Introduction
Welcome to Mathematics Lesson 1: Introduction to Pythagoras
Briefing
Today we will be taking a new approach to learning mathematics! We will be using the world wide web to learn all about Pythagoras and his theorem. Then we will use internet-based resources to complete a classwork task and learn new a few new skills.
Background
Pythagoras was a young boy when he first discovered his theorem. Legend says he was a Greek philosopher whose father owned a shipping boat. His father would transport tiles on his boat from suppliers to customers along the open waters and as a young boy, Pythagoras would tag along and learn the trade. One day while packing tiles he noticed a pattern when creating a right-angled shape triangle. He had placed three adjacent tiles opposite four tiles and five along the length of the triangle and realized that if he kept packing tiles till it formed a square, the sum of the two shorter sides would create the number of tiles on the longest side or Hypotenuse if you will. And so, the theorem came to life.
On arrival home, he saw two men trying to build a ladder to ascend a wall. However, they would measure the wall, build a ladder according to that measurement, and when they erected it, it would be too short to climb to the top of the wall. He used his knowledge of his theory to measure the wall, the distance between the wall and the base of the ladder where it will rest, and calculate the length that the ladder should be.
From this story, we can explain his theory in mathematical terms as:
A^2 + B^2 = C^2
in any right-angled triangle where A and B are the two shorter sides and C is the longest side or hypotenuse. This rule only applies to right-angled triangles and is used to calculate the measurements of the sides of the triangle.
If we look deeper into the theory of Pythagoras, we also notice how this concept can have a range of applications. If we had the hypotenuse and one side, we can use it to calculate the other by using subtraction. We can use it when determining the ratio of the triangle, we can use it when solving the angles inside (SIN, COS, TAN rules) and we can use it to solve shapes and angles attached to a triangle.
In nature, it is common to find perpendicular structures and the Pythagorean theorem can help us to measure these distances. If we look at a tree for example and we would like to create a ladder to go up a tree. Or a light pole that needs a securing strap. It is easy to measure the light pole and easy to measure distance on the ground but not so easy to measure a hypotenuse that does not exist. Therefore, we can use the Pythagorean theorem to do that.
Task
The task today deals with solving triangles using the Pythagorean theorem.
We will be handing out 6 triangular objects in class and you can use the following video from YouTube to learn how to solve for the hypotenuse, adjacent and opposite sides of a triangle:
Here are also some valuable links to check out other examples of Pythagorus:
Process
Process
We will divide the class into groups of 3. Each group will receive 6 triangular objects numbered from 1 to 6 and they will share the objects among themselves while presenting their findings. Some objects will have measurements on two sides, while some will have measurements on 1 side only.
Your job is to find the length of the missing sides with proof and reasons. You will be given 2 hours to complete the task during the class period and the final hour will be used to analyze what we have learned and which mistakes we have made if there are any.
Evaluation
Evaluation
Below you will find the table necessary to complete the task.
| Length 1 | Length 2 | |
| 1 |
|
|
| 2 |
|
|
| 3 |
|
|
| 4 |
|
N/A |
| 5 |
|
N/A |
| 6 |
|
N/A |
Rubric
| Thorough Understanding
The student demonstrates an above-average level of proficiency in meeting all criteria outlined below 3 pts |
Adequate Understanding
The student demonstrates an average level of proficiency in meeting most of the criteria outlined below 2 pts |
Partial Understanding
The student demonstrates a minimal level of proficiency in meeting some of the criteria outlined below 1 pts |
Needs Review
The student does not demonstrate a minimal level of proficiency 0 pts |
|
| Know parts of a right triangle
Students demonstrate knowledge of the appropriate parts of a right triangle. |
Thorough Understanding
|
Adequate Understanding
|
Partial Understanding
|
Needs Review
|
| Use Pythagorean Theorem
Students demonstrate knowledge of the correct use of the Pythagorean Theorem through the use of correct formulas, substitutions, and calculations. |
Thorough Understanding
|
Adequate Understanding
|
Partial Understanding
|
Needs Review
|
| Critical Thinking
The student demonstrates the ability to connect Pythagorean Theorem to other story problem examples. |
Thorough Understanding
|
Adequate Understanding
|
Partial Understanding
|
Needs Review
|
Conclusion
Conclusion
The final results should reflect as follows:
| Length 1 | Length 2 | |
| 1 |
4 cm |
5 cm |
| 2 |
9 cm |
11 cm |
| 3 |
7 cm |
7 cm |
| 4 |
13.5 cm |
N/A |
| 5 |
14 cm |
N/A |
| 6 |
22.5 cm |
N/A |
We will have a further discussion towards the end of the class and open the board for questions, suggestions and collaboration.
Credits
Teacher Page
Teacher: Ali Ryklief
Student Number: 216171652