Geometry

Number and Operations

Algebra

Geometry

Measurement

Data Analysis and Probability

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.

Content Standard 1.0: Number and Operations

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

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.

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.

Content 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.

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).

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.

Content 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.

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.

Sample Task:

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

Linkages: Mathematics: Measurement. Surveying and Art.

Content 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.

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.

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.

Content 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.

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.

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.