Download Common Core Learning Standards GRADE 8 Mathematics

Common Core Learning Standards
GRADE 8 Mathematics
GEOMETRY
Common Core Learning
Standards
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
8.G.1.
Verify experimentally the properties of
rotations, reflections, and translations:
8.G.1a.
Lines are taken to lines, and line segments to
line segments of the same length.
Concepts
Embedded Skills
Transformations Translate, rotate, and reflect lines and line
segments.
Explain the preservation of the sides of a figure
through a given transformation.
Identify corresponding parts between a figure and
its image using prime notation.
Show that lines are taken to lines and line
segments are taken to line segments.
Vocabulary
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Rotation
Reflection
Translation
Congruence
Transformation
Corresponding
parts
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Rigorous Sample Tasks
1) Part A: Given triangle ABC below, draw the image after a
transformation that preserves size and shape but NOT
orientation.
Scaffolded Sample Tasks
1.) Describe the given notation with its appropriate transformation.
a)
__________________________________________________
b)
__________________________________________________
c)
__________________________________________________
d)
__________________________________________________
e) T(-3,5)
__________________________________________________
2.)
Part B: Identify the transformation that you used to create your
image.
______________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
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What type of transformation is being demonstrated?
_________________________________
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Is orientation of triangle MATH preserved? Explain your
answer.
_________________________________________________________
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Is the size of triangle MATH preserved? Explain your answer.
_________________________________________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
Concepts
Embedded Skills
Transformations Translate, rotate, and reflect geometric shapes on
8.G.1.
Verify experimentally the properties of
rotations, reflections, and translations:
8.G.1b.
Angles are taken to angles of the same
measure.
a coordinate plane.
Measure angles using a protractor.
Identify corresponding parts between a figure and
its image using prime notation.
Show that angles are taken to angles of the same
measure.
Rigorous Sample Tasks
1) Part A: The triangle on the left was reflected across the
vertical line resulting in the triangle on the right. Using
what you know about line reflections, determine the value
of the missing angle, x.
Vocabulary
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Rotation
Reflection
Translation
Congruence
Properties
Transformation
Corresponding
parts
Scaffolded Sample Tasks
1)
Given the
clockwise rotation, label the appropriate vertices with
A’, B’, and C’.
Part B: The image on the right is then dilated with a scale
factor of
2
. Determine the value of the angle that corresponds to
3
angle x. Justify your result.
Explain why you chose the labels for each vertex.
_________________________________________________________
_________________________________________________________
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
Concepts
Embedded Skills
Transformations Translate, rotate, and reflect parallel lines on a
8.G.1.
Verify experimentally the properties of
rotations, reflections, and translations:
8.G.1c.
Parallel lines are taken to parallel lines.
Rigorous Sample Tasks
1)
coordinate plane.
Explain that the slope and distance between two
parallel lines will be preserved after a translation,
rotation, or reflection of the parallel lines.
Identify corresponding lines after a translation,
rotation, or reflection, by using prime notation.
Show that parallel lines are translated, rotated, or
reflected parallel lines.
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Parallel lines
Rotation
Reflection
Translation
Transformation
Slope/rate of
change
Corresponding
parts
Scaffolded Sample Tasks
1) What is the slope of the equation y 
line a
Vocabulary
5
x  10 ?
6
line b
Translate lines a and b, with the motion rule (x +2, y – 1). Label the
appropriate images lines a’ and b’.
Find the slopes of your translated images.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
slope of line a’ _________
slope of line b’ ___________
How do the slopes of your transformations compare with the slopes
of the given parallel lines?
______________________________________________________
_____________________________________________________
2.) Given the equation, y = 2x – 8, write the equation of a line that is
parallel and has a y-intercept of -7.
______________________________
3.) Given the lines below determine if they are parallel. Justify your
result.
y=
1
x  10
2
y = -2x +10
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
Concepts
Congruent
figures
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.
Embedded Skills
Explain the preservation of congruence when a
figure is rotated, reflected, and/or translated.
Describe the sequence of transformations that
occurred from the original 2D figure to the image.
Draw a reflection of an object.
Draw a translation of an object.
Draw a rotation of an object.
Name corresponding parts in congruent figures.
Vocabulary
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Transformation
Reflection
Rotation
Translation
Congruence
Corresponding
parts
sequence
SAMPLE TASKS
1) The two triangles in the picture below are congruent:
a) Give a sequence of rotations, translations, and/or reflections
which take
b) Is it possible to show the congruence in part (a) using only
translations and rotations? Explain.
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
Concepts
Embedded Skills
Transformations Dilate a two-dimensional figure using coordinates.
8.G.3.
Describe the effect of dilations, translations,
rotations, and reflections on two-dimensional
figures using coordinates.
Rigorous Sample Tasks
Describe the effect of dilating a two-dimensional
figure using coordinates.
Rotate a two-dimensional figure using coordinates.
Describe the effect of rotating a two-dimensional
figure using coordinates.
Translate a two-dimensional figure using
coordinates.
Describe the effect of translating a twodimensional figure using coordinates.
Reflect a two-dimensional figure using
coordinates.
Describe the effect of reflecting a two-dimensional
figure using coordinates.
Vocabulary
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Coordinate
Figure
Ordered pair
Reflect
Translate
Dilate
Rotate
Transformation
Prime
Image
X-axis
Y-axis
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
Concepts
Embedded Skills
Similarity with Explain the preservation of similarity when a figure
transformations is dilated, rotated, reflected, and/or translated.
Describe the sequence of transformations that
occurred from the original 2D figure to the image
to show the similarity.
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 two-dimensional
figures, describe a sequence that exhibits the
similarity between them.
Rigorous Sample Tasks
Vocabulary
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Rotation
Reflection
Translation
Dilation
Transformation
Similarity
Congruent
Similar
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Concepts
Embedded Skills
Understand congruence and
similarity using physical models,
transparencies, or geometry
software.
Prove/explain why the three angles of a triangle
equal 180°.
Prove/explain why the exterior angle theorem of a
triangle.
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.
Prove/explain why alternate interior angles are
congruent.
Prove/explain why alternate exterior angles are
congruent.
Rigorous Sample Tasks
Prove/explain why corresponding angles are
congruent.
Prove/explain why angle-angle criterion works to
prove similarity of two triangles.
Vocabulary
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Triangle
Similar
Parallel lines
Transversal
Congruent
Supplementary
Linear pair
Corresponding
Vertical
Alternate,
exterior,
interior angles
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand and apply the
Pythagorean Theorem.
Concepts
Pythagorean
theorem
8.G.6.
Explain a proof of the Pythagorean Theorem
and its converse.
Rigorous Sample Tasks
Embedded Skills
Explain a proof of Pythagorean theorem. (If a
triangle is a right triangle, then a2 + b2 = c2)
Explain a proof of the converse of Pythagorean
theorem. (If a2 + b2 = c2, then a triangle is a right
triangle)
Identify the legs and hypotenuse of a right triangle.
Solve multi-step equations.
Vocabulary
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Pythagorean
theorem
Converse
Proof
Legs
Hypotenuse
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand and apply the
Pythagorean Theorem.
Concepts
Embedded Skills
Pythagorean
theorem
Calculate the length of a leg of a right triangle using
Pythagorean theorem.
Calculate the length of the hypotenuse of a right
triangle using Pythagorean theorem.
Calculate the diagonal of a three-dimensional figure
using Pythagorean Theorem.
8.G.7.
Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in realworld and mathematical problems in two and
three dimensions.
Read and interpret a word problem involving
Pythagorean Theorem.
Solve word problems involving Pythagorean
Theorem.
Vocabulary
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Leg
Hypotenuse
Right angle
Pythagorean
theorem
Square root
Radical
Diagonals
Solve multi-step equations.
Identify the legs and hypotenuse of a right triangle.
Round to given place value.
Rigorous Sample Tasks
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Understand and apply the
Pythagorean Theorem.
8.G.8.
Apply the Pythagorean Theorem to find the
distance between two points in a coordinate
system.
Concepts
Pythagorean
theorem on a
coordinate
plane
Embedded Skills
Calculate the distance between two points in a
coordinate plane using the Pythagorean Theorem.
Plot points in a coordinate plane
Solve multi-step equations
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Identify the legs and hypotenuse of a right triangle
Solve Pythagorean Theorem
Rigorous Sample Tasks
Vocabulary
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Leg
Hypotenuse
Right angle
Pythagorean
theorem
Ordered pair
Coordinate
plane
Square root
Distance
formula
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.
Common Core Learning
Standards
Concepts
Solve real-world and mathematical
problems involving volume of
cylinders, cones, and spheres.
8.G.9.
Know the formulas for the volumes of cones,
cylinders, and spheres and use them to solve
real-world and mathematical problems.
Embedded Skills
Write and solve using the formula for the volume
of a cone.
Write and solve using the formula for the volume of
a cylinder.
Write and solve using the formula for the volume of
a sphere.
Solve word problems involving the volume of cones,
cylinders, and spheres.
Vocabulary
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Volume
Cone
Cylinder
Sphere
Area
Base
Formula
Solve a multi-step equation for a missing variable.
Rigorous Sample Tasks
Scaffolded Sample Tasks
Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.