Mathematics Cumulative Progress Indicators (CPIs) for the end of the designated grade span

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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.

A.     Geometric Properties

By the end of Grade 2, students will:

 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)

 3.         Describe, identify and create instances of line symmetry.

 4.         Recognize, describe, extend and create designs and patterns with geometric objects of different shapes and colors.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 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

 3.         Identify similar figures.

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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 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

 3.         Identify similar figures.

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

 5.         Compare properties of cylinders, prisms, cones, pyramids, and spheres.

 6.         Identify, describe, and draw the faces or shadows (projections) of three-dimensional geometric objects from different perspectives.

 7.         Identify a three-dimensional shape with given projections (top, front and side views).

 8.         Identify a three-dimensional shape with a given net (i.e., a flat pattern that folds into a 3D shape).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.         Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).

 2.         Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).

 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

B.     Transforming Shapes

By the end of Grade 2, students will:

 1.         Use simple shapes to make designs, patterns, and pictures.

 2.         Combine and subdivide simple shapes to make other shapes.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

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

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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

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

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

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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

  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

  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)

 Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

  1.         Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.

 2.         Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.

 3.         Determine whether two or more given shapes can be used to generate a tessellation.

 4.         Generate and analyze iterative geometric patterns.

·        Fractals (e.g., Sierpinski’s Triangle)

·        Patterns in areas and perimeters of self-similar figures

·        Outcome of extending iterative process indefinitely

C.     Coordinate Geometry

By the end of Grade 2, students will:

 1.         Give and follow directions for getting from one point to another on a map or grid.

 Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

 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.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

 1.         Create geometric shapes with specified properties in the first quadrant on a coordinate grid.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:

 1.         Create geometric shapes with specified properties in the first quadrant on a coordinate grid.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:

 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).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:

 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).

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

 1.         Use coordinate geometry to represent and verify properties of lines.

·        Distance between two points

·        Midpoint and slope of a line segment

·        Finding the intersection of two lines

·        Lines with the same slope are parallel

·        Lines that are perpendicular have slopes whose product is –1

 2.       Show position and represent motion in the coordinate plane using vectors.

·        Addition and subtraction of vectors

D.    Units of Measurement

 By the end of Grade 2, students will:

 1.         Directly compare and order objects according to measurable attributes.

·        Attributes – length, weight, capacity, time, temperature

 2.         Recognize the need for a uniform unit of measure.

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

·        Length – inch, foot, yard, centimeter, meter

·        Weight – pound, gram, kilogram

·        Capacity – pint, quart, liter

·        Time – second, minute, hour, day, week, month, year

·        Temperature – degrees Celsius, degrees Fahrenheit

 4.         Estimate measures.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:

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

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

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

·        Area – square inch, square centimeter

·        Weight – ounce

·        Capacity – fluid ounce, cup, gallon, milliliter

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

Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:

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

 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