Elementary Mathematics Benchmarks, Geometry Strand

Kindergarten

G.K.1 Create, explore, and describe shapes.

a. Identify common shapes such as rectangle, circle, triangle, and square.

• Draw a variety of triangles (equilateral, right, isosceles, scalene) in a variety of positions.
• Draw squares and rectangles of different proportions (tall, squat, square-like) both horizontally and vertically positioned. Recognize tipped squares and rectangles.

  Note: Squares are a type of rectangle.

  Note: Drawing of tipped rectangles is typically too hard for kindergarten.

• Describe attributes of common shapes (e.g., number of sides and corners).

b. Use geometric tiles and blocks to assemble compound shapes.

• Assemble rectangles from two congruent right triangular tiles.
• Explore two-dimensional symmetry using matching tiles.

c. Recognize and use words that describe spatial relationships such as above, below, inside, outside, touching, next to, far apart.

Grade 1

G.1.1 Recognize, describe, and draw geometric figures.

a. Identify and draw two-dimensional figures.

• Include trapezoids, equilateral triangles, isosceles triangle, parallelograms, quadrilaterals.

  Note: Be sure to include a robust variety of triangles as examples, especially ones that are very clearly not equilateral or isosceles.

• Describe attributes of two-dimensional shapes (e.g., number of sides and corners).

b. Identify and name three-dimensional figures.

• Include spheres, cones, prisms, pyramids, cubes, rectangular solids.
• Identify two-dimensional shapes as faces of three-dimensional figures.

c. Sort geometric objects by shape and size.

• Recognize the attributes that determined a particular sorting of objects and use them to extend the sorting.

  Example: Various L-shaped figures constructed from cubes are sorted by the total number of cubes in each. Recognize this pattern, then sort additional figures to extend the pattern.

• Explore simultaneous independent attributes.

  Example: Sort triangular tiles according to the four combinations of two attributes, such as right angle and equal sides.

G.1.2 Rotate, invert, and combine geometric tiles and solids.

a. Describe and draw shapes resulting from rotations and flips of simple two-dimensional figures.

• Identify the same (congruent) two-dimensional shapes in various orientations and move one on top of the other to show that they are indeed identical.
• Extend sequences that show rotations of simple shapes.

b. Identify symmetrical shapes created by rotation and reflection.

c. Use geometric tiles and cubes to assemble and disassemble compound figures.

• Count characteristic attributes (lines, faces, edges) before and after assembly.

  Examples: Add two right triangles to a trapezoid to make a rectangle; create a hexagon from six equilateral triangles; combine two pyramids to make a cube.

Grade 2

G.2.1 Recognize, classify, and transform geometric figures in two and three dimensions.

a. Identify, describe, and compare common geometric shapes in two and three dimensions.

• Define a general triangle and identify isosceles, equilateral, right, and obtuse special cases.

  Note: The goal of naming triangles is not the names themselves but to focus on important differences. Triangles (or quadrilaterals) are not all alike, and it is their differences that give them distinctive mathematical features.

• Identify various quadrilaterals (rectangles, trapezoids, parallelograms, squares) as well as pentagons and hexagons.

  Note: In this grade parallel is used informally and intuitively; it receives more careful treatment at a later grade.

  Note: A square is a special kind of rectangle (since it has four sides and four right angles); a rectangle is a special kind of parallelogram (since it has four sides and two pairs of parallel sides); and a parallelogram is a special kind of trapezoid (since it has four sides and at least one pair of parallel sides); and a trapezoid is a polygon (since it is a figure formed of several straight sides). So for example, contrary to informal usage, in mathematics, a square is a trapezoid.

• Understand the terms perimeter and circumference.

  Note: The primary meaning of both terms is the outer boundary of a two-dimensional figure; circumference is used principally in reference to circles. A secondary meaning for both is the length of the outer boundary. Which meaning is intended needs to be determined from context.

• Distinguish circles from ovals; recognize the circumference, diameter, and radius of a circle.
• In three dimensions, identify spheres, cones, cylinders, and triangular, and rectangular prisms.

b. Describe common geometric attributes of familiar plane and solid objects.

• Common geometric attributes include position, shape, size, and roundness, and numbers of corners, edges, and faces.
• Distinguish between geometric attributes and other characteristics such as weight, color, or construction material.
• Distinguish between lines and curves and between flat and curved surfaces.

c. Rotate, flip, and fold shapes to explore the effect of transformations.

• Use paper folding to find lines of symmetry.
• Recognize congruent shapes.
• Identify shapes that have been moved (flipped, slid, rotated), enlarged, or reduced.

G.2.2 Understand and interpret rectangular arrays as a model of multiplication.

a. Create square cells from segments of the discrete number line used as sides of a rectangle.

• Match cells to discrete objects lined up in regular rows of the same length.

b. Understand rectangular arrays as instances of repeated addition.

Grade 3

G.3.1 Recognize basic elements of geometric figures and use them to describe shapes.

a. Identify points, rays, line segments, lines, and planes in both mathematical and everyday settings.

• A line is a straight path traced by a moving point having no breadth nor end in either direction.

  Examples: Each figure on the left above represents a line; the arrows indicate that the lines keep going in the indicated directions without end. The number line with both positive and negative numbers is a line.

• A part of a line that starts at one point and ends at another is called a line segment. Line segments are drawn without arrows on either end because line segments end at points.

  Examples: The figure in the center above is a line segment. The edges of a desk or door or piece of paper are everyday examples of line segments.

• Part of a line that starts at one point and goes on forever in one direction is called a ray.

  Examples: The figure above on the right is a ray. The positive number line (to the right of 0) is a ray. On the other hand, none of the four examples below are lines:

  

  Caution: Not all sources distinguish carefully among the terms line, segment, or ray, nor do all sources employ the convention of arrowheads in exactly the manner described above. Often context is the best guide to distinguish among these terms.

• Know that a plane is a flat surface without thickness that extends indefinitely in every direction.

  Examples: Everyday examples that illustrate a part of a plane are the flat surfaces of a floor, desk, windowpane, or book. Examples that are not part of a plane are the curved surfaces of a light bulb, a ball, or a tree.

b. Understand the meaning of parallel and perpendicular and use these terms to describe geometric figures.

• Lines and planes are called parallel if they do not meet no matter how far they are extended.
• Lines and planes are called perpendicular if the corners formed when they meet are equal.
• Identify parallel and perpendicular edges and surfaces in everyday settings (e.g., the classroom).

  Examples: The lines on the page of a notebook are parallel, as are the covers of a closed book. Corners of books, walls, and rectangular desks are perpendicular, as are the top and side edges of a chalk board and a wall and a floor in a classroom.

• The corner where two perpendicular lines meet is called a right angle.

  Note: The general concept of "angle" is developed later; here the term is used merely as the name for this specific and common configuration.

• Understand and use the terms vertical and horizontal.
• Recognize that vertical and horizontal lines or planes are perpendicular, but that perpendicular lines or planes are not necessarily vertical or horizontal.

c. Use terms such as line, plane, ray, line segment, parallel, perpendicular, and right angle correctly to describe everyday and geometric figures.

G.3.2 Identify and draw perpendicular and parallel lines and planes.

a. Draw perpendicular, parallel, and non-parallel line segments using rulers and squares.

• Recognize that lines that are parallel to perpendicular lines will themselves be perpendicular.

  Example: Fold a piece of paper in half from top to bottom, then fold it in half again from left to right. This will give two perpendicular fold lines and four right angles.

  

• Edges of a polygon are called parallel or perpendicular if they lie on parallel or perpendicular lines, respectively.
• Similarly, faces of a three-dimensional solid are called parallel or perpendicular if they lie in parallel or perpendicular planes, respectively.

G.3.3 Explore and identify familiar two- and three-dimensional shapes.

a. Describe and classify plane figures and solid shapes according to the number and shape of faces, edges, and vertices.

• Plane figures include circles, triangles, squares, rectangles, and other polygons; solid shapes include spheres, pyramids, cubes, and rectangular prisms.
• Recognize that the exact meaning of many geometric terms (e.g., rectangle, square, circle and triangle) depends on context: Sometimes they refer to the boundary of a region and sometimes to the region contained within the boundary.

b. Know how to put shapes together and take them apart to form other shapes.

Examples: Two identical right triangles can be arranged to form a rectangle. Two identical cubes can be arranged to form a rectangular prism. See figures below.

c. Identify edges, vertices (corners), perpendicular and parallel edges and right angles in two-dimensional shapes.

Example: A rectangle has four pairs of perpendicular edges, two pairs of parallel edges and four right angles.

d. Identify right angles, edges, vertices, perpendicular and parallel planes in three-dimensional shapes.

G.3.4 Understand how to measure length, area, and volume.

a. Understand that measurements of length, area, and volume are based on standard units.

• Fundamental units are: a unit interval of length 1 unit, a unit square whose sides have length 1 unit, and a unit cube whose sides have length 1 unit.
• The volume of a rectangular prism is the number of unit cubes required to fill it exactly (with no space left over).

  

  Note: The common childhood experience of pouring water or sand offers a direct representation of volume.

• The area of a rectangle is the number of unit squares required to pave the rectangle—that is, to cover it completely without any overlapping.

  Note: Area provides a critical venue for developing the conceptual underpinnings of multiplication.

• The length of a line segment is the number of unit intervals that are required to cover the segment exactly with nothing left over.

b. Know how to calculate the perimeter, area, and volume of shapes made from rectangles and rectangular prisms.

• The perimeter of a rectangle is the number of unit intervals that are required to enclose the rectangle.
• Measure and compare the areas of shapes using non-standard units (e.g., pieces in a set of pattern blocks).
• Recognize that the area of a rectangle is the product of the lengths of its base and height (A = b × h) and that the volume of a rectangular prism is the product of the lengths of its base, width, and height (V = b × w × h).

  Example: Build solids with unit cubes and use the formula for volume (V = bwh) to verify the count of unit cubes; make similar comparisons with rectilinear figures in the plane that are created from unit squares.

• Find the area of a complex figure by adding and subtracting areas.
• Compare rectangles of equal area and different perimeter and also rectangles of equal perimeter and different area.
• Measure surface area of solids by covering each face with copies of a unit square and then counting the total number of units.

Grade 4

G.4.1 Understand and use the definitions of angle, polygon, and circle.

a. An angle in a plane is a region between two rays that have a common starting point.

Note: According to this definition, a right angle (as determined by perpendicular rays) is indeed an angle.

b. If angle A is contained in another angle B, then angle B is said to be bigger than angle A.

• The figures below illustrate how to determine whether an angle is larger than, smaller than, or close to a right angle.

  The angle is not contained in a right angle, so this tells us that it is larger than a right angle.

  

  The angle is contained in a right angle, so this tells us that it is smaller than a right angle.

  

  Note: When two rays come from the same point (see figure below), they divide the plane into two regions, giving two angles. Except where otherwise indicated, the angle determined by the two rays is defined, by convention, as the smaller region.

  

• Understand that shapes such as triangles, squares, and rectangles have angles.

  Note: Technically, polygons do not contain rays, which are required for the definition of angles. Their sides are line segments of finite length. Nonetheless, if we imagine the sides extending indefinitely away from each corner, then each corner becomes an angle.

  Example: Describe the difference between the two figures below:

  

• Identify acute, obtuse and right angles.

c. Know and use the basic properties of squares; rectangles; and isosceles, equilateral and right triangles.

• Identify scalene, acute, and obtuse triangles.
• Know how to mark squares, rectangles, and triangles appropriately:

  

  

d. Know what a polygon is and be able to identify and draw some examples.

• A polygon is a figure that lies in a plane consisting of a finite number of line segments called edges (or sides) with the properties that (a) each edge is joined to exactly two other edges at the end points; edges do not meet each other except at end points; and the edges enclose a single region.

  Example: The figures below are not polygons:

  

e. Know and use the basic properties of a circle.

• A circle is the set of points in a plane that are at a fixed distance from a given point.

  

• Know that a circle is not a polygon.

Grade 5

G.5.1 Measure angles in degrees and solve related problems.

a. Understand the definition of degree and be able to measure angles in degrees.

• A degree is one part of the circumference of a circle of radius 1 unit (a unit circle) that is divided into 360 equal parts. The measure of an angle in degrees is defined to be the number of degrees of the arc of the unit circle, centered at the vertex of the angle, that is intercepted by the angle.
• The measure of an angle in degrees can also be interpreted as the amount of counter-clockwise turning from one ray to the other.

  Note: Earlier (in grade 4), the angle determined by two rays was defined to be the smaller of the two options. For consistency, therefore, when an angle is measured by the amount of turning necessary to rotate one ray into another, it is important to start with the particular ray that will produce an angle measure no greater than 180°.

• The symbol ° is an abbreviation for "degree" (e.g., 45 degrees = 45°).
• As a shorthand, angles are called equal if the measures of the angles are equal.

b. Know and use the measures of common angles.

• Recognize that angles on a straight line add up to 180° and that angles around a point add up to 360°. An angle of 180° is called a straight angle.
• A right angle is an angle of 90°. An acute angle is an angle of less than 90°, while an obtuse angle is an angle of more than 90°.

  Note: Since a pair of perpendicular lines divides the plane into 4 equal angles, the measure of a right angle is 360°/4 = 90°.

  

c. Interpret and prepare circle graphs (pie charts).

G.5.2 Know how to do basic constructions using a straightedge and compass.

a. Basic constructions include (a) drop a perpendicular from a point to a line, (b) bisect an angle, (c) erect the perpendicular bisector of a line, and (d) construct a hexagon on a circle.

Note: A straightedge is a physical representation of a line, not a ruler that is used for measuring. The role of a straightedge in constructions is to draw lines through two points, just as the role of the compass is to draw a circle based on two points, the center and a point on the circumference.

Note: Students need extended practice with constructions, since constructions embody the elements of geometry—lines and circles—independent of numbers and measurement. Since constructions are so central to Euclidean geometry, they are often called Euclidean constructions.

Note: These constructions are basic in the sense that other important constructions introduced in later grades (e.g., of an equilateral triangle given one side; of a square inscribed in a circle) build on them.

• Use informal arguments such as paper folding to verify the correctness of constructions.

G.5.3 Recognize and work with simple polyhedra.

a. Represent and work with rectangular prisms by means of orthogonal views, projective views, and nets.

• A net is a flat (two-dimensional) pattern of faces that can be folded to form the surface of a solid.

  Note: Because a net represents the surface of a polyhedra spread out in two dimensions, the area of a net equals the surface area of the corresponding solid.

• Orthogonal views are from top, front, and side; picture views are either projective or isometric; and nets are plane figures that can be folded to form the surface of the solid.

  Example: An orthogonal view (a), a projective view (b), and a net (c) of the same rectangular prism:

  

b. Recognize the five regular ("Platonic") solids.

• Count faces, edges, and vertices, and make a table with the results.

G.5.4 Find the area of shapes created out of triangles.

a. Understand, derive, and use the formula A = ½bh for the area of a triangle.

• Arrange two identical right triangles with base b and height h to form a rectangle whose area is bh. Since the area of each right triangle is half that of the rectangle, A = ½bh.

  

• If triangle ABC is not a right triangle, then placing two copies together will form a parallelogram with base b and height h. This parallelogram can be transformed into a rectangle of area bh by moving a right triangle of height h from one side of the parallelogram to the other. So here too, A = ½bh.

  

• Alternatively, divide a general triangle ABC into two right triangles as shown below and combine the areas of the two parts:

  

  Note: As the diagrams show, there are two cases to consider: For an acute triangle (where all angles are smaller than a right angle), the parts are added together. For an obtuse triangle (where one angle is larger than a right angle), one right triangle must be subtracted from the other).

b. Find the area of a convex polygon by decomposing it into triangles.

• A polygon is called convex if a line segment joining any two points on the perimeter of the polygon will lie inside or on the polygon.
• Any convex polygon of n sides can be decomposed into (n – 2) triangles.

c. Find the area of other geometric figures that can be paved by triangles.

G.5.5 Interpret and plot points on the coordinate plane.

a. Associate an ordered pair of numbers with a point in the first (upper right) quadrant and, conversely, any such point with an ordered pair of numbers.

• Positions on the coordinate plane are determined in relation to the coordinate axes, a pair of number lines that are placed perpendicular to each other so that the zero point of each coincides.

  

  Note: The coordinate plane is a two-dimensional extension of the number line and builds on extensive (but separate) prior work with the number line and with perpendicular lines.

• Recognize the similarity between locating points on the coordinate plane and locating positions on a map.
• Recognize and use the terms vertical and horizontal.

b. Identify characteristics of the set of points that define vertical and horizontal line segments.

• Use subtraction of whole numbers, fractions, and decimals to find the length of vertical or horizontal line segments identified by the ordered pairs of its endpoints.

  Example: What is the length of the line segment determined by (3/5, 0) and (1.5, 0)?

Grade 6

G.6.1 Understand and use basic properties of triangles and quadrilaterals.

a. Understand and use the angle properties of triangles and quadrilaterals.

• The sum of angles in a right triangle is 180°, since two identical right triangles form a rectangle.

  

  Note: By definition, a rectangle has 4 right angles, so the sum of the angles of a rectangle is 4 x 90° = 360°. Each right triangle contains half 360°, or 180°.

  Note: Since one angle in a right triangle is 90°, the sum of the remaining two angles is also 90°.

• Since the sum of angles in a parallelogram is also 360°, the sum of angles in any triangle is also 360°/2 = 180°.

  Note: Following the line of argument used in grade 5 to find the area of a triangle, we note that (a) two identical copies of any triangle can be arranged to form a parallelogram, and (b) any parallelogram can be transformed into a rectangle with the same angle sum by moving a triangle from one side of the parallelogram to the other.

  Note: Alternatively, following the secondary argument offered in grade 5, one can drop a perpendicular to divide any triangle into two right triangles. The sum of the interior angles of each of these right triangles is 180°, but when put together they include two superfluous right angles. Subtracting these yields 180° + 180° – (90° + 90°) = 180° as the sum of the interior angles of any triangle.

  

• Since any quadrilateral can be divided into two triangles the sum of the angles in a quadrilateral is also 2 × 180° = 360°.

  

b. Use a protractor, ruler, square, and compass to draw triangles and quadrilaterals from data given in either numerical or geometric form.

• Draw a variety of triangles (right, isosceles, acute, obtuse) and quadrilaterals (squares, rectangles, parallelograms, trapezoids) of different dimensions.
• Verify basic properties of triangles and quadrilaterals by direct measurement.

  Note: Verification by measurement requires many examples, especially some with relatively extreme or uncommon dimensions.

  Examples: In parallelograms, opposite sides and opposite angles are equal; in rectangles, diagonals are equal.

  Example: Cut any triangle out of paper and tear it into three parts so that each part contains one of the triangle’s vertices. Notice that when the angles are placed together, the edge is straight (180°).

  

• Explore properties of triangles and quadrilaterals with dynamic geometry software.

  Note: Is a parallelogram a trapezoid? It depends on the definition of trapezoid. If a trapezoid is defined as a quadrilateral with at least one pair of parallel edges, parallelograms become special cases of trapezoids. However, dictionaries usually define a trapezoid as a quadrilateral with exactly one pair of parallel edges, thereby distinguishing between parallelograms and trapezoids. Mathematicians generally prefer nested definitions as conditions become more or less restrictive. For example, all positive whole numbers are integers, all integers are rational numbers, and all rational numbers are real numbers. So in the world of mathematics, squares are rectangles, rectangles are parallelograms, and parallelograms are trapezoids.

G.6.2 Understand and use the concepts of translation, rotation, reflection, and congruence in the plane.

a. Recognize that every rigid motion of a polygon in the plane can be created by some combination of translation, rotation, and reflection.

• Translation, rotation, and reflection move a polygonal figure in the plane from one position to another without changing its length or angle measurements.
• Explore the meaning of rotation, translation, and reflection through drawings and hands-on experiments.

  Example: The figure below illustrates how a rigid motion can be decomposed into a series of three steps: translation, reflection, and rotation.

  

b. Understand several different characterizations and examples of congruence.

• Two figures in the plane are called congruent if they have the same size and same shape.
• Two shapes are congruent if they can be made to coincide when superimposed by means of a rigid motion.
• Two polygons are congruent if they have the same number of sides and if their corresponding sides and angles are equal.

  Note: Historically—beginning with Euclid—congruence applied only to polygons and used this as the definition. Indeed, the important properties of congruence are typically only about polygonal figures.

  

• Congruent figures in the plane are those that can be laid on top of one another by rotations, reflections, and translations.

  Note: Using rotations, reflections, and translations to define congruence gives precise meaning to the intuitive idea of congruence as "same size and same shape," thus permitting a precise definition of congruence for shapes other than polygons.

  Note: Technological aids (transparencies, dynamic geometry programs) help greatly in studying rigid motions.

c. Identify congruent polygonal figures.

• Understand why the two triangles formed by drawing a diagnonal of a parallelogram are congruent.
• Understand why the two triangles formed by bisecting the vertex angle of an isosceles triangle are congruent.

G.6.3 Understand and use different kinds of symmetry in the plane.

a. Symmetries in the plane are actions that leave figures unchanged.

• Explore and explain the symmetry of geometric figures from the standpoint of rotations, reflections and translations.

b. Identify and utilize bilateral and rotational symmetry in regular polygons.

• A regular polygon is a polygon whose sides and angles are all equal.
• Bilateral symmetry means there is a reflection that leaves everything unchanged.
• Regular polygons have rotational symmetries.

c. Identify and utilize translational symmetry in tessellations of the plane.

• Most tessellations have translation symmetry.

d. Use reflections to study isosceles triangles and isosceles trapezoids.

• Understand why the base angles of an isosceles triangle are equal.
• Understand why the bisector of the angle opposite the base is the perpendicular bisector of the base.

  Note: Draw an angle bisector on an isosceles triangle. Fold the drawing along the angle bisector (that is, reflect across the angle bisector). Then the base vertices collapse on each other: Both angles are equal, thus the angle bisector also bisects the base.

  

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