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

 

By the end of Grade 2, students will:

 

A.     Geometric Properties

 1.           Identify and describe spatial relationships among objects in space and their relative shapes and sizes.

·        Inside/outside, left/right, above/below, between

·        Smaller/larger/same size, wider/ narrower, longer/shorter

·        Congruence (i.e., same size and shape)

 2.         Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes.

·        Vertex, edge, face, side

·        3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid

·        2D figures – square, rectangle, circle, triangle

·        Relationships between three- and two-dimensional shapes (i.e., the face of a 3D shape is a 2D shape)

 3.         Describe, identify and create instances of line symmetry.

 4.         Recognize, describe, extend and create designs and patterns with geometric objects of different shapes and colors.

 

B.     Transforming Shapes

 1.         Use simple shapes to make designs, patterns, and pictures.

 2.         Combine and subdivide simple shapes to make other shapes.

 

C.     Coordinate Geometry

 1.         Give and follow directions for getting from one point to another on a map or grid.

 

D.    Units of Measurement

 1.         Directly compare and order objects according to measurable attributes.

·        Attributes – length, weight, capacity, time, temperature

 2.         Recognize the need for a uniform unit of measure.

 3.         Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems.

·        Length – inch, foot, yard, centimeter, meter

·        Weight – pound, gram, kilogram

·        Capacity – pint, quart, liter

·        Time – second, minute, hour, day, week, month, year

·        Temperature – degrees Celsius, degrees Fahrenheit

 4.         Estimate measures.

 

E.     Measuring Geometric Objects

 1.         Directly measure the perimeter of simple two-dimensional shapes.

 2.         Directly measure the area of simple two-dimensional shapes by covering them with squares.

 

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 

A.     Geometric Properties

 1.         Identify and describe spatial relationships of two or more objects in space.

·        Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)

·        Relative shapes and sizes

 2.         Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.

·        Vertex, edge, face, side, angle

·        3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid

·        2D figures – square, rectangle, circle, triangle, pentagon, hexagon, octagon

 3.         Identify and describe relationships among two-dimensional shapes.

·        Same size, same shape

·        Lines of symmetry

 4.         Understand and apply concepts involving lines, angles, and circles.

·        Line, line segment, endpoint

 5.         Recognize, describe, extend, and create space-filling patterns.

 

B.     Transforming Shapes

 1.         Describe and use geometric transformations (slide, flip, turn).

 2.         Investigate the occurrence of geometry in nature and art.

 

C.     Coordinate Geometry

 1.         Locate and name points in the first quadrant on a coordinate grid.

 

D.    Units of Measurement

 1.         Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

 2.         Select and use appropriate standard units of measure and measurement tools to solve real-life problems.

·        Length – fractions of an inch (1/4, 1/2), mile, decimeter, kilometer

·        Area – square inch, square centimeter

·        Weight – ounce

·        Capacity – fluid ounce, cup, gallon, milliliter

 3.         Incorporate estimation in measurement activities (e.g., estimate before measuring).

 

E.     Measuring Geometric Objects 

 1.         Determine the area of simple two-dimensional shapes on a square grid.

 2.         Determine the perimeter of simple shapes by measuring all of the sides.

 3.         Measure and compare the volume of three–dimensional objects using materials such as rice or cubes.

 

 

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 

A.     Geometric Properties

 1.         Identify and describe spatial relationships of two or more objects in space.

·        Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)

·        Relative shapes and sizes

·        Shadows (projections) of everyday objects

 2.         Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.

·        Vertex, edge, face, side, angle

·        3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid

·        2D figures – square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon

·        Inclusive relationships – squares are rectangles, cubes are rectangular prisms

 3.         Identify and describe relationships among two-dimensional shapes.

·        Congruence

·        Lines of symmetry

 4.         Understand and apply concepts involving lines, angles, and circles.

·        Point, line, line segment, endpoint

·        Parallel, perpendicular

·        Angles – acute, right, obtuse

·        Circles – diameter, radius, center

 5.         Recognize, describe, extend, and create space-filling patterns.

 

B.     Transforming Shapes

 1.         Use simple shapes to cover an area (tessellations).

 2.         Describe and use geometric transformations (slide, flip, turn).

 3.         Investigate the occurrence of geometry in nature and art.

 

C.     Coordinate Geometry

 1.         Locate and name points in the first quadrant on a coordinate grid.

 2.         Use coordinates to give or follow directions from one point to another on a map or grid.

 

D.    Units of Measurement

 1.         Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

 2.         Select and use appropriate standard units of measure and measurement tools to solve real-life problems

·        Length – fractions of an inch (1/8, 1/4, 1/2), mile, decimeter, kilometer

·        Area – square inch, square centimeter

·        Volume – cubic inch, cubic centimeter

·        Weight – ounce

·        Capacity – fluid ounce, cup, gallon, milliliter

 3.         Develop and use personal referents to approximate standard units of measure (e.g., a common paper clip is about an inch long).

 4.         Incorporate estimation in measurement activities (e.g., estimate before measuring).

 5.         Solve problems involving elapsed time.

 

E.     Measuring Geometric Objects 

 1.         Determine the area of simple two-dimensional shapes on a square grid.

 2.         Distinguish between perimeter and area and use each appropriately in problem-solving situations.

 3.         Measure and compare the volume of three–dimensional objects using materials such as rice or cubes.

 

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 

A.     Geometric Properties

 1.         Understand and apply concepts involving lines and angles.

·        Notation for line, ray, angle, line segment

·        Properties of parallel, perpendicular, and intersecting lines

·        Sum of the measures of the interior angles of a triangle is 180°

 2.         Identify, describe, compare, and classify polygons.

·        Triangles by angles and sides

·        Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

·        Polygons by number of sides

·        Equilateral, equiangular, regular

·        All points equidistant from a given point form a circle

 3.         Identify similar figures.

 4.         Understand and apply the concepts of congruence and symmetry (line and rotational).

 

B.     Transforming Shapes

 1.         Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

 2.         Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.

 

C.     Coordinate Geometry

 1.         Create geometric shapes with specified properties in the first quadrant on a coordinate grid.

 

D.    Units of Measurement

 1.         Select and use appropriate units to measure angles and area.

 2.         Convert measurement units within a system (e.g., 3 feet = ___ inches).

 3.         Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).

 4.         Use measurements and estimates to describe and compare phenomena.

 

E.     Measuring Geometric Objects

 1.         Use a protractor to measure angles.

 2.         Develop and apply strategies and formulas for finding perimeter and area.

·        Square

·        Rectangle

 3.         Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.

 4.         Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot).

 

 

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 

A.     Geometric Properties

 1.         Understand and apply concepts involving lines and angles.

·        Notation for line, ray, angle, line segment

·        Properties of parallel, perpendicular, and intersecting lines

·        Sum of the measures of the interior angles of a triangle is 180°

 2.         Identify, describe, compare, and classify polygons and circles.

·        Triangles by angles and sides

·        Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

·        Polygons by number of sides.

·        Equilateral, equiangular, regular

·        All points equidistant from a given point form a circle

 3.         Identify similar figures.

 4.         Understand and apply the concepts of congruence and symmetry (line and rotational).

 5.         Compare properties of cylinders, prisms, cones, pyramids, and spheres.

 6.         Identify, describe, and draw the faces or shadows (projections) of three-dimensional geometric objects from different perspectives.

 7.         Identify a three-dimensional shape with given projections (top, front and side views).

 8.         Identify a three-dimensional shape with a given net (i.e., a flat pattern that folds into a 3D shape).

 

B.     Transforming Shapes

 1.         Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

 2.         Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.

 

C.     Coordinate Geometry

 1.         Create geometric shapes with specified properties in the first quadrant on a coordinate grid.

 

D.    Units of Measurement

 1.         Select and use appropriate units to measure angles, area, surface area, and volume.

 2.         Use a scale to find a distance on a map or a length on a scale drawing.

 3.         Convert measurement units within a system (e.g., 3 feet = ___ inches).

 4.         Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).

 5.         Use measurements and estimates to describe and compare phenomena.

 

E.     Measuring Geometric Objects

 1.         Use a protractor to measure angles.

 2.         Develop and apply strategies and formulas for finding perimeter and area.

·        Triangle, square, rectangle, parallelogram, and trapezoid

·        Circumference and area of a circle

 3.         Develop and apply strategies and formulas for finding the surface area and volume of rectangular prisms and cylinders.

 4.         Recognize that shapes with the same perimeter do not necessarily have the same area and vice versa.

 5.         Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot).

 

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 

A.     Geometric Properties

 1.         Understand and apply properties of polygons.

·        Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

·        Regular polygons

 2.         Understand and apply the concept of similarity.

·        Using proportions to find missing measures

·        Scale drawings

·        Models of 3D objects

 3.         Use logic and reasoning to make and support conjectures about geometric objects.

 

B.     Transforming Shapes

 1.         Understand and apply transformations.

·        Finding the image, given the pre-image, and vice-versa

·        Sequence of transformations needed to map one figure onto another

·        Reflections, rotations, and translations result in images congruent to the pre-image

·        Dilations (stretching/shrinking) result in images similar to the pre-image

 

C.     Coordinate Geometry

 1.         Use coordinates in four quadrants to represent geometric concepts.

 2.         Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).

 

D.    Units of Measurement

 1.         Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4’3” plus 7’10” equals 12’1”).

 2.         Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.

 3.         Recognize that all measurements of continuous quantities are approximations.

 

E.     Measuring Geometric Objects

 1.         Develop and apply strategies for finding perimeter and area.

·        Geometric figures made by combining triangles, rectangles and circles or parts of circles

·        Estimation of area using grids of various sizes

 2.         Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).

 

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 

A.     Geometric Properties

 1.         Understand and apply concepts involving lines, angles, and planes.

·        Complementary and supplementary angles

·        Vertical angles

·        Bisectors and perpendicular bisectors

·        Parallel, perpendicular, and intersecting planes

·        Intersection of plane with cube, cylinder, cone, and sphere

 2.         Understand and apply the Pythagorean theorem.

 3.         Understand and apply properties of polygons.

·        Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

·        Regular polygons

·        Sum of measures of interior angles of a polygon

·        Which polygons can be used alone to generate a tessellation and why

 4.         Understand and apply the concept of similarity.

·        Using proportions to find missing measures

·        Scale drawings

·        Models of 3D objects

 5.         Use logic and reasoning to make and support conjectures about geometric objects.

 

B.     Transforming Shapes

 1.         Understand and apply transformations.

·        Finding the image, given the pre-image, and vice-versa

·        Sequence of transformations needed to map one figure onto another

·        Reflections, rotations, and translations result in images congruent to the pre-image

·        Dilations (stretching/shrinking) result in images similar to the pre-image

 2.        Use iterative procedures to generate geometric patterns.

         ·        Fractals (e.g., the Koch Snowflake)

·        Self-similarity

·        Construction of initial stages

·        Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski’s Triangle)

 

C.     Coordinate Geometry

 1.         Use coordinates in four quadrants to represent geometric concepts.

 2.         Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).

 

D.    Units of Measurement

 1.         Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4’3” plus 7’10” equals 12’1”).

 2.         Use approximate equivalents between standard and metric systems to estimate measurements (e.g., 5 kilometers is about 3 miles).

 3.         Recognize that the degree of precision needed in calculations depends on how the results will be used and the instruments used to generate the measurements.

 4.         Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.

 5.         Recognize that all measurements of continuous quantities are approximations.

 6.         Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile).

 

E.     Measuring Geometric Objects

 1.         Develop and apply strategies for finding perimeter and area.

·        Geometric figures made by combining triangles, rectangles and circles or parts of circles

·        Estimation of area using grids of various sizes

·        Impact of a dilation on the perimeter and area of a 2-dimensional figure

 2.         Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).

 3.         Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure.

·        Volume - prism, cone, pyramid

·        Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base)

·        Impact of a dilation on the surface area and volume of a three-dimensional figure

 4.         Use formulas to find the volume and surface area of a sphere.

 

 

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