Grade 6
Geometry
Standard 6-4: The student will
demonstrate through the mathematical processes an understanding of shape, location, and
movement within a coordinate system; similarity, complementary, and supplementary
angles; and the relationship between line and rotational symmetry.
The indicators for this standard
are grouped by the following major concepts:
The
indicators that support each of those major concepts and an explanation of the
essential learning for each major concept follows.
Dimensional
Indicators
6-4.7 Compare the angles, side lengths, and perimeters of similar
shapes.
6-4.8 Classify shapes as similar.
6-4.9 Classify pairs of angles as either complementary or
supplementary.
Sixth
grade is a transition and application year for students’ knowledge of
geometry. The focus begins to move from an informal
experience of geometry to a
more formal deductive experience.
Students are expected to make conjectures,
describe, and justify geometric
concepts. It is extremely important for
teachers to
provide concrete experiences
through engaging activities. It is through models and
hands-on activities that students
develop an understanding of geometry and begin
to analyze concepts in a more formal
manner.
In
fifth grade the emphasis was on congruency and students classified shapes as
congruent by comparing the angles,
side lengths, and perimeters of an object. In
6th grade the emphasis
is on similarity and students should classify shapes as
similar by comparing the angles
and proportional relationships of the side lengths
and perimeters.
Sixth grade is the first time students
are formally introduced to the concept of similarity. Therefore, experiences
should actively engage and enable students to discover that similar figures
have the same shape, equal corresponding angle measures, and proportional
corresponding side lengths. This can be accomplished through the use of similar
geometric manipulatives and similar shapes formed on geoboards or dot paper to
compare angles, side lengths and perimeters. The exploration of similarity
provides the opportunity to review and apply measurement skills as students
measure side lengths and angles to determine if two shapes are similar.
While on the surface comparison of
angles, side lengths, and perimeters of similar shapes may appear to be a
simple concept, the indicator requires a more in-depth level of mathematical
understanding. One of the most common mistakes students make is comparing sides
or angles that are not
corresponding. Therefore, it is important that students understand the concept
of correspondence. Two polygons are in
correspondence when consecutive vertices of one are matched with consecutive
vertices of the other. It is critical that
students discover that the corresponding angle measures will be equal whereas
the corresponding side lengths and perimeters are proportional. With this knowledge
students should determine and justify similarity of shapes.
Sixth grades students should also
understand and be able to determine and apply the concepts of complementary and
supplementary in regards to angle measures.
The sum of the angle measures of a pair of complementary angles equals
90 degrees, whereas the sum of the angle measures of a pair of supplementary
angles equals 180 degrees. Students should have the opportunity to use
geometric manipulative shapes or to cut out angle measures on paper to create angle
pairs of 90 degrees and 180 degrees as they learn and explore these concepts. This allows students to create mental models
of the concept. The teacher can then move to the more abstract by giving
students drawings with angle measurements and asking them to determine if the
angles are complementary or supplementary.
Teacher Note: Students are not to
find missing sides of similar shapes in the 6th grade. Finding the missing side of a similar shape
is a 7th grade indicator.
Plane and Transformational
Indicators
6-4.1 Represent with ordered pairs of integers the location of points
in a coordinate grid.
6-4.2 Apply strategies and procedures to find the coordinates of the
missing vertex of a square, rectangle, or right triangle when given the
coordinates of the polygon’s other vertices.
6-4.3 Generalize the relationship between line symmetry and rotational
symmetry for two-dimensional shapes.
6-4.4 Construct two-dimensional shapes with line
or rotational symmetry.
6-4.5 Identify the transformation(s) used to move a polygon from one
location to another in the coordinate plane.
6-4.6 Explain how transformations affect the location of the original
polygon in the coordinate plane.
In 4th grade, students identified and
named points in the first quadrant only
of a coordinate plane/grid. (The terms coordinate
plane and coordinate grid can be used interchangeably. It is
important that students be exposed to both.) In 6th grade, students are to understand
integers and this understanding should be extended and applied as they are now
expected to represent with ordered pairs of integers the location of points in
a coordinate plane. Students must now
understand that the coordinate plane they used in the 4th grade was only the
first quadrant of a plane.
Because 6th grade is the first time
students are introduced to plotting and naming points in all four quadrants of
a coordinate plane, students should be actively engaged through the use of
concrete models to develop an in-depth understanding and competency in regards
to this indicator. One way of doing this
is to have students build a coordinate grid by creating two number lines (-
Having this understanding, students
should now apply strategies and procedures to find the coordinates of the
missing vertex of a square, rectangle, or right triangle when given the
coordinates of the polygon’s other vertices. This provides an opportunity to build on
students' prior knowledge of quadrilaterals (a 4th and 5th grade indicator) and
right triangles (a 3rd grade indicator). Polygons should all be oriented
horizontally on the coordinate plane with vertices at ordered pairs containing
integers. Therefore no polygon should appear diagonally or contain fractional
points. Please note also that a polygon
can be placed on the coordinate plane in such a way that it has vertices in
more than one quadrant.
In 5th grade, students predicted the
results of multiple transformations on a geometric shape when combinations of
translations, reflections, and rotations were used. Once 6th grade students
have attained competency with polygons in the coordinate plane, this knowledge
should be extended to identifying the transformation(s) used to move a polygon
from one location to another in the coordinate plane and to explain how
transformations affect the location of the original polygon in the coordinate
plane.
In a translation, (sometimes called a
slide) every point in the figure slides the same distance in the same
direction. A reflection (sometimes
called a flip) is when a figure is flipped over a line. Each point in a reflection image is the same
distance from the line as the corresponding point in the original shape. A rotation (sometimes called a turn) is when
an object is turned. Rotating a figure
means turning a figure around a point.
The point can be on the figure, but it doesn't have to be. The point is called the center of rotation. The
angle that the figure turns is called the angle
of rotation. When you describe the
rotation of a figure, you give the direction, the angle of rotation, and the
center of rotation. It is important for 6th
grade students to use correct terminology (translation, reflection, rotation)
to describe the change made to the figure or polygon. For example: The polygon was translated 2 units to the
right and 3 units down. Using the
coordinate grid provides reference points for students to use when describing
the transformation.
Fifth grade students analyzed shapes
to determine line symmetry and/or rotational symmetry. In 6th grade, students should construct
two-dimensional shapes with line and rotational symmetry and generalize the
relationship between line symmetry and rotational symmetry for two-dimensional
shapes. A two-dimensional shape has line
symmetry if you can fold the figure along a line so that it has two parts that
match exactly. A shape can have no lines of symmetry, one line of symmetry, or
more than one line of symmetry. The folded line(s) is called the line of symmetry. A shape that rotates onto itself before
turning 360o has rotational symmetry. It is important to note that
all regular (meaning all side lengths are equal lengths) polygons have
rotational symmetry. Through hands-on investigations students should be able to
describe and identify those shapes that have line symmetry, those that have
rotational symmetry and those that have both.