Author:
Mark Freed
Subject:
Mathematics
Material Type:
Teaching/Learning Strategy
Level:
Middle School
Tags:
License:
Creative Commons Attribution
Language:
English

Education Standards

OREGON MATH STANDARDS (2021): [8.GM]

OREGON MATH STANDARDS (2021): [8.GM]

Overview

The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards. 

Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.

2021 Oregon Math Guidance: 8.GM.A.1

ODE and Oregon Math Project Logo

Cluster: 8.GM.A - Understand congruence and similarity using physical models, transparencies, or geometry software.

STANDARD: 8.GM.A.1

 Standards Statement (2021):

Verify experimentally the properties of rotations, reflections, and translations.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

7.GM.A.2, 7.GM.B.4

8.GM.A.2, 8.GM.A.3

 N/A

8.G.A.1

8.GM.A Crosswalk

 Standards Guidance:

Clarifications

  • Understand that:
    • Lines are taken to lines, and line segments to line segments of the same length.
    • Angles are taken to angles of the same measure.
    • Parallel lines are taken to parallel lines.

Boundaries

  • Rotations can be limited to 90, 180, 270 and 360 degrees around the origin
  • Reflections can be limited to reflection over horizontal and vertical lines

Progressions

  • Students should get a sense that rigid motions are special transformations. They should encounter and experience transformations which do not preserve lengths, do not preserve angles, or do not preserve either. (Please reference page 9 in the Progression document).

Examples

2021 Oregon Math Guidance: 8.GM.A.2

 ODE and Oregon Math Project Logo

Cluster: 8.GM.A - Understand congruence and similarity using physical models, transparencies, or geometry software.

STANDARD: 8.GM.A.2

 Standards Statement (2021):

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.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

8.GM.A.1

8.GM.A.4, 8.GM.A.5, HS.GM.A.1

 N/A

8.G.A.2

8.GM.A Crosswalk

 Standards Guidance:

Clarification

  • Students describe a series of rigid transformations that map a two dimensional figure onto its image.

Terminology

  • Rigid transformations include translations (slides), reflections (flips), rotations (turns), or glide reflections.

Progressions

  • Two figures in the plane are said to be congruent if there is a sequence of rigid motions that takes one figure onto the other. It should be noted that if we find a sequence of rigid motions taking figure A to figure B, then we can also find a sequence taking figure B to figure A. In high school mathematics the topic of congruence will be developed in a coherent, logical way, giving students the tools to investigate many geometric questions. In Grade 8, the treatment is informal, and students discover what they can about congruence through experimentation with actual motions. (Please reference page 9 in the Progression document).

Examples

2021 Oregon Math Guidance: 8.GM.A.3

 ODE and Oregon Math Project Logo

Cluster: 8.GM.A - Understand congruence and similarity using physical models, transparencies, or geometry software.

STANDARD: 8.GM.A.3

 Standards Statement (2021):

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

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

6.GM.A.3, 8.GM.A.1

8.GM.A.4, HS.GM.A.1, HS.GM.A.2

6.NS.C.8

8.G.A.3

8.GM.A Crosswalk

 Standards Guidance:

Progressions

  • In Grade 7, students study scale drawings to as a prelude to the transition from “same shape” to similarity in Grade 8. In Grade 8, change in scale becomes understood in terms of transformations that expand or contract the plane and the previous work with scale drawings flows naturally into describing dilations in terms of coordinates. (Please reference pages 9 & 10 in the Progression document).

Examples

  • Given a triangle with given coordinates, give the new coordinates after a prescribed transformation.
  • The image of Triangle ABC with 𝐴=(−3,0), 𝐵=(−3,−2) and 𝐶=(4,−2) would have coordinates 𝐴’=(−3−3,0+2)=(−6,2), 𝐵’=(−3−3,−2+2)=(−6,0), and 𝐶’=(4−3,−2+2)=(1,0) following a translation 3 units to the left and 2 units up.
  • The center of dilation should be limited to a) the origin on the coordinate plane or b) one vertex of a figure such as a triangle.
  • Illustrative Mathematics:

2021 Oregon Math Guidance: 8.GM.A.4

 ODE and Oregon Math Project Logo

Cluster: 8.GM.A - Understand congruence and similarity using physical models, transparencies, or geometry software.

STANDARD: 8.GM.A.4

 Standards Statement (2021):

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/or dilations.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

8.GM.A.2, 8.GM.A.3

8.GM.A.5, HS.GM.A.2

 N/A

8.G.A.4

8.GM.A Crosswalk

 Standards Guidance:

Progressions

  • Students observe the properties of dilations by experimenting with them, just as they did with rigid motions. They notice that shape is preserved under dilations, but that size is not preserved unless r =1. This observation suggests that the idea of “same shape” can be made precise as similarity: Two figures are similar if there is a sequence of rigid motions and dilations that places one figure directly on top of the other. (Please reference page 10 in the Progression document).

Examples

2021 Oregon Math Guidance: 8.GM.A.5

 ODE and Oregon Math Project Logo

Cluster: 8.GM.A - Understand congruence and similarity using physical models, transparencies, or geometry software.

STANDARD: 8.GM.A.5

 Standards Statement (2021):

Use informal arguments to establish facts about interior and exterior angles of triangles and angles formed by parallel lines cut with a transversal.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

8.GM.A.2, 8.GM.A.4

HS.GM.A.1, HS.GM.A.2, HS.GM.B.6

 N/A

8.G.A.5

8.GM.A Crosswalk

 Standards Guidance:

Terminology

  • Including identify alternate exterior angles, alternate interior angles, linear pairs, same side interior angles, same side exterior angles, and corresponding angles.

Boundaries

  • This standard includes using the properties of  the angle sum of the interior angles of a triangle, exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles to find missing angle measures.

Progressions

  • Use informal (visual) construction with tools (patty paper, protractor, etc.) to discover the angle relationships between angles formed when two lines are cut by a transversal.
  • When using more than one transversal, tie into similar triangles and to set up problems using triangle sum relationships (angle sum). (Please reference page 10 in the Progression document).

Examples

 

2021 Oregon Math Guidance: 8.GM.B.6

  ODE and Oregon Math Project Logo

Cluster: 8.GM.B - Understand and apply the Pythagorean Theorem.

STANDARD: 8.GM.B.6

 Standards Statement (2021):

Distinguish between applications of the Pythagorean Theorem and its Converse in authentic contexts.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

8.GM.B.7

HS.GM.D.14, HS.GM.D.12

8.AEE.A.2

8.G.B.6

8.GM.B Crosswalk

 Standards Guidance:

Clarification

  • Analysis and justification can be done using a variety of representation including use of pictures, diagrams, narratives, or models.

Terminology

  • The Pythagorean Theorem states, that in any right triangle, the square length hypotenuse is equals the sum of the square length of the other two sides (e.g. a2+b2=c2).
  • The converse of the Pythagorean Theorem states that if a triangle has sides of length a, b, and c and if a2+b2=c2 then the angle opposite the side of length c is a right angle.

Teaching Strategies

  • Students should have the opportunity to explore one or more proofs of the Pythagorean Theorem, but are not required to prove the Theorem.
  • Geometric and spatial reasoning should be used when explaining the Pythagorean Theorem.

Progressions

  • Students learn that there are lengths that cannot be represented by a rational number. For example, by looking at areas of figures in the coordinate plane, students discover that the hypotenuse of a triangle with legs of length 1 is an irrational number. Students can continue this line of reasoning to explain a dissection proof of the Pythagorean Theorem. (Please reference page 11 in the Progression document).

Examples

  • Many ancient cultures used simple Pythagorean triples such as (3,4,5) in order to accurately construct right angles: if a triangle has sides of lengths 3, 4, and 5 units, respectively, then the angle opposite the side of length 5 units is a right angle.
  • The Pythagorean Theorem tells us that a certain relation holds amongst the side lengths of a right triangle. These ancient architects, however, do not have a right triangle but rather want to produce a right triangle. The converse of the Pythagorean Theorem enables them to do just this: they can conclude that an angle is a right angle provided a certain relationship holds between side lengths of a triangle.
  • Illustrative Mathematics:
  • Student Achievement Partners:

2021 Oregon Math Guidance: 8.GM.B.7

  ODE and Oregon Math Project Logo

Cluster: 8.GM.B - Understand and apply the Pythagorean Theorem.

STANDARD: 8.GM.B.7

 Standards Statement (2021):

Apply the Pythagorean Theorem in authentic contexts to determine unknown side lengths in right triangles.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

6.NS.C.8, 8.AEE.A.2

8.GM.B.6, 8.GM.B.8, HS.GM.D.12

 N/A

8.G.B.7

8.GM.B Crosswalk

 Standards Guidance:

Teaching Strategies

  • Geometric and spatial reasoning should be used to solve problems involving the Pythagorean theorem.
  • Models and drawings may be useful as students solve contextual problems in two- and three-dimensions.

Boundaries

  • Include authentic mathematical problems in two and three dimensions.

Examples

  • How tall is the Great Pyramid of Giza below?

  • Illustrative Mathematics:

 

 

2021 Oregon Math Guidance: 8.GM.B.8

  ODE and Oregon Math Project Logo

Cluster: 8.GM.B - Understand and apply the Pythagorean Theorem.

STANDARD: 8.GM.B.8

 Standards Statement (2021):

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

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

6.NS.C.8, 7.GM.B.5, 8.GM.B.7

HS.GM.D.14, HS.GM.D.13

 N/A

8.G.B.8

8.GM.B Crosswalk

 Standards Guidance:

Boundaries

  • The Distance Formula is NOT included in the 8th grade standard.
  • Students should apply their understanding of the Pythagorean Theorem to find the distance.  Use of the distance formula is not an expectation for this grade level.

Progressions

  • In Grade 6, students calculate distances in the coordinate plane between points lying on the same horizontal or vertical line. In particular, they calculate the lengths of the vertical and horizontal legs of a triangle corresponding to two points in the coordinate plane. In Grade 7, they can use the Pythagorean Theorem to calculate the length of its hypotenuse, which is the distance between the two points.
  • Calculating this distance as an application of the Pythagorean Theorem before doing so in high school as an application of the distance formula provides students an opportunity to look for and make use of structure in the coordinate plane (MP.7), and provides an opportunity for students to connect the distance formula to previous learning. (Please reference pages 11 & 12 in the Progression document).

Examples

  • There are two paths that Sarah can take when walking to school.  One path is to take A Street from home to the traffic light and then walk on B street from the traffic light to the school, and the other way is for her to take C street directly to the school.  How much shorter is the direct path along C Street?

 

2021 Oregon Math Guidance: 8.GM.C.9

   ODE and Oregon Math Project Logo

Cluster: 8.GM.C - Solve mathematical problems in authentic contexts involving volume of cylinders, cones, and spheres.

STANDARD: 8.GM.C.9

 Standards Statement (2021):

Choose and use the appropriate formula for the volume of cones, cylinders, and spheres to solve problems in authentic contexts.

  Connections:

Preceding Pathway Content (2021)

Subsequent Pathway Content (2021)

Cross Domain Connections (2021)

Common Core (CCSS)

(2010)

7.GM.B.3, 8.AEE.A.2

HS.GM.C.8, HS.GM.C.9, HS.GM.C.10, HS.GM.C.11

 N/A

8.G.C.9

8.GM.C Crosswalk

 Standards Guidance:

Boundaries

  • Memorizing the formulas is NOT included in this standard.

Teaching Strategies

  • Given the volume, solve for an unknown dimension of the figure. Students will need to be able to express the answer in terms of pi and as a decimal approximation.

Progressions

  • Students learn and use formulas for the volumes of cylinders, cones, and spheres. Explanations for these formulas do not occur until high school. However, Grade 8 students can look for structure in these formulas (MP.7). They know that the volume of a cube with sides of length s is s3.
  • A cube can be decomposed into three congruent pyramids, each of which has a square base, where the height is equal to the side length of the square. Each of these pyramids must have the volume (1/3)s3, suggesting that the volume of a pyramid whose base has area B and whose height is h might be (1/3)Bh. The volume formulas for cylinders and cones have an analogous relationship.
    • cylinder 
    • cone
  • Please reference page 12 in the Progression document.

Examples