Math 580Geometry for Middle Grade Teachers
Math 580: Geometry for Middle Grade Teachers
Humboldt State University
Spring Semester 2001
(Special Thanks to Dr. Laura Moss who team-taught the first version of this course with me during the Fall of 1999 and who authored the first version of these notes)
Course Information:
This course is designed to enable teachers and/or prospective teachers of middle school grades to develop an advanced viewpoint of geometric concepts found in the middle grades curriculum. This course should enhance a teachers’ understanding of the nature of geometry as well as laying a foundation for the development of mathematical reasoning through the study of geometry.
The content of the course includes selected topics in Euclidean geometry, non-Euclidean geometry, and related areas of measurement. Emphasis will be placed on the concepts of similarity and symmetry within two- and three- dimensional contexts. The topics studied are connected to the content standards for Geometry and Measurement for the middle grades, as found in the California Academic Content Standards in Mathematics.
The nature of the course is one of exploration, analysis, and reasoning. We will be using the software package, The Geometer’s Sketchpad, published by Key Curriculum Press, to aid us in our exploration. Learning how to use this tool does take some time, but I don’t want the focus of the course to be the tool In fact I chose it because doing so will strengthen your conceptual understanding of the interplay between fundamental mathematical objects – points, lines, and circles (objects that connect to the straightedge and compass of Greek Mathematicians) as well as help you visualize mathematical relationships between visual aspects (perpendicularity, parallelism, similarity, symmetry) and measurable aspects (length, area, degree) of figures in the Euclidean Plane.
I do expect that all of you – at one time in your life – either took a course in high school geometry OR that you have learned the foundations of geometry through teaching. This course is not a course on how to teach geometry. It is a course on improving your reasoning abilities in geometric contexts.
Course Grade:
Your course grade will be determined by weekly assignments. These assignments will consist of textbook type questions, responses to readings, activity questions, and computer lab inquiries.
Course outline:
The goal of the course is to help you build an advanced viewpoint of the ideas that are taught in middle grades geometry. Not all activities will be applicable to working with middle grade students. In this 10-mailings course we will be working with the following topical index:
1Definitions, classifications of triangles
2Constructions, classifications of quadrilaterals
3Properties and relationships of circles
4Proving Properties Geometrically
5Similarity
6Congruence
7Length, Area, and Volume Relationships
8Transformations
9Symmetry
10Patterns in the Plane
For your reference, I’ve included the following Geometry standards so that you have them in one place:
California Academic Content Standards in Mathematics
California Geometry Standards: Grade 5 through Geometry
These standards may be obtained under the mathematics framework from the web site:
Grade Five
1.0Students understand and compute the volumes and areas of simple objects:
1.1Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle. (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram.)
1.2Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.
1.3Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in.3], cubic yard [yd.3]) to compute the volume of rectangular solids.
1.4Differentiate between, and use appropriate units of measures for, two- and three-dimensional objects (i.e., find the perimeter, area, volume).
2.0Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures:
2.1Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software).
2.2Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.
2.3Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.
Grade Six
1.0Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems:
1.1Understand the concept of a constant such as ; know the formulas for the circumference and area of a circle.
1.2Know common estimates of (3.14; 22 ⁄7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.
1.3Know and use the formulas for the volume of triangular prisms and cylinders (area of base height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
2.0Students identify and describe the properties of two-dimensional figures:
2.1Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
2.2Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
2.3Draw quadrilaterals and triangles from given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle).
Grade Seven
1.0Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems:
1.1Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters).
1.2Construct and read drawings and models made to scale.
1.3Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.
2.0Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale:
2.1Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
2.2Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects.
2.3Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Under-stand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.
2.4Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft2] = [144 in2], 1 cubic inch is approximately 16.38 cubic centimeters or [1 in3] = [16.38 cm3]).
3.0Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures:
3.1Identify and construct basic elements of geometric figures (e.g., altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge.
3.2Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections.
3.3Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
3.4Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.
3.5Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones.
3.6Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect).
Geometry
1.0Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
2.0Students write geometric proofs, including proofs by contradiction.
3.0Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.
4.0Students prove basic theorems involving congruence and similarity.
5.0Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
6.0Students know and are able to use the triangle inequality theorem.
7.0Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
8.0Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
9.0Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.
10.0Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
11.0Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
12.0Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
13.0Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
14.0Students prove the Pythagorean theorem.
15.0Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
16.0Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.
17.0Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
18.0Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x))2 +(cos(x))2 = 1.
19.0Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
20.0Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.
21.0Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
22.0Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
An “advance perspective”:
What does it mean to build an advanced perspective on the Geometry standards in middle grades?
- Gaining an advanced viewpoint of geometric concepts means that you are consider these concepts in the context of higher order thinking skills.
- Gaining an advanced viewpoint of geometric concepts means that you are able to place these concepts in a broader context or mathematical system.
Let’s expand these two points below:
Higher order thinking skills in Geometry – The van Hiele Levels of Geometric Thinking.[1]
Dina and Pierre van Hiele are two Dutch educators who were concerned about the difficulties that their students were having in geometry. This concern motivated their research aimed at understanding students’ levels of geometric thinking to determine the kinds of instruction that can best help students.
The five levels that are described below are not age-dependent, but, instead, are related more to the experiences students have had. The levels are sequential; that is, students must pass through the levels in order as their understanding increases. The descriptions of the levels are in terms of “students” – and remember that we are all students in some sense.
Level 0 – Visualization
Students recognize shapes by their global, holistic appearance.
Students at level 0 think about shapes in terms of what they resemble and are able to sort shapes into groups that “seem to be alike.” For example, a student at this level might describe a triangle as a “clown’s hat.” The student, however, might not recognize the same triangle if it is rotated so that it “stands on its point.”
Level 1 – Analysis
Students observe the component parts of figures (e.g., a parallelogram has opposite sides that are parallel) but are unable to explain the relationships between properties within a shape or among shapes.
Student at level 1 are able to understand that all shapes in a group such as parallelograms have the same properties, and they can describe those properties.
Level 2 – Informal deduction (relationships)
Students deduce properties of figures and express interrelationships both within and between figures.
Students at level 2 are able to notice relationships between properties and to understand informal deductive discussions about shapes and their properties.
Level 3 – Formal deduction
Students can create formal deductive proofs.
Students at level 3 think about relationships between properties of shapes and also understand relationships between axions, definitions, theorems, corollaries, and postulates. At this level, students are able to “work with abstract statements about geometric properties and make conclusions based more on logic than intuition” (Van de Walle).
Level 4 – Rigor
Students rigorously compare different axiomatic systems.
Students at this level think about deductive axiomatic systems of geometry. This is the level that college mathematics majors think about Geometry.
In general, most elementary school students are at levels 0 or 1; some middle school students are at level 2. The CA standards are written to begin the transition from levels 0 and 1 to level 2 as early as 5th grade “Students identify, describe, draw and classify properties of, and
relationships between, plane and solid geometric figures.” (5th grade, standard 2 under Geometry and Measurement) This emphasis on relationships is magnified in the 6th and 7th grade standards.
Interestingly, the sixth National Assessment of Educational Progress report (1997) reported that “most of the students at all three grade levels (fourth, eight, and twelfth) appear to be performing at the ‘holistic’ level (level 0) of the van Heile levels of geometric thought.”
In this course, our goal is increase your experiences with geometry at and above level 2.
Broader mathematics Context – Mathematical (Geometric) Systems
We have already begun building our “big picture” of geometry. To do so requires that we look at
- The process of classification
- The nature of postulates, definitions, propositions, and proof
- The consequences of alternative assumptions
As we do so, systemic views of geometry will emerge:
- Euclidean Geometry in 2 and 3 dimensions (one emphasis)
- Hyperbolic (“Pringel’s Potato chip” surface) and Elliptic (Sphere surface) geometry
- Transformational Geometry
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[1] Cathcart, et al. Learning Mathematics in Elementary and Middle Schools. p.282-3.