Transcript of GEOMETRY IN THE UPPER ELEMENTARY AND MIDDLE GRADES Presented by Dr. Paul Gray Chief Curriculum...
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- GEOMETRY IN THE UPPER ELEMENTARY AND MIDDLE GRADES Presented by
Dr. Paul Gray Chief Curriculum Officer Cosenza & Associates,
LLC pgray73@sbcglobal.netpgray73@sbcglobal.net or pgray@uh.edu
pgray@uh.edu Session #32 National Council of Teachers of
Mathematics Regional Conference Richmond, Virginia November 13,
2014
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- WHY IS GEOMETRY IMPORTANT? National Math Panel (2008) three
areas are Critical Foundations of Algebra Fluency with whole
numbers Fluency with fractions Particular aspects of geometry and
measurement 2 Particular aspects of geometry and measurement are:
Similar triangles Formulas for perimeter, area, volume, and surface
area Finding unknown lengths, angles, and areas Particular aspects
of geometry and measurement are: Similar triangles Formulas for
perimeter, area, volume, and surface area Finding unknown lengths,
angles, and areas
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- VERTICAL FLOW OF GEOMETRY: ANGLES
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- MATHEMATICAL PROCESS STANDARDS A. apply mathematics to problems
arising in everyday life, society, and the workplace; B. use a
problem-solving model that incorporates analyzing given
information, formulating a plan or strategy, determining a
solution, justifying the solution, and evaluating the
problem-solving process and the reasonableness of the solution; C.
select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including
mental math, estimation, and number sense as appropriate, to solve
problems; D. communicate mathematical ideas, reasoning, and their
implications using multiple representations, including symbols,
diagrams, graphs, and language as appropriate; E. create and use
representations to organize, record, and communicate mathematical
ideas; F. analyze mathematical relationships to connect and
communicate mathematical ideas; and G. display, explain, and
justify mathematical ideas and arguments using precise mathematical
language in written or oral communication. 4
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- MEASUREMENT (ANGLES): VERTICAL PROGRESSION 5 4.MDGeometric
measurement: understand concepts of angle and measure angles. 5.
Recognize angles as geometric shapes that are formed wherever two
rays share a common endpoint, and understand concepts of angle
measurement: a. An angle is measured with reference to a circle
with its center at the common endpoint of the rays, by considering
the fraction of the circular arc between the points where the two
rays intersect the circle. An angle that turns through 1/360 of a
circle is called a one-degree angle, and can be used to measure
angles. b.An angle that turns through n one-degree angles is said
to have an angle measure of n degrees. 6. Measure angles in
whole-number degrees using a protractor. Sketch angles of specified
measure. 7. Recognize angle measure as additive. When an angle is
decomposed into non- overlapping parts, the angle measure of the
whole is the sum of the angle measures of the parts. Solve addition
and subtraction problems to find unknown angles on a diagram in
real world and mathematical problems, e.g., by using an equation
with a symbol for the unknown angle measure.
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- GEOMETRY (ANGLES): VERTICAL PROGRESSION 6 4.GDraw and identify
lines and angles, and classify shapes by properties of their lines
and angles. 1 Draw points, lines, line segments, rays, angles
(right, acute, obtuse), and perpendicular and parallel lines.
Identify these in two-dimensional figures. 5.GClassify
two-dimensional figures into categories based on their properties.
3 Understand that attributes belonging to a category of two-
dimensional figures also belong to all subcategories of that
category. For example, all rectangles have four right angles and
squares are rectangles, so all squares have four right angles.
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- GEOMETRY (ANGLES): VERTICAL PROGRESSION 7 7.GDraw, construct,
and describe geometrical figures and describe the relationships
between them. 2 Draw (freehand, with ruler and protractor, and with
technology) geometric shapes with given conditions. Focus on
constructing triangles from three measures of angles or sides,
noticing when the conditions determine a unique triangle, more than
one triangle, or no triangle. 7.GSolve real-life and mathematical
problems involving angle measure, area, surface area, and volume. 5
Use facts about supplementary, complementary, vertical, and
adjacent angles in a multi-step problem to write and solve simple
equations for an unknown angle in a figure. 8.GUnderstand
congruence and similarity using physical models, transparencies, or
geometry software. 5 Use informal arguments to establish facts
about the angle sum and exterior angle of triangles, about the
angles created when parallel lines are cut by a transversal, and
the angle-angle criterion for similarity of triangles. For example,
arrange three copies of the same triangle so that the sum of the
three angles appears to form a line, and give an argument in terms
of transversals why this is so.
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- SAMPLE ACTIVITIES: EXTENDING UNDERSTANDING OF ANGLE
RELATIONSHIPS
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- ANGLE RELATIONSHIPS Use the picture to identify supplementary,
complementary, and vertical angles. Write a multi-step equation
that you could use to solve to determine the measure of an angle in
the picture. 9 Photo is of Port Arthur, Texas
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- ANGLES IN PARALLEL LINES 10 Use patty paper to investigate:
Corresponding angles Alternate interior angles Same-side interior
angles Alternate exterior angles Same-side exterior angles Vertical
angles Which pair(s) of angles are congruent? Which pair(s) of
angles are supplementary?
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- INTERIOR ANGLES OF A TRIANGLE 11
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- INVESTIGATING INTERIOR ANGLES 1. Construct a triangle and color
each interior angle a different color. 2. Cut out your triangle and
remove each angle from the triangle. 3. Use patty paper to
investigate: What do you notice about the relationship between the
measures of the interior angles? What do you notice about the
relationship between the measure of an exterior angle and its
remote interior angles? 12
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- INVESTIGATING INTERIOR ANGLES 13
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- INVESTIGATING INTERIOR ANGLES What do you notice about the sum
of the interior angle measures for any triangle? What do you notice
about the angle measures in an isosceles triangle? What do you
notice about the angle measures in an equilateral triangle? What do
you notice about the measures of the two acute angles in a right
triangle? 14
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- DIFFERENTIATION Recommendation 5 Intervention materials should
include opportunities for students to work with visual
representations of mathematical ideas and interventionists should
be proficient in the use of visual representations of mathematical
ideas. (p. 30) 15 Google: what works clearinghouse
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- PAUSE AND REFLECT How does this lesson support differentiated
instruction? How does this lesson integrate mathematical process
standards? How does this lesson build on previous grade level
content? 16
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- Tips for a great conference! Rate this presentation on the
conference app www.nctm.org/confapp Download available presentation
handouts from the Online Planner! www.nctm.org/planner Join the
conversation! Tweet us using the hashtag #NCTMRichmond
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- Tips for a great conference! Rate this presentation on the
conference app www.nctm.org/confapp Download available presentation
handouts from the Online Planner! www.nctm.org/planner Join the
conversation! Tweet us using the hashtag #NCTMHouston
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- REFERENCES National Mathematics Advisory Panel. Foundations for
Success: The Final Report of the National Mathematics Advisory
Panel, U.S. Department of Education: Washington, DC, 2008. Gersten,
R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R.,
& Witzel, B. (2009). Assisting students struggling with
mathematics: Response to Intervention (RtI) for elementary and
middle schools (NCEE 2009-4060). Washington, DC: National Center
for Education Evaluation and Regional Assistance, Institute of
Education Sciences, U.S. Department of Education. Retrieved from
http://ies.ed.gov/ncee/wwc/publications/practiceguides/. 19
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- QUESTIONS, COMMENTS, OR SUCCESS STORIES? Dr. Paul Gray
pgray73@sbcglobal.net pgray@uh.edu PowerPoint and Handout available
on www.cosenzaassociates.com www.cosenzaassociates.com (Events and
Conferences section) 20