Download 4.G.A.1

4.G.A.1
*This standard is part of an additional cluster
Standard
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular
and parallel lines. Identify these in two-dimensional figures.
Unpacked
Students describe, analyze, compare, and classify two-dimensional shapes by their
properties including explicit use of angle sizes 4.G.A.1 and the related geometric
properties of perpendicularity and parallelism. Such mathematical terms are useful in
communicating geometric ideas, but more important is that constructing examples of
these concepts, such as drawing angles and triangles that are acute, obtuse, and
right,4.G.A.1 help students form richer concept images connected to verbal definitions.
That is, students have more complete and accurate mental images and associated
vocabulary for geometric ideas (e.g., they understand that angles can be larger than 90_
and their concept images for angles include many images of such obtuse angles).
Similarly, students see points and lines as abstract objects: Lines are infinite in extent and
points have location but no dimension. Grids are made of point and lines and do not end
at the edge of the paper.
Students also learn to apply these concepts in varied contexts (MP4). For example, they
learn to represent angles that occur in various contexts as two rays, explicitly including
the reference line, e.g., a horizontal or vertical line when considering slope or a “line of
sight” in turn contexts. They understand the size of the angle as a rotation of a ray on the
reference line to a line depicting slope or as the “line of sight” in computer environments.
Students might solve problems of drawing shapes with turtle geometry.• Analyzing the
shapes in order to construct them (MP1) requires students to explicitly formulate their
ideas about the shapes (MP4, MP6). For instance, what series of commands would
produce a square? How many degrees would the turtle turn? What is the measure of the
resulting angle? What would be the commands for an equilateral triangle? How many
degrees would the turtle turn? What is the measure of the resulting angle? Such
experiences help students connect what are often initially isolated ideas about the concept
of angle.
This standard asks students to draw two-dimensional geometric objects and to also
identify them in two-dimensional figures. This is the first time that students are exposed
to rays, angles, and perpendicular and parallel lines. Examples of points, line segments,
lines, angles, parallelism, and perpendicularity can be seen daily. Students may not easily
identify lines and rays because they are more abstract. Thus, students develop explicit
awareness of and vocabulary for many concepts they have been developing, including
points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Such mathematical terms are useful in communicating geometric ideas, but
more important is that constructing examples of these concepts, such as drawing angles
and triangles that are acute, obtuse, and right, help students form richer concept images
connected to verbal definitions. That is, students have more complete and accurate mental
images and associated vocabulary for geometric ideas (e.g., they understand that angles
can be larger than 90 and their concept images for angles include many images of such
obtuse angles). Similarly, students see points and lines as abstract objects: Lines are
infinite in extent and points have location but no dimension. Grids are made of points and
lines and do not end at the edge of the paper.
Points, lines, segments, rays, and angles are the building blocks of the geometry. Point
and line are undefined terms because they do not have definitions. We can understand
these terms by thinking of examples of what a point and line might look like. A point can
be a tip of a pencil; it has position but no dimension. Euclid described a line by saying
that through any two points there is always a line and every line contains at least two
points. Line segment is part of a line and it contains two endpoints meaning it has a
beginning and endpoints. A line contains an infinite number of points and has no
endpoints and goes on and on forever. A ray is part of a line that has one endpoint and
extends forever in only one direction. Parallel lines are lines that never cross and are the
same distance apart. Perpendicular lines intersect to form right angles.
Analyzing the shapes in order to construct them requires students to explicitly formulate
their ideas about the shapes. For instance, what series of commands would produce a
square? How many degrees are the angles? What is the measure of the resulting angle?
What would be the commands for an equilateral triangle? How many degrees are the
angles? What is the measure of the resulting angle? Such experiences help students
connect what are often initially isolated ideas about the concept of angle.
(Progressions for the CCSSM, Geometry, CCSS Writing Team, June 2012, page 14)
Questions to check for understanding/ increase the rigor:
• Look at these pictures below; what points, line segments, rays, angles, parallel
lines, and perpendicular lines do you see? Do you see any two-dimensional
figures?
•
•
Why do you think points, lines, line segments, rays, angles, parallel lines, and
perpendicular lines are considered to be the building blocks of geometry?
Where in life might you find parallel lines? Where might you find perpendicular
lines?
•
Level 1
Entering
Identify point,
line, line
segment, ray,
right angle,
acute angle,
obtuse angle,
perpendicular
lines, and
parallel lines
on drawings
of 2D figures
by pointing to
the correct
picture when
given a word.
Level 2
Emerging
Identify
point, line,
line
segment,
ray, right
angle, acute
angle,
obtuse
angle,
perpendicul
ar lines, and
parallel lines
on drawings
of 2D
figures,
matching
terms to
pictures
using a word
bank when
teacher
points to a
term.
Level 3
Developing
Identify
point, line,
line
segment,
ray, right
angle, acute
angle,
obtuse
angle,
perpendicul
ar lines, and
parallel
lines on
drawings of
2D figures
by matching
labels to
pictures.
Level 4
Expanding
Identify
point, line,
line
segment,
ray, right
angle, acute
angle,
obtuse
angle,
perpendicul
ar lines, and
parallel lines
on drawings
of 2D figures
by matching
pictures to
given terms.
Level 5
Bridging
Identify
point, line,
line
segment,
ray, right
angle, acute
angle,
obtuse
angle,
perpendicul
ar lines, and
parallel lines
on drawings
of 2D figures
by matching
terms to
given
pictures.
Level 6 Reaching
Speaking
•
•
Using one rubber band on a geoboard, create a square using the corner pegs. If
you were to only move one of the corners, what other shapes could you make?
How many different shapes can you make using one rubber band on the
geoboard?
Draw two different types of quadrilaterals that have two pairs of parallel sides?
Is it possible to have an acute right triangle? Justify your reasoning using pictures
and words.
ELD Standard #3: English Language Learners communicate information, ideas,
and concepts necessary for academic success in the content area of
Mathematics.