Transcript Teaching Geometry-dj

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 crossdisciplinary knowledge
www.brainybetty.com
20
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
www.brainybetty.com
21
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
www.brainybetty.com
22
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
D
B
C
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
A
D
B
C
Theorem 6.4: The diagonals
of a parallelogram bisect
each other.
Parallelograms (4)
•Draw a parallelogram :
construction paper.
ABCD on a piece of
•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.
If AD  BC and AB  DC,
then ABCD is a parallelogram
A
D
Theorem 6.6: If both pairs of opposite
angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
B
C
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
D
B
C
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...
Both pairs of opposite sides
are parallel. (Definition)
Both pairs of opposite sides
are congruent. (Theorem 6.5)
Both pairs of opposite angles are
congruent. (Theorem 6.6)
Diagonals bisect each other. (Theorem 6.7)
A pair of opposite sides is both parallel and
congruent. (Theorem 6.8)
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)
h
b
The area of a region is the sum of the areas of
all its non-overlapping parts. (Do example 3 p. 531)
Rectangles
A rectangle is a quadrilateral with four
right angles.
Opp. angles in rectangles are congruent
(they are right angles) therefore rectangles
are parallelograms with all their properties.
Theorem 6-9 : If a parallelogram is a rectangle,
then its diagonals are congruent.
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
Squares and
Rhombi(2)
If a rhombus has an area of A square
units and diagonals of d1 and d2
units, then A = ½ d1d2.
If a quadrilateral is both, a rhombus
and a rectangle, is a square
h
Area of a
triangle:
b
If a triangle has an area of A square units
a base of b units and corresponding
height of h units, then A = ½bh.
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.
Trapezoids (2)
A
AB  CD
C
AC & BD
are non
parallel
B
D
AB = base
CD = base
AC = leg
BD = leg
<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)
The median of a trapezoid is the segment
that joints the midpoints of the legs (PQ).
A
P
C
B
Q
D
Theorem 6-16: The median of a trapezoid is
parallel to the bases, and its measure is onehalf the sum of the measures of its bases.
Area of Trapezoids
B
A
h
C
D
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.
This powerpoint was kindly donated to
www.worldofteaching.com
http://www.worldofteaching.com is home to over a
thousand powerpoints submitted by teachers. This is a
completely free site and requires no registration. Please
visit and I hope it will help in your teaching.
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
KHAN Academy
http://www.khanacademy.org/
http://www.khanacademy.org/
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 http://www.worldofteaching.com
www.brainybetty.com
46
Contact Information
• Presenters: Denise Johnson
LaShondia McNeal, Ed.D.
www.brainybetty.com
47