SECTION 210:15-3-58. Standard Three: Geometry and Measurement

(a)   Statement of the standard. Students will focus on number and operations to develop fluency with an importance of understanding numbers, ways of representing numbers, relationships among numbers, relationships among number systems, and meanings of operations and how they relate to one another. Students will place an emphasis on the development of estimation to determine the reasonableness of solutions and answers and to compute efficiently and proficiently.

(b)   Standard Three objectives for Pre-Kindergarten. The following objectives apply for students in Pre-Kindergarten:

(1)   Identify common shapes.

(A)   Objective 1. Identify circles, squares, rectangles, and triangles by pointing to the shape when given the name.

(2)   Describe and compare measureable attributes.

(A)   Objective 1. Identify measureable attributes of objects. Describe them as little, big, long, short, tall, heavy, light, or other age appropriate vocabulary.

(B)   Objective 2. Directly compare two objects with a common measureable attribute using words such as longer/shorter; heavier/lighter; or taller/shorter.

(C)   Objective 3. Sort objects into sets by one or more attributes.

(c)   Standard Three objectives for Kindergarten. The following objectives apply for students in Kindergarten:

(1)   Recognize and sort basic two-dimensional shapes and use them to represent real-world objects.

(A)   Objective 1. Recognize squares, circles, triangles, and rectangles.

(B)   Objective 2. Sort two-dimensional objects using characteristics such as shape, size, color, and thickness.

(C)   Objective 3. Identify attributes of two-dimensional shapes using informal and formal geometric language interchangeably.

(D)   Objective 4. Use smaller shapes to form a larger shape when there is an outline to follow.

(E)   Objective 5. Compose free-form shapes with blocks.

(F)   Objective 6. Use basic shapes and spatial reasoning to represent objects in the real world.

(2)   Compare and order objects according to location and measurable attributes.

(A)   Objective 1. Use words to compare objects according to length, size, weight, position, and location.

(B)   Objective 2. Order up to 6 objects using measureable attributes, such as length and weight.

(C)   Objective 3. Sort objects into sets by more than one attribute.

(D)   Objective 4. Compare the number of objects needed to fill two different containers.

(3)   Tell time as it relates to daily life.

(A)   Objective 1. Develop an awareness of simple time concepts using words such as yesterday, today, tomorrow, morning, afternoon, and night within his/her daily life.

(d)   Standard Three objectives for Grade 1. The following objectives apply for students in Grade 1:

(1)   Recognize, compose, and decompose two- and three-dimensional shapes.

(A)   Objective 1. Identify trapezoids and hexagons by pointing to the shape when given the name.

(B)   Objective 2. Compose and decompose larger shapes using smaller two-dimensional shapes.

(C)   Objective 3. Compose structures with three-dimensional shapes.

(D)   Objective 4. Recognize three-dimensional shapes such as cubes, cones, cylinders, and spheres.

(2)   Select and use nonstandard and standard units to describe length and volume/capacity.

(A)   Objective 1. Use nonstandard and standard measuring tools to measure the length of objects to reinforce the continuous nature of linear measurement.

(B)   Objective 2. Illustrate that the length of an object is the number of same-size units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other.

(C)   Objective 3. Measure the same object/distance with units of two different lengths and describe how and why the measurements differ.

(D)   Objective 4. Describe a length to the nearest whole number and a unit.

(E)   Objective 5. Use standard and nonstandard tools to identify volume/capacity. Compare and sort containers that hold more, less, or the same amount.

(3)   Tell time to the half and full hour.

(A)   Objective 1. Tell time to the hour and half-hour (analog and digital).

(e)   Standard Three objectives for Grade 2. The following objectives apply for students in Grade 2:

(1)   Analyze attributes of two-dimensional figures and develop generalizations about their properties.

(A)   Objective 1. Recognize trapezoids and hexagons.

(B)   Objective 2. Describe, compare, and classify two-dimensional figures according to their geometric attributes.

(C)   Objective 3. Compose two-dimensional shapes using triangles, squares, hexagons, trapezoids, and rhombi.

(D)   Objective 4. Recognize right angles and classify angles as smaller or larger than a right angle.

(2)   Understand length as a measurable attribute and explore capacity.

(A)   Objective 1. Explain the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.

(B)   Objective 2. Explain the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest whole unit.

(C)   Objective 3. Explore how varying shapes and styles of containers can have the same capacity.

(3)   Tell time to the quarter hour.

(A)   Objective 1. Read and write time to the quarter-hour on an analog and digital clock. Distinguish between a.m. and p.m.

(f)   Standard Three objectives for Grade 3. The following objectives apply for students in Grade 3:

(1)   Use geometric attributes to describe and create shapes in various contexts.

(A)   Objective 1. Sort three-dimensional shapes based on attributes.

(B)   Objective 2. Build a three-dimensional figure using unit cubes when picture/shape is shown.

(C)   Objective 3. Classify angles as acute, right, obtuse, and straight.

(2)   Understand measurable attributes of real-world and mathematical objects using various tools.

(A)   Objective 1. Find perimeter of polygon, given whole number lengths of the sides, in real-world and mathematical situations.

(B)   Objective 2. Develop and use formulas to determine the area of rectangles. Justify why length and width are multiplied to find the area of a rectangle by breaking the rectangle into one unit by one unit squares and viewing these as grouped into rows and columns.

(C)   Objective 3. Choose an appropriate measurement instrument and measure the length of objects to the nearest whole centimeter or meter.

(D)   Objective 4. Choose an appropriate measurement instrument and measure the length of objects to the nearest whole yard, whole foot, or half inch.

(E)   Objective 5. Using common benchmarks, estimate the lengths (customary and metric) of a variety of objects.

(F)   Objective 6. Use an analog thermometer to determine temperature to the nearest degree in Fahrenheit and Celsius.

(G)   Objective 7. Count cubes systematically to identify number of cubes needed to pack the whole or half of a three-dimensional structure.

(H)   Objective 8. Find the area of two-dimensional figures by counting total number of same size unit squares that fill the shape without gaps or overlaps.

(3)   Solve problems by telling time to the nearest 5 minutes.

(A)   Objective 1. Read and write time to the nearest 5-minute (analog and digital).

(B)   Objective 2. Determine the solutions to problems involving addition and subtraction of time in intervals of 5 minutes, up to one hour, using pictorial models, number line diagrams, or other tools.

(g)   Standard Three objectives for Grade 4. The following objectives apply for students in Grade 4:

(1)   Name, describe, classify, and construct polygons and three-dimensional figures.

(A)   Objective 1. Identify points, lines, line segments, rays, angles, endpoints, and parallel and perpendicular lines in various contexts.

(B)   Objective 2. Describe, classify, and sketch quadrilaterals, including squares, rectangles, trapezoids, rhombuses, parallelograms, and kites. Recognize quadrilaterals in various contexts.

(C)   Objective 3. Given two three-dimensional shapes, identify similarities, and differences.

(2)   Understand angle, length, and area as measurable attributes of real-world and mathematical objects. Use various tools to measure angles, length, area, and volume.

(A)   Objective 1. Measure angles in geometric figures and real-world objects with a protractor or angle ruler.

(B)   Objective 2. Find the area of polygons that can be decomposed into rectangles.

(C)   Objective 3. Using a variety of tools and strategies, develop the concept that the volume of rectangular prisms with whole-number edge lengths can be found by counting the total number of same-sized unit cubes that fill a shape without gaps or overlaps. Use appropriate measurements such as cm.

(D)   Objective 4. Choose an appropriate measurement instrument and measure the length of an object to the nearest whole centimeter or quarter-inch.

(E)   Objective 5. Solve problems that deal with measurements of length, when to use liquid volumes, when to use mass, temperatures above zero and money using addition, subtraction, multiplication, or division as appropriate (customary and metric).

(3)   Determine elapsed time and convert between units of time.

(A)   Objective 1. Determine elapsed time.

(B)   Objective 2. Solve problems involving the conversion of one measure of time to another.

(h)   Standard Three objectives for Grade 5. The following objectives apply for students in Grade 5:

(1)   Describe, classify, and draw representations of two- and three-dimensional figures.

(A)   Objective 1. Describe, classify and construct triangles, including equilateral, right, scalene, and isosceles triangles. Recognize triangles in various contexts.

(B)   Objective 2. Describe and classify three-dimensional figures including cubes, rectangular prisms, pyramids by the number of edges, faces or vertices, as well as the shapes of faces.

(C)   Objective 3. Recognize and draw a net for a three-dimensional figure (e.g. cubes, rectangular prisms, pyramids).

(2)   Understand how the volume of rectangular prisms and surface area of shapes with polygonal faces are determined by the dimensions of the object and that shapes with varying dimensions can have equivalent values of surface area or volume.

(A)   Objective 1. Recognize that the volume of rectangular prisms can be determined by the number of cubes (n) and by the product of the dimensions of the prism (axbxc = n). Know that rectangular prisms of different dimensions (p, q, and r) can have the same volume if axbxc = pxqxr = n.

(B)   Objective 2. Recognize that the surface area of a three-dimensional figure with rectangular faces with whole numbered edges can be found by finding the area of each component of the net of that figure. Know that three-dimensional shapes of different dimensions can have the same surface area.

(C)   Objective 3. Find the perimeter of polygons and create arguments for reasonable values for the perimeter of shapes that include curves.

(3)   Understand angle and length as measureable attributes of real-world and mathematical objects. Use various tools to measure angles and lengths.

(A)   Objective 1. Measure and compare angles according to size.

(B)   Objective 2. Choose an appropriate instrument and measure the length of an object to the nearest whole centimeter or 1/16-inch.

(C)   Objective 3. Recognize and use the relationship between inches, feet, and yards to measure and compare objects.

(D)   Objective 4. Recognize and use the relationship between millimeters, centimeters, and meters to measure and compare objects.

(i)   Standard Three objectives for Grade 6. The following objectives apply for students in Grade 6:

(1)   Calculate area of squares, parallelograms, and triangles to solve real-world and mathematical problems.

(A)   Objective 1. Develop and use formulas for the area of squares and parallelograms using a variety of methods including but not limited to the standard algorithm.

(B)   Objective 2. Develop and use formulas to determine the area of triangles.

(C)   Objective 3. Find the area of right triangles, special quadrilaterals, and polygons that can be decomposed into triangles and other shapes to solve real-world and mathematical problems.

(2)   Understand and use relationships between angles in geometric figures.

(A)   Objective 1. Solve problems using the relationships between the angles (vertical, complementary, and supplementary) formed by intersecting lines.

(B)   Objective 2. Develop and use the fact that the sum of the interior angles of a triable is 180° to determine missing angle measures in a triangle.

(3)   Choose appropriate units of measurement and use ratios to convert within measurement systems to solve real-world and mathematical problems.

(A)   Objective 1. Estimate weights, capacities and geometric measurements using benchmarks in customary and metric measurement systems with appropriate units.

(B)   Objective 2. Solve problems in various real-world and mathematical contexts that require the conversion of weights, capacities, geometric measurements, and time within the same measurement systems using appropriate units.

(4)   Use translations, reflections, and rotations to establish congruency and understand symmetries.

(A)   Objective 1. Predict, describe, and apply translations (slides), reflections (flips), and rotations (turns) to a two-dimensional figure.

(B)   Objective 2. Recognize that translations, reflections, and rotations preserve congruency and use them to show that two figures are congruent.

(C)   Objective 3. Use distances between two points that are either vertical or horizontal to each other (not requiring the distance formula) to solve real-world and mathematical problems about congruent two-dimensional figures.

(D)   Objective 4. Identify and describe the line(s) of symmetry in two-dimensional shapes.

(j)   Standard Three objectives for Grade 7. The following objectives apply for students in Grade 7:

(1)   Develop and understand the concept of surface area and volume of rectangular prisms.

(A)   Objective 1. Using a variety of tools and strategies, develop the concept that surface area of a rectangular prism with rational-valued edge lengths can be found by wrapping the figure with same-sized square units without gaps or overlap. Use appropriate measurements such as cm.

(B)   Objective 2. Using a variety of tools and strategies, develop the concept that the volume of rectangular prisms with rational-valued edge lengths can be found by counting the total number of same-sized unit cubes that fill a shape without gaps or overlaps. Use appropriate measurements such as cm.

(2)   Determine the area of trapezoids and area and perimeter of composite figures.

(A)   Objective 1. Develop and use the formula to determine the area of a trapezoid to solve problems.

(B)   Objective 2. Find the area and perimeter of composite figures to solve real-world and mathematical problems.

(3)   Use reasoning with proportions and ratios to determine measurements, justify formulas, and solve real-world and mathematical problems involving circles and related geometric figures.

(A)   Objective 1. Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is pi and can be approximated by rational numbers such as 22/7 and 3.14.

(B)   Objective 2. Calculate the circumference and area of circles to solve problems in various contexts, in terms of pi and using approximations for pi.

(4)   Analyze the effect of dilations, translations, and reflections on the attributes of two-dimensional figures on and off the coordinate plane.

(A)   Objective 1. Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors resulting from dilations.

(B)   Objective 2. Apply proportions, ratios, and scale factors to solve problems involving scale drawings and determine side lengths and areas of similar triangles and rectangles.

(C)   Objective 3. Graph and describe translations and reflections of figures on a coordinate plane and determine the coordinates of the vertices of the figure after the transformation.

(k)   Standard Three objectives for Pre-Algebra. The following objectives apply for students in Pre-Algebra:

(1)   Solve problems involving right triangles using the Pythagorean Theorem.

(A)   Objective 1. Informally justify the Pythagorean Theorem using measurements, diagrams, or dynamic software and use the Pythagorean Theorem to solve problems in two and three dimensions involving right triangles.

(B)   Objective 2. Use the Pythagorean Theorem to find the distance between any two points in a coordinate plane.

(2)   Calculate surface area and volume of three-dimensional figures.

(A)   Objective 1. Calculate the surface area of a rectangular prism using decomposition or nets. Use appropriate measurements such as cm.

(B)   Objective 2. Calculate the surface area of a cylinder, in terms of pi and using approximations for pi, using decomposition or nets. Use appropriate measurements such as cm.

(C)   Objective 3. Develop and use the formulas V = lwh and V = Bh to determine the volume of rectangular prisms. Justify why base area (B) and height (h) are multiplied to find the volume of a rectangular prism. Use appropriate measurements such as cm.

(D)   Objective 4. Develop and use the formulas V = πr^2h and V = Bh to determine the volume of right cylinders, in terms of pi and using approximations for pi. Justify why base area (B) and height (h) are multiplied to find the volume of a right cylinder. Use appropriate measurements such as cm.

(l)   Standard Three objectives for Algebra 1. The following objectives apply for students in Algebra 1:

(1)   Understand functions as descriptions of covariation (how related quantities vary together) in real-world and mathematical problems.

(A)   Objective 1. Distinguish between relations and functions.

(B)   Objective 2. Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts.

(C)   Objective 3. Write linear functions, using function notation, to model real-world and mathematical situations.

(D)   Objective 4. Given a graph modeling a real-world situation, read and interpret the linear piecewise function (excluding step functions).

(2)   Recognize functions and understand that families of functions are characterized by their rate of change.

(A)   Objective 1. Distinguish between linear and nonlinear (including exponential) functions arising from real-world and mathematical situations that are represented in tables, graphs, and equations. Understand that linear functions grow by equal intervals and that exponential functions grow by equal factors over equal intervals.

(B)   Objective 2. Recognize the graph of the functions (x) = x and (x) = |x| and predict the effects of transformations [f(x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators.

(3)   Represent functions in multiple ways and use the representation to interpret real-world and mathematical problems.

(A)   Objective 1. Identify and generate equivalent representations of linear equations, graphs, tables, and real-world situations.

(B)   Objective 2. Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems.

(C)   Objective 3. Add, subtract, and multiply functions using function notation.

(m)   Standard Three objectives for Geometry. The following objectives apply for students in Geometry:

(1)   Reasoning & Logic. Use appropriate tools and logic to evaluate mathematical arguments.

(A)   Objective 1. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs.

(B)   Objective 2. Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive.

(C)   Objective 3. Assess the validity of a logical argument and give counterexamples to disprove a statement.

(2)   Two-Dimensional Shapes. Discover, evaluate and analyze the relationships between lines, angles, and polygons to solve real-world and mathematical problems; express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts, or illustrations.

(A)   Objective 1. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs.

(B)   Objective 2. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs.

(C)   Objective 3. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs.

(D)   Objective 4. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs.

(E)   Objective 5. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments.

(F)   Objective 6. Apply the properties of polygons to solve real-world and mathematical problems involving perimeter and area (e.g. triangles, special quadrilaterals, regular polygons up to 12 sides, composite figures).

(G)   Objective 7. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logic reasoning.

(H)   Objective 8. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS).

(I)   Objective 9. Use numeric, graphic and algebraic representations of transformations in two dimensions, such as reflections, translations, dilations, and rotations about the origin by multiples of 90 degrees, to solve problems involving figures on a coordinate plane and identify types of symmetry.

(3)   Three-Dimensional Shapes. Solve real-world and mathematical problems involving three-dimensional figures.

(A)   Objective 1. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate.

(B)   Objective 2. Use ratios derived from similar three-dimensional figures to make conjectures, generalize, and to solve for unknown values such as angles, side lengths, perimeter or circumference of a face, area of a face, and volume.

(4)   Circles. Solve real-world and mathematical problems using the properties of circles.

(A)   Objective 1. Apply the properties of circles to solve problems involving circumference and area, approximate values and in terms of pi, using algebraic and logical reasoning.

(B)   Objective 2. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic reasoning.

(C)   Objective 3. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x - h)squared + (y - k)squared = rsquared, with and without graphs.

(D)   Objective 4. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form.

(5)   Right Triangle Trigonometry. Develop and verify mathematical relationships of right triangles and trigonometric ratios to solve real-world and mathematical problems

(A)   Objective 1. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples).

(B)   Objective 2. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning.

(C)   Objective 3. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles.

(D)   Objective 4. Apply the trigonometric functions as ratios (singe, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems.

(n)   Standard Three objectives for Algebra 2. The following objectives apply for students in Algebra 2:

(1)   Functions. Understand functions as descriptions of covariation (how related quantities vary together).

(A)   Objective 1. Use algebraic, interval, and set notations to specify the domain and range of functions of various types and evaluate a function at a given point in its domain.

(B)   Objective 2. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology.

(C)   Objective 3. Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology.

(D)   Objective 4. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically.

(E)   Objective 5. Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease.

(F)   Objective 6. Graph a rational function and identify the x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology (excluding slant or oblique asymptotes and holes).

(G)   Objective 7. Graph a radical function (square root and cube root only) and identify the x- and y-intercepts using various methods and tools that may include a graphing calculator or other appropriate technology.

(H)   Objective 8. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant.

(2)   Functions. Analyze functions through algebraic combinations, compositions, and inverses, if they exist.

(A)   Objective 1. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions.

(B)   Objective 2. Combine functions by composition and recognize that (x) = (f raised to the power of negative 1)(x), the inverse function of f(x), if and only if f(g(x)) = g(f(x)) = x.

(C)   Objective 3. Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f raised to the power of negative 1, and the range of the function f is the domain of the inverse function f raised to the power of negative 1.

(D)   Objective 4. Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another.

[Source: Added at 33 Ok Reg 1324, eff 9-11-16]