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Cube-n-ometry Unit

Cube-n-ometry Unit. Inspired by: MathScience Innovation Center (Permission Granted to Modify) Modified for: Wake County’s 7 th Grade Math Classes. Part 1: Questions to Ponder. As you view the next few slides, ask yourself the following questions: What can be seen in the pictures?

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Cube-n-ometry Unit

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  1. Cube-n-ometry Unit Inspired by: MathScience Innovation Center (Permission Granted to Modify) Modified for: Wake County’s 7th Grade Math Classes

  2. Part 1: Questions to Ponder • As you view the next few slides, ask yourself the following questions: • What can be seen in the pictures? • How were the pictures taken? • Where would you need to be to take the pictures? • Why are these pictures important to the specified professions?

  3. What would you take a picture of if you were an… • Architect? • Aerial view of Middle Creek High School • Buildings, land, streets, etc.

  4. What would you take a picture of if you were a… • Cartographer or map maker? • Aerial view of Raleigh, NC • Views of buildings, streets, roads, etc.

  5. What would you take a picture of if you were a… • Geologist? • Cross sectional view of Earth’s main layers

  6. What would you take a picture of if you were a… • Doctor? • X-rays of humans (surgeons and dentists) or even of animals (veterinarians)

  7. Questions to Answer • So, let’s answer the questions now: • What could be seen in the pictures? • How were the pictures taken? • Where would you need to be to take the pictures? • Why are these pictures important to the specified professions (map-maper, architect, scientist, doctor, etc)

  8. Introduction to Visualizing Figures • As you can see, POINTS OF VIEW are extremely important in all careers and professions. • In this unit – “Cube-n-ometry” – we’re going to: • Strengthen our ability to visualize three-dimensional objects, and • Explore methods for representing cubic structures.

  9. So…Let’s Explore Cubes • What makes a cube from the beginning… Square Cube • That is, what shape makes up a cube and how many are there in a cube? • A cube is made up of six (6) squares!

  10. Exploring Dimensions • What differences can you see in the two figures? Three-dimensional Two - dimensional Square Cube The major difference between two-dimensional and three-dimensional objects is that three-dimensional objects have depth/width.

  11. 3-Dimensional Versus 2-Dimensional Figures • Let’s explore solids and faces… Face A face is the TWO-DIMENSIONAL surface of a solid. A cube is made up of 6 identical faces, all of which are squares. Solid A solid is a THREE-DIMENSIONAL figure (such as a CUBE). THREE-DIMENSIONAL figures have length, width, and depth. Face

  12. Exploring Cubic Arrangements • Today we are going to explore possible arrangements of: • 3 Cubes, • 4 Cubes, and • Upon each task, we’ll create ISOMETRIC DRAWINGS of our 3D figures using isometric dot paper.

  13. Part 2: Isometric Drawings • What’s an ISOMETRIC DRAWING? • An isometric drawing is a perspective drawing of a three-dimensional figure. • Hence, isometric drawings show the 3RD dimension – “width” or “depth.” • ISOMETRIC DRAWINGS are drawn from the CORNER of a three-dimensional figure and can be placed onto isometric dot paper.

  14. Example of Isometric Dot Paper

  15. Arrangements of 3 Cubes • 3 Cubes are arranged so that at least one face of each cube must meet the face of another cube. • You cannot have a second story on a house, unless there is a first story below it- it needs support! • Rotations of the same arrangement do not count as different houses. If you have to PICK UP or DROP an arrangement, it is considered a different house.

  16. Arrangements of 3 Cubes • How many arrangements of 3 cubes are there? • There are FOUR arrangements of 3 cubes.

  17. Create Isometric Drawings of Your Cubic Arrangements • Using isometric dot paper, record each arrangement by creating an ISOMETRIC DRAWING of the cubic arrangement.

  18. Create Isometric Drawings of Your Cubic Arrangements

  19. Arrangements of 4 Cubes • New scenario! • Four cubes are arranged so that at least one face of each cube must meet the face of another cube. • How many arrangements of 4 cubes are possible?

  20. Arrangements of 4 Cubes • How many arrangements of 4 cubes are there? • A few examples are…

  21. Create Isometric Drawings of Your Cubic Arrangements

  22. How Many Arrangements of 4 Cubes Are There? • There are FIFTEEN arrangements of 4 cubes!

  23. Part 3: Orthogonal Drawings • What’s an ORTHOGONAL DRAWING? • An orthogonal drawing is a two-dimensional sketch of a three-dimensional figures’ face. • Orthogonal drawings are the FRONT, TOP, and SIDE views of a three-dimensional figure.

  24. Using 3D Figures to Generate Orthogonal Drawings • Views of a three-dimensional block figure can be broken down into three views: • Front View • Top View • Side View • These views are referred to as ORTHOGONAL DRAWINGS • Orthogonal drawings are two-dimensional representations of 3-D figures

  25. Construct Building 1 • Build this three dimensional block figure. • What does it look like from the: • Front?Top?Side? Sketch what you see.

  26. Watch the Views Light Up on Building 1

  27. Orthogonal Views of Building 1

  28. Construct Building 2 • If you were to snap a picture of this object what do you think it would look like from the: • Front? • Top? • Side?

  29. Orthogonal Views of Building 2

  30. Building 3 • Build this block figure and draw the front, top, and side views using your grid paper.

  31. Orthogonal Views of Building 3

  32. Build this block figure and draw the front, top, and side views using your grid paper. Building 4

  33. Orthogonal Views of Building 4

  34. Part 4: Building 3D Figures Using Orthogonal Drawings • PowerPoint Presentation – Building 3D figures using orthogonal views • Designing Spaces Section 3: “Seeing all Possibilities” • Construct Homes from Orthogonal Views • Drawing 3-D Views Packet • Figures 1-6 (Classroom Strategies) • Orthogonal to 3-D to Isometric • 4 Buildings

  35. Building 3D Figures Using Orthogonal Drawings • Now. . .let’s do the opposite! • Given the orthogonal drawings - front, top, and side views - can you build the three-dimensional block figures using the least number of cubes possible. • Hint: Once you’ve constructed the 3-D figure, see if one or more cubes can be removed to generate the same orthogonal drawings or views.

  36. Views of Building 1 • Given the views below, can you build the three-dimensional block figure? Front Top Side

  37. Picture of Building 1 Good Job!

  38. Views of Building 2 • Given the views below, can you build the three-dimensional block figure? Front Top Side

  39. Picture of Building 2

  40. Number of Cubesfor Building 2 (Option 1) • How many cubes are needed to construct this building? • 8 Cubes Front Top Side

  41. Least Number of Cubesfor Building 2 (Option 2) • What is the least number of cubes needed to construct this building? • Hint: Remove cubes to obtain the same orthogonal views. Front Top Side

  42. Picture of Building 2 withthe Least Number of Cubes • ONLY 6 Cubes are needed!

  43. Views of Building 3 • Given the views below, can you build the three-dimensional block figure?

  44. Picture of Building 3

  45. Least Number of Cubes for Building 3 • What is the least number of cubes needed to construct this building? • 10 cubes minus 1 unnecessary cube = 9 cubes!

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