STANDARD 4.2: Mathematics

Standard 4.2 Geometry and Measurement All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.
Big Idea Geometry: Spatial sense and geometric relationships are a means to solve problems and make sense of a variety of phenomena. Big Idea Measurement: Measurement is a tool to quantify a variety of phenomena.
4.2.5 A. Geometric Properties
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Essential Questions	Enduring Understandings
- How can spatial relationships be described by careful use of geometric language? - How do geometric relationships help in solving problems and/or make sense of phenomena?	- Geometric properties can be used to construct geometric figures. (4.5D1; 4.5D2; 4.5E3) - Geometric relationships provide a means to make sense of a variety of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Understand and apply concepts involving lines and angles. · Notation for line, ray, angle, line segment · Properties of parallel, perpendicular, and intersecting lines · Sum of the measures of the interior angles of a triangle is 180°	"Understand and apply" here means "define, recognize, and apply."
2. Identify, describe, compare, and classify polygons. · Triangles by angles and sides · Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi · Polygons by number of sides · Equilateral, equiangular, regular · All points equidistant from a given point form a circle
3. Identify similar figures.
4. Understand and apply the concepts of congruence and symmetry (line and rotational).
4.2.5 B. Transforming Shapes
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Essential Questions	Enduring Understandings
- What situations can be analyzed using transformations and symmetries? (4.5E1; 4.5E2; 4.5E3)	- Shape and area can be conserved during mathematical transformations..
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.
2. Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.
4.2.5 C. Coordinate Geometry
Descriptive Statement: Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry.
Essential Questions	Enduring Understandings
- How can geometric/algebraic relationships best be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5)	- Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4; 4.5D5; 4.5F5) - Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Create geometric shapes with specified properties in the first quadrant on a coordinate grid.	Sample SCR Item: Three vertices of a parallelogram are at the points (0, 0), (2, 4), and (6, 0). What are the coordinates of the fourth vertex? (Answer: (8,4) or (-4,4) or (4, -4). Although not expected to find either of the answers out of the first quadrant, a student would not be penalized for finding such a vertex.)
4.2.5 D. Units Of Measurement
Descriptive Statement: Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. -What we measure affects how we measure it. (4.5A4; 4.5A6) - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Select and use appropriate units to measure angles and area.	Sample MC Item: What units would most likely be used to measure the area of a classroom floor? a. Square inches * b. Square feet c. Cubic feet d. Cubic yards
2. Convert measurement units within a system (e.g., 3 feet = ___ inches).	Sample ECR Item: Two students measured the same book shelf. Debbie said the measurement is 3. Tim said the measurement is 36. How can both students be correct? Explain your reasoning.
3. Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).	"Know approximate equivalents" means that students should be able to recognize or produce approximate equivalents. Sample ECR Item: Carol measured her height to be 1.5. How can this be possible? Explain your reasoning.
4. Use measurements and estimates to describe and compare phenomena.	This CPI will infrequently be measured independently, but will provide a context for measuring other CPIs.
4.2.5 E. Measuring Geometric Objects
Descriptive Statement: This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply formulas, it is important they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. - What we measure affects how we measure it. (4.5A4; 4.5A6)** - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
1. Use a protractor to measure angles.
2. Develop and apply strategies and formulas for finding perimeter and area. · Square · Rectangle
3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.	Assessment of this CPI is generally within the context of CPI 4.2.5E2. Assessment Focus: • Students are expected to solve problems (4.5A2)** involving this recognition.
4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot).