Download Analytic Geometry Unit 1 - Teaching Plan - Word

CCSD Summer Institute
Analytic Geometry Unit 1
May 28-29, 2015
STANDARDS
Understand similarity in terms of similarity transformations
G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
1. The dilation of a line not passing through the center of the dilation results in a parallel line and leaves a
line passing through the center unchanged.
2. The dilation of a line segment is longer or shorter according to the ratio given by the scale factor.
G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if
they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality
of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be
similar.
Prove theorems involving similarity
G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the
other two proportionally, (and its converse); the Pythagorean Theorem proved using triangle similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Understand congruence in terms of rigid motions
G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given
rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to
decide if they are congruent.
G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if
and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motions. (Extend to include HL and AAS.)
Prove geometric theorems
G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a
transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent;
points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to
180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a
triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are
parallelograms with congruent diagonals.
Make geometric constructions
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an
angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular
bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.
Use coordinates to prove simple geometric theorems algebraically
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove
that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point
(1, √3) lies on the circle centered at the origin and containing the point (0, 2). (Focus on quadrilaterals, circles,
right triangles, and parabolas.)
STANDARDS FOR MATHEMATICAL PRACTICE
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Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
AGENDA
1. Spotlight Introductory Activity
a. Write the characteristics of special triangles & quadrilaterals in order to get at the major
ideas in constructions.
2. We really should discuss/teach congruence before we do constructions. Why?
3. FAL: Analyzing Congruency Proofs
a. http://map.mathshell.org/materials/download.php?fileid=1302
b. Discuss congruence cards
c. May NOT complete this – may be too abstract – we can come back to it after next task
4. Triangle Congruence with Spaghetti
a.
http://oak.ucc.nau.edu/smg224/155pdfs/Activities/Triangle%20Congruence%20with%20Spaghetti.pdf
b. SHOULD get the point across that SAS, SSS, ASA, & AAS are congruence theorems
c. SHOULD get the point across that AAA is similarity theorem – won’t call it that yet
5. FAL: Representing & Combining Transformations
a. http://map.mathshell.org/lessons.php?unit=8310&collection=8
b. This is 8th grade – Coordinate Alg appropriate about transformations in general. We apply
it to our standards as congruence between pre-image & image with rigid motion.
c. The shape being moved here is an “L” but we can easily create triangle cards.
6. Walk thru main constructions & discuss why they work
a. Copy a line segment – we will just discuss why this one works & make the point that we
should draw the entire circle instead of just the arc (as we usually do in our constructions)
so that we can see the congruent circles that would have congruent radii.
b. Copy an angle – one way to prove this is that we will have SSS triangle congruence so that
by CPCTC, the angles are also congruent.
c. Bisect a segment – one way to show this is that we can construct the perpendicular
bisector of the segment. In doing so, we create a rhombus. A rhombus’s diagonals are
perpendicular & bisect each other.
d. Bisect an angle – one way to show this is that we create a kite where the diagonal of the
kite is the bisector of the angle. This diagonal is the one that bisects the other diagonal and
it bisects both angles that it intersects.
e. Construct a parallel line – here you start with a line and a point not on the line. We draw
any line through the point that intersects the line, which will serve as the transversal of our
parallel lines. One way to prove that two lines are parallel is to show that corresponding
angles are congruent. So, we just need to copy the angle created by the original line &
transversal at the point.
f. Construct a perpendicular line – here you start with a line and a point not on the line.
We can choose two arbitrary points on the given line that lay on either side of the point not
on the line to use as endpoints of a segment. Construct a circle centered on each endpoint
that goes through the point not on the line. The circles will probably not be congruent (and
do not need to be). We have created a kite where the given line is one diagonal. The other
diagonal must be perpendicular to this.
7. Patty Paper
a. Angle bisector
b. Perpendicular bisector of line segment
c. Perpendicular line from given point to given line
d. Parallel line from given point to given line
8. Constructions Inscribed in a Circle – use color pencils to draw all 3 figures on the same circle
a. Construct hexagon in a circle (take radius length & mark it around the edge of the circle) –
why does it work? (hexagon is made up of 6 equilateral triangles so you copy the radius of
the circle around the edge of the circle (you are constructing 6 congruence circles around
the original circle))
b. Construct equilateral triangle in a circle (same as hexagon but connect every other
intersection
c. Construct square in a circle (draw the diameter & construct perpendicular bisector &
connect 4 points)
9. EOC construction items
10. Triangle & Parallelogram Proofs – we aren’t going to do these but this task is a good resource.
a. Show a couple from frameworks for reference
11. Triangle Similarity –
a. We often hear the word similar in and out of school. What does it mean, in general, and
give some examples. (From: Good questions for math teaching: Why ask them and what to
ask grades 5–8. Math Solutions Publications: Sausalita, CA. Schuster, L. and Andeson, N.C.
(2005). )
b. These figures are mathematically similar. What is it that makes the larger polygon similar
to the smaller one? Look at angles and sides. Measure them with a protractor and a ruler.
(From: Mathematics for Elementary School Teachers. (4th Edition). Brooks/Cole, Cengage
Learning. Bassarear, T. (2008). )
c. For each of the following, answer yes or no. In either case, justify your response.
i. Are all isosceles triangles similar?
ii. Are all equilateral triangles similar?
iii. Are all squares similar?
iv. Are all rhombuses similar?
v. Are congruent polygons similar?
(From: Mathematics for Elementary School Teachers. (4th Edition). Brooks/Cole, Cengage
Learning. Bassarear, T. (2008). )
d. The coordinates of the vertices of triangle ABC are (2,1), (4,1), and (3,4). Give the
coordinates for the triangle DEF so that it is similar to triangle ABC. Identify the scale
factor from triangle ABC to triangle DEF. (From: Good questions for math teaching: Why
ask them and what to ask grades 5–8. Math Solutions Publications: Sausalita, CA. Schuster,
L. and Andeson, N.C. (2005). ) This is a great problem for dilations NOT centered on
the origin!!
e. I drew a set of ten triangles. Five were similar to one another, three were congruent, and
two were neither similar nor congruent to any other. What might these triangles have
looked like? Draw the triangles and label their side lengths. (From: Good questions for
math teaching: Why ask them and what to ask grades 5–8. Math Solutions Publications:
Sausalita, CA. Schuster, L. and Andeson, N.C. (2005). )
f. What are the main ideas behind triangle similarity?
i. Scale factor
ii. Lead in to Trigonometry!!
iii. ALL 30-60-90 & ALL 45-45-90 & ALL 20-70-90 etc are SIMILAR – THEREFORE, the
ratios are ALWAYS the same
iv. Sine 30 = Opp/Hyp = ½  ALWAYS