1 / 51

Big Ideas in Geometry: Classifying and Sorting Geometric Figures

This text explores the big ideas in geometry, including classifying, distinguishing, and sorting geometric figures based on their properties. It also discusses the use of formulas and mathematical statements to describe and solve problems related to geometric figures.

garver
Download Presentation

Big Ideas in Geometry: Classifying and Sorting Geometric Figures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geometry

  2. Big Ideas in Geometry • 2- and 3-dimensional figures can be classified and distinguished by their properties or attributes • 2-dimensional figures are viewed in the rectangular coordinate plane; transformations of 2-dimensional figures within the plane may produce figures that are similar and/or congruent to the original figure

  3. (π r2 h) = 3v Big Ideas in Geometry • Mathematical statements can be used to describe geometric relationships within and between geometric figures • Formulas that describe attributes of geometric figures can be utilized for indirect measurements and problem solving

  4. Sorting Geometric Figures • What properties or attributes could be used to sort these figures?

  5. Geometric Figures • What do you see? • What disadvantages are there for students in learning about attributes of figures when their experiences are in a pictorial mode rather than with models?

  6. In the 2-D Rectangular Plane

  7. Objectives into Big Ideas • Decide within your groups which big idea relates to each objective in the Standard Course of Study • Write the grade level and objective on the charts under the appropriate big idea

  8. Triangle Thinking • On your handout draw 5 different triangles • After the first triangle, each new triangle must be different in some way from those already drawn • Number the triangles and write why you think each is different

  9. Triangle Thinking • What are the thought processes you used in completing this activity? • What conversations and types of activities that promote this thinking among out students?

  10. Geometric Thinking • With a partner examine the student work and respond to the guiding questions • What do you think helps students be successful in geometry?

  11. Habits of Mind Framework Driscoll’s framework describes components of geometric thinking • Seeking relationships • Generalizing geometric ideas • Checking transformation effects • Balancing exploration with deduction

  12. Seeking Relationships Actively looking for and applying relationships, within and between geometric figures, in one, two, and three dimensions • How are these figures are alike? • How are these figures different? • What would I have to do to this object to make it like that object? • Have I found all the ones that fit this description? • What if I think about this relationship in a different dimension?

  13. Generalizing Geometric Ideas Wanting to understand and describe the “always” and the “every” related to geometric phenomena • Does this happen in every case? • Why would this happen in every case? • Can I think of examples when this is not true? • Would this apply in other dimensions?

  14. Checking Transformation Effects Noticing and analyzing effects when transformations act on geometric objects • How did that get from here to there? • What changes? Why? • What stays the same? Why?

  15. Balancing Exploration& Deduction Trying various ways to approach a problem and regularly stepping back to take stock • What happens if I… draw a picture, add to/take apart this picture, work backwards from the ending place, etc.? • What did that action tell me?

  16. Try the Circle Task handout with a partner Try the Quadrilateral Task handout with a partner Circles and Quadrilaterals Make lists of all the attributes of triangles, circles, and quadrilaterals What habits of mind did you use in completing the tasks and lists?

  17. Shapes & Properties

  18. Thinking about Formulas • Formulas that describe attributes of geometric figures can be utilized for indirect measurements and problem solving • Two dimensional figures are viewed in the rectangular coordinate plane; transformations of two dimensional figures within the plane may produce figures that are similar and/or congruent to the original figure

  19. Shrinking a Circle • Fold the piece of string in half and place your finger on the 2 loose ends on the mark at the center of the paper • Draw a circle about that fixed point • Fold the string in half again and draw a second circle with the same procedure, around the same fixed point

  20. Shrinking a Circle • How do the radii of the two circles relate? • How do the circumferences of the two circles relate? Justify your answer with a formula. • How do the areas of the two circles related? Justify your answer with a formula. • Are the circles similar, congruent, both, or neither? How do you know?

  21. Follow the directions on the handout to enlarge this drawing. Enlarging a Drawing 21

  22. Enlarging a Drawing Which features of the drawing stayed the same? Which features of the drawing changed? 22

  23. Comparing New to Old What do you notice about the lengths of the sides? What do you notice about the angle measures? How do the areas compare? What do you notice about the shapes of the figures? 23

  24. Under the original drawing of the figure, write how you know when figures are similar, when they are congruent, and when they are neither Similarity and Congruence 24

  25. Two figures are similar if all of the corresponding angles are congruent and corresponding sides are proportional Two figures are congruent if the all the corresponding angles are congruent and the ratio of the corresponding sides is 1:1 Van de Walle (2006, p. 199) Similarity and Congruence 25

  26. Similar Triangles Activity Arrange the smaller triangles to form two similar triangles. How do you know they are similar? Which of the smaller triangles are congruent? How do you know? Which smaller triangles are similar? How do you know? 26

  27. Tools for investigating What are the types of tools we can use to help students engage in explorations of transformations? • Paper folding, cutting • Miras or mirrors • Patty paper or wax paper • Computer software & applets

  28. Similar Triangles Activity What types of geometric thinking do students need to engage in to solve this task? How could you modify or extend this task to incorporate more geometric thinking? 28

  29. Blue Box Task • Individually complete the Blue Box worksheet then compare your answers with your table Dimensions: Height = 12 cm Width = 10 cm Depth = 4 cm Radius of cut out circle = 2 cm

  30. What do you think? • Identify the mathematics that is needed to solve this task • What are the volume and surface area? • What are possible nets? • What additional questions did you write?

  31. Habits of Mind • How might giving students tasks like this promote “habits of mind”? List everything you know that could help you determine the volume and surface area of the box and draw its net • Is there a relationship between success in geometry and habits of mind?

  32. Transformations Match Use the cards to match the term and its definition Write the words for your matches on the recording sheet As the terms are discussed add personal notes to make clearer the vocabulary 32

  33. What are transformations? Transformations are about the movement of figures If all the points of a geometric figure are moved according to the same rules, the figure is transformed and a new figure is created Transformations are 1-to-1 relationships from points in a pre-image to points in the image 33

  34. Pre-image: the points of the original figure Image: the points of the new figure One-to-one implies that for every point in the image there is exactly one point in the pre-image Images and Pre-images 34

  35. Rigid Transformations Translation: points in a figure are slid across a translation vector the same distance along parallel paths Rotation: turns a set of points about the center of rotation an identical number of degrees Reflection: points in a figure are reflected across a line (mirror image) 35

  36. Non-rigid Transformation Dilation Every point of a figure is scaled using a scale factor from the center The scale factor reduces or enlarges a figure The shape is preserved, but not the size 36

  37. Type of Transformations • What types of transformations were involved in… • The Shrinking Circle? • Enlarging a Drawing? • Similar Triangles Task?

  38. Movin’ On Reflect the shape over the x-axis What will the new coordinates be? 38

  39. Movin’ On Now reflect the new shape over the y-axis What will the coordinates be? 39

  40. Movin’ On How far away from the original point C is the final location for C? Explain your reasoning 40

  41. Movin’ On Rotate the pre-image (original shape) 180 degrees Dilate the final shape by a factor of 2 41

  42. Movin’ On What is the perimeter of the new shape? the area? 42

  43. Levels of Geometric Thought Pierre and Dina van Hiele were secondary mathematics teachers concerned with their students’ struggle to reach high levels of thinking in geometry • Read through the handout describing van Hiele Levels of Geometric Thought

  44. van Hiele Levels of Geometric Thought* • Level 0: Visualization/Recognition • Level 1: Descriptive/Analytic • Level 2: Abstract/Relational • Level 3: Formal Deduction • Level 4: Rigor *Sequential Levels—These are not dependent on age; rather they are dependent on experiences

  45. Sorting Geometric Figures • Think about attributes, defining and sorting figures through the lens of van Hiele

  46. Reflection • What are instructional implications for helping students acquire knowledge to move them up the levels? • What experiences do students need to understand properties and definitions in geometry? • What are the roles of language and experience versus memorization of definitions?

  47. Reflecting on MS Geometry • What ideas from this professional development can help you better understand your students’ geometric thinking? • In what ways do the objectives in the geometry strand for your grade relate to objectives in the measurement strand?

  48. DPI Mathematics Staff Partners for Mathematics Learning is a Mathematics-Science Partnership Project funded by the NC Department of Public Instruction. Permission is granted for the use of these materials in professional development in North Carolina Partner school districts. Partners for Mathematics Learning

  49. PML Consultants

  50. PML Writers Please give appropriate credit to the Partners for Mathematics Learning project when using these materials. Permission is granted for their use in professional development in North Carolina Partner school districts. Jeane Joyner, Project Director

More Related