School Curriculum: Geometry and Measurement
This page is designed to enable parents to understand what their child should be learning, when they should be learning it, and what degree of mastery the child should have attained (at a median level) by a certain grade level. For Homeschoolers, we hope that this page will serve as a valuable asset in establishing a baseline curriculum. For parents whose children attend public or private schools (or for the inquisitive student) this page should give some guidance as to whether or not the school curriculum and methods are providing students with an adequate standard of education.
What is meant by "Geometry and Measurement," why is it important, and how is it approached ? Below is a description of the core discipline and its components, and the answers to why-how-when these components are taught. Geometry and Measurement components have median level goals to be attained by the end of Grade 2, by the end of Grade 3, by the end of Grade 4, by the end of Grade 5, by the end of Grade 6, by the end of Grade 7, by the end of Grade 8, and by the end of Grade 12.
This page does not contain articles for education in this discipline. For educational articles, go to "Mathematics"
Geometry and Measurement: A. Geometric Properties, B. Transforming Shapes, C. Coordinate Geometry, D. Units of Measurement, E. Measuring Geometric Objects
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.
Fostering respect for the power of mathematics. All students should learn that mathematics is integral to the development of all cultures and civilizations, and in particular to the advances in our own society. They should be aware that the adults in their world (parents, relatives, mentors, community members, role models) use mathematics on a daily basis. And they should know that success in mathematics may be a critical gateway to success in their careers, citizenship, and lives.
Setting high expectations. All students should have high expectations of themselves. These high expectations should be fostered by their teachers, administrators, and parents all of whom should themselves believe that all students can and will succeed in mathematics. This belief in his or her abilities often makes it possible for a child to succeed.
Providing opportunities for success. High expectations can only be achieved if students are provided with the appropriate opportunities. At all grade levels, students should receive instruction by teachers who have had the training and professional development appropriate for their grade level. Students should receive prompt and appropriate services essential to ensure that they can learn the mathematical skills and concepts included in the core curriculum, and to ensure that their weaknesses do not result in trapping them in a cycle of failure. Students should receive equitable treatment without regard to gender or ethnicity, and should not be conditioned to fail by predetermined low expectations.
Encouraging all students to go beyond the standards. Teachers should help students develop a positive attitude about mathematics by engaging them in exploring and solving interesting mathematical problems, by using mathematics in meaningful ways, by focusing on concepts and understanding as well as on rules and procedures, and by consistently expecting them to go beyond repetition and memorization to problem solving and understanding. Every effort should be made to ensure that all students are continuously encouraged, nurtured, and challenged to maximize their potential at all grade levels and to become prepared for college-level mathematics. Students who have achieved the standards should be encouraged to go beyond the standards. If schools challenge all students at lower grade levels, they will attain the goal of having advanced mathematics classrooms whose students reflect the diversity of the school’s total population.
Strands and 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-imageC. 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 conjecturesB. 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 indefinitelyC. 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 vectorsD. 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 measurementsE. 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
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