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Unit Name:
Creating, Comparing, and Analyzing Geometric Figures
Time Frame: 5-6 Weeks
Standards Addressed:
Ch 11 Congruence, Similarity, and Transformation
7.G.2, 7.G5, 8.G.1, 8.G.1a, 8.G.1b, 8.G.1c,
8.G.2, 8.G.3, 8.G.4, 8.G.5
Ch 12 Volume and Surface Area
7.G.3, 7.G.4, 7.G.6, 8.G9
Instructional Guidelines:
The standards document identifies ratios and proportional reasoning as one of the critical areas in 6 th and 7th grade standards. 7.G.1 which
deals with solving problems involving scale drawings, mirrors this focus on proportional reasoning. 7.G.2 introduces students to informal
geometric constructions, asking them to use understandings developed in previous grades to help them draw and classify geometric shapes.
Teachers may find it beneficial to use dynamic geometry software when teaching this standard, as the technology can help students focus on
the results of changing the conditions of the figures rather than one the process of creating the drawings. 7.G.3 requires work with threedimensional figures, which students must relate to two-dimensional figures using cross sections. The use of geometric models or dynamic
geometry software can aid this visualization process. Interpreting or drawing different views of structures may also help students as they
develop this skill.
In 3rd grade, students are introduced to the concepts of the area and liquid volume. In 4th and 5th grades, they begin to apply the area
formula for rectangles to problems and recognize volume as an attribute of a three-dimensional shape. In 5th and 6th grades, students
develop an understanding of and apply the formula for volume, and in 6th grade, they also get their first exposure to the concept of surface
area.
7.G.4 shows, by the time students reach 7th grade, they are expected not only to know and use the formulas associated with circles but also to
be able to illustrate an informal derivation of the relationship between circumference and area. This emphasis on successfully explaining a
formula or relationship rather than just memorizing and using it is a common theme throughout the Common Core standards document.
Consider that prior to the 7th grade; students calculated the area and perimeter of polygons only. Sensibly, finding the area of a circle is
delayed until after the focus on ratios and proportions begun in 6th grade and continued in 7th grade enables students to grasp the
relationship between circumference and area in a meaningful way.
7.G.5 focuses on angle measurement, introducing the idea of angles formed by intersecting lines. Looking back over the related standards in
prior grade levels, we can see that while students learn to classify shapes based on angles very early on, they are not asked to understand
angle measurement until 4th grade. That is the point at which they learn what an angle is, how to measure it and the fact that an angle
measurement is additive. Fourth graders are also asked to solve addition and subtraction problems to find unknown angles on a diagram by
using an equation with a symbol for the unknown measure. This is similar to what 7th grades are asked to do here in 7.G.5 and that students
use facts about supplementary, complementary, vertical and adjacent angles.
In 7th grade, students learned to reason about relationships among two-dimensional figures using scale drawings and informal geometric
constructions. An understanding of these concepts along with practice visualizing geometric shapes, prepares students to grasp ideas of
congruence and similarity—a critical concept they will continue to explore in high school Geometry. 8.G.2 and 8.G.4 extend students’
understanding of rigid motions described in 8.G.1 , which is fundamental to a definition of congruence and similarity transformations. In 7th
grade, students also explored the relationships between angles formed by intersecting lines. They extend that understanding to create
informal arguments about a variety of geometric constructions. Although the use of mathematical arguments is a mathematical practice
standard, 8.G.5 is the first content standard to use the term argument. Because students in 8th grade may not be familiar with the
construction of a mathematical argument, teachers should discuss the elements of a strong mathematical argument, teachers should discuss
the elements of a strong mathematical argument, as described in the mathematical practice standards section of the standards document—
namely, the sue of stated assumptions, definitions and previously established results; a logical progression of statements; justification and
communication of conclusions; and responding to the arguments of others. Practicing these components will assist students as they move
forward into the more formal argumentation processes that will be required in high school mathematics courses.
In order to understand and apply the Pythagorean Theorem and its converse, students must be able to integrate several concepts—
exponents, ratios and square roots and irrational numbers.. In 5th grade, students were introduced to exponents as they began to explore the
place value system and learn how to denote powers of 10. The 6th grade standards extend this understanding of exponents to include
numerical expressions involving exponents and also introduced ratios to include proportions and integer exponents. Another of the 8th
grade geometry standards introduces students to square roots using the square root symbol to represent solutions to equations, and the
idea that the square root of 2 is irrational.
8.G.6, 7 and 8 ask students to muster all these understandings. It is somewhat unusual for a standards document to require that students
explain a proof of the Pythagorean Theorem. However, given the emphasis the Common Core standards place on conceptual understanding
rather than just rote memorizations, the requirement is not entirely surprising.
In 6th grade, students learned to solve real-world problems concerning the volume and surface areas of right rectangular prisms; in 7th grade
they worked with problems related to the area of a circle. From this foundation, 8.G.9 asks 8th graders to expand their skills to include
working with three-dimensional shapes that have a circular component, namely, cones, cylinders, and spheres’ this work completes
students’ learning about volume.
Test 11
7.G.5, 8.G5
CC Standard
8.G.5
CC Standard
Lesson 1: Angle and Line Relationships
Section Title
Lesson 2: Triangles
8.G.3
Lessons 4: Translations and Reflections
on the Coordinate Plane
CC Standard
Section Title
8.G.3
Plane
8.G.5
CC Standard
8.G.2
Transformations
CC Standard
Section Title
Lesson 3: Polygons
CC Standard
Lesson 5: Rotations on the Coordinate
Section Title
Section Title
Lesson 6: Congruence and
Section Title
8.G.3
PlaneLAB
CC Standard
8.G.4
CC Standard
Lesson 7: Dilations on the Coordinate
Section Title
Lesson 8: Similarity and Transformation
Section Title
Test 12
7.G.4
CC Standard
7.G.4
CC Standard
Lesson 1: Circles and Circumference
Section Title
Lesson 2: Area of Circles
Section Title
8.G.9
CC Standard
8.G.9
Spheres
CC Standard
7.G.6
CC Standard
CC Standard
Section Title
7.G.6
CC Standard
Lesson 7: Volume of Pyramids, Cones, and
Section Title
Lesson 8: Surface Area of Prisms
Section Title
Lesson 4: Three-Dimensional Figures
Section Title
7.G.4
CC Standard
7.G.6
Section Title
Lesson 3: Area of Composite Figures
CC Standard
7.G.3
Lesson 6: Volume of Cylinders
Lesson 9: Surface Area of Cylinders
Section Title
Lesson 5: Volume of Prisms
Section Title
7.G.6
Cones
CC Standard
Lesson 10: Surface Area of Pyramids and
Section Title
CC Math Standard Achievement Level Descriptors
Geometry
Draw, construct, and describe
geometrical figures and describe the
relationships between them.
7.G.1: Solve problems involving scale
drawings of geometric figures, including
computing actual lengths and areas from
a scale drawing and reproducing a scale
drawing at a different scale.
7.G.2: Draw (freehand, with ruler and
protractor, and with technology)
geometric shapes with given conditions.
Focus on constructing triangles from
three measures of angles or sides,
noticing when the conditions determine
a unique triangle, more than one
triangle, or no triangle.
7.G.3: Describe the two-dimensional
figures that result from slicing threedimensional figures, as in plane sections
of right rectangular prisms and right
rectangular pyramids.
Level 1
Draw or construct
geometric shapes with
given conditions by
freehand, with ruler
and protractor, and by
using technology.
Level 2
Describe the
relationship between a
geometric
figure and its scale
drawing by finding the
scale factor
between them.
Determine whether
or not a set of any three
given angle or sidelength
measures can result in
a unique triangle, more
than
one triangle, or no
triangle at all.
Describe geometric
shapes with given
conditions.
Level 3
Level 4
Compute actual lengths
and areas from a scale
drawing and reproduce
a scale drawing using a
different scale.
Describe the twodimensional figures
that result from slicing
prisms and pyramids
by planes that are
parallel to a face.
Describe the twodimensional figures
that result from slicing
cones, spheres,
cylinders, or other
three-dimensional
figures with
rectangular
or triangular faces by
planes that are not
parallel to a given face.
Solve real-life and mathematical
problems involving angle measure, area,
surface area, and volume.
7.G.4: Know the formulas for the area
and circumference of a circle and use
them to solve problems; give an
informal derivation of the relationship
between the circumference and area of a
circle.
7.G.5: Use facts about supplementary,
complementary, vertical, and adjacent
angles in a multi-step problem to write
and solve simple equations for an
unknown angle in a figure.
7.G.6: Solve real-world and
mathematical problems involving area,
volume and surface area of two- and
three-dimensional objects composed of
triangles, quadrilaterals, polygons,
cubes, and right prisms
Level 1
Level 2
Level 3
Level 4
Identify appropriate
formulas for the
area and circumference
of a circle.
Classify pairs of angles
as
supplementary,
complementary,
vertical, or adjacent.
Measure
angles with
appropriate tools.
Calculate the
circumference of a
circle. Calculate the
area of
circles.
Use supplementary,
complementary,
vertical, or adjacent
angles to solve
problems with angles
expressed as numerical
measurements in
degrees.
Use formulas for the
area and circumference
of a circle to solve
problems.
Use supplementary,
complementary,
vertical, and adjacent
angles to solve one- or
two-step problems
with angle measures
expressed as variables
in degrees.
Use supplementary,
complementary,
vertical, and adjacent
angles to solve multistep problems with
angle measures
expressed as variables
in degrees.
Calculate the area of
quadrilaterals, and
polygons. Calculate
the volume of right
rectangular prisms.
Solve problems
involving the area of
polygons, the surface
area of threedimensional objects
composed of triangles
and/or quadrilaterals,
and the volume of right
prisms.
Solve problems
involving surface area
and volume of threedimensional figures
with polygonal faces.
Calculate the area of
triangles and
rectangles and the
volume of cubes.
Geometry
Understand congruence and similarity
using physical models, transparencies, or
geometry software.
8.G.1: Verify experimentally the properties
of rotations, reflections, and translations.
a. Lines are taken to lines, and line
segments to line segments of the same
length
b. Angles are taken to angles of the same
measure.
c. Parallel lines are taken to parallel lines.
Level 1
Level 1 students
should be able to
identify reflections,
rotations, and
translations and the
result of these
rigid motions on
figures.
Level 2
Level 3
Level 2 students
should be able to
construct
reflections and
translations of figures
in a coordinate
plane.
They should also be
able to
construct rotations
and dilations of figures
in a
coordinate plane.
8.G.2: Understand that a two-dimensional
figure is congruent to another if the second
can be obtained from the first by a
sequence of rotations, reflections, and
translations; given two congruent figures,
describe a sequence that exhibits the
congruence between them.
8.G.3: Describe the effect of dilations,
translations, rotations, and reflections on
two-dimensional figures using coordinates.
Identify dilations and
the results of dilations
on figures.
They should be able to
use or describe a
sequence of
transformations to
determine or exhibit
the
congruence of two
figures.
Level 3 students
should be able to
understand and
describe the impact of
a transformation on a
figure
and its component
parts with or without
coordinates.
Level 4
Level 4 students
should be able to
describe a sequence
that exhibits the
similarity between two
shapes and
understand that the
angle measures
are unchanged.
8.G.4: Understand that a two-dimensional
figure is similar to another if the second can
be obtained from the first by a sequence of
rotations, reflections, translations, and
dilations; given two similar twodimensional figures describe a sequence
that exhibits the similarity between them.
8.G.5: Use informal arguments to establish
facts about the angle sum and exterior
angle of triangles, about the angles created
when parallel lines are cut by a transversal,
and the angle-angle criterion for similarity
of triangles. For example, arrange three
copies of the same triangle so that the sum
of the three angles appears to form a line,
and give an argument in terms of
transversals why this is so.
Understand and apply the Pythagorean
Theorem.
8.G.6: Explain a proof of the Pythagorean
Theorem and its converse.
8.G.7: Apply the Pythagorean Theorem to
determine unknown side lengths in right
triangles in real-world and mathematical
problems in two and three dimensions.
Level 1
Level 1 students
should be able to
identify the
hypotenuse and the
legs of a right triangle
given the side lengths
or an image of a right
triangle.
Level 1 students
should be able to
identify the
hypotenuse and the
legs
of a right triangle given
the side
lengths or an image of
a right
triangle.
Level 2
Level 3
Level 2 students
should be able to apply
the Pythagorean
theorem to determine
whether or not a
given triangle is a right
triangle, given its side
lengths.
Level 3 students
should be able to apply
the Pythagorean
theorem to determine
the unknown side
lengths of right
triangles.
Level 4
8.G.8: Apply the Pythagorean Theorem to
find the distance between two points in a
coordinate system.
Level 1 student should
be able to identify the
hypotenuse and the
legs of a right triangle
given the side lengths
or an image of a right
triangle.
They should be able to
find the distance
between two points on
a horizontal or vertical
line in a two
dimensional
coordinate system.
Find the distance
between two points in
a coordinate system in
two dimensions.
Level 4 students
should be able to
apply the Pythagorean
theorem to
find the distance
between two points
in a coordinate system
in three dimensions.
Solve real-world and mathematical
problems involving volume of cylinders,
cones, and spheres.
Level 1
Level 2
Level 3
Level 4
Level 2 students
should be able to
identify the
appropriate formula
for the volumes of a
cone, a cylinder, and a
sphere and should be
able to connect the key
dimensions to the
appropriate locations
in the formula.
Level 3 students
should be able to
calculate the volumes
of cones, cylinders, and
spheres in direct and
familiar mathematical
and real-world
problems.
Level 4 students
should be able to solve
unfamiliar or multistep problems
involving volumes of
cones, cylinders, and
spheres.
Level 1 students
should be able to
8.G.9: Know the formulas for the volumes of identify the key
cones, cylinders, and spheres and use them dimensions (i.e., radii,
to solve real-world and mathematical
heights,
problems.
circumferences, and
diameters) of cones,
cylinders, and spheres.
Student Misconceptions/Errors
Chapter 11 -- Congruence, Similarity, and Transformation
Lesson 2: All angles are equal in an equilateral triangle
Lesson 3: Students may just divide each number by 180 to find the number of sides. Remind students that 2 is subtracted from the
number of sides when they find the sum of the interior angle measures, so they need to add 2 after dividing.
Lesson 4: Remind students that the reflection of 0 over the y-axis is 0, and the reflection of any point on the
axis remains on the y-axis.
y-axis over the y-
Lesson 5: Remind students to use mathematical symbols and an analysis to describe the appearance of the figure rotated about
the origin.
Lesson 6: Remind students to use the transformations in the order specified.
Lesson 7: Remind students to be sure to multiply both coordinates of vertex by the scale factor to determine the coordinates of its
image.
Student Misconceptions/Errors
Chapter 12 -- Volume and Surface Area
Lesson 1: Students create a table of values and a graph in the coordinate plane, and use verbal analysis to relate the slope of the
graphed line to the ratio of the circumference of a circle to its radius.
Lesson 2: Students use a table to show how the area of a circle changes when the radius is changed.
Lesson 4: Remind students to use a table of values and numberical analysis to write an equation relating vertices, faces, and edges
of polyhedral figures.
Lesson 5: Encourage students to first identify the base of the triangular prism, then substitute the values into the volume formula,
ie: a triangular prism’s base is a triangle and not a rectangular.
Lesson 8: Remind students that a cube is a rectangular prism where all six faces have the same area.
Lesson 9: Remind students that the lateral area of the half-cylinder will be half the lateral area of a complete cylinder plus the area
of the rectangular surface.