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Common Core Learning StandardsGRADE 8 Mathematics

GEOMETRY

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Understand congruence and similarity using physical models, transparencies, or geometry software.

Transformations Translate, rotate, and reflect figures. Rotation Reflection Translation Congruence Transformation Corresponding

parts

Explain the preservation of the sides of a figure through a given transformation.

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.

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.

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.



Transformations Translate, rotate, and reflect geometric shapes on a coordinate plane.

Rotation Reflection Translation Congruence Properties Transformation Corresponding

parts

Measure angles using a protractor.

8.G.1.Verify experimentally the properties of rotations, reflections, and translations:8.G.1b.Angles are taken to angles of the same measure.

Identify corresponding parts between a figure and its image using prime notation.Show that angles are taken to angles of the same measure.

SAMPLE TASKS




Transformations Translate, rotate, and reflect geometric shapes on a coordinate plane.

Parallel lines Rotation Reflection Translation Transformation Slope/rate of

change Corresponding

parts

Explain what happens to parallel lines after a given transformation.

8.G.1.Verify experimentally the properties of rotations, reflections, and translations:8.G.1c.Parallel lines are taken to parallel lines.

Identify corresponding parts between a figure and its image using prime notation.Show that parallel lines are taken to parallel lines.

SAMPLE TASKS




Congruent figures

Explain the preservation of congruence when a figure is rotated, reflected, and/or translated.

Transformation Reflection Rotation Translation Congruence Corresponding

parts sequence

Describe the sequence of transformations that occurred from the original 2D figure to the image.

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.

SAMPLE TASKS




Transformations Dilate a two-dimensional figure using coordinates. Coordinate Figure Ordered pair Reflect Translate Dilate Rotate Transformation Prime Image X-axis Y-axis

Describe the effect of dilating a two-dimensional figure using coordinates.

8.G.3.Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures 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 two-dimensional figure using coordinates.Reflect a two-dimensional figure using coordinates.Describe the effect of reflecting a two-dimensional figure using coordinates.

SAMPLE TASKS




Similarity with transformations

Explain the preservation of similarity when a figure is dilated, rotated, reflected, and/or translated.

Rotation Reflection Translation Dilation Transformation Similarity Congruent Similar

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.

SAMPLE TASKS




Prove/explain why the three angles of a triangle equal 180°.

Triangle Similar Parallel lines Transversal Congruent Supplementar

y Linear pair Corresponding Vertical Alternate,

exterior, interior angles

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.Prove/explain why corresponding angles are congruent.Prove/explain why angle-angle criterion works to prove similarity of two triangles.

SAMPLE TASKS



Understand and apply the Pythagorean Theorem.

Pythagorean theorem

Explain a proof of Pythagorean theorem. (If a triangle is a right triangle, then a2 + b2 = c2)

Pythagorean theorem

Converse Proof Legs Hypotenuse

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.

8.G.6.Explain a proof of the Pythagorean Theorem and its converse.

SAMPLE TASKS Given triangle ABC with side lengths of 10 cm, 8 cm, and 5 cm prove whether or not the sides form a right triangle. The inside edges of three fields meet to form a right triangle. The first field has an edge of 30 yards, the second field has an edge of

40 yards, and the third field has an edge of 50 yards. Use this to show the Pythagorean Theorem is true. Jacob needs to construct a right triangle using drinking straws. If he has straw lengths of 5 cm, 12 cm and 15 cm, can he construct a

right triangle? Explain why or why not.




Pythagorean theorem

Calculate the length of a leg of a right triangle using Pythagorean theorem.

Leg Hypotenuse Right angle Pythagorean

theorem Square root Radical Diagonals

Calculate the length of the hypotenuse of a right triangle using Pythagorean theorem.

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.

Calculate the diagonal of a three-dimensional figure using Pythagorean Theorem.Read and interpret a word problem involving Pythagorean Theorem.Solve word problems involving Pythagorean Theorem.Solve multi-step equations.Identify the legs and hypotenuse of a right triangle.Round to given place value.

SAMPLE TASKS

In Juanita’s Little League championship game the bases are all 60 feet apart forming a square. Juanita is on first base planning on stealing second, how far does the catcher need to throw the ball from home plate to second base in order to make the out?

Simon leans a 20-foot ladder against the side of his house so that the base of the ladder is 5 feet from the house. About how high up the side of the house does the ladder reach? Round your answer to the nearest tenth of a foot.

A cylindrical can of soda has a height of 5 inches and a diameter of 2 inches as shown. What is the shortest straw that can be used in this can so that it doesn’t fall into the can?


5 in

2 in



Pythagorean theorem on a

coordinate plane

Calculate the distance between two points in a coordinate plane using the Pythagorean Theorem.Plot points in a coordinate planeSolve multi-step equationsIdentify the legs and hypotenuse of a right triangleSolve Pythagorean Theorem

Leg Hypotenuse Right angle Pythagorean

theorem Ordered pair Coordinate

plane Square root Distance

formula

8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

SAMPLE TASKS

Line segment has endpoints at (6, 1) and (-6, 6). What is the length of ? A team of archaeologists fenced off an ancient ruin they are exploring. They created a grid to represent the area, so they could label

the locations of several artifacts. If each unit on the grid represents 2 meters, approximately how many meters apart were the pitcher and the pot found.

What is the approximate length of shown on the right?


A

B


Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Write and solve using the formula for the volume of a cone.

Volume Cone Cylinder Sphere Area Base Formula

Write and solve using the formula for the volume of a cylinder.

8.G.9.Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Write and solve using the formula for the volume of a sphere.Solve word problems involving the volume of cones, cylinders, and spheres.Solve a multi-step equation for a missing variable.

SAMPLE TASKS How much ice cream can fit exactly inside a cone that has a diameter of 8 centimeters and a height of 9 centimeters? A pitcher holds 1,614.7 in3 of liquid. Each can of punch is 15 inches tall with a diameter of 8 inches. How many full cans will the

pitcher hold? Explain. A cylindrical can has a height of 24 inches and a volume of 864 cubic inches. What is its diameter? To the nearest cubic inch, how much space is there inside a hamster ball with a diameter of 10 inches? Two companies manufacture tanks that store water. Company A sells cylindrical tanks like the one shown below for $500, and

company B sells spherical tanks like the one shown below also for $500. Which company should you purchase your tank from in order to maximize your water storage?


8 ft

3 ft

r = 2.5


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