Teaching Geometry: To See it Like a Mathematician

Presenters: Denise Johnson & LaShondia McNeal, Ed.D.

April 27-28, 20121

Objectives:• To recognize that visual forms in the practice of math have the

potential to improve student learning

• To discuss building of the five strands of mathematical proficiency as an intervention in teaching concepts of geometry and model various activities teachers can implement in their classrooms

• Broaden Instructional Strategies as it relates to Geometry (balance, symmetry, Area, Volume)

• Identify key elements of success skills in passing the GED and vocational tests

• Introduce Technology Tools for GED Math Teachers

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Visual forms in the practice of Math

“Experience and the current literature show that at least four of the intelligences play an important role in the practice of mathematics. In particular, some mathematicians and some students rely essentially of the spatial/visual and kinesthetic intelligences.”

By Walter Whiteley7

Teaching to see in mathematics.

Teach these skills:• A first step is an evolving awareness of how visuals are or could be

used, and an explicit encouragement of their uses.

• A second step is paying attention to when students don’t see what we see, seeking those occasions out and exploring them.

• A third step is developing and sharing diverse examples, and diverse ways to see individual examples, along with tools which let students experience what we are seeing.

“Too often, we do not teach the skills, or even explicitly model the skills in a way that the students can observe and imitate.”

Walter Whiteley8

Building Math Proficiency 

Mathematical proficiency is developed through five interwoven and interdependent strands: • conceptual understanding• procedural fluency• strategic competence • adaptive reasoning • productive disposition

“Mathematical proficiency cannot be achieved by focusing on only one or two strands”

Kilpatrick, et.al.9

Building a Visual Guide to Math

Let’s make a HEXASTIX “a geometric form that deals with patterns and relationships derived from classical ideals of balance and symmetry” George Hart

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Mathematics as a Way of Knowing

• Essential to becoming a productive citizen

• Informs decision making

• Must be considered as a source of cross-disciplinary knowledge

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The New GED Standards

• The new GED is scheduled to be released in January 2014

• The New GED Mathematics test is being designed using the Common Core State Standards (CCSS) for Mathematics

• The CCSS have been adopted by 48 states and providences. However, Texas has not adopted the CCSS

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Texas Career and College Readiness Standards

• Currently Texas is working under the Texas Career and College Readiness Standards (TCCRS)

• When comparing the TCCRS to the CCSS, the same material is covered but TCCRS appear to have more rigorous performance indicators

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Technology in the Classroom

• Technology such as calculators and computers is essential for teaching, learning, and doing math

• Enable students to collect, organize, and analyze data

• Enables students to view dynamic images of mathematical models

• Enables students to perform computations with accuracy and efficiency

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Parallelograms

Quadrilaterals are four-sided polygons

Parallelogram: is a quadrilateral with both pairs of opposite sides parallel.

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Parallelograms (2)

Theorem 6.1 : Opposite sides of a parallelograms are congruent

Theorem 6.2: Opposite angles of a parallelogram are congruent

Theorem 6.3: Consecutive angles in a parallelogramare supplementary.

A

D C

B

AD BC and AB DC

<A <C and <B <D

m<A+m<B = 180°m <B+m<C = 180° m<C+m<D = 180°m<D+m<A = 180°

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Parallelograms (3)

Diagonals of a figure: Segments that connect any to vertices of a polygon

Theorem 6.4: The diagonals of a parallelogram bisect each other.

A B

CD

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Parallelograms (4)• Draw a parallelogram : ABCD on a piece of

construction paper.

• Cut the parallelogram.

• Fold the paper and make a crease from A to C and from B to D.

• Fold the paper so A lies on C. What do you observe?

• Fold the paper so B lies on D. What do you observe? • What theorem is confirmed by these Observations?

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Tests for Parallelograms

Theorem 6.5 :If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 6.6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

A

D C

BIf AD BC and AB DC, then ABCD is a parallelogram

If <A <C and <B <D, then ABCD is a parallelogram 19

Tests for Parallelograms 2

Theorem 6.7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

A

D C

B

Theorem 6.8: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

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A quadrilateral is a parallelogram if...

Diagonals bisect each other. (Theorem 6.7)

A pair of opposite sides is both parallel and congruent. (Theorem 6.8)

Both pairs of opposite sides are congruent. (Theorem 6.5)

Both pairs of opposite angles are congruent. (Theorem 6.6)

Both pairs of opposite sides are parallel. (Definition)

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Area of a parallelogram

If a parallelogram has an area of A square units, a base of b units and a height of h units, then A = bh. (Do example 1 p. 530)

The area of a region is the sum of the areas of all its non-overlapping parts. (Do example 3 p. 531)

b

h

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RectanglesA rectangle is a quadrilateral with four right angles.

Theorem 6-9 : If a parallelogram is a rectangle, then its diagonals are congruent.

Opp. angles in rectangles are congruent (they are right angles) therefore rectangles are parallelograms with all their properties.

Theorem 6-10 : If the diagonals of a parallelogrmaare congruent then the parallelogram is a rectangle.23

Rectangles (2)If a quadrilateral is a rectangle, then the following properties hold true:

• Opp. Sides are congruent and parallel

• Opp. Angles are congruent

• Consecutive angles are supplementary

• Diagonals are congruent and bisect each other

• All four angles are right angles24

Squares and RhombiA rhombus is a quadrilateral with four congruent

sides. Since opp. sides are , a rhombus is a parallelogram with all its properties.

Special facts about rhombi

Theorem 6.11: The diagonals of a rhombus are perpendicular.

Theorem 6.12: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Theorem 6.13: Each diagonal of a rhombus bisects a pair of opp. angles

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Squares and Rhombi(2)

If a quadrilateral is both, a rhombus and a rectangle, is a square

If a rhombus has an area of A square units and diagonals of d1 and d2 units, then A = ½ d1d2.

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Area of a triangle:

If a triangle has an area of A square units a base of b units and corresponding height of h units, then A = ½bh.

h

b

Congruent figures have equal areas.

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Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The nonparallel sides are called legs.At each side of a base there is a pair of base angles.

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Trapezoids (2)

A

C D

B AB = baseCD = baseAC = legBD = leg

AB CD

AC & BD are non parallel

<A & <B = pair of base angles<C & <D = pair of base angles

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Trapezoids (3)

Isosceles trapezoid: A trapezoid with congruent legs.Theorem 6-14: Both pairs of base angles of an isosceles trapezoid are congruent.Theorem 6-15: The diagonals of an isosceles trapezoid are congruent.

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Trapezoids (4)

A

C D

B

The median of a trapezoid is the segment that joints the midpoints of the legs (PQ).

QP

Theorem 6-16: The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of its bases.

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Area of Trapezoids

A

C D

B

Area of a trapezoid: If a trapezoid has an area of A square units, bases of b1 and b2 units and height of h units, then A = ½(b1 + b2 )h.

h

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Technology for Teacher Preparation

• The internet contain thousands of Web sites devoted to mathematics– History of math– Basic operations– Trigonometry, Calculus,– Imaginary numbers and beyond

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References

• Common Core State Standards Initiative, Preparing America’s students for College and Career Drivers of Persistence, http://www.corestandards.org/

• Texas College and Career Readiness, http://txccrs.org/index.htm

• Khan Academy http://www.khanacademy.org/• World of Teaching

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http://www.worldofteaching.com