TDOE: Teaching and Learning: Curriculum: Math

Geometry

Course description:

Geometry uses problem situations, physical models, and appropriate technology to investigate and justify geometric concepts and relationships. Problem-solving situations provide all students an environment that promotes communication, engages student reasoning, and fosters connections within mathematics, to other disciplines, and to the real world. Students will use physical models to represent, explore, develop, and apply abstract concepts. The use of appropriate technology will help students develop mathematics needed in an increasingly technological world. The concepts and topics emphasized in the course include measurement, geometric patterns, coordinate geometry, two- and three-dimensional figures, transformational geometry, congruence, similarity, inductive and deductive reasoning, logic, and proof.

Standard Number 1.0: Number and Operations

Students will recognize, order, represent, and graph rational and irrational numbers, including absolute value notation.

Learning Expectations:

The student will:

1.1 demonstrate an understanding of the relative size of rational and irrational numbers;
1.2 choose and use appropriate notations for rational and irrational numbers, including graphic representations;
1.3 demonstrate an understanding of absolute value.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

order a set of rational and irrational numbers;
find an integral power of a positive rational number (exponents 1-3).

At Level 2, the student is able to

use absolute value to express the distance between two points on a number line and vice versa;
simplify a radical (radicand less than 1000);
match a given irrational number to the appropriate point on a number line and vice versa (e.g., Ö2, Ö30, pi).

There are no state-assessed performance indicators at Level 3.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

estimate the value of an irrational number expressed as a radical (radicand less than 1000);
approximate pi given a table of values for the circumference and diameter of circles.

At Level 2, the student is able to

use radicals and decimal approximations of irrational numbers to indicate calculated lengths or distances;
discuss the relative accuracy of radicals and their decimal approximations.

At Level 3, the student is able to

represent irrational numbers as lengths of lines in the coordinate plane (e.g. ,Ö5 is the length of the diagonal of a rectangle with base 1 and height 2).

Sample Task:

Students will compute the hypotenuse of a given right triangles and arrange themselves in order from smallest to largest.

Linkages:

Mathematics: Estimation, Measurement, and Algebra.

Standard 2.0: Algebra

Students will recognize, extend, create, and analyze a variety of geometric, spatial, and numerical patterns; solve real-world problems related to algebra and geometry; and use properties of various geometric figures to analyze and solve problems.

Learning Expectations:

The student will:

2.1 recognize, extend, and create geometric, spatial, and numerical patterns;
2.2 analyze mathematical patterns related to algebra and geometry in real-world problem solving;
2.3 solve problems connecting geometry with number theory, probability and statistics, and measurement and estimation using algebraic thinking and symbolism;
2.4 apply coordinate geometry to analyze and solve problems;
2.5 apply ratio and proportion to problems involving similar figures.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

extend or find missing element(s) in a geometric pattern;
solve multistep linear equations to find length, width, perimeter, and area of geometric figures;
apply the concept of rate of change to solve a real-world problem given a pattern of data;
determine the slope given a graph of a linear equation and vice versa;
determine the distance, midpoint, or slope when given the coordinates of two points (answers must be given as decimals to the nearest hundredth).

At Level 2, the student is able to

determine the equation of a line parallel or perpendicular to a given line, from given information (e.g., equations of lines, graphs of lines, or two points);
apply ratio and proportion to solve real-world problems involving polygons, (e.g., scale drawings, similar figures);
apply the triangle inequality property to determine which sets of side lengths determine a triangle;
determine the perimeter, area, or volume given the ratio of two similar polygons or rectangular solids;
apply the Triangle Sum Theorem or Exterior Angle Theorem to determine the measures of the angles of a given triangle with the angle measures expressed algebraically.

At Level 3, the student is able to

determine the equation of a circle given coordinates or the graph of the circle (e.g., the center, the endpoints of the diameter).

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

apply the line of best fit given real-world data from geometric figures using technology (e.g., finding the interior angle sum of polygons when given the number of sides; find the circumference of circles when given the diameter).

At Level 2, the student is able to

explore patterns in geometric situations (e.g., Fibonacci sequence and Golden Ratio);
use manipulatives to determine relationships between linear, square, or cubic measures when one of the measures of the object has changed and represent algebraically.

At Level 3, the student is able to

recognize complete and incomplete networks;
graph plane figures on a coordinate plane and solve problems algebraically.

Sample Task:

Students construct designs using basic geometric constructions. Then they transfer the design to a piece of 8"”X 11"”pane of plexiglass and paint the pane to create a “stained glass.”

Linkages:

Mosaic Tiling.

Standard 3.0: Geometry

Students will investigate, model, and apply geometric properties and relationships and use indirect reasoning to make conjectures; deductive reasoning to draw conclusions; and both inductive and deductive reasoning to establish the truth of statements.

Learning Expectations:

The student will:

3.1 analyze relationships among corresponding parts of similar or congruent geometric figures;
3.2 apply geometric properties of solids, polygons, and circles to solve real-world problems;
3.3 justify conclusions and solve problems using deductive reasoning;
3.4 use inductive reasoning to make conjectures and solve problems;
3.5 communicate position using spatial sense with two- and three-dimensional coordinate systems;
3.6 demonstrate an understanding of transformations of geometric figures (i.e., translations, rotations, dilations, and reflections);
3.7 apply right triangle relationships including the Pythagorean Theorem, the distance formula, and trigonometric ratios;
3.8 describe geometric objects and recognize minimal conditions necessary to define the geometric objects;
3.9 apply reflexive, transitive, and symmetric properties when appropriate;
3.10 demonstrate understanding of geometric properties of congruence, similarity, perpendicularity, and parallelism;
3.11 recognize and articulate relationships among families of geometric figures (e.g., quadrilaterals, prisms);
3.12 use logic and proof to establish the validity of conjectures and theorems.

Performance Indicators State:

As documented through state assessment:

At Level 1, the student is able to

identify corresponding parts of similar and congruent geometric figures given a diagram.
determine the length of a missing side in a right triangle when given two sides (answers must be given as simplified radicals).

At Level 2, the student is able to

identify properties of plane figures from information given in a diagram;
identify chords, inscribed angles, or central angles of circles given a diagram;
determine congruence or similarity relations between triangles or quadrilaterals given a diagram;
determine whether a plane figure has been translated, dilated, reflected, or rotated given a diagram and vice versa;
solve problems involving complementary, supplementary, congruent, vertical, or adjacent angles given angle measures expressed algebraically;
determine the trigonometric ratio for a right triangle needed to solve a real-world problem given a diagram;
find a missing side length in a 30-60-90 or 45-45-90 degree triangle without rationalizing the denominator
apply properties of quadrilaterals to solve a real-world problem given a diagram (opposite sides and angles, consecutive sides and angles, or diagonals);
solve real-world problems involving measures of interior or exterior angles of regular polygons;
identify the appropriate segment of a triangle given a diagram and vice versa (i.e. median, altitude, angle bisector, perpendicular bisector);
determine which three-dimensional solid is represented by a given net and vice versa (two-dimensional drawing);
determine the area of indicated regions involving circles, squares, rectangles, and/or triangles;
justify triangle congruence given a diagram (i.e., ASA, SSS, AAS, SAS, or Hypotenuse/ Leg);
determine if a triangle is a right triangle given the length of all the sides of a triangle.

At Level 3, the student is able to

solve problems involving the properties of arcs, chords, tangents, or secants;
find the area of a sector of a circle given a diagram.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

investigate the Pythagorean Theorem by constructing right triangles using a variety of manipulatives and/or technologies;
construct parallelograms, rectangles, rhombi, and squares using physical materials, manipulatives, or technology.

At Level 2, the student is able to

apply reflexive, transitive, or symmetric properties of equality or congruence;
investigate the properties of angles, arcs, chords, tangents, and/or secants using technology or manipulatives;
use inductive and deductive reasoning to make conjectures, draw conclusions, and solve problems;
recognize and articulate relationships among families of geometric figures (e.g., quadrilaterals, prisms);
write and defend indirect and direct proofs;
use logical reasoning to solve problems in the real world;
use manipulatives to explore the geometric mean of similar triangles;
use appropriate tools or technology to develop geometric and spatial concepts;
construct three-dimensional objects using physical materials and manipulatives;
compare and construct quadrilateral properties using a variety of models (e.g., Venn diagrams, family trees, manipulative mobiles).

At Level 3, the student is able to

use coordinates to communicate the location of a three-dimensional figure that has been rotated or reflected.

Sample Task:

Students construct and use a hypsometer to measure several tall structures on the school grounds.

Linkages: Mathematics: Measurement. Surveying and Art.

Standard 4.0: Measurement

Students will apply appropriate units of measurement; develop effective estimation and computation strategies for solving real world problems involving length, area, and volume; and choose appropriate techniques and tools to measure quantities in order to meet specifications for precision, accuracy, and tolerance.

Learning Expectations:

The student will:

4.1 use concepts of length, area, and volume to estimate and solve real-world problems;
4.2 apply measurement concepts and relationships in algebraic and geometric problem-solving situations;
4.3 choose appropriate techniques and tools to measure quantities in order to meet specifications for precision, accuracy, and tolerance.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

determine the perimeter or area of a triangle or rectangle when the dimensions are given as first degree binomials in one variable;
solve real world problems involving perimeter or area of three or four sided plane figures.

At Level 2, the student is able to

determine the volume or surface area of a rectangular solid or cylinder in a real-world situation.

At Level 3, the student is able to

determine whether a reading falls within an acceptable tolerance range.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

determine the measure of an angle using a protractor.

At Level 2, the student is able to

construct bisectors of angles and line segments, perpendicular lines, congruent line segments and angles, and perpendicular bisectors using a variety of methods (e.g., patty paper, technology);
solve problems involving volume of 3-dimensional figures, e.g. right prisms, pyramids, cones, cylinders, and spheres;
solve problems involving surface area of prisms and cylinders.

At Level 3, the student is able to

choose appropriate techniques and tools to measure quantities in order to meet specification for precision, accuracy, and tolerance;
locate the irrational numbers Ö2 and Ö3 on a number line by using the Pythagorean relationship and a straightedge and compass, manipulatives, or technology;
solve problems involving surface area of pyramids, cones, and spheres.

Sample Task:

Students construct designs using basic geometric constructions. Then they transfer the design to a piece of 8" X 11" pane of plexiglass and paint the pane to create a “stained glass.” Students construct one of the regular 3-dimensional solid and compute the volume and surface area.

Linkages:

Mathematics – Geometry and Number & Operations. Surveying, construction, and architecture. .Mosaic Tiling.

Standard 5.0: Data Analysis and Probability

The student will investigate, explore, and apply geometric representations to calculate theoretical probability; and will use data from geometric figures to investigate relationships.

Learning Expectations:

The student will:

5.1 apply geometric representations to calculate theoretical probability;
5.2 use data analysis to investigate geometric relationships.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

make a prediction from a geometric representation of a real-world data set;

At Level 2, the student is able to

determine the probability of an event represented as a subset of the area of a two-dimensional geometric figure.

There are no performance indicators for Level 3 of Data Analysis and Probability.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

explain and justify the given geometric representation of the probability of an event.

At Level 2, the student is able to

use hands-on activities to model geometric representations of probability;
collect and analyze data to make conjectures about geometric relationships.

At Level 3, the student is able to

analyze and debate the validity of claims made based on the given theoretical probability of a real-world situation.

Sample Task:

Construct two 1’ X 1’ dart boards and draw circular targets on each that are externally tangent to each adjacent circle and to the edge of the board. Draw two circles on one dartboard and three on the other. Throw randomly and count the throws that hit the board to determine which board yields the highest probability of a dart’s landing in a circle. Calculate the probability for each bard.

Linkages:

Game theory.