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

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.       Identify and describe spatial relationships among objects in space and their relative shapes and sizes.

·        Inside/outside, left/right, above/below, between

·        Smaller/larger/same size, wider/ narrower, longer/shorter

·        Congruence (i.e., same size and shape)

2.         Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes.

·        Vertex, edge, face, side

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

·        2D figures – square, rectangle, circle, triangle

·        Relationships between three- and two-dimensional shapes (i.e., the face of a 3D shape is a 2D shape)

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

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, pentagon, hexagon, octagon

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

·        Same size, same shape

·        Lines of symmetry

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

·        Line, line segment, endpoint

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

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.

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°

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

 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°

 2.         Identify, describe, compare, and classify polygons and circles.

·        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

4.         Understand and apply the concepts of congruence and symmetry (line and rotational).

1.         Understand and apply properties of polygons.

 ·        Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

·        Regular polygons

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.

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

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

·        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.

3.         Apply the properties of geometric shapes.

·        Parallel lines – transversal, alternate interior angles, corresponding angles

·        Triangles

a.      Conditions for congruence

b.      Segment joining midpoints of two sides is parallel to and half the length of the third side

c.      Triangle Inequality

·        Minimal conditions for a shape to be a special quadrilateral

·        Circles – arcs, central and inscribed angles, chords, tangents

·        Self-similarity

4.         Use reasoning and some form of proof to verify or refute conjectures and theorems.

·        Verification or refutation of proposed proofs

·        Simple proofs involving congruent triangles

·        Counterexamples to incorrect conjectures

4.2 B. Transforming Shapes

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 simple shapes to make designs, patterns, and pictures.

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

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

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.

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.

1.         Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

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

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

·