# OREGON MATH STANDARDS (2021): [HS.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: HS.GM.A.1

**Cluster: HS.GM.A - Apply geometric transformations to figures through analysis of graphs and understanding of functions. **

## STANDARD: HS.GM.A.1

### Standards Statement (2021):

Apply definitions of rotations, reflections, and translations to transform a figure and map between two figures in authentic contexts.

### 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, HS.GM.A.4 | N/A | 8.AFN.A.1, HS.AFN.A.2, HS.AFN.D.9 | HSG.CO.A.2 HSG.CO.A.4 HSG.CO.A.5 HS.GM.A Crosswalk |

### Standards Guidance:

#### Clarifications

- Students should be able to determine congruency by identifying the rigid transformation(s) that produced the image of a figure.
- Opportunities should be provided for students to write statements of congruency.
- Given two polygons, students should be able to use the definition of congruence in terms of rigid motions to verify congruence if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
- Students should be able to use function notation to represent transformations in the coordinate plane.

#### Terminology

- A rigid transformation that preserves size and shape (e.g. translation, rotation, or reflection).

#### Boundaries

- Draw the transformation (rotation, reflection, or translation) for a given geometric figure.
- Students should be able to apply definitions of reflections across any line in context or on a coordinate grid.
- Students should be able to apply definitions of rotations around any point of any degree in context or on a coordinate grid.

#### Teaching Strategies

- Students should have ample opportunities to use geometric tools and/or technology to explore figures created from translations, reflections, and rotations.
- Students should be able to determine images created by a given translations, reflections, or rotations.

#### Examples

# 2021 Oregon Math Guidance: HS.GM.A.2

**Cluster: HS.GM.A - Apply geometric transformations to figures through analysis of graphs and understanding of functions.**

## STANDARD: HS.GM.A.2

### Standards Statement (2021):

Verify experimentally the properties of a dilation given a center and a scale factor. Solve problems in authentic contexts involving similar triangles or dilations.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

8.GM.A.3, 8.GM.A.4, 8.GM.A.5 | HS.GM.A.1, HS.GM.A.3, HS.GM.D.12 | HS.AFN.D.9 | HSG.SRT.B.5, HSG.SRT.A.1, HSG.SRT.A.2, HSG.SRT.A.3 HS.GM.A Crosswalk |

### Standards Guidance:

#### Clarifications

- Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
- Students should be able to identify dilation as reduction or enlargement depending on scale factor.
- Students should be given opportunities to draw a dilated image given any center and scale factor in context or on a coordinate grid.
- Students should be able to describe properties of dilations, such as center, scale factor, angle measure, parallelism, and collinearity.

#### Terminology

- A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
- The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

#### Teaching Strategies

- Triangles can be shown to be similar using transformations and triangle similarity theorems. Apply theorems of AA similarity, SSS similarity, and SAS similarity to prove similarity of two given triangles.
- Dilations should be limited to those centered at the origin.

#### Progressions

- Model with mathematics to use similarity to solve authentic problems to measure lengths and distances indirectly.
- Use the properties of similarity transformations could be used to establish the Angle-Angle (AA) criterion for two triangles to be similar.

#### Examples

- A high school student visits a giant cedar tree near the town of Elk River, Idaho and the end of his shadow lines up with the end of the tree’s shadow. The student is 6 feet tall and his shadow is 8 feet long. The cedar tree’s shadow is 228 feet long. How tall is the cedar tree?

# 2021 Oregon Math Guidance: HS.GM.A.3

**Cluster: HS.GM.A - Apply geometric transformations to figures through analysis of graphs and understanding of functions.**

## STANDARD: HS.GM.A.3

### Standards Statement (2021):

Use the slopes of segments and the coordinates of the vertices of triangles, parallelograms, and trapezoids to solve problems in authentic contexts.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

HS.GM.A.2 | N/A | 8.AEE.B.5, 8.AEE.B.6, 8.AEE.C.8 | HSG.GPE.B.5 HS.GM.A Crosswalk |

### Standards Guidance:

#### Clarifications

- Students should have opportunities to analyze and apply theorems about lines and angles from the context of parallel lines cut by a transversal to make sense of relationships between lines and angles in quadrilaterals and triangles.
- Students should be familiar with triangle congruence theorems (SSS, SAS, ASA, AAS, or HL) to solve problems and to prove relationships in geometric figures by applying geometric and algebraic reasoning.

#### Progressions

- Possible applications include using slopes to determine parallel sides in parallelograms and trapezoids, perpendicular diagonals in rhombuses, perpendicular sides in a rectangle
- Use slope and coordinates to verify mid-segment properties in triangles and trapezoids.
- Use coordinates of vertices for lengths of sides and diagonals to classify quadrilaterals and triangles.

#### Examples

# 2021 Oregon Math Guidance: HS.GM.A.4

**Cluster: HS.GM.A - Apply geometric transformations to figures through analysis of graphs and understanding of functions.**

## STANDARD: HS.GM.A.4

### Standards Statement (2021):

Use definitions of transformations and symmetry relationships to justify the solutions of problems in authentic contexts.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

4.GM.A.1, 4.GM.A.2, 4.GM.A.3, 4.GM.C.7 | HS.GM.A.1 | HS.AFN.D.9 | HSG.CO.A.1 HS.GM.A Crosswalk |

### Standards Guidance:

#### Clarifications

- Students should be able to define and identify figures as preimages and images.
- Students use definitions to identify lines of symmetry and angles of rotation to map a figure onto itself.
- Students use definitions to identify angles of rotation, lines of reflection, and directions of translations to map a preimage onto its image.
- Students use definitions to experiment with transformations represented on and off the coordinate plane.

#### Terminology

- Definitions of geometric figures and geometric relationships could include definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

#### Boundaries

- Definitions should include angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

#### Examples

- Illustrative Mathematics:

# 2021 Oregon Math Guidance: HS.GM.B.5

**Cluster: HS.GM.B - Construct and communicate geometric arguments through use of proofs, logical reasoning, and geometric technology. **

## STANDARD: HS.GM.B.5

### Standards Statement (2021):

Apply and justify triangle congruence and similarity theorems in authentic contexts.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

7.GM.A.2 | HS.GM.B.6 | N/A | HSG.CO.B.7 HSG.CO.B.8 HS.GM.B Crosswalk |

### Standards Guidance:

#### Clarifications

- Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure;
- Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
- Students should be able to apply properties of congruence to solve problems with missing values involving corresponding parts. Opportunities should also be available for students to understand when conditions do not result in congruence.

#### Boundaries

- The focus here is to develop an understanding of techniques for proving that two triangles are congruent.
- Advanced courses could include explanations for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions (HSG.CO.B.8).

#### Terminology

- Logic statements include conditional, converse, inverse, and contrapositive statements.

#### Teaching Strategies

- Use of triangle congruence theorems (SSS, SAS, ASA, AAS, or HL) should be used to solve problems in authentic contexts.
- Students’ ways of communicating triangle congruence could possibly include formal methods such as: logic statements, two-column proofs, paragraph proofs, and flow proofs.

#### Examples

- Construct viable arguments and critique the reasoning of others when showing that two triangular roof trusses must be congruent.

# 2021 Oregon Math Guidance: HS.GM.B.6

**Cluster: HS.GM.B - Construct and communicate geometric arguments through use of proofs, logical reasoning, and geometric technology. **

## STANDARD: HS.GM.B.6

### Standards Statement (2021):

Justify theorems of line relationships, angles, triangles, and parallelograms; and use them 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.4, 8.GM.A.5, HS.GM.B.5 | HS.GM.D.12 | N/A | HSG.CO.C.9 HSG.CO.C.10 HS.GM.B Crosswalk |

### Standards Guidance:

#### Clarification

- Students should be given opportunities to explore using visual tools in order to precisely prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent.

#### Boundaries

- Angle and line relationship theorems include:
- when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and conversely use to justify lines are parallel;
- points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
- vertical angles are congruent;

- Triangle Theorems include:
- Sum of interior angles 180 degrees
- Properties of special triangles (isosceles, equilateral, and right).
- Angle sums formed by polygons

- Parallelogram theorems include:
- Properties of special quadrilaterals (sides, angles, and diagonals), and
- Properties of special triangles (isosceles, equilateral, and right).

#### Clarifications

- Students should be provided opportunities to build a conceptual understanding of a point, line, line segment, plane, arc, and angle through modeling and exploration of authentic phenomena.
- Students should use symbolic notation for point, line, plane, line segment, angle, circle, arc, perpendicular line, and parallel line.

#### Progressions

- Construct viable arguments and critique the reasoning of others when justifying the congruence of diagonals in a rectangle that is built by a contractor installing a rectangular window.

#### Examples

# 2021 Oregon Math Guidance: HS.GM.B.7

**Cluster: HS.GM.B - Construct and communicate geometric arguments through use of proofs, logical reasoning, and geometric technology. **

## STANDARD: HS.GM.B.7

### Standards Statement (2021):

Perform geometric constructions with a variety of tools and methods.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

7.GM.A.2 | N/A | N/A | HSG.CO.D.12 HS.GM.B Crosswalk |

### Standards Guidance:

#### Clarifications

- Student should be able to:
- Copy a segment and angle.
- Bisect a segment and angle.
- Construct perpendicular lines, including the perpendicular bisector of a line segment.
- Construct a line parallel to a given line through a point not on the line.

#### Teaching Strategies

- Tools to include compass and straightedge, string, reflective devices, paper folding, and/or dynamic geometric software.
- Constructions to include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

#### Progressions

- Use appropriate tools strategically when choosing the physical method and appropriate procedures for performing a construction

#### Examples

- Illustrative Mathematics:

# 2021 Oregon Math Guidance: HS.GM.C.8

**Cluster: HS.GM.C - Solve problems and interpret solutions of area and volume of shapes by applying concepts of congruence, similarity, symmetry in authentic contexts. **

## STANDARD: HS.GM.C.8

### Standards Statement (2021):

Solve authentic modeling problems using area formulas for triangles, parallelograms, trapezoids, regular polygons, and circles.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

6.GM.A.1, 7.GM.B.3, 7.GM.B.5, 8.GM.C.9 | HS.GM.C.9 | N/A | HSG.GMD.A.1 HS.GM.C Crosswalk |

### Standards Guidance:

#### Teaching Strategies

- Students should give informal arguments for area formulas, and combine them to solve problems with composite figures.
- Students should be able to choose the appropriate geometric polygon to approximate the area of irregular objects.

#### Examples

- Model with Mathematics can be used here to solve a variety of problems involving area.

- Illustrative Mathematics:

# 2021 Oregon Math Guidance: HS.GM.C.9

**Cluster: HS.GM.C - Solve problems and interpret solutions of area and volume of shapes by applying concepts of congruence, similarity, symmetry in authentic contexts. **

## STANDARD: HS.GM.C.9

### Standards Statement (2021):

Use volume and surface area formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and apply to authentic contexts.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

7.GM.B.5, 8.GM.C.9, HS.GM.C.8 | N/A | N/A | HSG.GMD.A.3 HS.GM.C Crosswalk |

### Standards Guidance:

#### Teaching Strategies

- Students should be able to choose the appropriate geometric figure or solid to approximate density of irregular objects in a geometric situation.
- Students should give informal arguments for area and volume formulas, and combine them to solve problems with composite figures. This standard is limited to right solids.

#### Examples

- Make sense of problems and persevere in solving them when finding the volume of prisms and pyramids with regular polygon bases (possibly using trigonometry)
- Persons per square mile, fish per cubic feet of a fish tank

- Illustrative Mathematics:

# 2021 Oregon Math Guidance: HS.GM.C.10

**Cluster: HS.GM.C - Solve problems and interpret solutions of area and volume of shapes by applying concepts of congruence, similarity, symmetry in authentic contexts. **

## STANDARD: HS.GM.C.10

### Standards Statement (2021):

Use geometric shapes, their measures, and their properties to describe real world objects, and solve related authentic modeling and design problems.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

7.GM.A.1, 7.GM.B.3, 7.GM.B.5, 8.GM.C.9 | N/A | HS.NQ.B.3, HS.NQ.B.4, HS.NQ.B.5 | HSG.MG.A.1 HSG.MG.A.3 HS.GM.C Crosswalk |

### Standards Guidance:

#### Clarifications

- This includes the use of volume formulas for prisms, cylinders, pyramids, cones, and spheres.
- Students should be able to verify experimentally the formulas for the volume of a cylinder, pyramid, sphere, prism and cone; emphasize volume as the product of the area of the base and the height for both prisms and cylinders.
- Students should find the volume of solids and composite solids to explain real-life phenomena.

#### Terminology

- Prism – a solid figure that has the same cross section all along its length

#### Examples

- Model with Mathematics can be used here to solve a variety of problems such as designing a real world object with CAD design tools for 3D printing or CNC machining.

# 2021 Oregon Math Guidance: HS.GM.C.11

**Cluster: HS.GM.C - Solve problems and interpret solutions of area and volume of shapes by applying concepts of congruence, similarity, symmetry in authentic contexts. **

## STANDARD: HS.GM.C.11

### Standards Statement (2021):

Apply concepts of density based on area and volume in authentic modeling situations.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

8.GM.C.9 | N/A | HS.AFN.A.3, 7.RP.A.1, HS.NQ.B.4, HS.NQ.B.5 | HSG.MG.A.2 HS.GM.C Crosswalk |

### Standards Guidance:

#### Clarifications

- The focus is on geometric probability and proportional reasoning.
- This should include an understanding of the ratios of areas (area ratio = (scale factor)^2) and volumes (volume ratio = (scale factor)^3) of similar figures.

#### Examples

- Model with Mathematics to compute persons per square miles, BTUs per cubic foot, or specimens per acre.

# 2021 Oregon Math Guidance: HS.GM.D.12

**Cluster: HS.GM.D - Apply concepts of right triangle trigonometry in authentic contexts to solve problems and interpret solutions. **

## STANDARD: HS.GM.D.12

### Standards Statement (2021):

Apply sine, cosine, and tangent ratios, and the Pythagorean Theorem, to solve problems in authentic contexts.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

8.GM.B.6, 8.GM.B.7, HS.GM.A.2, HS.GM.B.6 | N/A | N/A | HSG.SRT.C.8 HSG.SRT.C.6 HSG.SRT.C.7 HS.GM.D Crosswalk |

### Standards Guidance:

#### Clarifications

- In seventh grade, students write and solve equations using supplementary, complementary, vertical, and adjacent angles.
- Explain and use the relationship between the sine and cosine of complementary angles (e.g. sin(30) = cos(60) = 0.5).

#### Teaching Strategies

- Demonstrate understanding that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
- Students should be able to use sine, cosine, and tangent to solve real-life problems that require them to find missing side and angle measurements.

#### Progressions

- Applications should involve finding angle and side measures of right triangles.

#### Examples

- Illustrative Mathematics:

# 2021 Oregon Math Guidance: HS.GM.D.13

**Cluster: HS.GM.D - Apply concepts of right triangle trigonometry in authentic contexts to solve problems and interpret solutions.**

## STANDARD: HS.GM.D.13

### Standards Statement (2021):

Apply the Pythagorean Theorem in authentic contexts, and develop the standard form for the equation of a circle.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

8.GM.B.8 | HS.GM.D.14 | HS.AEE.A.2, HS.AEE.D.9 | HSG.GPE.A.1 HS.GM.D Crosswalk |

### Standards Guidance:

#### Terminology

- The standard form of the equation for a circle is (x-h)
^{2}+ (y-k)^{2}= r^{2}.

#### Clarifications

- Students should be able to identify the center and radius of a circle from an equation in standard form or from the graph of a circle.
- Students should be able to write the equation of a circle in standard form given the graph of the circle.
- Students should be able to graph a circle from the standard form equation of a circle.

#### Teaching Strategies

- Given the coordinates of the center and length of the radius, write the equation of the circle in standard form.
- Given the equation of a circle in standard form, determine the coordinates of its center and the length of its radius.

#### Progressions

- Use the Pythagorean Theorem to develop and apply the distance formula
- Look for and make use of structure to make connections to the Pythagorean Theorem and distance formula.

#### Examples

- Illustrative Mathematics:

# 2021 Oregon Math Guidance: HS.GM.D.14

**Cluster: HS.GM.D - Apply concepts of right triangle trigonometry in authentic contexts to solve problems and interpret solutions.**

## STANDARD: HS.GM.D.14

### Standards Statement (2021):

Use the coordinate plane to determine parallel and perpendicular relationships, and the distance between points.

### Connections:

Preceding Pathway Content (2021) | Subsequent Pathway Content (2021) | Cross Domain Connections (2021) | Common Core (CCSS) (2010) |

8.GM.B.6, 8.GM.B.8, HS.GM.D.13 | N/A | N/A | HSG.GPE.B.4 HS.GM.D Crosswalk |

### Standards Guidance:

#### Clarifications

- Students should be able to classify quadrilaterals as parallelograms (including rectangles, rhombi, and squares) using sides and diagonals.
- Students should be familiar with the distance formula when calculating the area and perimeter of quadrilaterals and triangles.

#### Terminology

- Cartesian coordinates refer to (x,y) system on a rectangular grid with the x-coordinate representing horizontal distance from the origin, and the y-coordinate representing vertical distance from the origin.

#### Boundaries

- Course level expectation is limited to use of a rectangular (Cartesian) coordinate system.

#### Teaching Strategies

- Applications include the use of coordinates to compute perimeters of polygons and areas of triangles and rectangles. The distance formula will play an important role in these applications.
- Students apply their understanding of linear relationships to derive definitions and to solve problems related to distance, midpoint, slope, area, and perimeter.

#### Progressions

- Use slope and length of line segments to classify quadrilaterals in the coordinate plane.
- Calculate the area and perimeter of parallelograms, triangles, and regular polygons in the coordinate plane.

#### Examples

- Use appropriate tools strategically to choose between tools such as the slope formula, distance formula, midpoint formula, or Pythagorean Theorem.
- Find the length of a line segment plotted on the coordinate plane.

- Illustrative Mathematics: