 # Using the Core Learning Goals: Mathematics  ## CLG Toolkit

Tools aligned to CLG expectations and/or indicators.

• Skill Statements
Describes how a student demonstrates an understanding of an indicator
• Public Release Items
HSA items and annotated student responses as appropriate
• Lesson Plans
Written by Maryland educators for teaching the concepts

## Goal 2: Geometry, Measurement, And Reasoning

The student will demonstrate the ability to solve mathematical and real-world problems using measurement and geometric models and will justify solutions and explain processes used.

### Expectation

2.1 The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.

### Indicator

• 2.1.1 The student will analyze the properties of geometric figures.

###### Assessment limit:

Essential properties, relationships, and geometric models include the following:

• Congruence and similarity
• line/segment/plane relationships (parallel, perpendicular, intersecting, bisecting, midpoint, median, altitude)
• point relationships (collinear, coplanar)
• angles and angle relationships (vertical, adjacent, complementary, supplementary, obtuse, acute, right, interior, exterior)
• angle relationships with parallel lines
• polygons (regular, non-regular, composite, equilateral, equiangular)
• geometric solids (cones, cylinders, prisms, pyramids, composite figures)
• circle/sphere (tangent, radius, diameter, chord, secant, central/inscribed angle, inscribed, circumscribed).

### Indicator

• 2.1.2 The student will identify and/or verify properties of geometric figures using the coordinate plane and concepts from algebra.

###### Assessment limits:
• “Verify properties” means to justify solutions using definitions and/or mathematical principles.
• Properties, relationships, and geometric models include the following:
• Congruence and similarity
• line/segment relationships (parallel, perpendicular, intersecting, bisecting, midpoint, median, altitude)
• point relationships (collinear)
• angles and angle relationships (obtuse, acute, right)
• polygons (regular, non-regular, equilateral, equiangular)
• circle (tangent, radius, diameter, chord).
• Items for this indicator may be set on the coordinate plane or may just have coordinates identified with no grid.
• Concepts from algebra include applications of the distance, midpoint, and slope formulas.

### Indicator

• 2.1.3 The student will use transformations to move figures, create designs, and/or demonstrate geometric properties.

###### Assessment limits:
• Transformations include reflections, rotations, translations, and dilations.
• Items should go beyond the identification of transformations.
• Essential properties and relationships include the following: congruence, similarity, and symmetry.
• The student’s explanation of a transformation must include the following:
• translation – distance and direction
• reflection – line of reflection
• rotation – center of rotation, angle measure, direction (clockwise or counterclockwise)
• dilation – center and scale factor
• Paper folding and the use of MirasTM and mirrors are appropriate methods for performing transformations, and their use must be referenced.

### Indicator

• 2.1.4 The student will construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology.

###### Assessment limits:
• “Validate properties” in this indicator, means justifying solutions using definitions, mathematical principles and/or measurement.
• Students may use a compass, straightedge, patty paper, a MiraTM, and/or a mirror as construction tools. Using a ruler or protractor cannot be part of the strategy.
• Students may use a compass, ruler, patty paper, a MiraTM, a mirror and/or a protractor as drawing tools.
• It is acceptable to do a construction when the item asks for a drawing.
• Paper folding and the use of MirasTM and mirrors are appropriate methods for representing, constructing, and/or analyzing figures, and their use must be referenced.
• Constructions and drawings are limited to the two-dimensional relationships listed in 2.1.1.

### Expectation

2.2 The student will apply geometric properties and relationships to solve problems using tools and technology when appropriate.

### Indicator

• 2.2.1 The student will identify and/or verify congruent and similar figures and/or apply equality or proportionality of their corresponding parts.

###### Assessment limits:
• Students will demonstrate geometric reasoning and justify conclusions. Although the focus is on geometric theory, answers to some items may include a numeric answer.
• Corresponding measurements include length, angle measure, perimeter, circumference, area, volume, surface area and lateral area.

### Indicator

• 2.2.2 The student will solve problems using two-dimensional figures and/or right-triangle trigonometry.

###### Assessment limits:
• Students will demonstrate geometric reasoning and justify conclusions.
• Trigonometric functions may be used to find sides or angles.
• Trigonometric functions will be limited to sine, cosine, and tangent and their inverses.

### Indicator

• 2.2.3 The student will use inductive or deductive reasoning.

###### Assessment limits:
• Students are expected to demonstrate their geometric reasoning and justify conclusions. Although the focus is on geometric theory, answers to some questions may include a numeric answer.
• Items may include geometric applications, patterns, and logic, including syllogisms.
• Narrative, flow chart, or two-column proof may be used as a valid argument.

### Expectation

2.3 The student will apply concepts of measurement using tools and technology when appropriate.

### Indicator

• 2.3.1 The student will use algebraic and/or geometric properties to measure indirectly.

###### Assessment limits:
• “Measure indirectly” means to use mathematical concepts such as congruence, similarity, and ratio and proportion to calculate measurements.
• Similarity and congruence will be directly stated or implied (scale drawings, enlargements).
• Items may require the student to make comparisons.
• This indicator may incorporate measuring.
• This indicator does not include right-triangle trigonometry.

### Indicator

• 2.3.2 The student will use techniques of measurement and will estimate, calculate, and/or compare perimeter, circumference, area, volume, and/or surface area of two-and three-dimensional figures and their parts.

###### Assessment limits:
• Two-dimensional shapes include polygons, circles, and composite figures.
• Three-dimensional shapes include cubes, prisms, pyramids, cylinders, cones, spheres, and composite figures.
• Formulas will be provided.
• No oblique solids will be used.
• Items may involve applications of geometric properties and relationships.
• Students may be required to make comparisons which do not require calculations. 