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.
4.2.7 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 properties of polygons. · Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi · Regular polygons	Instructional/Assessment Focus: • "Understand and apply" here means "define, recognize, and apply" • It is assumed that students will be familiar with and be able to use the notation for "parallel" and "perpendicular."
2. Understand and apply the concept of similarity. · Using proportions to find missing measures · Scale drawings · Models of 3D objects
3. Use logic and reasoning to make and support conjectures about geometric objects.
4.2.7 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. Understand and apply transformations. · Finding the image, given the pre-image, and vice-versa · Sequence of transformations needed to map one figure onto another · Reflections, rotations, and translations result in images congruent to the pre-image · Dilations (stretching/shrinking) result in images similar to the pre-image
4.2.7 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. Use coordinates in four quadrants to represent geometric concepts.
2. Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).
4.2.7 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. Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4’3” plus 7’10” equals 12’1”).	Sample ECR Item: Given an 8-foot by 10-foot wall, how many 6-inch square tiles would be needed to cover the wall? Explain or show how you got your answer. Sample MC Item: Mr. Hernandez wants to tile his table-top with rectangular tiles, each measuring 3 inches by 4 inches. Which of the following represents the least number of tiles he can use if the table-top is a square measuring three feet on each side? a. 36 b. 72 * c. 108 d. 144 Sample MC Item: Beth is covering an 8-foot by 10-foot wall with cork tiles. Each tile is a 6-inch by 6-inch square. How many tiles will she need to cover the wall? a. 80 b. 160 * c. 320 d. 640
2. Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.	Assessment of this CPI is frequently within the context of one or more of the other content CPIs, and in an open-ended or extended constructed response item.
3. Recognize that all measurements of continuous quantities are approximations.	Assessment of this CPI is generally within the context of one or more of the other content CPIs.
4.2.7 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. Develop and apply strategies for finding perimeter and area. · Geometric figures made by combining triangles, rectangles and circles or parts of circles · Estimation of area using grids of various sizes
2. Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).	Sample MC Item: The cone below has the same height and same size base as the cylinder. The cone has a total volume of 12 cu. in. [Insert appropriate diagram] What is the volume of the cylinder? a. 4 cu. in. b. 12 cu. in. c. 24 cu. in. * d. 36 cu. in.