mr. small's eighth grade math class -...

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 lines and line segments.

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.

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 Scaffolded Sample Tasks1) Part A: Given triangle ABC below, draw the image after a

transformation that preserves size and shape but NOT orientation.

Part B: Identify the transformation that you used to create your image.

1.) Describe the given notation with its appropriate transformation.

a)

__________________________________________________

b)

__________________________________________________

c)

__________________________________________________

d)

__________________________________________________

e) T(-3,5)

__________________________________________________

2.)


______________________

What type of transformation is being demonstrated?

_________________________________

Is orientation of triangle MATH preserved? Explain your answer.

_________________________________________________________

Is the size of triangle MATH preserved? Explain your answer.

_________________________________________________________




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.

Rigorous Sample Tasks Scaffolded Sample Tasks1) 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.

1)


Part B: The image on the right is then dilated with a scale

factor of . Determine the value of the angle that corresponds to

angle x. Justify your result.

Given the clockwise rotation, label the appropriate vertices with

A’, B’, and C’.

Explain why you chose the labels for each vertex.

_________________________________________________________

_________________________________________________________




Transformations Translate, rotate, and reflect parallel lines on a coordinate plane.

Parallel lines Rotation Reflection Translation Transformation Slope/rate of

change Corresponding

parts

Explain that the slope and distance between two parallel lines will be preserved after a translation, rotation, or reflection of the parallel lines.

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

Rigorous Sample Tasks Scaffolded Sample Tasks1) 1) What is the slope of the equation ?


line a 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.

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 =

y = -2x +10




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.

Draw a reflection of an object.Draw a translation of an object.Draw a rotation of an object.Name corresponding parts in congruent figures.

SAMPLE TASKS1) 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.



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.

Rigorous Sample Tasks Scaffolded 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.





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.




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)

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

Identify the legs and hypotenuse of a right triangle.Solve multi-step equations.





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.





Pythagorean theorem on a

coordinate plane

Calculate the distance between two points in a coordinate plane using the 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.

Plot points in a coordinate plane

Solve multi-step equations

Identify the legs and hypotenuse of a right triangle

Solve Pythagorean Theorem




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.



mr. small's eighth grade math class -...

Documents