ALEX Lesson Plan

     

Distance "The Pythagoras Way"

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  This lesson provided by:  
Author:Morgan Boyd
Organization:Retirement
  General Lesson Information  
Lesson Plan ID: 35589

Title:

 Distance "The Pythagoras Way"

Overview/Annotation:

In this lesson, the teacher will demonstrate how to use the Pythagorean Theorem to find distance between two points in the coordinate system. In the coordinate plane, the difference in the x- and y-values will determine the numbers to calculate the distance. This lesson will use online graphing tools as well as graph paper to plot the points. This lesson can also be used to show the relationship between the distance formula and the Pythagorean Theorem.

This lesson results from the ALEX Resource Gap Project.

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
27. Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given real-world and mathematical problems that can be represented on a coordinate plane, apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths (distances between points) in right triangles within the problem contexts.
Teacher Vocabulary:
  • Pythagorean Theorem
Knowledge:
Students know:
  • The Pythagorean Theorem.
  • The operations and labeling within a coordinate system.
Skills:
Students are able to:
  • Solve equations involving one variable and square root.
  • Represent real-world and mathematical contexts involving right triangles in a variety of formats (drawings on coordinate planes, equations).
  • Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse).
Understanding:
Students understand that:
  • The properties of right triangles can be used to solve problems.
  • Theorems represent general relationships that are true for all shapes that fit certain criteria.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.27.1: Recall how to name points on a Cartesian plane using ordered pairs.
M.8.27.2: Recognize ordered pairs (x, y).
M.8.27.3: Solve problems using the Pythagorean Theorem, with or without a calculator.
M.8.27.4: Identify right triangles.
M.8.27.5: Demonstrate how to find square roots, with or without a calculator.
M.8.27.6: Solve problems with exponents, with or without a calculator.

Prior Knowledge Skills:
  • Define area, special quadrilaterals, right triangles, and polygons.
  • Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
  • Apply area formulas to solve real-world mathematical problems.
  • Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
  • Explain how to find the area for rectangles.
  • Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
  • Recognize and demonstrate that two right triangles make a rectangle.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.27 Use the pythagorean theorem to find the hypotenuse when given the measures of two legs in a real-world context. Limit to Pythagorean triples.

Local/National Standards:

  

Primary Learning Objective(s):

The student will apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

The student will be able to calculate the distance between two points in the coordinate plane.

The student will be able to use technology to represent two points in the coordinate plane.

 

Additional Learning Objective(s):

The student will be able to manipulate the GEOGEBRA website with the given interactive controls.

 Preparation Information 

Total Duration:

31 to 60 Minutes

Materials and Resources:

Notebook paper

Pencil

iPad, Chromebook, or MacBook for each student

Desktop computer and interactive whiteboard for lesson

Graph Paper (make 3 copies for each student)

Ruler or another device with a straight edge

Graph paper in the attachment section (Make copies front and back, each student will need two copies) 

Worksheet from the attachment section titled "Points for Pythagorean Theorem" (make copies for each student)

Worksheet from the attachment section titled "Exit Slip Pythagorean" (make copies for each student)

Worksheet from the attachment section titled "Distance to the Moon" (make enough copies so that you will be able to give to the students that finish the primary assignment early)

 

Technology Resources Needed:

Website for interactive graphing in the coordinate plate (https://www.geogebra.org/m/jmMXpzuM)

iPad, Chromebook, MacBook for each student to follow the steps as the teacher explains the process of calculating the distance

Desktop computer and interactive whiteboard for teacher and student use during the lesson

Background/Preparation:

Teacher:

The teacher will need to understand how to use the screenshot mode of the interactive whiteboard. The interactive whiteboard will be able to take a picture of the coordinate plane. The teacher will need to save the graph to the desktop of the computer. The teacher and the students will need to be able to plot points on a graph. The teacher will need to use the correct mathematical terminology related to this lesson. The teacher will need to practice with the website “GEOGEBRA” prior to teaching the lesson. As the teacher introduces the website, he or she needs to allow extra time for the students to learn how to manipulate the website. Finally, the teacher will need to randomly select graph points to use as examples.

Student:

The student will need to have colored pencils, especially blue and red. The student will need to understand the distance between two points on a graph by counting and by finding the difference between numbers on the coordinate plane. The student will demonstrate knowledge of correct mathematical vocabulary. The terms that students will use in this lesson are difference, squared, square root, and sum. The student should be able to round to the given place value. Finally, the student should be able to plot points on a coordinate plane.

  Procedures/Activities: 

Before:

  1. Type the following scenario on the interactive whiteboard before the students enter the room.  “Steve asked his parent if he could borrow the car to go to Jim’s house to study. His parents said that he could, but he could only drive to Jim’s house and back. They went to the movies instead of studying. The next morning Steve was grounded! How do you think that Steve’s parents knew that he went to the movies?” While students are working to answer the question, hand out two sheets of graph paper to each student.
  2. Tell the students to write down their answers but to not share their answers with anyone.
  3. After two minutes, tell the students to discuss their answers with one person without getting out of their seat.
  4. After two minutes, call the students back together.
  5. Call on a student to write the possible answers on the interactive whiteboard.
  6. The correct answer is: ”Steve's parents knew the number of miles from their house to Jim’s. They multiplied by 2 and added the result to the car's previous mileage. Guess what? Steve put more miles on the car than he should have...busted!”
  7. The teacher can relate this problem to the concept that subtracting the values is part of the distance formula. In comparison, the two formulas are identical, but different names.
  8. State the lesson's objective: to be able to demonstrate the Pythagorean Theorem in the coordinate plane.

During:

  1. Ask the students to go to the website https://www.geogebra.org/m/jmMXpzuM
  2. Let the students practice putting points on the graph. The graph has three boxes at the top left corner. You can put points on the graph and connect the points to make lines. The arrow box will change the cursor back to the plus sign.
  3. As you demonstrate the process to students, make sure they are writing the information from the graph on their graph paper.
  4. Randomly pick two point on the graph [such as (-2, 3) and (4, 6)].
  5. Put the two points on the graph
  6. Change the point to a line and connect the two dots. Tell the students to make this line red on their paper.
  7. Make a horizontal line including (-2, 3) and make it blue.
  8. Make a vertical line including (4, 6) and make it blue.
  9. The students should see where the two blue lines cross. They should also know this is a right angle.
  10. The teacher can ask questions about right triangles as an informal assessment.
  11. Ask the students to count the distance around the right angle. Write the answer beside each line.
  12. Ask the question “Why would this not work for a slanted line?” Answer: The lines go though the boxes on the graph.
  13. Using the Pythagorean Theorem (a2 + b2 = c2), Fill in ‘a’ and ‘b’ with the distance and solve the equation.
  14. Guide the student though more points that you make up. Then, use the worksheet “Points for Pythagorean Theorem” to practice applying this equation.
  15. After working number one as a class, allow the students to work on 2 through 5 independently. Call on students to write the answers on the interactive whiteboard. The teacher should have a picture of the graph paper saved to the desktop of his or her computer so that the students can show how they got their answers.

 After:

  1. Students will complete problems 6 through 10 individually. The teacher can take this assignment up for a formal assessment.
  2. If some are falling behind, the teacher should put them with a partner or reteach the concept in small groups.
  3. If students finish early, then they can complete the acceleration activity called “Distance to the Moon” (see attachment).

Exit Slip: “Exit Slip Pythagorean”

While students are working on problems 6 through 10, the teacher should distribute the worksheet titled “Exit Slip Pythagorean.” This is another method of formal assessment.

Students should turn in the exit slip at the end of class.


  Assessment  

Assessment Strategies

The teacher will make informal assessments while guiding the students the steps in the lesson. The teacher will evaluate the students before moving to the next procedure. The teacher will allow the students to come to the interactive whiteboard and work out the given steps.

The teacher will make a formal assessment when the students have concluded the lesson by reviewing the students' work on the "Points for Pythagorean Theorem" handout.

The teacher can make a formal assessment with the exit slip from the attachment section ("Exit Slip Pythagorean").

Acceleration:

Students requiring acceleration will work on the problem "What is the distance to the moon?"

Worksheet is in the attachment section (Distance to the Moon)

Intervention:

Students that will need intervention can receive help with one-on-one reteaching with the teacher. Students may also work in groups of two or three. The teacher can assign a peer tutor to help with the small group instruction.


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.
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