Download Geometry standards

Geometry standards
MATH 124
Kindergarten
Identify and describe shapes.
• CCSS.Math.Content.K.G.A.1
Describe objects in the environment using names of
shapes, and describe the relative positions of these
objects using terms such as above, below, beside, in
front of, behind, and next to.
• CCSS.Math.Content.K.G.A.2
Correctly name shapes regardless of their
orientations or overall size.
• CCSS.Math.Content.K.G.A.3
Identify shapes as two-dimensional (lying in a plane,
"flat") or three-dimensional ("solid").
Analyze, compare, create, and compose shapes.
• CCSS.Math.Content.K.G.B.4
Analyze and compare two- and three-dimensional shapes, in
different sizes and orientations, using informal language to
describe their similarities, differences, parts (e.g., number of
sides and vertices/"corners") and other attributes (e.g., having
sides of equal length).
• CCSS.Math.Content.K.G.B.5
Model shapes in the world by building shapes from
components (e.g., sticks and clay balls) and drawing shapes.
• CCSS.Math.Content.K.G.B.6
Compose simple shapes to form larger shapes. For example,
"Can you join these two triangles with full sides touching to
make a rectangle?"
Analysis
• Students are expected to be at Van Hiele level 0.
▫ Students recognize and name shapes.
▫ What shapes “look like”
▫ They may not understand that orientation does
not affect the shape.
▫ Students at this level can begin to understand
classifications of shapes.
First grade
Reason with shapes and their attributes.
• CCSS.Math.Content.1.G.A.1
Distinguish between defining attributes (e.g., triangles are closed and threesided) versus non-defining attributes (e.g., color, orientation, overall size);
build and draw shapes to possess defining attributes.
• CCSS.Math.Content.1.G.A.2
Compose two-dimensional shapes (rectangles, squares, trapezoids,
triangles, half-circles, and quarter-circles) or three-dimensional shapes
(cubes, right rectangular prisms, right circular cones, and right circular
cylinders) to create a composite shape, and compose new shapes from the
composite shape.1
• CCSS.Math.Content.1.G.A.3
Partition circles and rectangles into two and four equal shares, describe the
shares using the words halves, fourths, and quarters, and use the phrases
half of, fourth of, and quarter of. Describe the whole as two of, or four of
the shares. Understand for these examples that decomposing into more
equal shares creates smaller shares.
Analysis
• A few of you made the observation about the
relationship between geometry and fractions in
the standards. This is probably just
acknowledgment that when you are breaking up
a square, circle, or rectangle into equal parts,
you are also using geometric reasoning.
• There is a neat progression of this
understanding between grades 1-3: halves and
fourths in first grade, thirds in second grade,
and other parts in third.
• First grade still seems to be at Ven Hiele level 0,
but moving toward level 1. The focus seems to be
on manipulating shapes and developing
visualization. However, understanding defining
versus non-defining characteristics is level 1.
Second grade
Reason with shapes and their attributes.
• CCSS.Math.Content.2.G.A.1
Recognize and draw shapes having specified attributes, such
as a given number of angles or a given number of equal faces.1
Identify triangles, quadrilaterals, pentagons, hexagons, and
cubes.
• CCSS.Math.Content.2.G.A.2
Partition a rectangle into rows and columns of same-size
squares and count to find the total number of them.
• CCSS.Math.Content.2.G.A.3
Partition circles and rectangles into two, three, or four equal
shares, describe the shares using the words halves, thirds, half
of, a third of, etc., and describe the whole as two halves, three
thirds, four fourths. Recognize that equal shares of identical
wholes need not have the same shape.
Analysis
• The focus in second grade is still on using
geometry to develop number and operation
sense, so area us used to help understand
multiplication, and students split regions into
thirds and not just halves.
• Recognizing attributes is Van Hiele level 1, but
no great geometric development happens in
grade 2.
Third grade
Reason with shapes and their attributes.
• CCSS.Math.Content.3.G.A.1
Understand that shapes in different categories (e.g.,
rhombuses, rectangles, and others) may share attributes (e.g.,
having four sides), and that the shared attributes can define a
larger category (e.g., quadrilaterals). Recognize rhombuses,
rectangles, and squares as examples of quadrilaterals, and
draw examples of quadrilaterals that do not belong to any of
these subcategories.
• CCSS.Math.Content.3.G.A.2
Partition shapes into parts with equal areas. Express the area
of each part as a unit fraction of the whole. For example,
partition a shape into 4 parts with equal area, and describe
the area of each part as 1/4 of the area of the shape.
Analysis
• In third grade, it seems that students should be
at Van Hiele level 1, progressing to level 2.
• At level 1, students begin to notice properties of
shapes, as they are supposed to do in third grade
with quadrilaterals.
• However, students at level 1 do not necessarily
understand that different quadrilaterals are
subclasses of one another, which is why third
grade standards progress toward level 2.
Fourth grade
Draw and identify lines and angles, and classify shapes by
properties of their lines and angles.
• CCSS.Math.Content.4.G.A.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines. Identify these in twodimensional figures.
• CCSS.Math.Content.4.G.A.2
Classify two-dimensional figures based on the presence or absence
of parallel or perpendicular lines, or the presence or absence of
angles of a specified size. Recognize right triangles as a category,
and identify right triangles.
• CCSS.Math.Content.4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line
across the figure such that the figure can be folded along the line
into matching parts. Identify line-symmetric figures and draw lines
of symmetry.
Analysis
• In fourth grade, many new terms are defined
and used, but still with the intention of
classifying shapes.
• No great geometric developments are made in
grade 4, and students are still expected to be at
level 1.
Fifth grade
Classify two-dimensional figures into
categories based on their properties.
• CCSS.Math.Content.5.G.B.3
Understand that attributes belonging to a category
of two-dimensional figures also belong to all
subcategories of that category. For example, all
rectangles have four right angles and squares are
rectangles, so all squares have four right angles.
• CCSS.Math.Content.5.G.B.4
Classify two-dimensional figures in a hierarchy
based on properties.
Analysis
• In fifth grade, all the work from K-4 grade comes
together. Quadrilaterals and triangles are fully
classified based on their properties.
• According to the standards, fifth grade students
should be at Van Hiele level 2: they think about
properties of geometric objects and are able to
reason about them.
• Some of you noted that these standards are
demanding.
• It might be unreasonable to expect that students will
go from level 0 to level 2 in five years. It is possible if
they are exposed to a lot of hands-on activities
where they can interact with objects of study, but
unfortunately geometry is not the primary focus in
elementary classrooms. It helps that some
connections are made between geometry and
operation standards.
• My recommendation: use a lot of hands-on activities
with young children to help them progress through
the Van Hiele levels.
Van Hiele level 3
• High school geometry classes are taught at level
3, and if students have not progressed to level 2
in K-8 education, they will not be able to succeed
in such a class.
Coordinate geometry?
Graph points on the coordinate plane to solve real-world
and mathematical problems.
• CCSS.Math.Content.5.G.A.1
Use a pair of perpendicular number lines, called axes, to define a
coordinate system, with the intersection of the lines (the origin)
arranged to coincide with the 0 on each line and a given point in the
plane located by using an ordered pair of numbers, called its
coordinates. Understand that the first number indicates how far to
travel from the origin in the direction of one axis, and the second
number indicates how far to travel in the direction of the second
axis, with the convention that the names of the two axes and the
coordinates correspond (e.g., x-axis and x-coordinate, y-axis and ycoordinate).
• CCSS.Math.Content.5.G.A.2
Represent real world and mathematical problems by graphing
points in the first quadrant of the coordinate plane, and interpret
coordinate values of points in the context of the situation.
Activity (Level 2)
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If it is a square, then it is a rhombus.
All squares are rectangles.
Some parallelograms are rectangles.
All parallelograms have congruent diagonals.
If it has exactly two lines of symmetry, it must be
a quadrilateral.
• All pyramids have square bases.
Area and geometry?
• Starting in 6th grade, geometry standards start
talking about area. Why not before?
• Because area does come up before, but in the
measurement strand (starting in third grade).
Geometry and measurement are related but
different strands of mathematics. Both are
taught in geometry courses.
• We will look at area in some detail in the next
few weeks, and will revisit the measurement
standards next week.
Congruence and similarity
• Because these are middle school topics, we are
leaving them for the last week of classes as
optional material.
Anything else?