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.
4.2.8 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, angles, and planes. · Complementary and supplementary angles · Vertical angles · Bisectors and perpendicular bisectors · Parallel, perpendicular, and intersecting planes · Intersection of plane with cube, cylinder, cone, and sphere
2. Understand and apply the Pythagorean theorem.
3. Understand and apply properties of polygons. · Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi · Regular polygons · Sum of measures of interior angles of a polygon · Which polygons can be used alone to generate a tessellation and why
4. Understand and apply the concept of similarity. · Using proportions to find missing measures · Scale drawings · Models of 3D objects
5. Use logic and reasoning to make and support conjectures about geometric objects.
6. Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology). • Congruent angles or line segments • Midpoint of a line segment	This CPI was added by the State Board of Education on January 9, 2008 and is an area of focus in grade 8.
7. Create two-dimensional representations (e.g., nets or projective views) for the surfaces of three-dimensional objects.	This CPI was added by the State Board of Education on January 9, 2008 and is an area of focus in grade 8.
4.2.8 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	This is an area of focus in grade 7 and may be assessed at a higher level of understanding in grade 8.
2. Use iterative procedures to generate geometric patterns. · Fractals (e.g., the Koch Snowflake) · Self-similarity · Construction of initial stages · Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski’s Triangle)
4.2.8 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).
4.2.8 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: You are purchasing a wallpaper border that will go around the top of a room. The room measures 8 feet 9 inches by 13 feet 8 inches. If the border is sold by the yard, how many whole yards will you need to buy? Explain your reasoning to support your answer. Sample Short Constructed Response (SCR) Item: A tire is 25 inches in diameter. How many times will it turn in traveling a mile?
2. Use approximate equivalents between standard and metric systems to estimate measurements (e.g., 5 kilometers is about 3 miles).	Assessment of this CPI is generally within the context of one or more of the other content CPIs.
3. Recognize that the degree of precision needed in calculations depends on how the results will be used and the instruments used to generate the measurements.
4. Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.
5. 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.
6. Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile).
4.2.8 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 · Impact of a dilation on the perimeter and area of a 2-dimensional figure
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).	Assessment of this CPI is generally within the context of CPI 4.2.8E3.
3. Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure. · Volume - prism, cone, pyramid · Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base) · Impact of a dilation on the surface area and volume of a three-dimensional figure
4. Use formulas to find the volume and surface area of a sphere.