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.
2.1 The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.
- 2.1.1 The student will analyze the properties of geometric figures.
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).
- 2.1.2 The student will identify and/or verify properties of geometric figures using the coordinate plane and concepts from algebra.
- “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.
- 2.1.3 The student will use transformations to move figures, create designs, and/or demonstrate geometric properties.
- 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.
- 2.1.4 The student will construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology.
- “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.
2.2 The student will apply geometric properties and relationships to solve problems using tools and technology when appropriate.
- 2.2.1 The student will identify and/or verify congruent and similar figures and/or apply equality or proportionality of their corresponding parts.
- 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.
- 2.2.2 The student will solve problems using two-dimensional figures and/or right-triangle trigonometry.
- 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.
- 2.2.3 The student will use inductive or deductive reasoning.
- 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.
2.3 The student will apply concepts of measurement using tools and technology when appropriate.
- 2.3.1 The student will use algebraic and/or geometric properties to measure indirectly.
- “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.
- 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.
- 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.