Standard 1: Develop an understanding of the Pythagorean Theorem and angle relationships involving triangles.

General Information
Number: MA.8.GR.1
Title: Develop an understanding of the Pythagorean Theorem and angle relationships involving triangles.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Grade: 8
Strand: Geometric Reasoning

Related Benchmarks

This cluster includes the following benchmarks.

Related Access Points

This cluster includes the following access points.

Access Points

MA.8.GR.1.AP.1
Find the hypotenuse of a two-dimensional right triangle using the Pythagorean Theorem.
MA.8.GR.1.AP.2
Given the Pythagorean Theorem, determine lengths/distances between two points in a coordinate system by forming right triangles, with natural number side lengths.
MA.8.GR.1.AP.3a
Measure the sides of triangles to establish facts about the Triangle Inequality Theorem (i.e., the sum of two side lengths is greater than the third side).
MA.8.GR.1.AP.3b
Substitute the side lengths of a given figure into the Pythagorean Theorem to determine if a right triangle can be formed.
MA.8.GR.1.AP.4
Identify supplementary, complementary, vertical or adjacent angle relationships.
MA.8.GR.1.AP.5
Given an image, solve simple problems involving the relationships of interior and exterior angles of a triangle.
MA.8.GR.1.AP.6
Use tools to calculate the sum of the interior angles of regular polygons when given the formula.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Formative Assessments

Distance Between Two Points:

Students are asked to find the distance between two points in the coordinate plane.

Type: Formative Assessment

Three Dimensional Diagonal:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

Type: Formative Assessment

Pythagorean Squares:

Students are asked to explain how a pair of figures demonstrates the Pythagorean Theorem and its converse.

Type: Formative Assessment

New Television:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

Type: Formative Assessment

Justifying the Exterior Angle of a Triangle Theorem:

Students are asked to apply the Exterior Angle of a Triangle Theorem and provide an informal justification.

Type: Formative Assessment

How Far to School:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

Type: Formative Assessment

What Is Your Angle?:

Students are asked use knowledge of angle relationships to write and solve an equation to determine an unknown angle measure.

Type: Formative Assessment

Sides of Triangles:

Students are asked to determine if given lengths will determine a triangle.

Type: Formative Assessment

Drawing Triangles SSS:

Students are asked to draw a triangle with given side lengths, and explain if these conditions determine a unique triangle.

Type: Formative Assessment

Straight Angles:

Students are asked to write and solve equations to determine unknown angle measures in supplementary angle relationships.

Type: Formative Assessment

Solve for the Angle:

Students are asked to write and solve equations to determine unknown angle measures in supplementary and complementary angle pairs.

Type: Formative Assessment

Find the Angle Measure:

Students are asked to use knowledge of angle relationships to write and solve equations to determine unknown angle measures.

Type: Formative Assessment

Distance on the Coordinate Plane:

Students are asked to find the distance between two points in the coordinate plane.

Type: Formative Assessment

Coordinate Plane Triangle:

Students are asked to determine the lengths of the sides of a right triangle in the coordinate plane given the coordinates of its vertices.

Type: Formative Assessment

Calculate Triangle Sides:

Students are asked to determine the length of each side of a right triangle in the coordinate plane given the coordinates of its vertices.

Type: Formative Assessment

Lesson Plans

Discovering and Using the Pythagorean Theorem:

Students will complete a hands-on activity to discover a geometric proof of the Pythagorean Theorem, and they will use and apply the Pythagorean Theorem to solve examples and real-world situations.

Type: Lesson Plan

Discovering and Using the Pythagorean Theorem:

Students will complete a hands-on activity to discover a geometric proof of the Pythagorean Theorem, and they will use and apply the Pythagorean Theorem to solve examples and real-world situations.

Type: Lesson Plan

A Hypotenuse is a WHAT????:

Students are guided through a short history of Pythagoras and a discovery of the Pythagorean Theorem using the squaring of the sides of a right triangle.

Type: Lesson Plan

Discovering and Using the Pythagorean Theorem:

Students will complete a hands-on activity to discover a geometric proof of the Pythagorean Theorem, and they will use and apply the Pythagorean Theorem to solve examples and real-world situations.

Type: Lesson Plan

Deriving and Applying the Law of Sines:

Students will be introduced to a derivation of the Law of Sines and apply the Law of Sines to solve triangles.

Type: Lesson Plan

Triangle Mid-Segment Theorem:

The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.

Type: Lesson Plan

The Laws of Sine and Cosine:

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

Type: Lesson Plan

Pythagorean Perspective:

This lesson serves as an introductory lesson on the Pythagorean Theorem and its converse. It has a hands-on discovery component. This lesson includes worksheets that are practical for individual or cooperative learning strategies. The worksheets contain prior knowledge exercises, practice exercises and a summative assignment.

Type: Lesson Plan

Parallel Thinking Debate:

Students prove theorems related to parallel lines using vertical, corresponding, and alternate interior angles.

Type: Lesson Plan

Sine and Cosine Relationship between Complementary Angles:

This is a lesson on the relationship between the Sine and Cosine values of Complementary Angles.

Type: Lesson Plan

Airplanes in Radar's Range:

For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.

Type: Lesson Plan

What's the Problem:

Students solve problems using triangle congruence postulates and theorems.

Type: Lesson Plan

Proving and Using Congruence with Corresponding Angles:

Students, will prove that corresponding angles are congruent. Directions for using GeoGebra software to discover this relationship is provided.

Type: Lesson Plan

Pondering Points Proves Puzzling Polygons:

In a 55 minute class, students use whiteboards, Think-Pair-Share questioning, listen to a quadrilateral song, and work individually and in groups to learn about and gain fluency in using the distance and slope formulas to prove specific polygon types.

Type: Lesson Plan

Sine, Sine, Everywhere a Sine:

Students discover the complementary relationship between sine and cosine in a right triangle.

Type: Lesson Plan

How Do You Measure the Immeasurable?:

Students will use similar triangles to determine inaccessible measurements. Examples include exploring dangerous caves and discovering craters on Mars.

Type: Lesson Plan

Triangles: To B or not to B?:

Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a triangle.

Type: Lesson Plan

Will You Survive?:

Students are stranded on a desert island and will need to use the law of sines in order to find the quickest path to a rescue vessel.

Note: This is not an introductory lesson for the standard.

Type: Lesson Plan

My Geometry Classroom:

Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

Type: Lesson Plan

What's the Point?:

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

Note: This is not an introductory lesson for this standard.

Type: Lesson Plan

The Copernicus' Travel:

This lesson uses Inverse Trigonometric Ratios to find acute angle measures in right triangles. Students will analyze the given information and determine the best method to use when solving right triangles. The choices reviewed are Trigonometric Ratios, The Pythagorean Theorem, and Special Right Triangles.

Type: Lesson Plan

Discovering Triangle Sum:

This lesson is designed to address all levels and types of learners to improve understanding of the triangle sum theorem from the simplest perspective and progress steadily by teacher led activities to a more complex level. It is intended to create a solid foundation in geometric reasoning to help students advance to higher levels in confidence.

Type: Lesson Plan

Let's Prove the Pythagorean Theorem:

Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles.

Type: Lesson Plan

The Trig Song:

This lesson is a group project activity designed to reinforce the concepts of sine and cosine. The lesson begins with a spiral review of the concepts, which will move into the group project - writing an original song to demonstrate understanding and application of sine and cosine ratios.

Type: Lesson Plan

Following the Law of Sine:

This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

Type: Lesson Plan

Geometer Sherlock: Triangle Investigations:

The students will investigate and discover relationships within triangles; such as, the triangle angle sum theorem, and the triangle inequality theorem.

Type: Lesson Plan

How Tall am I?:

Students will determine the height of tall objects using three different calculation methods. They will work in groups to gather their data and perform calculations. A whole-class discussion is conducted at the end to compare results and discuss some of the possible errors.

Type: Lesson Plan

Proofs of the Pythagorean Theorem:

This lesson enriches the students' perspective of geometric proofs that are not in two-column form. Students will apply algebaic skills and geometric properties to prove the Pythagorean Theorem in a variety of ways. This unit is designed to help you identify and assist students who have difficulties in:

  • Interpreting diagrams.
  • Identifying mathematical knowledge relevant to an argument.
  • Linking visual and algebraic representations.
  • Producing and evaluating mathematical arguments.

Type: Lesson Plan

As the Crow Flies:

This two-day lesson teaches students to use the Pythagorean Theorem with simple right triangles on the first day, then progresses to using the theorem to find the distance between two points on a coordinate graph.

Type: Lesson Plan

Angles, angles, everywhere!:

This is an introductory lesson that allows students to explore complementary and supplementary angles and their relationships. Students will measure the degrees for sets of angles. They will then use their knowledge of straight, right, obtuse, and acute angles to make connections to complementary, supplementary, adjacent, and vertical angles. This lesson provides opportunities for kinesthetic, auditory, and visual learning.

Type: Lesson Plan

Origami Boats - Pythagorean Theorem in the Real-World Part 2:

Students will create origami boats and use them to make a net drawing. The drawing will be labeled with measurements, based on the number of squares on the graph for units, such that the students will use the Pythagorean Theorem to find several of the lengths. This is part 2 of a lesson plan for the Pythagorean Theorem. The resource, Applying the Pythagorean Theorem Part 1, with ID #48973, lays the groundwork for this exercise.

Type: Lesson Plan

Shape It Up:

Students will derive the formula for the sum of the interior angles of a polygon by drawing diagonals and applying the Triangle Sum Theorem. The measure of each interior angle of a regular polygon is also determined.

Type: Lesson Plan

Just Plane Ol' Area!:

Students will construct various figures on coordinate planes and calculate the perimeter and area. Use of the Pythagorean theorem will be required.

Type: Lesson Plan

The Ins and Outs of Polygons:

In this lesson, students will explore how to find the sum of the measures of the angles of a triangle, then use this knowledge to find the sum of the measures of angles of other polygons. They will also be able to find the sum of the exterior angles of triangles and other polygons. Using both concepts, students will be able to find missing measurements.

Type: Lesson Plan

Triangles: Finding Interior Angle Measures:

The lesson begins with a hands-on activity and then an experiment with a GeoGebra-based computer model to discover the Triangle Angle Sum Theorem. The students write and solve equations to find missing angle measures in a variety of examples.

Type: Lesson Plan

Triangle Inequality Investigation:

Students use hands-on materials to understand that only certain combinations of lengths will create closed triangles.

Type: Lesson Plan

Applying the Pythagorean Theorem, Part 1:

This lesson applies the Pythagorean Theorem and teaches the foundations of the Pythagorean Theorem. It is part 1 of 2 lessons. The second lesson, Origami Boats - Pythagorean Theorem in the real world, Resource ID 49055, provides an application to use the Pythagorean Theorem for distance in the coordinate plane.

Type: Lesson Plan

Bike Club Trip:

In this activity the students will rank different locations for a bike club's next destination. In order to do so, the students must use Pythagorean Theorem and well as analyze data of the quantitative and qualitative type.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Special Angle Pairs Discovery Activity:

This lesson uses a discovery approach to identify the special angles formed when a set of parallel lines is cut by a transversal. During this lesson, students identify the angle pair and the relationship between the angles. Students use this relationship and special angle pairs to make conjectures about which angle pairs are considered special angles.

Type: Lesson Plan

Pythagoras - You Clever Dog:

This lesson starts with an introduction of the Pythagorean Theorem. It introduces vocabulary, formulas and concepts related to right triangles and the use of the Pythagorean Theorem in the real world. Students will learn the basics through real world application.

Type: Lesson Plan

How Many Degrees?:

This lesson facilitates the discovery of a formula for the sum of the interior angles of a regular polygon. Students will draw all the diagonals from one vertex of various polygons to find how many triangles are formed. They will use this and their prior knowledge of triangles to figure out the sum of the interior angles. This will lead to the development of a formula for finding the sum of interior angles and the measure of one interior angle.

Type: Lesson Plan

Alas, Poor Pythagoras, I Knew You Well! #2:

Using different activities, students will find real life uses for the Pythagorean Theorem.

Type: Lesson Plan

A Hypotenuse is a WHAT????:

Students are guided through a short history of Pythagoras and a discovery of the Pythagorean Theorem using the squaring of the sides of a right triangle.

Type: Lesson Plan

Kissing “V”s:

This lesson uses a paper-cutting activity to teach vertical angles. The lesson provides examples in which students must solve equations to find missing measures.

Type: Lesson Plan

What's Your Angle?:

Through a hands-on-activity and guided practice, students will explore parallel lines intersected by a transversal and the measurements and relationships of the angles created. They will solve for missing measurements when given a single angle's measurement. They will also use the relationships between angles to set up equations and solve for a variable.

Type: Lesson Plan

Original Student Tutorials

Pythagorean Theorem: Part 3:

Use the Pythagorean Theorem to find the legs of a right triangle in mathematical and real worlds contexts in this interactive tutorial.

This is part 3 in a 3-part series. Click below to explore the other tutorials in the series. 

Type: Original Student Tutorial

Pythagorean Theorem: Part 2:

Use the Pythagorean Theorem to find the hypotenuse of a right triangle in mathematical and real worlds contexts in this interactive tutorial.

This is part 2 in a 3-part series. Click below to explore the other tutorials in the series. 

Type: Original Student Tutorial

Pythagorean Theorem: Part 1:

Learn what the Pythagorean Theorem and its converse mean, and what Pythagorean Triples are in this interactive tutorial.

This is part 1 in a 3-part series. Click below to explore the other tutorials in the series. 

Type: Original Student Tutorial

Playground Angles Part 1:

Explore complementary and supplementary angles around the playground with Jacob in this interactive tutorial.

This is Part 1 in a two-part series. Click HERE to open Playground Angles: Part 2.

Type: Original Student Tutorial

Playground Angles: Part 2:

Help Jacob write and solve equations to find missing angle measures based on the relationship between angles that sum to 90 degrees and 180 degrees in this playground-themed, interactive tutorial.

This is Part 2 in a two-part series. Click HERE to open Playground Angles: Part 1.

Type: Original Student Tutorial

Applying the Pythagorean Theorem to Solve Mathematical and Real-World problems:

Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Expert

Measuring a Grid for Underwater Archeology:

Don't be a square! Learn about how even grids help archaeologists track provenience!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiast

What's the Distance from Here to the Middle of Nowhere?:

Find out how math and technology can help you (try to) get away from civilization.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Perspectives Video: Teaching Idea

KROS Pacific Ocean Kayak Journey: Kites, Geometry, and Vectors:

Set sail with this math teacher as he explains how kites were used for lessons in the classroom.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set [.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth [.KML]

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Teaching Idea

Presentation/Slideshow

Pythagoras' Theorem:

This resource can be used to introduce the Pythagorean Theorem to students. It provides sketches, applets, examples and easy-to-understand visual proofs as well as an algebra proof for the theorem.

It also includes interactive multiple choice practice questions on solving for a side of a right triangle, and applications involving right triangles, as well as a hands-on activity for students to do that allows them to create their own proof.

Type: Presentation/Slideshow

Problem-Solving Tasks

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

The Shortest Line Segment from Point P to Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Type: Problem-Solving Task

Slopes and Circles:

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever angle AXB is a right angle.

Type: Problem-Solving Task

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Type: Problem-Solving Task

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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. In this case the grid is also a map, and the street names can be viewed as defining a coordinate system (although the coordinate system is not needed to solve the problem).

Type: Problem-Solving Task

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. This helps pave the way toward what students will see later in trigonometry but some guidance will likely be needed in order to get students started on this path.

Type: Problem-Solving Task

Find the Angle:

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.

Type: Problem-Solving Task

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. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Type: Problem-Solving Task

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task

Area of a Trapezoid:

The purpose of this task is for students to use the Pythagorean Theorem to find the unknown side-lengths of a trapezoid in order to determine the area. This problem will require creativity and persistence as students must decompose the given trapezoid into other polygons in order to find its area.

Type: Problem-Solving Task

Applying the Pythagorean Theorem in a Mathematical Context:

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure used shows some of the dimensions but is not drawn to scale. Apply the Pythagorean Theorem to prove whether the figure is a right triangle.

Type: Problem-Solving Task

Areas of Geometric Shapes with the Same Perimeter:

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes. For example, of all triangles, the one with fixed perimeter P and largest area is the equilateral triangle whose side lengths are all P3 but this is difficult to show because it is not easy to find the area of triangle in terms of the three side lengths (though Heron's formula accomplishes this). Nor is it simple to compare the area of two triangles with equal perimeter without knowing their individual areas. For quadrilaterals, a similar problem arises: showing that of all rectangles with perimeter P the one with the largest area is the square whose side lengths are P4 is a good problem which students should think about. But comparing a square to an irregularly shaped quadrilateral of equal perimeter will be difficult.

Type: Problem-Solving Task

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

Type: Problem-Solving Task

Text Resource

Pythagoras Explained:

This informational text resource is intended to support reading in the content area. The text describes a method for predicting the win-loss record for baseball teams based on runs scored and runs allowed, using the "Pythagorean Expectation" formula invented by Bill James. The text goes on to show the relationship of the prediction formula to the Pythagorean theorem, pointing out a very cool application of the theorem to the world of sports.

Type: Text Resource

Tutorials

Finding Missing Angle Measures:

In this video, we find missing angle measures from a variety of examples.

 

Type: Tutorial

Finding the Measure of Complementary Angles:

The video will use algebra to find the measure of two angles whose sum equals 90 degrees, better known as complementary angles.

Type: Tutorial

Find Measure of Complementary Angles:

Watch as we use algebra to find the measure of two complementary angles. 

Type: Tutorial

Find Measure of Supplementary Angles:

Watch as we use algebra to find the measure of supplementary angles, whose sum is 180 degrees.

Type: Tutorial

Find Measure of Vertical Angles:

This video uses knowledge of vertical angles to solve for the variable and the angle measures.

Type: Tutorial

Introduction to Vertical Angles:

This video uses facts about supplementary and adjacent angles to introduce vertical angles.

Type: Tutorial

Find Measure of Angles in a Word Problem:

This video demonstrates solving a word problem involving angle measures.

Type: Tutorial

Construct a Right Isosceles Triangle:

This video discusses constructing a right isosceles triangle with given constraints and deciding if the triangle is unique.

Type: Tutorial

Construct a Triangle with Given Side Lengths:

This video demonstrates drawing a triangle when the side lengths are given.

Type: Tutorial

Complementary and Supplementary Angles:

The video will demonstrate the difference between supplementary angles and complementary angles, by using the given measurements of angles.

Type: Tutorial

Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

Original Student Tutorials

Pythagorean Theorem: Part 3:

Use the Pythagorean Theorem to find the legs of a right triangle in mathematical and real worlds contexts in this interactive tutorial.

This is part 3 in a 3-part series. Click below to explore the other tutorials in the series. 

Type: Original Student Tutorial

Pythagorean Theorem: Part 2:

Use the Pythagorean Theorem to find the hypotenuse of a right triangle in mathematical and real worlds contexts in this interactive tutorial.

This is part 2 in a 3-part series. Click below to explore the other tutorials in the series. 

Type: Original Student Tutorial

Pythagorean Theorem: Part 1:

Learn what the Pythagorean Theorem and its converse mean, and what Pythagorean Triples are in this interactive tutorial.

This is part 1 in a 3-part series. Click below to explore the other tutorials in the series. 

Type: Original Student Tutorial

Playground Angles Part 1:

Explore complementary and supplementary angles around the playground with Jacob in this interactive tutorial.

This is Part 1 in a two-part series. Click HERE to open Playground Angles: Part 2.

Type: Original Student Tutorial

Playground Angles: Part 2:

Help Jacob write and solve equations to find missing angle measures based on the relationship between angles that sum to 90 degrees and 180 degrees in this playground-themed, interactive tutorial.

This is Part 2 in a two-part series. Click HERE to open Playground Angles: Part 1.

Type: Original Student Tutorial

Applying the Pythagorean Theorem to Solve Mathematical and Real-World problems:

Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Expert

Measuring a Grid for Underwater Archeology:

Don't be a square! Learn about how even grids help archaeologists track provenience!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiast

What's the Distance from Here to the Middle of Nowhere?:

Find out how math and technology can help you (try to) get away from civilization.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Tasks

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

The Shortest Line Segment from Point P to Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Type: Problem-Solving Task

Slopes and Circles:

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever angle AXB is a right angle.

Type: Problem-Solving Task

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Type: Problem-Solving Task

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Find the Angle:

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.

Type: Problem-Solving Task

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. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Type: Problem-Solving Task

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

Type: Problem-Solving Task

Tutorials

Finding Missing Angle Measures:

In this video, we find missing angle measures from a variety of examples.

 

Type: Tutorial

Finding the Measure of Complementary Angles:

The video will use algebra to find the measure of two angles whose sum equals 90 degrees, better known as complementary angles.

Type: Tutorial

Find Measure of Complementary Angles:

Watch as we use algebra to find the measure of two complementary angles. 

Type: Tutorial

Find Measure of Supplementary Angles:

Watch as we use algebra to find the measure of supplementary angles, whose sum is 180 degrees.

Type: Tutorial

Find Measure of Vertical Angles:

This video uses knowledge of vertical angles to solve for the variable and the angle measures.

Type: Tutorial

Introduction to Vertical Angles:

This video uses facts about supplementary and adjacent angles to introduce vertical angles.

Type: Tutorial

Find Measure of Angles in a Word Problem:

This video demonstrates solving a word problem involving angle measures.

Type: Tutorial

Construct a Right Isosceles Triangle:

This video discusses constructing a right isosceles triangle with given constraints and deciding if the triangle is unique.

Type: Tutorial

Construct a Triangle with Given Side Lengths:

This video demonstrates drawing a triangle when the side lengths are given.

Type: Tutorial

Complementary and Supplementary Angles:

The video will demonstrate the difference between supplementary angles and complementary angles, by using the given measurements of angles.

Type: Tutorial

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Perspectives Video: Expert

Measuring a Grid for Underwater Archeology:

Don't be a square! Learn about how even grids help archaeologists track provenience!

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Perspectives Video: Professional/Enthusiast

What's the Distance from Here to the Middle of Nowhere?:

Find out how math and technology can help you (try to) get away from civilization.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Tasks

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

The Shortest Line Segment from Point P to Line L:

This is a foundational geometry task designed to provide a route for students to develop some fundamental geometric properties that may seem rather obvious at first glance. In this case, the fundamental property in question is that the shortest path from a point to a line meets the line at a right angle which is crucial for many further developments in the subject.

Type: Problem-Solving Task

Slopes and Circles:

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever angle AXB is a right angle.

Type: Problem-Solving Task

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Type: Problem-Solving Task

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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. In this case the grid is also a map, and the street names can be viewed as defining a coordinate system (although the coordinate system is not needed to solve the problem).

Type: Problem-Solving Task

Find the Angle:

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.

Type: Problem-Solving Task

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. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Type: Problem-Solving Task

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task

Area of a Trapezoid:

The purpose of this task is for students to use the Pythagorean Theorem to find the unknown side-lengths of a trapezoid in order to determine the area. This problem will require creativity and persistence as students must decompose the given trapezoid into other polygons in order to find its area.

Type: Problem-Solving Task

Applying the Pythagorean Theorem in a Mathematical Context:

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure used shows some of the dimensions but is not drawn to scale. Apply the Pythagorean Theorem to prove whether the figure is a right triangle.

Type: Problem-Solving Task

Areas of Geometric Shapes with the Same Perimeter:

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes. For example, of all triangles, the one with fixed perimeter P and largest area is the equilateral triangle whose side lengths are all P3 but this is difficult to show because it is not easy to find the area of triangle in terms of the three side lengths (though Heron's formula accomplishes this). Nor is it simple to compare the area of two triangles with equal perimeter without knowing their individual areas. For quadrilaterals, a similar problem arises: showing that of all rectangles with perimeter P the one with the largest area is the square whose side lengths are P4 is a good problem which students should think about. But comparing a square to an irregularly shaped quadrilateral of equal perimeter will be difficult.

Type: Problem-Solving Task

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

Type: Problem-Solving Task