Unit 4

The Number System

 

Grade 8 Math 

Unit Length and Description:

 

30 days

 

In this unit, students analyze two- and three-dimensional space and figures using congruence and similarity.  Students study congruence by experimenting with rotations, reflections, and translations of geometrical figures. This work then prepares students for the more complex work of understanding the effects of dilations on geometrical figures in their study of similarity.

 

Students use similar triangles to solve unknown angle, side length and area problems. They revisit the Pythagorean Theorem from the perspective of similar triangles. In this unit, students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines.  The angle-angle criterion for triangle similarity underlies the fact that a non-vertical line in the coordinate plane has equation y = mx + b. Therefore, students must do work with congruence and similarity in order to be able to justify the connections among proportional relationships, lines, and linear equations.

 

Standards:

 

8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation 𝑦= mx+𝑏 for a line intercepting the vertical axis at 𝑏.

 

8.G.A.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.

 

  • Standards Clarification:  The skill of transforming geometric items as well as the properties of transforming these items will extend to develop and establish the criteria for figure congruence and figure similarity. This standard does not include the transformation of figures.
  • Standards Clarification:  Translations, reflections, and rotations are called rigid transformations because they do not change the size or shape of an item.  Characteristics such as the length of line segments, angle measures, and parallel lines are unchanged by these three types of transformations.

 

8.G.A.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.

  • Standards Clarification:  Because size and shape are preserved under translations, reflections, and rotations, the result of these transformations is an exact copy of the original figure.  When two figures have the exact same size and shape, they are called congruent figures.

 

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

 

  • Standards Clarification:  When you apply transformations to figures in the coordinate plane, you can describe the results of the transformation by giving the coordinates of the vertices of the figures.  For some of these transformations, it is easy to write a general rule that describes what happens to each coordinate under the transformation.

 

8.G.A.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.

 

  • Standards Clarification:  A dilation changes the size of a figure but not its shape.  When two figures have the same shape but different sizes, they are called similar figures.

 

8.G.A.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.

 

  • Standards Clarification:  Students will learn about the special angle relationships formed when parallel lines are intersected by a third line called a transversal.  Students will learn that the sum of the angle measures in a triangle is the same for all triangles.  Students will learn one way, angle-angle criterion, to determine whether two triangles are similar.

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

8.EE.B.6

·         I can find the slope of a line between a pair of distinct points.

·         I can determine the y-intercept of a line (interpreting unit rate as the slope of the graph is included in 8.EE).

·         I can analyze patterns for points on a line through the origin.

·         I can derive an equation of the form y=mx for a line through the origin.

·         I can analyze patterns for points on a line that does not pass through or include the origin.

·         I can derive an equation of the form y=mx + b for a line intercepting the vertical axis at b (the y-intercept).

·         I can use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.

 

8.G.A.1a

·         I can define and identify rotations, reflections, and translations.

·         I can identify corresponding sides and corresponding angles of similar figures.

·         I can understand prime notation to describe an image after a translation, reflection, or rotation.

·         I can identify center of rotation.

·         I can identify direction and degree of rotation.

·         I can identify line of reflection.

·         I can use physical models, transparencies, or geometry software to verify the properties of rotations, reflections, and translations (i.e. lines are taken to lines and line segments to line segments of the same length.)

 

8.G.A.1b

·         I can define and identify rotations, reflections, and translations.

·         I can identify corresponding sides and corresponding angles of similar figures.

·         I can understand prime notation to describe an image after a translation, reflection, or rotation.

·         I can identify center of rotation.

·         I can identify direction and degree of rotation.

·         I can identify line of reflection.

·         I can use physical models, transparencies, or geometry software to verify the properties of rotations, reflections, and translations (i.e. angles are taken to angles of the same measure.)

 

8.G.A.1c

·         I can define and identify rotations, reflections, and translations.

·         I can identify corresponding sides and corresponding angles of similar figures.

·         I can understand prime notation to describe an image after a translation, reflection, or rotation.

·         I can identify center of rotation.

·         I can identify direction and degree of rotation.

·         I can identify line of reflection.

·         I can use physical models, transparencies, or geometry software to verify the properties of rotations, reflections, and translations (i.e. parallel lines are taken to parallel lines.)

 

8.G.A.2      

·         I can define congruency.

·         I can identify symbols for congruency.

·         I can apply the concept of congruency to write congruent statements.

·         I can reason that a 2-D figure is congruent to another if the second can be obtained by a sequence of rotation, reflections, and translation.

·         I can describe the sequence of rotations, reflections, translations that exhibits the congruence between 2-D figures using words.

·         I can justify congruence of figures using a series of transformations.

 

8.G.A.3

·         I can define dilations as a reduction or enlargement of a figure.

·         I can identify scale factor of the dilation.

·         I can describe the effects of dilations, translations, rotations, and reflections on 2-D figures using words and coordinates.

 

8.G.A.4

·         I can define similar figures as corresponding angles are congruent and corresponding side lengths are proportional.

·         I can recognize the symbol for similar.

·         I can apply the concept of similarity to write similarity statements.

·         I can reason that a 2-D figure is similar to another if the second can be obtained by a sequence of rotations, reflections, translation or dilation.

·         I can describe the sequence of rotations, reflections, translations, or dilations that exhibits the similarity between 2-D figures using words and/or symbols.

·         I can justify similarity of figures using a series of transformations.

 

8.G.A.5

·         I can define similar triangles.

·         I can define and identify transversals.

·         I can identify angles created when a parallel line is cut by transversal (alternate interior, alternate exterior, corresponding, vertical, adjacent, etc.).

·         I can justify that the sum of the interior angles equals 180. (For example, arrange three copies of the same triangle so that the three angles appear to form a line).

·         I can justify that the exterior angles of a triangle is equal to the sum of the two remote interior angles.

·         I can use Angle-Angle Criterion to prove similarity among triangles. (Give an argument in terms of transversals why this is so).

·         I can recognize the angles formed by two parallel lines and a transversal.

·         I can find the measure of angles using transversals, the sum of angles in a triangle, the exterior angles of triangles.

 

Enduring Understandings:

 

·         Those transformations that modify an item so that it is in a different position, but still has the same shape, size, segment lengths, and angle measures are translations, reflections, and rotations.

·         Translations (slides), reflections (flips), and rotations (turns) are transformation actions that produce congruent geometric items.

·         Congruent figures have the same size and shape.

·         A dilation (stretching/shrinking) is a transformation that changes the size of a figure but not the shape, thus producing similar figures not congruent figures (unless the scale factor is equal to 1).

·         If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement, and if the scale factor is less than 1, the image is a reduction.

·         A two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of transformations.

·         Two shapes are similar if the lengths of all the corresponding sides are proportional and all the corresponding angles are congruent.

·         Two similar figures are related by a scale factor, which is the ratio of the lengths of corresponding sides.

·         Congruent figures have the same size and shape. If the scale factor of a dilation is equal to 1, the image resulting from the dilation is congruent to the original figure.

·         When parallel lines are cut by a transversal, corresponding angles, alternate interior angles, alternate exterior angles, and vertical angles are congruent.

·         A non-vertical line in the coordinate plane has equation y = mx + b.

 

Essential Questions:

 

·         How can we best show or describe the change in position of a figure?

·         What are transformations and what effect do they have on a two-dimensional figure?

·         What are the similarities and differences between the pre-image (the original figure) and the image (the translated figure) generated by translations?

·         How can transformations be used to determine congruency?

·         Does a two-dimensional figure change dimensions when transformed?

·         How can you use coordinates to describe the result of a translation, reflection, or rotation?

·         What properties of a two-dimensional figure are preserved under a translation, reflection, or rotation?

·         What is the relationship of the points, lines, line segments, angles, etc. between the pre-image (the original figure) and the item’s image (the translated figure) generated by translations?

·         Where is the origin on a coordinate grid?

·         What is meant by “rotate about a point”?

·         What is meant by “rigid motion”?

·         How can translations be applied to real-world situations?

·         What does the scale factor of a dilation convey?

·         What are the similarities and differences between the pre-image (the original figure) and the image (the translated figure) generated by dilations?

·         How can transformations be used to determine similarity?

·         Does a two-dimensional figure change dimensions when dilated?

·         How can you use coordinates to describe the result of a dilation?

·         What properties of a two-dimensional figure are preserved under a dilation?

·         What is the relationship of the points, lines, line segments, angles, etc. between the pre-image (the original figure) and the item’s image (the transformed figure) generated by transformations?

·         What angle relationships are formed by a transversal?

·         Can two figures be both congruent and similar?

·         How can dilations be applied to real-world situations?