Technical Geometry

Course description:

Technical Geometry incorporates the same core geometric concepts required in a standard geometry course but includes additional topics that focus on career and technical applications.  These concepts will be taught using practical applications in a contextual style of teaching, including labs and projects. The structure of the course will include teaching groups of skills and concepts followed by their incorporation in a real world application and setting. 

 Standard 1.0: Number and Operations

Students will recognize, order, represent, and graph rational and irrational numbers.

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

·         order a set of rational numbers (e.g., determine the sizing of electrical wire by gauge);

·         estimate the value of an irrational number expressed as a radical (radicand less than 1000) (e.g., order lengths determined by a carpenter’s square);

·         approximate pi given a table of values for the circumference and diameter of circles.

At Level 2, the student is able to 

·         estimate the distance between two points on a line;

·         discuss the accuracy of radicals and their decimal approximations in contexts such as carpentry.

 At Level 3, the student is able to

·         Represent irrational numbers as lengths of lines in the coordinate plane (e.g., √5 is the length of a diagonal brace of a rectangle frame with a base of  1 and a height of  2).

Sample Task:

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

Linkages:

Mathematics: Estimation, Measurement, and Algebra.  Construction, auto mechanics, electrical, and plumbing.

 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 to make predictions and describe trends (e.g., quality control sampling, marketing sales of specific products, demographics in an area);

·         translate given data into algebraic expressions (e.g., feasibility studies, cost vs. profit/production, and consumer costs for products);

·         find powers and roots of numbers in problem solving using appropriate technology (e.g., apply concept to bacterial growth, amortization of a house).

At Level 2, the student is able to

·         explore patterns in real-world situations (e.g., Golden Ratio, Pythagorean Triples, and Tiling);

·         use manipulatives to determine relationships between linear, square, or cubic measures when one of the measures of the object has changed;

·         simplify radicals to estimate irregular areas (e.g., building design, building braces, and grade of a slope);

·         find regression equations for data sets using technology;

·         solve problems involving indirect ratios, such as gear ratios.

At Level 3, the student is able to

·         recognize complete and incomplete networks (e.g., delivery routes, mapping, electrical and plumbing applications);

·         apply the Law of Sines and Law of Cosines to triangles (e.g., surveying, architecture, plotting location, welding).

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

·         use indirect reasoning to make conjectures and solve problems (e.g., crop analysis, inventory);

·         investigate the Pythagorean Theorem using various technologies;

·         construct parallelograms, rectangles, rhombi, and squares using physical materials, manipulatives, or technology (e.g., building construction, automobile design, quilting, pattern design).

At Level 2, the student is able to

·         use inductive and deductive reasoning to draw a conclusion (e.g., diagnostics in automotives, health science nutrition science);

·         recognize and articulate relationships among families of geometric figures (e.g., metal fabrication, floral design, landscaping);

·         investigate the properties of angles, arcs, chords, tangents, and/or secants using technology or manipulatives;

·         use logical reasoning to solve problems in the real world (e.g., health science, agri-science, criminal justice, nutrition);

·         use manipulatives to explore the geometric mean of similar triangles (e.g., plumbing, electrical wiring);

·         use appropriate technology to develop geometric and spatial concepts;

·         construct three-dimensional objects using physical materials and manipulatives (e.g., packaging, cake decorating, building construction, sculptural design, and mobile creation in child care);

·         identify the three basic trig ratios and their graphs;

·         recognize and apply reflexive, symmetric, and transitive properties of equality, similarity, and congruence.

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 (e.g., systems, diagnostics, CAD);

·         apply the three basic trig ratios to solving problems (e.g., angle of elevation, grade of a road, bearings).

Sample Task:

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

Linkages:

Mathematics: Measurement. Surveying, Design, Road Construction, 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 (angles of elevation and grade of a road in surveying);

·         recognize vector quantities;

·         determine the best estimate for a given measurement.

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 such as patty paper and technology (e.g., tailoring of clothes, dentistry, carpentry); draw auxiliary diagrams to help solve for an unknown dimension or an unknown angle;

·         choose appropriate techniques and tools to measure quantities in order to meet specification for precision, accuracy, and tolerance (quality control assurance, jewelry, tool and die);

·         solve problems involving volume of three dimensional figures, e.g., right prisms, pyramids, cones, cylinders and spheres;

·         solve problems involving surface area of prisms and cylinders;

·         make simple scale drawings (e.g., blueprints, models, publishing);

·         find the magnitude and direction of a vector (e.g., location, headings).

At Level 3, the student is able to

·         locate the irrational numbers Ö2 and Ö3 on a number line by using the Pythagorean relationship and a straightedge and compass (e.g., surveying);

·         solve problems involving surface area of pyramids, cones and spheres;

·         solve problems involving signed numbers and vectors (e.g., work, force, bearings).

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:

·         apply geometric representations to calculate theoretical probability;

·         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 (sales projections).

At Level 2, the student is able to

·         use hands-on activities to model geometric representations of probability (polling results, inventory control).

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 (defective parts in a sample of products, analysis of the validity of survey results).