Teaching Geometry: To See it Like a Mathematician Presenters: Denise Johnson LaShondia McNeal, Ed.D. April 27-28, 2012

Why Should we learn Geometry? “Those who complain about its impracticality ignore that Math teaches the mind how to think.” David Eggenshwiler

Objectives: Broaden Instructional Strategies as it relates to Geometry (balance, symmetry, Area, Volume) 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 Identify key elements of success skills in passing the GED and vocational tests Introduce Technology Tools for GED Math Teachers

Visual forms in the practice of Math

Teaching what we see. We need to show students what we are attending to Find ways to direct (and shift) their attention to track effective visual thinking around a diagram

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 Whiteley

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

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

Materials: 1.A supply of sticks in 4 colours. This hexastix uses 18 sticks of each colour, but you need a few extras for breakage and stuff. 2.8 small elastics. 3.A poker thingy, like a pointed skewer. 4.(optional, not shown) white glue, water, ziplock bag, human powered centrifuge.

Steps: 1.Fasten 7 orange sticks together with an elastic at each end. I double my elastics so they stay on, but aren’t tight. 2.Now add four blue sticks 3.The blue sticks go between the orange ones, separating orange into sets of 2, 3, 2. Fasten the blue sticks together with doubled elastics at each end.

Steps: 4.Add four purple sticks 5.The blue sticks separated the orange ones into a 2,3,2 configuration. The purple sticks will do that too, but in a different direction. The old 2,3,2 separation is shown with blue lines, and the new 2,3,2 separation (by purple) is shown with black lines. 6.The purple sticks also go between the blue sticks, separating them into 2,2 7.Here it is again from the ends of orange, and the ends of blue. 8.Put an elastic around each end of purple.

Steps: 9.THIS IS THE KEY STEP. Look down the end of orange and identify the 2,3,2 separations. One of them has blue sticks between the sets, another has purple sticks, and there is one more, shown in white, with no sticks separating the sets of 2,3,2. There is one more way to split the orange sticks into sets of 2,3,2, and that is what the pink sticks do.

Steps: 10.The pink sticks need to alternate properly with blues, and purples too, so there is a trick to inserting them. Rotate the above configuration away from you, so that you now hold orange vertically, looking down the future pink direction. Put your thumbs on the ends of the blue and purple sticks nearest you, and your middle fingers on the bottom end of orange. Pull orange toward you, and push down on the blue and purple sticks. This should reveal six-sided holes for inserting pink!

Steps: 11.Insert 7 pink sticks into the holes. It does not matter which holes you choose, and it’s OK that some holes are open on one side. Make sure that all pink sticks are parallel to each other. 12.The Put elastics around the ends of pink.

Steps: 13.Complete the blue and purple colours by adding three of each stick 14.To add a blue stick, tuck it under the elastic, and push it into a hole. As long as it is parallel to the other blues, it is in a valid hole. You need to lift the elastic on the other end with a skewer to tuck the stick under that one too.

Steps: 15.Here is the finished blue. Some blue sticks are not really in “holes”, but that’s OK because they are held in place by the elastics.

Steps: 16.After you have done the same thing with purple, you have constructed a Level 1 Hexastix. Congratulations!

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

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

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

USING TECHNOLOGY WITH MATH

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

Parallelograms Quadrilaterals are four-sided polygons Parallelogram: is a quadrilateral with both pairs of opposite sides parallel.

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 parallelogram are supplementary. A DC 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°

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 C D

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?

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 DC B If AD  BC and AB  DC, then ABCD is a parallelogram If <A  <C and <B  <D, then ABCD is a parallelogram

Tests for Parallelograms 2 Theorem 6.7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram A DC B Theorem 6.8: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

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)

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

Rectangles A 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 parallelogrma are congruent then the parallelogram is a rectangle.

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 angles

Squares and Rhombi A 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 C

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 d 1 and d 2 units, then A = ½ d 1 d 2.

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.

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. C

Trapezoids (2) C A C D BAB = base CD = base AC = leg BD = leg AB  CD AC & BD are non parallel <A & <B = pair of base angles <C & <D = pair of base angles

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.

Trapezoids (4) C 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.

Area of Trapezoids C A C D B Area of a trapezoid : If a trapezoid has an area of A square units, bases of b 1 and b 2 units and height of h units, then A = ½(b 1 + b 2 )h. h

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References Common Core State Standards Initiative, Preparing America’s students for College and Career Drivers of Persistence, Texas College and Career Readiness, Khan Academy World of Teaching

Contact Information Presenters: Denise Johnson LaShondia McNeal, Ed.D.