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Clarifications
Clarification 1: Instruction includes exploring right triangles with natural-number side lengths to illustrate the Pythagorean Theorem.Clarification 2: Within this benchmark, the expectation is to memorize the Pythagorean Theorem.
Clarification 3: Radicands are limited to whole numbers up to 225.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Converse of Pythagorean Theorem
- Theorem
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 6, students worked with right triangles, with a focus on how the area of a right triangle is determined by its side lengths. In grade 8, students use the Pythagorean Theorem to determine side lengths of right triangles. In Geometry, students will use trigonometry to continue their work with right triangles, and they extend the understanding of the Pythagorean Theorem to create the equation of a circle.- While it is not the expectation of this benchmark, instruction includes building understanding of the Pythagorean Theorem (² + ² = ²) by proving it in various ways. Many of them connect the side lengths of right triangles to the areas of associated squares.
- Instruction includes using exploration activities that allow students to see how to use areas of squares to find missing sides. Students can use grid paper to draw right triangles from given measures and represent and compute the areas of the squares on each side. Below is an example of what students can visualize in order to understand the conceptual understanding within the Pythagorean Theorem.
- Data can be recorded in a chart such as the one below, allowing for students to conjecture about the relationship among the areas of squares and side lengths of right triangles.
- Instruction includes using exploration activities that allow students to see how to use areas of squares to find missing sides. Students can use grid paper to draw right triangles from given measures and represent and compute the areas of the squares on each side. Below is an example of what students can visualize in order to understand the conceptual understanding within the Pythagorean Theorem.
- When introducing the Pythagorean Theorem, some students may not be able to visualize the side lengths and the connection to the values of , and . Using colors to color code the sides and hypotenuse will allow students to see the connection and identify with the , and used to represent the sides.
- While solving real-world problems, students should be encouraged to draw diagrams where they can see the right triangles being used. Students will need to understand the ideas within the Triangle Inequality Theorem to help differentiate between the legs and the hypotenuse of a right triangle.
Common Misconceptions or Errors
- Students may make errors in calculations when using the Pythagorean Theorem and finding square roots.
- Students may not be able to spatially visualize triangles within the real-world problems. To address this misconception, instruction includes models for these problems with triangles and drawings to help students orient the ideas within the tasks.
- Students may misidentify the side lengths and hypotenuse when connecting to the formula of ² + ² = ². To support students as they are developing the conceptual understanding of this benchmark, using the idea of ² + ² = ² as a transition to using the formula.
Strategies to Support Tiered Instruction
- Instruction includes modeling the differences between doubling and squaring a radius. Doubling a radius would be represented by multiplying the given length, whereas squaring a number would be represented by the area of a square with the given radius.
- For example, students can be given the table below to show how the left column doubles a length whereas the right column squares a length.
- For example, students can be given the table below to show how the left column doubles a length whereas the right column squares a length.
- Teacher creates and posts an anchor chart for calculating the Pythagorean Theorem with visuals focused on solving for the variable and finding the square root.
- Instruction includes providing students with a right triangle as a visual in the context of a real-world problem. Teacher provides instruction using the information from the real-world problem to label the visual representation before solving.
- Instruction includes co-constructing a graphic organizer for the square root of perfect squares from 0 to 225 to provide students with the opportunity to determine benchmark numbers for non-perfect squares.
- Instruction includes color-coding and labeling a right triangle or a rectangular prism to provide a visual representation of variables, side lengths and hypotenuse.
- Instruction includes co-constructing a model with students and completing a graphic organizer to make the connection between the side lengths of right triangles to the area of the associated square.
- Instruction includes including time for reviewing solving for variables and finding square roots prior to instruction on this benchmark.
- Instruction includes models for problems with triangles and drawings to help students orient the ideas within the tasks.
- Instruction includes the use of the idea of ² + ² = ² as a transition to using the formula to assist in developing a conceptual understanding of the benchmark for students that misidentify the side lengths and hypotenuse when connecting the formula of ² + ² = ² (laminating formulas on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
Instructional Tasks
Instructional Task 1 (MTR.2.1, MTR.4.1, MTR.7.1)The bases on a baseball diamond are 90 feet apart on a standard baseball field.
- Part A. Draw a model of the baseball diamond.
- Part B. What is the distance, in feet, for the catcher to throw from home plate to second base?
- Part C. What is the distance, in feet, from first base to third base?
Instructional Task 2 (MTR.1.1)
You are wrapping a gift for your teacher’s birthday. It is a very long and skinny pencil. You want to wrap it in a box so that your teacher cannot tell what shape it is. Your friend has a shoe box that measures 10 inches by 7 inches by 5 inches.
- Part A. What questions would still need to be answered to approach this problem? Do you need all of the measurements provided in the problem? Explain your answer.
- Part B. If the pencil measures 13 inches long, will it fit in the shoe box with the lid closed? Explain your answer.
- Part C. What are the possible dimensions of a box that can just barely fit a pencil measuring 9 inches long?
Instructional Items
Instructional Item 1The bottom of a ladder must be placed 3 feet from a wall. The ladder is 10 feet long. How far above the ground does the ladder touch the wall?
Instructional Item 2
Using the figure below, find the value of the length of side in meters.
Instructional Item 3
If a right triangle’s legs are both the same length, , and the hypotenuse of the triangle is 25 feet, what is the value of , in feet?
*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 Tutorials
Perspectives Video: Expert
Perspectives Video: Teaching Idea
Problem-Solving Tasks
Text Resource
MFAS Formative Assessments
Students are asked to determine the length of a side of a right triangle in a real-world problem.
Students are asked to determine the length of a side of a right triangle in a real-world problem.
Students are asked to explain how a pair of figures demonstrates the Pythagorean Theorem and its converse.
Students are asked to determine the length of a side of a right triangle in a real-world problem.
Original Student Tutorials Mathematics - Grades 6-8
Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.
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.
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.
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.
Student Resources
Original Student Tutorials
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
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
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
Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.
Type: Original Student Tutorial
Perspectives Video: Expert
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
Problem-Solving Tasks
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
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
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: Expert
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
Problem-Solving Tasks
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
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
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
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
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
