the given directions into an accurate picture, and persevere in solving a multi-step problem: They must calculate segment lengths, requiring the use of the Pythagorean theorem, and either know or derive trigonometric properties of isosceles right triangles.
• Pythagorean Explorer
This applet challenges the student to find the length of the third side of a triangle when given the two sides and the right angle.
• Pythagorean Theorem
In this lesson students will be able to use the Pythagorean Theorem to find side lengths of right triangles, the areas of right triangles, and the perimeter and areas of triangles.
• 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.
• Reflecting reflections
The goal of this task is to give students an opportunity to experiment with reflections of triangles on a coordinate grid.
• Reflection
In this animated video from UEN (Utah Education Network) students learn about one type of movement for geometric shapes - reflection. In the accompanying activity students demonstrate understanding by creating a geometric figure on a plane and then reflecting it into another quadrant. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Reflections, Rotations, and Translations
The goal of this task is to use technology to visualize what happens to angles and side lengths of a polygon (a triangle in this case) after a reflection, rotation, or translation.
• 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.
• Rotation
In this animated Math Shorts video from the Utah Education Network, learn about rotation, which describes how a geometric shape turns around a point, called the center of rotation. In the accompanying classroom activity, students are given two rotations from a handout and work in pairs to try to determine whether one figure is a rotation of the other figure around the given point. Note: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Rotation Symmetry
Exploring this interactive students are able to predict and find the angle of rotation for various figures by using rotation symmetry. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Running on the Football Field
Students need to reason as to how they can use the Pythagorean Theorem to find the distance ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance, but on seeing how you can set up right triangles to apply the Pythagorean Theorem to this problem.
• 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.
• Scaling
An interactive from Annenberg asks students to scale a picture by using the math strategies of multiplicative and additive relationships. Students then use those strategies to compare photocopies and rectangles in different scales. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Scaling angles and polygons
The goal of this task is to gather together knowledge and skills from the seventh grade in a context which prepares students for the important eighth grade notion of similarity.
• Shipping Rolled Oats
Given different scenarios, students will generate dimensions of boxes and calculate the different surface areas.
• 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.
• Sizing up Squares
The goal of this task is for students to check that the Pythagorean Theorem holds for two specific examples. Although the work of this task does not provide a proof for the full Pythagorean Theorem, it prepares students for the area calculations they will need to make as well as the difficulty of showing that a quadrilateral in the plane is a square.
• Spiderbox
The purpose of this task is for students to work on their visualization skills and to apply the Pythagorean Theorem.
• Squaring the Triangle
Students can manipulate the sides of a triangle in this applet in order to better understand the Pythagorean Theorem.
• Street Intersections
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.
In this task, students will determine how many matchsticks can be made from a tree with a trunk with a base radius of 1 foot and a height of 80 feet.
In this task, students will create a design using rotations and reflections.
• Student Task: Circles and Squares
In this task, students must solve a problem about circles inscribed in squares
In this task, students will find the volumes of different shaped drinking glasses.
The Hopewell people were Native Americans whose culture flourished in the central Ohio Valley about 2000 years ago. They constructed earthworks using right triangles. In this task, the student will look at some of the geometrical properties of a Hopewell earthwork.
In this task, students will need to work out the actual dimensions of TV screens, which are sold according to their diagonal measurements.
• Student Task: Proofs Of The Pythagorean Theorem?
In this task, students will look at three different attempts to prove the Pythagorean theorem and determine which is the best "proof".
In this task, the student will investigate Pythagorean Triples.
A set of short tasks for grades 7 & 8 dealing with geometry.
During the Edo period (1603-1867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. Here is one such problem for students to investigate.
• The Largest Container: Problems Using Volume and Shape
By using a single sheet of paper this interactive leads students to construct shapes, calculate volume, and think about the relationships between different shapes. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• The Number System (8.NS) - 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 - The Number System.
• The Pythagorean Theorem and 18th-Century Cranes
A video from Annenberg Learner Learning Math shows how the Pythagorean Theorem was useful in the reconstruction of an 18th century crane. The classroom activity asks students to apply the theorem and understand its usefulness in construction and design. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• The Pythagorean Theorem: Square Areas
This lesson unit is intended to help educators assess how well students are able to use the area of right triangles to deduce the areas of other shapes, use dissection methods for finding areas, organize an investigation systematically and collect data, and deduce a generalizable method for finding lengths and areas (The Pythagorean Theorem.)
• 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.
• Translation
This lesson opens with an animated short from the Utah Education Network to explain the concept of translation when a geometric figure changes location on a plane. Students are then asked to solve practice translation problems and explain their solution. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Translations, Reflections, and Rotations
Students are introduced to the concepts of translation, reflection and rotation in this lesson plan.
• Triangle congruence with coordinates
This task gives students a chance to explore several issues relating to rigid motions of the plane and triangle congruence.
• 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.
• Two Triangles' Area
This task requires the student to draw pictures of the two triangles and also make an auxiliary construction in order to calculate the areas (with the aid of the Pythagorean Theorem). Students need to know, or be able to intuitively identify, the fact that the line of symmetry of the isosceles triangle divides the base in half, and meets the base perpendicularly. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.