A Theorem is an evident pattern in nature or natural phenomenon that can be shown to exist (proven) with mathematics. Theorems that apply to the 1st, second or third dimension can be seen in our physical/material world. Other dimensions or realities beyond our 3rd dimension can be established with mathematical theory, but we cannot sense them within the 3rd dimension.
Prior to this lesson the students should know what a dimension is, and be able to express the first 3 dimensions (1st, 2nd, 3rd), points, lines, shapes and objects as well as having the ability to use ruler and compass. Students should by now know that mathematics is the study of patterns and the purpose of science. (see earlier postings.) In science, students would have recently studied and practiced making hypothesis and understand how hypothesis are used to gain knowledge of the world around us. This lesson plan for Pascals Theorem of the mystical hexagon teaches about the hexagon polygon, and can be used at any grade. This lesson can be taught as part of Common Core State Standard 6th Grade Geometry CCSS G.6.1: Solve real world and mathematical problems involving area, surface area and volume. Find the area of right triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and other shapes, apply these techniques in the context of solving real-world and mathematical problems
Review: Because we have not discovered all of the patterns of nature in the science of mathematics, so mathematicians are always looking to discover new patterns, to prove them and to apply them to improve our lives in countless ways. Once they have been tested with mathematical proof, they are called a Theorem.
A scientist makes a hypothesis based on his question or curiosity and what he already knows. This is the starting place for his investigation. Then he or she does the experiments to discover if the hypothesis is true. Once a hypothesis has been tested many times and cannot be proven wrong, it is called a Theory. In Mathematics, it is called a Theorem.
A Theorem is developed based on observable facts. For example, a line is composed of points, everyone can see this clearly, it is obvious and evident to everyone; a fact that no one, so far, has been able to disprove. This is a basic truth we have observed. From this fact, we can understand and observe patterns and mathematically test new, more advanced possibilities using calculations beginning with what we already know.
For example, Pascal’s Theorem, called Pascal’s Mystical Hexagon, is expressed as follows:
Pascal’s projective theorem, Also known as Pascals Mystical Hexagon
The 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.
You can see this for yourself by putting the 6 points on a circle, drawing the lines as above and noting the intersecting points. They are co-linear or all on the same line.
This pattern is easily observable in nature, it is reflected in art and even music and musical rhythm. This fact can be drawn on graph paper, vector sketched or cut from wood or plastic. This is a very valuable piece of information for any kind of design using triangles or circles, including manufacturing design and packaging, fashion design, equipment design, video design and more. For those who are musically inclined, or for a dancer, they may hear or move using the stop and start rhythms that match the length of each line.
In nature the best place to find the hexagonal (six-sided) system is in the structure of certain crystals.
Lesson Using Applied Kinesiology (applied science of movement): Educational Kinesiology: How does it apply to life and living things?
Materials: Graph paper, compass, pencils, natural environment (trees, flowers, plants), rulers. blocks, if possible, carpentry shop where designs can be made, or manipulative that has a circle with the triangles that fit inside, following the Pascal’s Hexagon pattern.
Kinesthetically (and visually), Pascal’s Theorem of the Mystical Hexagon can be expressed by drawing the circle and the lines on a grid and forming various geometric patterns using compass. It can be found by discovering nature’s patterns and proportions through measurement and examination of phenomenon. It can be demonstrated in the carpentry shop with rulers and block designs, it can be shown in paper designs. Model with examples
Students will come to understand that Pascal’s theorem is held true until proven wrong and so far it has not been done: no one can find anything otherwise than this theorem to be evident in mathematical drawings and in nature.
Students can continue their research by building designs, creating art with designs, discovering this pattern in music (certain rhythm expresses the same relationship as the triangle line lengths).It can be expressed physically by any angular shape made such as with the hands; there are few but they do exist can can be discovered and measured.
Is Pascal’s Theorem and Pascal’s Triangle, the same thing?
In a word, no. One of our favorite websites, mathisfun.com provides an excellent explanation of Pascal’s Triangle. More on Pascal’s triangle, later.
About me: I am Tara, the writer of these blogs about education and specifically my notes and lesson plans using kinesthetic methods to teach basic principles so they will be remembered and used for a lifetime. I began this blog to record my notes and lesson plans. According to research 60% of children learn kinesthetically, like I do. I hope to be an Educator who has developed a way to provide this service of kinesthetic learning for others.
“God is universal harmony perceived through numbers.”
xo Tara Pelton