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 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

By the end of Grade 4:

1.       Identify and describe spatial relationships of two or more objects in space.

·        Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)

·        Relative shapes and sizes

·        Shadows (projections) of everyday objects

2.         Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.

·        Vertex, edge, face, side, angle

·        3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid

·        2D figures – square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon

·        Inclusive relationships – squares are rectangles, cubes are rectangular prisms

3.        Identify and describe relationships among two-dimensional shapes.

·      Congruence

·      Lines of symmetry

4.         Understand and apply concepts involving lines, angles, and circles.

·        Point, line, line segment, endpoint

·        Parallel, perpendicular

·        Angles – acute, right, obtuse

·        Circles – diameter, radius, center

5.         Recognize, describe, extend, and create space-filling patterns.

Instructional/Assessment Focus:
• This is an area of focus in grade 3 and may be assessed at a higher level of understanding in grade 4.

4.2 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

By the end of Grade 4:

1.        Use simple shapes to cover an area (tessellations).

Suggested Instructional/Assessment Strategies:
• This content provides an opportunity to integrate mathematics with the visual arts. Students can:
• view prints by M.C. Escher and see how tessellations can become a famous art form;
•engage in problem solving as they discover the different ways they can tessellate polygons from pattern blocks or geoblocks;
•tessellate shapes using slides, rotations, and reflections; or
•explore various figures (including, but not limited to, those mentioned in CPIs 4.2.3A2 and 4.2.4A2) as they try to tessellate kites, ovals, parallelograms, rhombi, triangles, pentagons, hexagons, circles, or rectangles.

2.         Describe and use geometric transformations (slide, flip, turn).

Instructional/Assessment Focus:
• This is an area of focus in grade 3 and may be assessed at a higher level of understanding in grade 4.

3.         Investigate the occurrence of geometry in nature and art.

Instructional/Assessment Focus:
• This is an area of focus in grade 3 and may be assessed at a higher level of understanding in grade 4.

4.2 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

By the end of Grade 4:

1.        Locate and name points in the first quadrant on a coordinate grid.

2.         Use coordinates to give or follow directions from one point to another on a map or grid.

4.2 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

By the end of Grade 4:

1.         Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

Instructional/Assessment Focus:
• This is an area of focus in grade 3 and may be assessed at a higher level of understanding in grade 4.

2.         Select and use appropriate standard units of measure and measurement tools to solve real-life problems

·        Length – fractions of an inch (1/8, 1/4, 1/2), mile, decimeter, kilometer

·        Area – square inch, square centimeter

·        Volume – cubic inch, cubic centimeter

·        Weight – ounce

·        Capacity – fluid ounce, cup, gallon, milliliter

Sample Assessment Item:
• MC: What is the most reasonable estimate of the length of a city’s swimming pool?
a. 1 meter

* b. 25 meters

c. 1 kilometer

d. 25 kilometers

3.         Develop and use personal referents to approximate standard units of measure (e.g., a common paper clip is about an inch long).

Instructional/Assessment Focus:
• This CPI is largely an instructional CPI. Assessment of this CPI is generally within the context of one or more of the other content CPIs.

Suggested Instructional/Assessment Strategy:
• Students identify parts of their body that are the same length as 10 centimeters and use them to measure the length of their pencil.

4.         Incorporate estimation in measurement activities (e.g., estimate before measuring).

Instructional/Assessment Focus:
• This is an area of focus in grade 3 and may be assessed at a higher level of understanding in grade 4.

5.         Solve problems involving elapsed time.

4.2 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

By the end of Grade 4

1.         Determine the area of simple two-dimensional shapes on a square grid.

2.         Distinguish between perimeter and area and use each appropriately in problem-solving situations.

Sample Assessment Item:
• ECR: Veronica is making a rectangular garden. She plans to put a fence around the garden using 28 feet of fencing, and she wants the garden to be 8 feet long.
• How wide will Veronica's garden be? Show your work or explain how you got your answer.
• If Veronica is going to put fence posts two feet apart around the outside of the garden, how many fence posts will she need? Show your work or explain your answer.

3.         Measure and compare the volume of three–dimensional objects using materials such as rice or cubes

Instructional/Assessment Focus:
• Students are expected to solve problems (4.5A2)** involving this recognition.
• While the emphasis in grade 3 was be on the “measure,” the emphasis in grade 4 is on the “compare.”