MASTA Proof and Reasoning 9-12 Module Handout - Page 1 of 12
Geometric and Spatial Reasoning
Symbolic and Algebraic Reasoning Review
1.What did you learn most from the Symbolic and Algebraic Reasoning baseline and summative assessments?
2.Describe at least two ways that you helped your students use Symbolic and Algebraic reasoning.
3.Describe one classroom situation where you saw a student exhibit growth in Symbolic or Algebraic reasoning.
Geometric Reasoning
Van Hiele Levels
4.The levels provide ______
______
______.
5.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:
6.Level 1: Analysis
◦students can consider all shapes within a class rather than a specific shape
◦focus on properties
◦Example:
7.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:
8.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:
9.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:
10.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.
—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?
11.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?
Relations:Van Hiele Level:
12.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.
13.Can a Picture Prove Something?
Yes/No
Examples:
14.M. Giaquinto (noted math philosopher) noted that there is a distinction
between ______.
15.Fact: 1/4 + 1/16 + 1/64 + … = 1/3
Algebraic Proof:
Visual Proof?
16.Visual proof that 64 = 65:
17.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.
18.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.
Proof #1:
What other details are necessary for this to be a proof?
Proof #2:
Proof #3:
Spatial Reasoning
19.Build the following with the centimeter cubes at your table.
20.Four representations:
- 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 ______
______.
21.