Arizona Standards

Mathematics Standards
(Adopted 8/26/96)

Standard 4: Geometry

Students use geometric methods, properties and relationships as a means to recognize, draw, describe, connect, and analyze shapes and representations in the physical world.

Students know and are able to do the following:

READINESS (Kindergarten)

  • 4M-R1.Identify, compare, classify, draw and make models of shapes
  • 4M-R2.Recognize geometry in their surroundings

FOUNDATIONS (Grades 1-3)

  • 4M-F1.Relate geometric concepts to number and measurement ideas (e.g., dividing a rectangle into parts to represent multiplication)

Note: - two-dimensional shapes: square, rectangle, triangle, circle

- three-dimensional figures: sphere, cube, rectangular prism (box), cone, pyramid

- attributes: size; shape; the number of sides, corners and faces

PO 1. Identify two-dimensional shapes by name and attribute

PO 2. Draw two-dimensional shapes

PO 3. Identify three-dimensional figures by name and/or attribute

PO 4. Compare attributes of two-dimensional shapes

PO 5. Compare attributes of three-dimensional figures

PO 6. Use a rectangular array to represent a multiplication fact (e.g., put 12 tiles in a rectangular array; make a 3 x 4, 6 x 2, and 12 x 1 array)

  • 4M-F2.Predict how shapes can be changed by combining or dividing them

PO 1. Build geometric shapes with other common shapes (e.g., tangrams, pattern blocks, geoboards)

ESSENTIALS (Grades 4-8)

  • 4M-E1.Visualize and draw two- and three-dimensional geometric figures with special attention to analyzing and reasoning informally about their properties (e.g., parallelism, perpendicularity and congruence)

PO 1. Classify two-dimensional shapes and three-dimensional figures by their properties

    1. by sight (Grades 4-5)
    2. by properties (Grades 6-8)

PO 2. Identify the properties of geometric figures using appropriate terminology and vocabulary (e.g., parallelism, perpendicularity and congruency)

    1. two-dimensional shapes (three- and four-sided polygons) (Grades 4-5)
    2. three-dimensional figures (prisms) (Grades 6-8)

PO 3. Draw or build two-dimensional shapes (Grades 4-5) and three-dimensional figures (Grades 6-8) by applying significant properties of each (e.g., draw a rectangle with two sets of parallel sides and four right angles)

  • 4M-E2.Apply geometric properties and relationships such as congruence, similarity, angle measure, parallelism and perpendicularity to real-world situations

PO 1. Design or draw a model (e.g., designing a playhouse, garden) that demonstrates basic geometric relationships, such as

    1. parallelism, perpendicularity, similarity (Grades 4-5)
    2. all of the above and proportionality and congruency (Grades 6-8)

PO 2. Classify triangles by their angles and sides (e.g., equilateral, acute, isosceles . . .) (Grades 4-5)

PO 3. Label corresponding, supplementary and complementary angles (Grades 6-8)

PO 4. Measure and label specified angles (e.g., alternate interior, obtuse, acute, right, corresponding . . .) (Grades 6-8)

PO 5. Identify lines that are parallel and perpendicular (Grades 4-5)

PO 6. Distinguish shapes that are congruent (Grades 4-5)

  • 4M-E3.Perform elementary transformations (e.g., tessellations, flips, slides, rotations)

PO 1. Demonstrate slide, flip or turn using concrete geometric figures (Grades 4-5)

PO 2. Illustrate, using concrete or pictorial models,

    1. slide, flip or turn (e.g., quilts) (Grades 4-5)
    2. reflections, rotations and translations (e.g., tessellations) (Grades 6-8)

PO 3. Draw or build a shape that

    1. has symmetry (Grades 4-5)
    2. has two or more lines of symmetry (Grades 6-8)
  • 4M-E4.Represent and solve problems relating to size, shape, area and volume using geometric models

PO 1. Solve problems using given formulas for

    1. simple area and perimeter (Grades 4-5)
    2. area, perimeter/circumference of various circles/polygons (Grades 6-8)
    3. volume of prisms (Grades 6-8)

PO 2. Identify a variety of shapes having the same perimeter and area (Grades 4-5)

PO 3. Draw or build a variety of shapes having the same perimeter and area (Grades 6-8)

PROFICIENCY (Grades 9-12)

  • 4M-P1.Interpret and draw three-dimensional objects

PO 1. Sketch prisms, pyramids, cones, cylinders and spheres

PO 2. Classify prisms, pyramids, cones, cylinders and spheres by base shape, lateral surface shape, related surface area and volume formulas

PO 3. Recognize the three-dimensional figure represented by a two-dimensional drawing (e.g., "What figures are represented by given nets, sketches, photographs?")

  • 4M-P2.Represent problem situations with geometric models and apply properties of figures

PO 1. Calculate surface areas and volumes of three-dimensional geometric figures, given the required formulas

PO 2. Solve applied problems using angle, side length, and triangle inequality relationships

PO 3. Solve applied problems using the Pythagorean theorem (e.g., determine whether a wall is square)

PO 4. Solve applied problems using congruence and similarity relationships of triangles (e.g., estimate the height of a building, using shadows)

PO 5. Make a model of a three-dimensional figure from a two-dimensional drawing and make a two-dimensional representation of a three-dimensional object (models and representations include scale drawings, perspective drawings, blueprints or computer simulations)

PO 6. Calculate distance and midpoint between points within a coordinate system representative of a practical application

PO 7. Find the area of a geometric figure composed of a combination of two or more geometric figures, given an appropriate real-world situation and the formulas

PO 8. Solve problems involving complementary, supplementary and congruent angles

  • 4M-P3.Deduce properties of figures using transformations in coordinate systems, identifying congruency and similarity

PO 1. Determine whether a figure is symmetric with respect to a line or a point

PO 2. Give the new coordinates of a transformed geometric figure

PO 3. Determine the effects of a transformation on linear and area measurements of the original figure

PO 4. Sketch the figure that is the result of a given transformation

  • 4M-P4.Deduce properties of, and relationships between, figures from given assumptions

PO 1. Find similarities and differences among geometric shapes and designs using a given attribute (e.g., height, area, perimeter, diagonals, angle measurements)

PO 2. Identify arcs, chords, tangents and secants of a circle

PO 3. State valid conclusions using given definitions, postulates and theorems

  • 4M-P5.Translate between synthetic and coordinate representations (e.g., a straight line is represented by the algebraic equation Ax + By = C)

PO 1. Determine the relative placement of two lines on a coordinate plane by examining the algebraic equations representing them

PO 2. Verify characteristics of a given geometric figure using coordinate formulas such as distance, mid-point, and slope to confirm parallelism, perpendicularity, and congruency

  • 4M-P6.Recognize and analyze Euclidean transformations (e.g., reflections, rotations, dilations and translations)

PO 1. Classify transformations based on whether they produce congruent or similar non-congruent figures

PO 2. Determine whether a given pair of figures on a coordinate plane represents a translation, reflection, rotation and/or dilation

PO 3. Apply transformational principles to practical situations (e.g., enlarge a photograph)

DISTINCTION (Honors)

  • 4M-D1.Deduce properties of figures using vectors
  • 4M-D2.Apply transformations, coordinates and vectors in problem solving