End-of-Course - Geometry

Sites to help students practice skills needed for the Geometry exam

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Standard 1 Number Sense and Number Theory The student will recognize, order, represent, and graph rational and irrational numbers.
Level 1
match a given irrational number to the appropriate point on a number line (e.g., pi)	Approximations of irrational numbers [this expired page is from the Internet Archive known as the Wayback Machine] Practice with approximations of irrational numbers [this expired page is from the Internet Archive known as the Wayback Machine] Rational and Irrational Numbers - from Tutor Vista Rational and Irrational Numbers on a Number Line - from Tutor Vista Teacher Hints for Irrational Approximations [this expired page is from the Internet Archive known as the Wayback Machine]
Level 2
order a set of rational and irrational numbers	Comparing Seven Digit Numbers - practice area followed by an interactive game Stretch your mind - try ordering a set of number names (in Spanish)
Sample Task
Every student will be given a different size right triangle. They will compute the hypotenuse and arrange themselves in order from smallest to largest.

Standard 2 Estimation, Measurement, and Computation The student 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.
Level 1
determine the perimeter or area of a triangle or rectangle when the dimensions are given as binomials in one variable
determine the perimeter or area of a triangle or rectangle (including squares) in a real-world situation given the dimensions as linear algebraic expressions in one variable	Areas of Triangles, Trapezoids, and Kites - discover or demonstrate the formulas for the areas of triangles, trapezoids, and kites. (three activities)
Level 2
determine the volume or surface area of a rectangular solid in a real-world situation	Aquarium Arithmetic - measuring and calculating volumes Calculate Volume of a Rectangular Solid - a calculator Another Calculator
Level 3
apply the concepts of length, perimeter, area, surface area, and volume to two- and three-dimensional figures in real-world situations	Developing Estimation Strategies by Making Connections among Number, Geometry, Measurement, and Data Concepts: Estimating Scoops - NCTM lesson plan including video segments
Sample Task
Have the students construct a design using basic geometric constructions. The students will transfer the design to a piece of 8" x 11" plastic pane of glass. Students will paint the pane to create a stained glass.

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Standard 3 Patterns, Functions, and Algebraic Thinking The student will recognize, extend, and 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.
Level 1
extend a geometric pattern
solve multistep linear equations applied to geometric figures	Linear Function and Graph - observe pouring water into a tank to see the relation between time and volume
solve systems of two linear equations with integral coefficients to find length, width, perimeter, and area of geometric figures
solve systems of two linear equations with integral coefficients to determine if the lines are parallel, intersecting, or coinciding
choose the equations of parallel lines or perpendicular lines given the coordinates (equations written in both slope-intercept and standard form)	Parallel Lines - If the slopes are the same, the two lines are parallel. Try graphing both equations using the Graph-It applet Angles and Parallel Lines - A line passing through two or more lines in a plane is called a transversal. Use these interactive investigation to investigate a transversal cutting two lines Problem of Angles and Parallel Lines (problem 1) Problem of Angles and Parallel Lines (problem 2)
choose the equations of parallel lines or perpendicular lines given the graphs (equations written in both slope-intercept and standard form)	Perpendicular Lines - Two lines are perpendicular if the product of their slopes is -1. Also, the two intersecting lines form right angles. Use the applet to investigate perpendicular lines.
apply the concept of rate of change to solve a real-world problem given a pattern of data	Stressed Out - Slope as Rate of Change
determine the slope given a graph of a linear equation	Linear Function and Graph - observe pouring water into a tank to see the relation between time and volume
determine the distance, midpoint, or slope when given the coordinates of two points (answers must be given in simplified, radical form)
Level 2
apply ratio and proportion to solve real-world problems involving polygons, (e.g., scale drawings, similar triangles)	Create polygons using this interactive exploration Follow instructions which use mathematical language to construct objects using polygons Identification of Polygons - listing of characteristics and sample drawings Finding Hidden Polygons - four drawings are presented as challenges Minimum/Maximum Problems Related To Triangles - Only problems having purely geometrical solutions are presented on this page. Percent - graphical examples and practice problems Percent and Proportion - unit 10 from Ms. Glosser's Math Goodies Polygon classification game using Venn diagrams Further explore the properties of polygons using Venn diagrams Properties of Polygons - Polygons can be divided into groups according to certain properties. Proportional Thinking and Probability - here you will find the Customizable Proportion Problems and much more Quick Images - In the following Interactive Activity, you'll stretch both your visualization and drawing muscles. (click on Show Shape to begin the activity) Ratio and Proportion - four stage lesson includes examples and a set of workout problems Similar Figures - nine java applets for investigating similar figures Similar Triangles - investigate some shortcuts for triangle similarity (three activities) The Exterior Angle Sum of a Polygon - investigate the sum of the exterior angles of a polygon The Sum of Outer Angles of a Polygon
apply the triangle inequality property to determine if a triangle exists and arrange the sides and angles according to size	Triangle Inequality Property – Seven questions to use as an assessment of student understanding Triangle Inequality theorem from OnlineMath The Triangle Inequality - explore a relationship between the lengths of the three sides of a triangle
identify the graphical representation of the inequality that represents the possible lengths of a third side of a triangle when the other two sides are given
determine the perimeter, area, surface area, or volume given the ratio of two similar geometric figures	Similar Figures - nine java applets for investigating similar figures
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
Level 3
apply the properties of angles, arcs, chords, tangents and/or secants to solve problems (with diagrams)	Intersecting Secants - find a relationship between the measure of an angle formed by two intersecting secants and the measures of the intercepted arcs Chord Properties - investigate the properties of chords (three explorations) Intersecting Chords - find a relationship between the measure of an angle formed by two intersecting chords and the measures of the intercepted arcs
determine the equation of a circle given coordinates or the graph of the circle (e.g., the center, the endpoints of the diameter)	Circles - twenty-three java applets for investigating circles
Sample Task
Have students make a Hypsometer. The students will use the Hypsometer to measure several tall objects on the school grounds.	Instructions for making a hypsometer (scroll to the bottom of the page) How to make, and use, a clinometer (hypsometer)

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Standard 4 Statistics and Probability The student will investigate, explore, and apply geometric representations to calculate theoretical probability.
Level 1
make a prediction from a geometric representation of a real-world data set
determine the probability of an event using a spinner and a circle graph
Level 2
determine the probability of an event represented as a subset of the area of a two-dimensional geometric figure
Sample Task
Construct two square dart boards which measure 1' x 1'. Circular targets are drawn on each board in such a way that they are all externally tangent to each adjacent circle and to the edge of the board. There are two circles on one dartboard and three circles on the other. Assuming you throw darts randomly and count only the throws that hit the board, which board yields the highest probability of a dart landing in a circle? Calculate the probability for each board.

Standard 5 Spatial Sense and Geometric Concepts The student 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.
Level 1
identify corresponding parts of similar and congruent geometric figures given a diagram	Similar Triangles - investigate some shortcuts for triangle similarity (three activities) Congruent Triangle activities - One | Two | Three | Four | Five Similar Figures - nine java applets for investigating similar figures
determine the length of a missing side in a right triangle when given two sides (answers must be given as simplified radicals)
identify basic geometric figures given a diagram
identify chords, inscribed angles, or central angles of circles given a diagram	Chord Properties - investigate the properties of chords (three explorations) Circles - twenty-three java applets for investigating circles Intersecting Chords - find a relationship between the measure of an angle formed by two intersecting chords and the measures of the intercepted arcs
apply reflexive, transitive, or symmetric properties of equality
Level 2
analyze and compare congruence or similarity relations between triangles or quadrilaterals given a diagram	Similar Figures - nine java applets for investigating similar figures Quadrilaterals and Conservation of Area - ten java applets for investigating quadrilaterals
determine whether a figure has been translated, dilated, reflected, or rotated given a diagram	Geometry in Action - various areas in which ideas from discrete and computational geometry (meaning mainly low-dimensional Euclidean geometry) meet some real world applications. 3-D Object Viewer - Students may explore a variety of 3-D objects and their accompanying 2-D views. Building Houses with Side View - student constructs a block figure (dynamic, perspective drawing) to match (10 different figures) Coloring 3-D sides - [UK spelling on this site] Find the red sides shown in a series of 2-D drawings and click on the right face of the 3-D model to color it red. 20 questions Coloring 2-D sides - Use the colored portion of the 3-D object to color the correct side of the 2-D drawing. Cube - Find out which colors will be on opposite faces of a cube whose faces are shown unfolded. Guess the View - Students are given a 3-D view of an object, and then given a 2-D view of the object. Students must choose which of 6 views is being displayed from a list. Plot Plans and Silhouettes - from Shape and Space in Geometry - the student task is to come up with plot plans that could match the given silhouettes. Background information is available at another page. Rotating Houses - Students are presented with a 3-D figure created with blocks that can be rotated and flipped using a mouse. The figure must be rotated until it matches a 2-D representation of one of the views.
solve problems involving complementary, supplementary, congruent, vertical, or adjacent angles given angle measures expressed algebraically	Angle Trisector - use this applet to determine the relationship between two angles
solve problems involving angles formed when parallel lines are cut by transversals	Special Angles on Parallel Lines - investigate the properties of the angles that are formed when a transversal cuts two parallel lines Problem of Angles and Parallel Lines (1) - use this applet to solve problems like " When the angle a=30degrees, b=45degrees, find the angle APB" Problem of Angles and Parallel Lines (2) - another java applet investigating the concept
determine the trigonometric ratio for a right triangle needed to solve a real-world problem given a diagram
solve real-world problems using 30-60-90 or 45-45-90 degree triangles (no irrational denominators)
apply properties of quadrilaterals to a solve real-world problem given a diagram (opposite sides and angles, consecutive sides and angles, or diagonals)	Linkage Strip Construction: Drawing Quadrilaterals - Try to build quadrilaterals with the given lengths. Use this linkage-strip Interactive Activity to answer a set of problems. Quadrilaterals and Conservation of Area - ten java applets for investigating quadrilaterals
solve real-world problems using measures of interior or exterior angles of regular polygons	Identification of Polygons - listing of characteristics and sample drawings Finding Hidden Polygons - four drawings are presented as challenges Regular m/n-Polygons - how to make regular m/n-polygon The Exterior Angle Sum of a Polygon - investigate the sum of the exterior angles of a polygon
identify the appropriate segment of a triangle given a diagram (i.e. median, altitude, angle bisector, perpendicular bisector)
determine which three-dimensional solid is represented by a given net (two-dimensional drawing)
determine the area of shaded regions involving circles, squares, rectangles, and/or triangles	Areas of Triangles, Trapezoids, and Kites - discover or demonstrate the formulas for the areas of triangles, trapezoids, and kites. (three activities) Circles - twenty-three java applets for investigating circles
justify triangle congruence given a diagram (i.e. ASA, SSS , AAS , SAS, or hypotenuse/leg)
determine if a triangle is acute, obtuse, or right given the length of all the sides of a triangle	Classifying Triangles - showing how triangles can be classified according to some of their features (followed by a four-question quiz) Linkage Strip Construction - A triangle has three sides, but not just any set of three lengths will make a triangle. Use this linkage-strip Interactive Activity to answer a set of problems. Triangle Homework - Eight problems to solve Three Types of Angles - explore three types of angles - right, acute, and obtuse; and begin to visually recognize the three types of angles
Level 3
solve problems using the properties of angles, arcs, chords, tangents, or secants	Intersecting Secants - find a relationship between the measure of an angle formed by two intersecting secants and the measures of the intercepted arcs
find the area of a sector or segment of a circle given a diagram	Circles - twenty-three java applets for investigating circles
choose the three-dimensional geometric object that has been rotated or reflected given a diagram
Sample Task
Research famous buildings. Choose one that depicts a design using parallel or perpendicular lines. Obtain a drawing or photograph of your chosen building. On one side of a sheet of plain paper give a short history of the building including when it was built, who designed it, and where it is located. On the other side of the paper draw a diagram of the building emphasizing its parallel and perpendicular lines.
Review Assistance
On-line sample tests	Timed Test (geometry concepts) - Improve your SAT scores by becoming familiar with the types of questions frequently asked