Standard #: MA.8.GR.1.2


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Apply the Pythagorean Theorem to solve mathematical and real-world problems involving the distance between two points in a coordinate plane.


Examples


The distance between (-2,7) and (0,6) can be found by creating a right triangle with the vertex of the right angle at the point (-2,6). This gives a height of the right triangle as 1 unit and a base of 2 units. Then using the Pythagorean Theorem the distance can be determined from the equation 1²+2²=c², which is equivalent to 5=c². So, the distance is begin mathsize 12px style square root of 5 end styleunits.

Clarifications


Clarification 1: Instruction includes making connections between distance on the coordinate plane and right triangles.

Clarification 2: Within this benchmark, the expectation is to memorize the Pythagorean Theorem. It is not the expectation to use the distance formula.

Clarification 3: Radicands are limited to whole numbers up to 225.



General Information

Subject Area: Mathematics (B.E.S.T.)
Grade: 8
Strand: Geometric Reasoning
Standard: Develop an understanding of the Pythagorean Theorem and angle relationships involving triangles.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Coordinate
  • Coordinate Plane

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 6, students used their understanding of the coordinate plane to plot rational-number ordered pairs in all four quadrants and on both axes, and they found the distances between ordered pairs with the same x-coordinate or the same y-coordinate represented on the coordinate plane. In grade 8, students find the distance between two points using the Pythagorean Theorem. In Geometry, students will use coordinate geometry to classify or justify definitions, properties and theorems involving circles, triangles or quadrilaterals. Additionally, students will extend this understanding to using coordinate geometry and trigonometry to solve mathematical and real-world problems involving lines, circles, triangles, quadrilaterals and finding the perimeter or area of polygons.
  • Instruction includes creating a right triangle from two given points and then using the Pythagorean Theorem to find the distance between the two given points. This work can be started by using Geoboards to see the triangle that is formed within the coordinate plane. Students can show how to make a right triangle using vertical and horizontal lines. From there they can build the area models of the Pythagorean Theorem to support understanding.
    Geoboards to see the triangle
  • Students should be given multiple opportunities to see the importance of using the coordinate plane to find the distance between two points.
  • Instruction includes providing students with a structure to support the organization of their work since using the Pythagorean Theorem may require multiple steps. Provide students with resources, including the coordinate plane and graph paper, as a way to plan out their work.

 

Common Misconceptions or Errors

  • Students may have the misconception that the Pythagorean Theorem will apply to any triangle.
  • Students may invert the x- and y-value of the point.
  • When finding distances that cross over an axis students may incorrectly use operations with integers.
    • For example, if given the points (−2, 0) and (3, 0), students may calculate the distance as 1 unit instead of 5 units.

 

Strategies to Support Tiered Instruction

  • Instruction includes the use of geometric software to explore the Pythagorean Theorem on obtuse, acute and right triangles.
  • Instruction includes students adding the absolute value of two x-coordinates or two y-coordinates when the given points cross over an axis.
    • For example, if the given points are (−4, 8) and (7, 8), students will add the absolute value of −4 and 7.
      |−4| + |7| = 11
  • Teacher provides opportunities for students to comprehend the context or situation by engaging in questions.
    • What do you know from the problem?
    • What is the problem asking you to find?
    • Can you create a visual model to help you understand or see patterns in your problem?
  • Instruction includes labeling the x- and y-value of a coordinate point before graphing to reinforce the process of graphing x- and y-values.
  • Instruction includes laying trace paper on top of a coordinate plane, tracing the points, drawing a number line through the two points, and counting the space between the points to find the distance.
  • Teacher creates an anchor chart while students create a similar own graphic organizer to include key features of a coordinate plane. Features include the x-axis, y-axis, origin, quadrants, numbered scales and an ordered pair.
  • Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
    • First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
    • Second, read the problem with the purpose of answering the question: What are we trying to find out?
    • Third, read the problem with the purpose of answering the question: What information is important in the problem?

 

Instructional Tasks

Instructional Task 1 (MTR.2.1, MTR.7.1)
Pineridge Middle School was given a grant from Home Helper Depot to create a triangular garden along a wall of the cafeteria for fresh vegetables. The length of the hypotenuse and the sides are being determined to see if it will fit in the space. On the model for the garden, the designer started by plotting the points (2, 2) and (6, 5) on a coordinate plane and connected the points with a line. She needs to complete the triangular model and determine all three side lengths.
  • Part A. Using a coordinate grid, complete the designer's drawing.
  • Part B. Calculate the side lengths of the triangular garden on the model.
  • Part C. What would be appropriate lengths for a triangular garden if the length of one side of the building is 20 feet? Use your model to help determine the side lengths.

 

Instructional Items

Instructional Item 1
On a coordinate plane, plot the points (−3, 4) and (0, −3). Using the Pythagorean Theorem, determine the distance between the two points.

Instructional Item 2
Using the Pythagorean Theorem, determine the distance from point (8, −6) to the origin.

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.


Related Courses

Course Number1111 Course Title222
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))


Related Access Points

Access Point Number Access Point Title
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.


Related Resources

Formative Assessments

Name Description
Distance Between Two Points

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

Distance on the Coordinate Plane

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

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.

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.

Lesson Plans

Name Description
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.

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.

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.

Triangle Mid-Segment Theorem

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

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.

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.

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.

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.

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.

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.

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.

Original Student Tutorial

Name Description
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.

Perspectives Video: Professional/Enthusiast

Name Description
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.

Problem-Solving Tasks

Name Description
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.

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.

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).

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.

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.

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.

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.

Text Resource

Name Description
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.

Student Resources

Original Student Tutorial

Name Description
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.

Perspectives Video: Professional/Enthusiast

Name Description
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.

Problem-Solving Tasks

Name Description
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.

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.

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.

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.



Parent Resources

Perspectives Video: Professional/Enthusiast

Name Description
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.

Problem-Solving Tasks

Name Description
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.

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.

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).

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.

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.

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.

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.



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