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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
units.
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.
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 -coordinate or the same -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.
- 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 - and -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 -coordinates or two -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
- For example, if the given points are (−4, 8) and (7, 8), students will add the absolute value of −4 and 7.
- 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 - and -value of a coordinate point before graphing to reinforce the process of graphing - and -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 -axis, -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 1On 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
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorial
Perspectives Video: Professional/Enthusiast
Presentation/Slideshow
Problem-Solving Tasks
Text Resource
STEM Lessons - Model Eliciting Activity
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.
MFAS Formative Assessments
Students are asked to determine the length of each side of a right triangle in the coordinate plane given the coordinates of its vertices.
Students are asked to determine the lengths of the sides of a right triangle in the coordinate plane given the coordinates of its vertices.
Students are asked to find the distance between two points in the coordinate plane.
Students are asked to find the distance between two points in the coordinate plane.
Original Student Tutorials Mathematics - Grades 6-8
Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.
Student Resources
Original Student Tutorial
Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.
Type: Original Student Tutorial
Perspectives Video: Professional/Enthusiast
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
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
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
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
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
Parent Resources
Perspectives Video: Professional/Enthusiast
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
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
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
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
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
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
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
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
