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UNIT 1: TOPIC LIST FOR CONJECTURING, WORDS, DEFINITIONS, AND QUADRILATERALS ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Can you use a given definition to identify all objects that will belong to this set? Can you determine why a particular object would not fit a given definition? Can you find a counterattack for a given definition (e.g. A triangle is a three sided figure)?

2. Do you know the definitions and properties (regarding angles, sides, and parallel-ness of sides) of quadriaterals (parallelograms, rectangles, squares, rhombi, kites, darts, trapezoids, isosceles trapezoids, concave, and convex quadrilaterals)? Can you categorize quadrilaterals based on their similarities or differences?

3. Given a few geometric figures or drawings, can you make (and clearly articulate) a conjecture that you believe will always be true? Can you come up with a conjecture that is interesting and or unexpected? (For example, noticing the angle formed by connect the two endpoints of a diameter to any point on a semi-cirlce is always 90 is surprsing!)

4. Can you prove our paper-folding conjecture? (When you fold the bottom of a piece of paper to form two adjacent triangles, the fold lines will form a right angle!)

5. Do you know the definition of a polygon? Do you know the definition of a circle?

6. Do you understand that we can use words to describe an object, but that the object might not exist (for example, a triangle with two parallel sides)? Can you determine if a given definition describes objects that exist or not?

7. Are you able to articulate that some objects are subsets of others? (For example, a square is a type of rhombus.) Do you understand that some sets of objects can be defined using larger sets? (For example, a square can be defined as a quadrilateral with four equal sides and four equal angles AND as a rhombus with four right angles.)

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UNIT 2: TOPIC LIST FOR EQUATIONS OF LINES AND QUADRILATERALS ON THE COORDINATE PLANE ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Do you understand what a solution to an equation is? Do you understand that a line is composed of infinitely many points, each a solution to a single equation?

2. Do you know how to write the equation of a line in slope-intercept form (y = mx + b) and in point-slope form (y y1 = m(x x1))? Do you know how to graph a line given its equation or given a point and slope? Can you use the equation of a line to find exact coordinates of points on the line (e.g. can you find the value of y at a particular x-value)?

3. Do you know how to calculate the slope of a line between two points?

4. Do you know the relationship between the slopes of parallel lines and perpendicular lines?

5. Do you know the Pythagorean Theorem (both the hypothesis and the conclusion)? Can you apply the Pythagorean Theorem and/or the distance formula to calculate distances between points on the coordindate plane?

6. Given some of the vertices of a triangle or quadrilateral, can you determine the coordinates of the missing vertex or vertices? For example, given two points can you find the other two vertices that would form a square? Can you find another pair that would make a different-sized square? Given three vertices, can you find three different points that could be the fourth vertices of a parallelogram?

7. Can you calculate distances to determine if a triangle is scalene, isosceles, or equilateral? Or if a quadrilateral is a rhombus?

8. Using distance and slope calculations, can you prove that four given points are indeed the vertices of a rectangle? A rhombus? A square? A parallelogram?

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UNIT 3: TOPIC LIST FOR BASIC CONGRUENCE ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Given two points, can you calculate the midpoint using the midpoint formula?

2. Given two points, can you find the vector connecting one point to the other?

3. Given the endpoints of a segment, can you find the points that divide the segment into n equal pieces? Can you clearly express not only the process to get those points, but why that process works?

4. Given a system of linear equations, can you immediately determine whether there is no solution, one solution, or infinitely many solutions (without solving the system)? Can you explain in words why there is no solution, one solution, or infinitely many solutions?

5. Can you solve a system of linear equations using substitution? Elimination? By doing an interpretive dance?

6. Can you clearly explain the three meanings/interpretations of the perpendicular bisector (its definition, a property about distance, and a property about reflection)?

7. Given the endpoints of a segment, can you write the equation of the perpendicular bisector of the segment? Given a point and its reflection across a line, can you give the equation of the line of reflection?

8. We have provided you with two fill in the blank formal writeups involving perpendicular bisectors (showing that all points on a perpendicular bisector of a segment are equidistant from the endpoints of that segment; given a point and a line of reflection, show that the line of reflection is the perpendicular bisector of the point and its reflected image). You will not be provided with a fill in the blank format on the assessment. Instead, you will be asked to write out a air-tight proof, so that it is clear to the reader. (Numbering each step, and your reason justifying each step may be helpful.) P.S. We will choose which proof you will have to do. Be ready for both!

9. Can you explain what properties are preserved through reflection and translation?

10. Can you explain why the three perpendicular bisectors of the sides of a triangle always meet at a single point?

Geogebra Skills

Can you create a vector? Can you create a regular polygon? Can you translate objects using a specific vector? Can you graph a perpendicular bisector? Can you write/use square roots in equations? Can you create a circle with a specified radius?

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UNIT 4: TOPIC LIST FOR ROTATIONS ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Given two points, can you find all centers of rotation that will swap the location of the two points? Can you articulate the reason why all these points are centers of rotation?

2. Given two congruent line segments, can you find the two centers of rotation that will swap the location of the two line segments? Can you articulate the reason why those two points are centers of rotation?

3. Given a figure with rotational symmetry, can you find all rotations that will return that figure to its original orientation?

4. Given a figure and a center of rotation, can you draw a new figure that has undergone a 90o rotation? What about a 180o rotation?

5. Given a complex figure and a rotation of this complex figure, can you use a ruler and protractor (and compass, if desired) to find the center of rotation? Can you determine the angle of rotation?

6. Given three non-collinear points, can you find the center of a circle that goes through all three points? Can you articulate the reason why any three non-collinear points can have a unique (meaning: only one) circle drawn through them?

Geogebra Skills

Can you rotate a figure Can you graph a perpendicular bisector Can you measure an angle

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UNIT 5: TOPIC LIST FOR CIRCLES AND BASIC TRANSFORMATIONS ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Know the derivation for the equation of a circle (given a center and a radius). Imagine you were asked to teach someone, from scratch, where this equation came from could you do this? [Note: this is when we used a right triangle and the Pythagorean theorem.]

2. Given the graph of a circle, write its equation; given an equation of a circle, graph it.

3. Given a circle and an x-coordinate of a point on the circle, algebraically be able to find the two points that are on the circle with that x-coordinate (exactly).

4. Given the endpoints of a diameter of a circle, write the equation for this circle. [Note: the diameter might not be horizontal or vertical.]

5. Given the center and one point on the circle, write the equation of the circle.

6. Determine whether a given point lies on a circle or not.

7. Graphically be able to determine if a line is ever a given distance to the origin.

8. Be able to come up with equations for lines that intersects a given circle no times. exactly one time, or exactly two times.

9. Explain why why the coordinates for the center of a circle do not satisfy the equation for a circle. [Note: Be sure to use the definition of a circle in your answer.]

10. Transform a figure by a given algebraic rule, determine if the transformation is rigid, and be able describe the transformation.

11. Know the basic algebraic rules for transformations. [Note: These are on pages 1 and 2 of Rigid Transformation Rules on the Coordinate Plane]

This assessment will not cover any material past page 2 in Rigid Transformation Rules on the Coordinate Plane. It will not include identifying multiple basic transformations that can be applied in order to yield a more complicated transfoormation. Geogebra Skills You will not have a geogebra part to this assessment, because you will be doing a lot with sliders on your Art&Geometry project.

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UNIT 6: TOPIC LIST FOR ADVANCED TRANSFORMATIONS AND AN INTRODUCTION TO REASONING ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Can you apply a given rule to a figure to see how the figure is transformed? Can you describe this what this transformation is doing visually? Can you identify if this is a rigid transformation or not, and justify your answer?

2. Do you know the rules for the four basic transformations (reflection over the x-axis, reflection over the y-axis, reflection over the line y=x, translation by a vector)? Can you apply multiple basic transformations (in the order that they are given to you) to find a rule for these multiple transformations? After applying this rule generated from multiple transformations to a figure, can you visually describe what this rule is doing to the figure? [Unless otherwise stated, rule means an algebraic rule that brings a point to another point, e.g. ( ), 1) ( 3,xx y y + ]

3. Given a figure and its reflection over a vertical or horizontal line (e.g. 3y = ), can you use the four basic transformations to come up with the rule for this reflection? Can you come up with a second way to use the four basic transformations to come up with a rule for this reflection?

4. We learned that if you perform multiple basic transformations, sometimes the order that you perform the transformations matters. In other words, if you have a figure and perform transformation P then transformation Q, the resulting figure may be different than if you had performed transformation Q then transformation P. We also saw that sometimes the order of the transformations doesnt matter. Can you come up with examples where the order of the transformations matters, and examples where the order of the transformations doesnt matter?

4. Do you understand how to reflect any point ( , )x y over any vertical line ( )x k= or any horizontal line ( y k= )? Given a blank sheet of paper, can you use words and diagrams to explain to someone how you came up with the coordinates of the reflected point? Make sure your explanation is clear, logical, and convincing. (Advice: break your explanation up into steps, use multiple diagrams)

5. Do you understand how you can generate regular polygons by rotating particular isoceles triangles? What must be true about the isosceles triangles for you to be able to generate a regular polygon through rotation?

6. Do you know how to factor a basic quadratic (a quadratic with a 1 coefficient in front of the 2x term)?

7. Do you understand how geometric diagrams are marked to show congruent segments, congruent angles, and parallel lines? Can you draw conclusions based on these markings? Can you draw diagrams with the appropriate markings given information that you know (e.g. draw a diagram for NERF which is a parallelogram)?

THERE IS MORE ON THE BACK SIDE OF THIS PAGE! TURN OVER!

8. Given two polygons are congruent (e.g. A GHIJKLBCDEF ), what conclusions can you draw?

9. Can you come up with descriptions of what inductive reasoning and what deductive reasoning are? Can you identify examples of inductive reasoning and deductive reasoning?

10. Can you prove that vertical angles are congruent? Note: In order to help you do well on #7 and #8, be sure to study the definitions of quadrilaterals, as well as their properties.

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UNIT 7: TOPIC LIST FOR BASIC PROOFS AND POLYGONAL ANGLES ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Given a completed flowchart proof, can you identify ways that it needs to be improved so that the explanation is clear, the flow of the argument (arrows) make sense, and all important steps are included?

2. Given an incorrect flowchart proof, can you identify errors? [You have not been given any of these; you should be able to read a flowchart proof and decide whether each step of the argument is valid]

3. Using given information and a statement to be proved, create a flowchart proof. This may be a proof youve seen before, or it may be different. You should definitely feel comfortable reproducing the arguments for the proofs weve done together.

4. Once you are able to prove a statement with a set of given information, can you articulate in words what youve proven? (Example: The sum of all exterior angles of a triangle is always 360 degrees)

5. In class, we measured the angles of a number of triangles (3-sided closed figures) and saw they always added up to a number close to 180 degrees. Think about all of the conversations we had about that including conversations about induction and deduction. Based on what we did in class, what do we truly know about the sum of the interior angles of triangles (3-sided closed figures)?

6. Can you explain graphically why the sum of the interior angles of an n-gon can be computed as ( 2)180on ? What about a graphical explanation for why the sum can also be computed as 180 360o on ?

7. If you have a regular n-gon, can you calculate each individual interior angle measure?

8. Given a diagram with some given angles marked, can you fill in the missing angles using what you have learned about the angles of polygons?

9. Given a dissection of a polygon into triangles, can you write and explain an expression for the sum of the interior angles of the polygon based solely on the dissected triangles (polygonal crystals)?

GEOGEBRA We will likely have a geogebra component to this assessment. However, you have not learned any new Geogebra skills. It will only require the skills youve already been working with.

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UNIT 8: TOPIC LIST FOR PARALLEL LINES AND INTERMEDIATE ALGEBRAIC PROOFS ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Can you show algebraically that the quadrilateral created by connecting consecutive midpoints of any quadrialteral (QMQ) is always a parallelogram?

2. Can you show algebrically that the segment that joins midpoints of two sides of a triangle is both parallel and half the distance of the triangles third side?

3. Given a diagram with two lines and a transversal, can you identify pairs of corresponding, alternate interior, alternate exterior, and same-side interior angles?

4. Given a diagram with corresponding, alternate interior, alternate exterior, and same-side interior angles, can you identify the two lines and the transversal that form these pairs?

5. Can you apply angle relationships of parallel lines to find the measures of other angles in a diagram that includes parallel lines?

6. Do you know different ways to deduce that two lines are parallel?

7. Can you articulate the difference between the postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent and the postulate If two lines cut by a transversal form congruent corresponding angles, then the two lines are parallel?

8. Can you deductively prove that the sum of the angles in a triangle is 180?

9. Can you deductively prove that the opposite angles in a parallelogram are congruent?

10. Can you apply the conjectures in #7 to deductively prove congruent and supplementary angle pairs when given parallel lines or prove lines parallel when given congruent or supplementary angles pairs?

Note: You may be given diagrams that include polygons, so it is important that you know how to find the interior angles of polygons.

NAME DATE BAND UNIT 9: TOPIC LIST FOR ANGLE BISECTORS, IMPOSSIBLE TRIANGLES, BASIC TRIANGLE CONSTRUCTION, &

TRIANGLE CONGRUENCE ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Can you explain clearly why the shortest distance between a point and a line is the length of a line segment from the point that is perpendicular to the line?

2. Given three side lengths of a triangle, can you identify if that triangle is possible to draw? If it is not possible, can you clearly explain why it is impossible (using a diagram and deductive logic to show it is true)? If it is possible, can you draw the triangle (you can use patty paper, a ruler, and a compass)?

3. If you are given two side lengths of a triangle, can you determine all possible lengths for the third side?

4. Given certain information about a triangle (certain angles and sides), can you draw a triangle that fits that information? (You will have access to patty paper, ruler, a protractor, and a compass.) If there are more triangles that can be constructed with given information, can you draw them? If there are not more triangles, can you explain why there is only one possible triangle?

5. Can you articulate why all points equidistant from the two rays making up an angle form the angle bisector?

6. Do you know how to use a compass, protractor, and ruler accurately?

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UNIT 10: TOPIC LIST FOR BASIC TRIANGLE CONSTRUCTION, ISOSCELES TRIANGLES, & SALT ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Do you know which configuration of given information about a triangle forces it to be rigid? Given information about two triangles, do you know if two triangles will be congruent (and by what reason)?

2. We found that when you were given certain information about triangles, there was a minimal amount that forced rigidity (e.g. SSS was enough for rigidity, but if you only had SS, you did not have rigidity). These triangles were forced to be rigid, and we saw this by doing constructions (with a compass, protractor, and ruler). Can you do a construction, given this minimal information, to illustrate that the triangle must be rigid, and write words explaining why your constructions shows that this triangle is forced?

3. Can you prove both theorems involve angle bisectors? These are: (1) Given an angle, any point equidistant to both rays of the angle lies on the angle bisector, (2) Given any angle and a point on the angle bisector, that point is equidistant to both rays of the angle.

4. Do you know, if you are not given enough information about a triangle to conclude rigidity, what additional information you could be given to force rigidity? Similarly, if you are given extra information for a triangle that is rigid, can you determine what information could be eliminated but still retain rigidity?

5. If given information about two triangles is insufficient to conclude they are congruent, can you draw an accurate counterexample that show the triangles are not congruent but retain the same given information

6. Can you complete a proof that requires you to prove that two triangles are congruent and then use their corresponding sides or angles to deduce other things?

7. Given a triangle with two congruent sides, can you prove that the base angles are congruent? Given a triangle with two angle congruent, can you prove that two sides are congruent? Can you use what you know about isosceles triangles to solve find the angles puzzles like in Jurgenson Section 4-4?

8. Can you prove why all three angle bisectors of a triangle meet at a single point? Can you explain why that point is the center of a circle that is tangent to all three sides of the triangle (we call the point an incenter)?

9. Can you anticipate what "lines"/ridges will be formed by pouring salt onto a particular shape (for example a triangle, rectangle, or other polygon)? Can you explain why these lines will form where they do?

10. Make sure you are comfortable understanding the problems we have discussed as a class from Isosceles Triangles and Congruent Triangles Challenge Problems. You may be presented with similar challenge problems.

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UNIT 11: TOPIC LIST FOR BASIC SIMILARITY, CHALLENGE PROBLEMS, AND THE SEMI-CIRCLE CONJECTURE ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Do you know how to write the equation of a circle? Do you know how to generate two different equations for semi-circles? Given a circle plotted on a coordinate plane, can you write its equation?

2. We proved that no matter which point on a semi-circle you choose, the angle formed by connecting a diameters endpoint to the point you chose to the diameters other endpoint would always be 90. Can you prove this geometrically? And do you know relevant theorem(s) that are required to do so? Can you prove this algebraically?

3. Do you understand the deductive process required to solve problem #6 from the Challenge Problems packet?

4. Do you know how to write missing coordinates of a vertex of a polygon given other coordiantes (both specific and general) of the polygon? Do you understand how to use information about midpoints to make conclusions about diagonals of a quadrilateral like you did to solve problem #5 from the Challenge Problems packet?

5. Do you know the definintion of similarity of polygons (congruence of all corresponding angles and all corresponding sides in proportion)? Can you draw examples of two polygons where only one of the two conditions for similarity is satisfied?

6. Can you apply the definition of similarity in order to determine if two polygons are similar, given two diagrams? Can you determine whether certain polygon types (e.g. rhombi) will always, sometimes, or never be similar to each other?

7. Given two similar figures, can you find the scale factor? Can you identify congruent angles and write proportions for corresponding sides? Can you use these proportions/the scale factor to find the lengths of unknown side lengths?

8. Can you plot transformations of given polygons on the coordinate plane and determine if these transformations yield a similar polygon or not?

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UNIT 12: TOPIC LIST FOR ADVANCED SIMILARITY AND GEOMETRIC MEAN ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Do you know what is necessary to prove two triangles similar to each other? Given two triangles, can you determine if they are similar to each other or not?

2. Can you set up and solve proportions involving similar triangles?

3. In class we saw there were many different approaches to the two pole problem, some involving coordinate geometry and the Pythagorean theorem. However, one elegant solution involved using two pairs of similar triangles. Do you understand how this problem can solved strictly using similar triangles? Could you solve the problem with different pole heights?

4. Can you prove the theorem: two inscribed angles that intercept the same arc are congruent? Can you apply this theorem to find missing angle measurements (like in our do now)? Can you apply this theorem to make conclusions about opposite angles of cyclic quadrilaterals? Can you identify which quadrilaterals are always cyclic, sometimes cyclic, or never cyclic?

5. Can you prove the circle-chord theorem? Can you apply the circle chord theorem to find missing segment lengths in a diagram?

6. Do you know what the definition of the geometric mean is? Can you calculate the geometric mean between two numbers?

7. In a right triangle where the altitude is drawn to the hypotenuse, there are many side lengths that could be found as the geometric means of other side lengths. Can you derive these relationships from similar triangles? Can you use these relationships to find various missing sides?

8. Given a number of points and a possible center of a circle, can you figure out if all the points lie on the circle with the given center or not?

9. Can you use similarity and the Platonic Right Triangles book to find missing side lengths? Missing angles?

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UNIT 13: TOPIC LIST FOR SIMILAR RIGHT TRIANGLES AND RIGHT TRIANGLE TRIGONOMETRY ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

1. Using the Book of Platonic Right Triangles and given an image of a right triangle with either two sides or an angle and a side, can you find all missing side lengths and angles? Using the Table of Right Triangle Ratios and given an image of a right triangle with either two sides or an angle and a side, can you find all missing side lengths and angles? Using the trigonometric functions on your calculator and given an image of a right triangle with either two sides or an angle and a side, can you find all missing side lengths and angles? Using the trigonometric functions on your calculator, do you know how to give your answers both exactly and approximately?

2. Given a horizontal distance from an object, the angle of elevation (measured by a clinometer, for example), and the eye height of the angle measurer, can you calculate the height of the object?

3. Can you articulate Platos argument that objects like triangles and circles cant exist in physical reality but can exisit in our minds (a.k.a. mathematical reality)?

4. Can you articulate why the ratio of any two sides of a right triangle corresponds to only one Platonic right triangle? And do you understand that the Platonic right triangles can be scaled to any size while preserving that ratio?

5. For any value given in the Table of Right Triangle Ratios, can you articulate what that numbers measures/means? Can you do this visually/geometrically?

6. Can you explain why we only need one ratio (e.g. leg opposite angle/hypotenuse) in order to identify which Platonic right triangle we have? Furthermore, considering we only need one ratio, can you explain why we have three different ratios in our Table of Right Triangle Ratios?

7. Can you articulate how the Book of Platonic Right Triangles and the Table of Right Triangle Ratios are the same? Are different? In what ways does the calculators use of sin, cos, tan, sin-1, cos-1, and tan-1 improve upon the Table of Right Triangle Ratios?

8. Can you draw connections among ratio of sides in a right triangle (such as the ones described in Similar Triangles #2 problems #7, 8, 9, 13 and in Similar Right Triangles #3 problems #6, 7, 8, 9)?

9. Can you estimate the shape of a right triangle given the value of a particular trigonometric ratio (like we did in our activity with the Placemat of Particular Platonic Plight Priangles)?