STANDARD 4.2     (GEOMETRY AND MEASUREMENT)     ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.

 

Descriptive Statement:  Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve describing the shapes we see all around us in art, nature, and the things we make.  Spatial sense, geometric modeling, and measurement can help us to describe and interpret our physical environment and to solve problems. 

 

            Geometric Properties.  This includes identifying, describing and classifying standard geometric objects, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.

 

            Transforming Shapes.  Analyzing how various transformations affect geometric objects allows students to enhance their spatial sense. This includes combining shapes to form new ones and decomposing complex shapes into simpler ones.  It includes the standard geometric transformations of translation (slide), reflection (flip), rotation (turn), and dilation (scaling).  It also includes using tessellations and fractals to create geometric patterns.

 

            Coordinate Geometry.  Coordinate geometry provides an important connection between geometry and algebra.  It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry.

 

            Units of Measurement.  Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them. 

 

            Measuring Geometric Objects.  This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement.  While students will eventually apply formulas, it is important that they develop and apply strategies that derive from their understanding of the attributes.  In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry.

 

Students of all ages should realize that geometry and measurement are all around them.  Through study of these areas and their applications, they should come to better understand and appreciate the role of mathematics in their lives.

 

Cumulative Progress Indicators

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 

A.     Geometric Properties

 1.         Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).

 2.         Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).

 3.         Apply the properties of geometric shapes.

·        Parallel lines – transversal, alternate interior angles, corresponding angles

·        Triangles

a.      Conditions for congruence

b.      Segment joining midpoints of two sides is parallel to and half the length of the third side

c.      Triangle Inequality

·        Minimal conditions for a shape to be a special quadrilateral

·        Circles – arcs, central and inscribed angles, chords, tangents

·        Self-similarity

 4.         Use reasoning and some form of proof to verify or refute conjectures and theorems.

·        Verification or refutation of proposed proofs

·        Simple proofs involving congruent triangles

·        Counterexamples to incorrect conjectures

 

B.     Transforming Shapes

 1.         Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.

 2.         Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.

 3.         Determine whether two or more given shapes can be used to generate a tessellation.

 4.         Generate and analyze iterative geometric patterns.

·        Fractals (e.g., Sierpinski’s Triangle)

·        Patterns in areas and perimeters of self-similar figures

·        Outcome of extending iterative process indefinitely

 

C.     Coordinate Geometry

 1.         Use coordinate geometry to represent and verify properties of lines.

·        Distance between two points

·        Midpoint and slope of a line segment

·        Finding the intersection of two lines

·        Lines with the same slope are parallel

·        Lines that are perpendicular have slopes whose product is –1

 2.       Show position and represent motion in the coordinate plane using vectors.

·        Addition and subtraction of vectors

 

D.    Units of Measurement

 1.         Understand and use the concept of significant digits.

 2.         Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation.

·        Degree of accuracy of a given measurement tool

·        Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements

 

E.     Measuring Geometric Objects

 1.         Use techniques of indirect measurement to represent and solve problems.

·        Similar triangles

·        Pythagorean theorem

·        Right triangle trigonometry (sine, cosine, tangent)

 2.         Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures.

·        Approximation of area using grids of different sizes

·        Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets

·        Estimation of area, perimeter, volume, and surface area

 

 

 

Link to Standard 4.2 Grade 8

 

Back to Main Page