Context for this Lesson
TOPIC: Practical applications of Pythagorean Theorem and Slope
GRADE: 8th Grade Math
- How do we use Pythagorean Theorem to solve everyday problems involving triangles?
- How/why is the Pythagorean Theorem useful in everyday life?
- How are pythagorean theorem and slope related
§111.24. Mathematics, Grade 8.
(4) Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:
- (C) use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.
(9) Measurement. The student uses indirect measurement to solve problems. The student is expected to:
- (A) use the Pythagorean Theorem to solve real-life problems;
- Letter from the Senator
- Rulers/Yard Sticks
- Note Paper
1. WARM UP - Considering Practical Applications
"Today we are going to consider some practical applications for Pythagorean Theorem and slope - a2 + b2 = c2"
- What does Pythagorean Theorem measure?
- What is the equation?
- In what sorts of real world situations might this equation be useful?
- Where/when would this equation come in handy?
- If we are measuring height over length (rise over run), what mathematical concept does that relate to?
- Why might knowing the exact length of something be useful in _____ situations? (discuss applications addressed by students)
- What might happen if we don’t have those measurements?
Consider this Practical Example:
Measuring a TV - “I really want to buy a TV, and the cabinet she has purchased to fit the TV will fit a television 33” tall and 56” wide, but when I go to the store, none of the televisions are measured by their height and width, but rather the diagonal measurement. What is the biggest TV I can buy?” (65” television)
2. ARTIFACT - Letter from the Senator
“Recently, our school received a letter from Senator Tom Harkin. As we read it, consider what the Sentator is asking us to do, and how we might use Pythagorean Theorem/Slope to help his cause.”
Read the following letter with students (either project the letter, or give each student a copy).
The state of Texas has been tasked by the United States Congress with auditing many of the middle schools, throughout the state, for not complying with regulations set forth by the Americans with Disabilities Act (ADA) of 1990. This law was put in place to prohibit any discrimination towards individuals with disabilities. The handicap accessible ramps located throughout middle school campuses must be measured to make sure they are in compliance with ADA regulations. The U.S. Congress feels it would be beneficial for the 8th graders of Covington Middle School in Austin, TX to apply their knowledge of the Pythagorean Theorem and Slope, to measure these ramps in order to make sure that their campus is in compliance with this law.
Your job during your mathematics period today is to locate the numerous handicap accessible ramps on your campus, measure the height and the base length of each of them, and calculate the slope at which these ramps sit. It is this slope that must be measured in order to determine if the ramps are compliant with the ADA. In order for you to successfully measure these ramps, you must know the current regulated measurement for the height and base length of ever ramp must adhere to the following rule: for every 1 inch of height, you must have 12 inches of length. Once you have completed these measurements, you will bring them back to your math teacher so that they may submit them to us for review. We appreciate your assistance with this endeavor. Good Luck!
Senator Tom Harkin (D-IA)
- What did we learn from this letter?
- What is the main point of this letter?
- Why is the lenght/height of a ramp important? (What happens if it’s too steep?)
- What is our goal as a class?
- What are the steps we will need to complete to determine the height/length (slope) of the ramps?
- How can we check to see if the height/length proportion is in compliance with the ADA laws?
- How might using Pythagorean Theorem help us in this situation?
“In a moment we are going to divide into groups and measure the ramps around our campus” Let students know which ramps they will be measuring.
3. FINDING THE SLOPE OF THE RAMP
Locate the wheelchair ramps in the school. Have students measure the two most accessible sides (often the ramp, or “c” and the height of the ramp or “a”. Use those measurements to determine the horizontal length. Use the length and the height of the ramp to determine the slope of the ramp (height/length) then compare the calculations to the 1/12 inches slope required for ramps."
4. A LETTER TO THE SENATOR
Once students determine the calculations, they will write a brief letter to Senator Tom Harkin. They will include the following information:
- The slope of each ramp measured
- Whether or not the ramp adheres to code
- A statement about whether or not their campus is accessible, and what they propose to do moving forward (at their school and other schools)
This lesson can also be applied to creating new ramps for the school. Using the 1/12 height/length measurement, students can measure the height of steps, multiply that height by 12 and then, using Pythagorean Theorem, determine the necessary length of the ramp.