Getting into Shapes
"Philosophy is written in that vast book which stands forever open before our eyes. I mean the universe; but it cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometric figures, without which means it is impossible to comprehend a single word." --Galileo, Il Saggiatore, 1623.
To make sense of the world, the human mind relies heavily on the perception of shapes and patterns. The artifacts and natural forms around us possess geometric properties. Although real-world objects rarely perfectly match an ideal mathematically-defined geometric figure, their properties will be similar and mathematical models can help us understand the possibilities of the real objects.
What Geometry Means to Me
You Either Have It . . . Or You Don't!
- Two-dimensional circles, squares, rectangles, and triangles, as well as three-dimensional spheres, cylinders, cubes, and prisms, can be observed in nature and in artifacts people create.
- Objects in nature and created objects can be described and classified in terms of their shape or a combination of the shapes of their parts.
- Shapes can compared using terms and concepts such as parallel and perpendicular line segments, acute and obtuse angles, and symmetry, congruence, and similarity.
The Frayer Model for Math Definitions
Game Time: Quadrilateral Quest
The Quadrilateral is the Key!
Challenge designed by Mrs. Anna Alexander
Think of a space you would like to design. It may be the perfect bedroom, a living room movie and game center, playground, city park, music store, office building, library, the exterior of your dream house from our first weeks, the building you used in the Hunt for Shapes challenge, or any other designed space people can visit.
You will create this space in your sketchbook, with a final design on graph paper, artist's paper using a straight edge, Google Drawing, or Scratch. If there is another program or method you would like to use for a final product, let me know.
The challenge: You must include all 5 of the classifications of quadrilaterals we have been exploring in M3 Math in your final design: parallelogram, rectangle, rhombus, square, and trapezoid.
Can you use more polygons or other two-dimensional shapes in addition to the quadrilaterals? Yes, so long as you have made use of the five quadrilaterals as well. Be creative and have fun!
Your design must include a KEY. In your key you must list the five quadrilaterals (trapezoid, rectangle, rhombus, square, and parallelogram) and any other two-dimensional forms you decided to use. You must list all the properties of those shapes in your key. Your key must also tell us what the shape represents in your sketch.
For example, if you designed a classroom and you drew squares for desks, your key would have a square and beside it you would write “desks.” If you have more than one square in your sketch use the right colors to tell us what every item is in your design through your key. You could say this is your HOUSE KEY . . .
While your sketchbook is a fine place to play with ideas and sharpen your imaginative vision, the final sketch on graph paper or in a computer program must use precision. Using the graph paper grid system and a straight edge will help you in this challenge. The key should also show thought and planning in arranging the shapes, differentiating with shading or color, and taking care with printing, cursive, or clear font choices to communicate with people who will see your design.
Autograph your work with pride, and include the date so you can look back later and know where you were in your artist's, architect's, or geometer's journey when you created this artifact.
You will create this space in your sketchbook, with a final design on graph paper, artist's paper using a straight edge, Google Drawing, or Scratch. If there is another program or method you would like to use for a final product, let me know.
The challenge: You must include all 5 of the classifications of quadrilaterals we have been exploring in M3 Math in your final design: parallelogram, rectangle, rhombus, square, and trapezoid.
Can you use more polygons or other two-dimensional shapes in addition to the quadrilaterals? Yes, so long as you have made use of the five quadrilaterals as well. Be creative and have fun!
Your design must include a KEY. In your key you must list the five quadrilaterals (trapezoid, rectangle, rhombus, square, and parallelogram) and any other two-dimensional forms you decided to use. You must list all the properties of those shapes in your key. Your key must also tell us what the shape represents in your sketch.
For example, if you designed a classroom and you drew squares for desks, your key would have a square and beside it you would write “desks.” If you have more than one square in your sketch use the right colors to tell us what every item is in your design through your key. You could say this is your HOUSE KEY . . .
While your sketchbook is a fine place to play with ideas and sharpen your imaginative vision, the final sketch on graph paper or in a computer program must use precision. Using the graph paper grid system and a straight edge will help you in this challenge. The key should also show thought and planning in arranging the shapes, differentiating with shading or color, and taking care with printing, cursive, or clear font choices to communicate with people who will see your design.
Autograph your work with pride, and include the date so you can look back later and know where you were in your artist's, architect's, or geometer's journey when you created this artifact.
The Hunt for Shapes
Challenge designed by Ms. Jeanne Blackburn
Architects use a variety of shapes in designing buildings, combining various shapes, modifying a single shape, repeating a single shape, or repeating combinations of shapes.
We can analyze buildings through the lens of geometric shapes to identify the variety and patterns of shapes designers used.
We can analyze buildings through the lens of geometric shapes to identify the variety and patterns of shapes designers used.
Using an image of a structure from an architectural era of your choice, identify three or more shapes or combinations of shapes using your sketching skills or Google Drawing tools to communicate.
http://rsd2-alert-durden-connections.weebly.com/sketchbook-exercises.html
http://rsd2-alert-durden-reading-room.weebly.com/the-designed-world.html
http://rsd2-alert-durden-connections.weebly.com/sketchbook-exercises.html
http://rsd2-alert-durden-reading-room.weebly.com/the-designed-world.html
Basilica Hunt for Shapes Example: Google Drawing File
Art Deco Hunt for Shapes: Google Drawing Example
Math Reasoning and Communication Challenge
Do You See What I See? Translations, Rotations, and Reflections
Henry Ernest Dudeney's Mathematical Play:
Hinged Polygons Transform into Other Figures for Ease of Measurement and Comparison
Adam Hillman, Qtip-top Shape, 2017
Adam Hillman, Clip Cubes, 2017
What do we mean when we talk about dimensions?
• The 0-dimensional point (no length, width, or height) is a shadow of a line…
• The 1-dimensional line (length or width or height) is a shadow of a square…
• The 2-dimensional square (length and width, or width and height) is a shadow of a cube…
• The 3-dimensional cube (length and width and height) is the shadow of an (unfolded) hypercube.
“Shadows of Higher Dimensions” by Paul Micarelli
Polyhedra: 3D Multifaceted forms Inside Out!
3D Cardstock Net and Model PDFs
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
The Geometry of Architecture:
Even Contemporary Complex Structures Can Be Viewed
As a Collection of Smaller Simple Prisms or Curved Forms
On the Grid
René Descartes and Cartesian Coordinates
"But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science."--René Descartes, La Géometrié, 1637.
In the Mind's Eye: Spatial Visualization
Continuous and Discrete Values in Graphing
Graphing Stories
Two types of numerical data scientists, engineers, and leaders encounter when creating graphs to communicate trends are continuous and discrete.
Continuous data may have a range of values, and include fractions and decimals:
Continuous data is often a measured quantitative value, and is displayed as a continuous line segment on a graph or chart, with the axis labeled in units (centimeters, kilograms, liters, minutes).
|
Discrete data can only take certain values:
Discrete data is often a counted quantitative value, using whole numbers, and is displayed as a series of steps or separated line segments on a graph or chart. The axis recording the discrete value will often be labeled in whole numbers.
|
The Kids Should See This: Tiling the Plane with Pentagons
Egypt: The Golden Ratio,
Right Triangles, and Pyramid Volumes
Babylonian Right Triangle Explorations
Greece: The School of Pythagoras
Greece: Plato, Platonic Solids,
and "Discovering" Math
Euler's Rule
The ancient Greeks discovered and defined five regular shapes, the Platonic solids in 300 BCE. Two thousand years later, Swiss mathematician Leonhard Euler discovered and defined a simple rule:
the number of faces, plus the number of vertices (corners), equals the number of edges plus 2.
f + v = e + 2
f + v - e = 2
the number of faces, plus the number of vertices (corners), equals the number of edges plus 2.
f + v = e + 2
f + v - e = 2
Archimedes and Approximations
Topology: Seven Bridges of Königsberg
How Physical Objects Are Connected
The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and to the two mainland portions of the city, by seven bridges. The problem was to devise a walk* through the city that would cross each of those bridges once and only once.
* No swimming, boating, ice-skating, or ballooning can be part of the solution.
* No swimming, boating, ice-skating, or ballooning can be part of the solution.
Leonhard Euler
4-Dimensional Geometry
Impossible Shapes
The Penrose-Reutersvärd triangle is an impossible object. It was first created by the Swedish artist Oscar Reutersvärd in 1934. The English mathematician son Roger Penrose independently devised and popularized it in the 1950s. It is featured prominently in the works of Netherlands artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it.