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Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
Sample Assessment
Connections
Critical for Mastery at Grade 10:
10.09.01
Identify possible types of two- or threedimensional figures that would match a set
of given conditions. (9A)
10.09.02
Determine if a triangle is possible using side
lengths and the triangle inequality. (9A)
10.09.03
Create and critique arguments concerning
geometric ideas and relationships such as
properties of circles, triangles and
quadrilaterals. (9C)
Make a chart with properties of a special
parallelogram ( four sides congruent,
opposite angles congruent, etc) across the
top and types of special parallelograms down
side. Ask students to check the correct
properties or attributes. Use sometimes, always,
never.
Connect to science: Analyze
three-dimensional figures in nature.
Consider lines of symmetry, rotational
patterns.
Make available several sets of three straws of
different lengths. Ask students to connect them
end to end to form a triangle. Find the minimum
and maximum length for a possible third side.
Construct a triangle given an adjacent side
and opposite side to a given angle.
Give students several printed circles. Ask
them to draw several chords and their
perpendicular bisectors in each circle. Find the
center of the circle.
Connect to social studies: Research how
archeologists reconstruct broken plates
found in ruins.
Connect the midpoints of the sides of a
quadrilateral and conjecture about the shape.
10.09.04
Develop a formal proof for a given geometric
situation on the plane. (9C)
Prove the properties of an isosceles triangle.
Look at pattern generated by the
Fibonacci sequence.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.05
Describe and apply properties of a polygon
or a circle in a problem-solving situation. (9A)
10.09.06
Classify angle relationships for two or more
parallel lines crossed by a transversal. (9A)
Sample Assessment
Connections
Make a sketch showing how to tile a floor
using both squares and regular octagons.
Connect to science: Note how often regular
polygons are found in nature.
Cross two parallel lines with a transversal.
Mark a pair of alternate interior angles with
measures of (3x+20) degrees and (x+50)
degrees.
Analyze and discuss the appearance of
non-parallelism: for example, examine
pictures of railroad tracks.
Research Rhiemann and Lobochevsky
geometry.
10.09.07
Represent, solve, and explain numerical and
algebraic relationships using geometric
concepts. (9C)
10.09.08
Provide examples or counter-examples to
either illustrate or disprove conjectures about
geometric characteristics. (9C)
10.09.09
Determine and justify the side length
relationships present in 45o-45o-90o triangles
and 30o-60o-90o triangles. (9D)
Using graph paper, draw two parallel lines and
cross with a transversal. Show that the lines
are parallel, the slopes are the same, the
corresponding angles are congruent.
Prove or disprove that an obtuse triangle
cannot contain a right angle.
Given the altitude of an equilateral triangle,
find the length of the side of the equilateral
triangle. Repeat for isosceles right triangle.
Draw a triangle on a deflated balloon.
Blow up the balloon. Discuss triangle
angle sum.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.10
Solve problems using 45o-45o-90o and
30o-60o-90o triangles. (9D)
10.09.11
Identify and apply properties of medians,
altitudes, angle bisectors, perpendicular
bisectors, and mid segments of a triangle.
(9A)
10.09.12
Solve problems using triangle congruence
and similarity of figures. (9B)
10.09.13
Extend knowledge of plane figure
relationships to relationships within and
between geometric solids. (9B)
10.09.14
Identify relationships among circles, arcs,
chords, tangents, and secants. (9B)
Sample Assessment
Connections
Each side of a regular hexagon measures
10 inches. Find the lengths of the diagonals
from one vertex. Repeat for a square.
Find the center of gravity in a triangle.
Analyze the Sierpinski Triangle.
Analyze the nine-point circle.
A utility pole is supported by two guy wires
of equal length. Reason why or why not the
wires are fastened to the ground at the same
distance from the base of the pole.
Connect to environmental values: Connect
to the career of wind prospectors.
Determine if a cube can have the same
number of cubic inches in volume as
square inches in total area.
Build a tetrahedral kite; note the space
between the tetrahedrons.
Two circles of different radii are tangent.
Find the length of their common external
tangent.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.15
Create and critique arguments concerning
geometric ideas and relationships, such as
congruence, similarity, the Pythagorean
relationship, or formulas for surface areas or
volume of simple three-dimensional objects.
(9C)
10.09.16
Describe the difference between an inductive
argument and a deductive argument. (9C)
10.09.17
Determine distances and angle measures
using indirect measurement and properties of
right triangles. (9D)
10.09.18
Create and analyze scale models using
proportional reasoning. (9B)
Sample Assessment
Connections
Given two similar cones with a sixty degree
angle made by the slant height and axis.
Discuss and compare the ratio of axes, ratio
of surface areas and ratio of volumes.
Instruct students to measure the angles of several
triangles using a protractor, and then form a
conclusion about the sum of angles in a triangle.
(inductive) Have students use parallel lines to
prove that the sum of the angles of a triangle is
180. Discuss the two methods.
Connect to a detective murder mystery.
Play the game of Clue.
A lumberjack stands 60 feet from the base
of a tree and measures the angle of elevation
to the top of the tree to be 60 degrees. Is the
tree more than 100 feet tall?
Create a problem about the line of descent
for an airplane and the runways at an
airport.
Draw a replica of a garden with dimensions
48 ft. by 45 ft. The scale choices are
1 in. = 2 ft., 1 in. = 3 ft., and 1 in. = 4 ft.
Decide which choice is most reasonable and
explain.
Analyze a blueprint of your home.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
Sample Assessment
Connections
Significant to Develop at Grade 10:
10.09.19
Solve problems in, and gain insights into,
other disciplines and other areas of interest
such as art and architecture using geometric
ideas. (9B)
Find the length of a golden rectangle given
the width.
Look at pictures of buildings
constructed by the ancient Greeks.
Find the golden rectangles in
architecture of the past and of today.
Using coordinate geometry, prove that the
diagonals of a square are perpendicular.
Learn about taxicab geometry as it
relates to city planning.
10.09.21
Analyze the results of a combination of
reflections, rotations, and translations of a
figure, and determine alternate motions that
could produce the same results. (9A)
Draw a polygon in the corner of a piece of
patty paper. Fold the paper in half and then
in half again. Trace the resulting image.
Discuss the rotation and reflection.
Determine the minimum length of
a full length mirror necessary to see
your entire image.
10.09.22
Design a net that will create a given figure
when folded. (9A)
Construct a net that will form a rectangular
based box with given dimensions.
How much wrapping paper is needed
to wrap a large boxed package.
Research the Japanese method of
wrapping.
.
Connect to art and culture: Create origami
animals or other shapes.
10.09.20
Represent and analyze the properties of
geometric shapes using coordinate
geometry. (9A)
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.23
Recognize Pythagorean Triples. (9D)
Sample Assessment
Show that a triangle with sides that are
multiples of 9, 40, and 41 is a right triangle.
Explain why the sides of a right triangle cannot
be 5, 6, 7.
10.09.24
Develop conjectures about geometric
situations with and without technology. (9C)
Using Geometer’s Sketchpad, list conjectures
about the properties of a kite.
10.09.25
Identify the basic trigonometric ratios in
terms of lengths of the sides of a right
triangle and an acute angle. (9D)
Given a 3,4,5 triangle, write the sine, cosine
and tangent of each of the acute angles.
10.09.26
Solve for missing side lengths using the
trigonometric ratios in right triangles. (9D)
10.09.27
Determine the ratio of lengths of sides of a
right triangle with given measures for its
acute angles using appropriate technologies.
(9D)
A guy wire is attached to the top of a 75 foot
pole. The wire makes an angle of 63
degrees with the ground. Find the length of
the wire.
A building casts a 20 foot shadow. The
angle of elevation (angle of depression, angle of
inclination) of the sun is 40 degrees. Find the
height of the building.
Connections
Connect to history of mathematics and
technology: Research the history of triples.
Write a calculator program that
generates Pythagorean Triples.
Research the job of a surveyor. Connect
the career to geometry.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.28
Visualize a three-dimensional object from
different perspectives and describe their
cross sections. (9A)
Sample Assessment
Connections
Find the volume of a frustum given the
parent pyramid.
Find examples of frustums in
pictures in magazines.
Find the area of a great circle.
Connect to social studies: Relate the great
circle with latitude and longitude lines
on a globe.
Useful to Work On at Grade 10:
10.09.29
Construct a perpendicular bisector of a segment.
(9A)
10.09.30
Combine simple construction techniques to
construct squares, equilateral triangles, or
other simple combinations of equal
segments, angles, etc. (9A)
10.09.31
Solve problems using constructions. (9A)
10.09.32
Analyze properties of a shape that enable it
to tessellate the plane. (9A)
Use Geometer’s Sketchpad, if available to
construct the perpendicular bisector. Prove that
every point that lies on the perpendicular
bisector is equidistant from the endpoints of the
segment.
Use any convenient length, construct an
equilateral triangle. Explain how to construct
a 30 degree angle.
Use a map. Find the shortest path to a
distant road from a marked town using a
construction.
Analyze a simple translation tessellation
by M. C. Escher. Make your own.
Connect to maps in geography.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.33
Analyze properties of a combination of
shapes that enable them to tessellate the
plane. (9B)
10.09.34
Analyze and describe the transformations
that lead to successful tessellations of one
or more figures. (9B)
Sample Assessment
Make a tessellation of the block letter F. Find
other block letters that will tessellate.
Connections
Make tessellations in art class.
Analyze pictures of tessellations by M. C. Escher.
Draw a non-specific quadrilateral. Find the
midpoint of one of its sides. Rotate the
original figure around the midpoint and describe
what happens.
10.09.35
Represent transformations of an object in the
plane using sketches, coordinates, and
vectors. (9A)
Sketch the image of a figure and give the
coordinates of the vertices of the image
if the figure is reflected over the y=x line.
10.09.36
Examine the congruence or similarity of
objects using transformations. (9B)
Prove that the diagonals of a parallelogram
bisect each other. (Hint: use half-turns)
10.09.37
Justify constructions using geometric
properties. (9C)
Justify the construction of a segment
congruent to a given segment.
Connect to technology and art: Create a
logo using transformations of your
school’s name.
Archdiocese of Chicago: Mathematics Curriculum Framework
Grade 10: Geometry Focus
State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space.
Learning Standard/Outcome
10.09.38
Justify the simple construction methods used
to produce angle bisectors, perpendicular
lines, and equilateral triangles. (9C)
Sample Assessment
Justify the construction of an angle bisector.
Connections
Discuss the importance of accurate
measurements in building a bridge. Build
a bridge using only constructions, no
ruler measurements.
Connect to Physics: Enter bridge building
contest.
10.09.39
Solve problems using the Law of Sines and
Law of Cosines. (9D)
Given an oblique triangle with lengths of
three sides given, find the measures of
the three angles.
Find the area of a plot of land in the
shape of a non-specific quadrilateral.