Geometric and Spatial Reasoning

1. Geometric and SpatialReasoning

2. Symbolic and Algebraic Reasoning: Review & Sharing Geometric Reasoning Can a Picture Prove? Spatial Reasoning Baseline Assessment Today’s Agenda

3. Symbolic and Algebraic Reasoning Review Symbolic and Algebraic Reasoning is interwoven throughout the MN math standards, but specifically, students should: “Justify steps in generating equivalent expressions by identifying the properties used...” (Minnesota Math Standard 9.2.3.7, Grade 9-11, MN Dept of Ed, 2007)

4. Reasoning Review What did you learn most from the Symbolic and Algebraic Reasoning baseline and summative assessments? Describe at least two ways that you helped your students use Symbolic and Algebraic reasoning. Describe one classroom situation where you saw a student exhibit growth in Symbolic or Algebraic reasoning.

5. Geometric Reasoning

6. Geometric Reasoning “Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations.” (Minnesota Math Standard 9.3.2.4, Grade 9-11, MN Dept of Ed, 2007)

7. Van Hiele Levels of Geometric Understanding: • Levels provide a way to characterize student understanding • Level 0: Visualization • students are able to recognize and name figures based on visual characteristics • students can make measurements • groupings are made based on appearances and not necessarily on properties • Example: Students see squares turned on their corner as “diamonds”.

8. Van Hiele levels: • Level 1: Analysis • students can consider all shapes within a class rather than a specific shape • focus on properties • Example: Students see rectangles as having right angles and parallel sides. But they may insist that a square is not a rectangle.

9. Van Hiele levels: • Level 2: Informal Deduction • students can develop relationships between and among properties • proofs arise here, informally • focus on relationships among properties of geometric objects • Students can recognize relationships between types of shapes. • Example: They can recognize that all squares are rectangle, but not all rectangles are squares.

10. Van Hiele levels: • Level 3: Deduction • students can use logic to establish conjectures made at Level 2 • student is able to work with abstract statements about geometric properties and make conclusions based on logic rather than intuition • focus on deductive axiomatic systems for geometry such as Euclidean • Example: Students can prove that the base angles in an isosceles triangle are congruent.

11. Van Hiele levels: • Level 4: Rigor • students appreciate the distinctions and relationships between different axiomatic systems • focus on comparisons and contrasts among different axiomatic systems of geometry • Example: Students understand the parallel postulate and its meaning in the Euclidean system compared to its meaning in the spherical geometry system.

12. Using Geometry Logic • Can you determine the shape that satisfies all of the following clues? • It is a closed figure with straight sides. • It has only two diagonals. • Its diagonals are perpendicular. • Its diagonals are not congruent. • It has a diagonal that lies on a line of symmetry. • It has a diagonal that bisects the angles it joins. • It has a diagonal that bisects the other diagonal. • It has a diagonal that does not bisect the other diagonal. • It has no parallel sides. • It has two pairs of consecutive congruent sides.

13. What Van Hiele Level? • Discuss what Van Hiele level you think the preceding geometry logic problem was. • What grade level for students? • How does this problem compare to the logic puzzle from the Math Reasoning Session, where you were determining the construction sequence for city buildings?

14. Developing Reasoning Via Open-Ended Problems • Open-ended problems encourage reasoning from students (NCTM Book on Open-Ended Problems) • Students use geometric reasoning in making observations about a figure • Students then use reasoning in formulating arguments to support their observations • Their observations can indicate their level of understanding

15. Open-Ended Problem In the figure below, BF and CD are angle bisectors of the isosceles triangle ABC. CF is the angle bisector of exterior angle ACH. Step 1: Find as many relations as you can. Step 2: What van Hiele level is required for each?

16. A Construction Problem You are given two intersecting straight lines and a point P marked on one of them, as in the figure below. Show how to construct, using straightedge and compass, a circle that is tangent to both lines and that has the point P as its point of tangency.

17. Can a Picture Prove Something? • Discuss your answer to the question above with your colleagues. • OK, why or why not? • Any examples to support your claim?

18. Does a picture prove? • Debate exists over this in the mathematics community. • M. Giaquinto (noted math philosopher) noted that there is a distinction between discovery and demonstration. • Discovery (of new theorems, facts, etc.) is often very visual for the expert. • Demonstration (proof) can only be accomplished visually if it’s clear the order of the statements being expressed.

19. A Mathematical Fact • Fact: 1/4 + 1/16 + 1/64 + … = 1/3 • Normally justified in a calculus class using the geometric series formula , since the series has the form a + ar + ar2 + … where a is ¼ and r is also ¼. • How else to show this?

20. Its Proof?

21. Visual Proof that 64 = 65?

22. Visual proof necessary conditions (Hanna & Sidoli, 2007, p. 75): • Reliability – that the underlying means of arriving at the proof are reliable and that the result is unvarying with each inspection • Consistency – That the means and end of the proof are consistent with other known facts, beliefs, and proofs. • Repeatability – That the proof may be confirmed by or demonstrated to others. • Discuss the conditions above with your colleagues. Do you agree?

23. Let’s use the prior discussion about conditions for visual proofs to revisit an old friend…

24. The Pythagorean Theorem! • The sum of the squares of the lengths of the legs on a right triangle is equal to the square of the length of the hypotenuse. That is, if a is the length of one leg, and b is the length of the other leg, and c is the length of the hypotenuse, then a2 + b2 = c2. • It is the ultimate marriage between geometry and algebra! • Let’s look at some possibilities for proving it to determine their efficacy…

25. Pythagorean Theorem Proof #1 Is this a correct proof of the Pythagorean Theorem? Why?

26. Pythagorean Theorem Proof #1 More (algebraic) detail added here. The area of the big square is c2. This equals the area of the four right triangles plus the area of the smaller inside square. Algebraically:

27. Pythagorean Theorem Proof #2 Is this a correct proof of the Pythagorean Theorem? Why?

28. Pythagorean Theorem Proof #3 Is this a correct proof of the Pythagorean Theorem? Why?

29. Spatial Reasoning

30. Spatial Reasoning • We have used spatial reasoning in several activities so far but these have been all geometric. • Q: What is an activity which uses spatial reasoning but is not so dependent upon standard geometry? • A: Isometric views!

31. Isometric Views • What is an isometric view? • It is a 2-d picture which portrays a 3-d shape. • The standard isometric view shows three faces of the 3-d object, top, right side and left side. • Example: Cube

32. More with Isometric Views • Cubes can be arranged and stacked to form 3-d landscapes • Example:

33. Spatial Reasoning with Isometric Views There are four representations at work here: • Physical manipulatives: centimeter blocks • Mat plans: 2-d blueprints • Top, bottom and side views of the 3-d shape • Isometric view Students use spatial reasoning in shifting among these representations. • Today we will focus mostly on shifting from isometric views to mat plans

34. Isometric View to Mat Plan - I • The mat plan for the given isometric view is below. • The value in each box refers to the number of cubes stacked on that square • Notice that the values are arranged in an “L”, like the iso view Front Front

35. Isometric View to Mat Plan - I • Create the mat plan for the given isometric view • Discuss your plan with your colleagues. • Can there be different mat plans for one isometric view? Why? Front Front

36. Extending Spatial Reasoning • The book pictured uses Cuisenaire Rods to encourage spatial reasoning • Three views are given, students then construct the shape with the rods • Position, color and length are all used as clues

37. BaselineAssessment

38. Baseline Assessment – Item 1 What is the shape described below? Clue 1: It is a closed figure with 4 straight sides. Clue 2: It has 2 long sides and 2 short sides. Clue 3: The 2 long sides are the same length. Clue 4: The 2 short sides are the same length. Clue 5: One of the angles is larger than one of the other angles. Clue 6: Two of the angles are the same size. Clue 7: The other two angles are the same size. Clue 8: The 2 long sides are parallel. Clue 9: The 2 short sides are parallel.

39. Baseline Assessment - Item 2 Mat Plan Front Front

40. Baseline Assessment – Item 3 Why is this proof of the Pythagorean Theorem incorrect?

41. Baseline Assessment – Item 3 Why is this proof of the Pythagorean Theorem incorrect?