3
Subject Class Calendar Spring 2008
Grade Leader Krauss
______
Subject M$F Teachers Goldberg
Date / Day / Lesson / HW / CompleteLesson #1 AIM: What are ratios and proportions?
Lesson #2 AIM: How do we prove triangles similar?
Lesson #3 AIM: What are other methods for proving triangles similar?
Lesson #4 AIM: How can we prove proportions involving line segments?
Lesson #5 AIM: How can we prove that products of line segments are equal?
Lesson #6 Aim: What are the properties of the centroid of a triangle?
Lesson #7 Aim: What is the Right-Triangle Altitude Theorem?
Lesson #8 Aim: How do we apply the Right-Triangle Altitude Theorem?
Lesson #9 AIM: How do we write the equation of a circle?
Lesson #10 AIM: How do we find a common solution to a quadratic-linear system of equations graphically?
1/30 / W / Lesson #11 Aim: What are the parts of a circle?
Lesson #12 AIM: What are the properties of the four centers of a triangle?
1/31 / Th / Lesson #13 Aim: How do we prove arcs congruent?
2/1 / F / Lesson #14 Aim: How do we prove chords congruent?
2/4 / M / Lesson #15 Aim: What relationships exist if a diameter is perpendicular to a chord?
2/5 / T / Lesson #16 Aim: How do we measure an inscribed angle?
2/6 / W / Lesson #17 Aim: What relationships exist when tangents to a circle are drawn?
2/7 / Th / Lesson #18 Aim: How do we measure an angle formed by a tangent and a chord?
2/8 / F / Test
2/11 / M / Lesson #19 Aim: How do we measure angles formed by two tangents, a tangent and a secant, or two secants to a circle?
2/12 / T / Lesson #20 Aim: How do we measure angles formed by two chords intersecting within a circle?
2/13 / W / Lesson #21 Aim: How do we apply angle measurement theorems to circle problems?
2/14 / Th / Lesson #22 Aim: How do we apply angle measurement theorems to more complex circle problems?
2/15 / F / Test
Lesson # 23 Aim: How do we use similar triangles to find the measure of segments of chords intersecting in a circle?
2/25 / M / Lesson # 24 Aim: How do we use similar triangles to find the measure of line segments formed by a tangent and secant to circle?
2/26-2/27 / T/W / Lesson #25 Aim: How do we find the measures of secants and their external segments drawn to a circle?
2/28 / Th / Review
2/29 / F / Test 2
3/3 / M / Lesson #26 Aim: How do we apply segment measurement relationships to problems involving circles?
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Subject M$G Teachers Goldberg
Lesson #27 AIM: How do we determine a probable locus?Lesson #28 AIM: How do we solve problems using compound loci?
Lesson #29 AIM: How do we find the equation of the locus of points at a given distance from a given point?
Lesson #30 AIM: How do we write linear equations that satisfy given locus conditions?
Lesson #31 AIM: How do we find the points in the coordinate plane which satisfy two different conditions?
Lesson #32 AIM: How are images and pre-images related under line reflections?
Lesson #33 AIM: How are images and pre-images related under point reflections and translations?
Mar 20 / Mon / Lesson #34 AIM: How are images and pre-images related under rotations?
Marc 21 / Tues / Lesson #35 AIM: How are images and pre-images related under dilations?
Lesson #36 AIM: How do we find an image under a composition of transformations?
Lesson #37 AIM: Which transformations are isometries?
Lesson #38 Aim: How do we apply the properties of transformations to geometric proofs?
Lesson #39 Aim: What is solid geometry?
Lesson #40 Aim: How do we determine a plane?
Lesson #41 Aim: When is a line perpendicular to a plane?
Lesson #42 Aim: When are planes perpendicular?
Lesson #43 Aim: When are planes parallel?
Lesson #44 Aim: How do we find the volume and surface area of prisms and cylinders?
Lesson #45 Aim: How do we find the volume and surface area of pyramids and cones?
Lesson #46 Aim: What are the properties of a sphere?
Lesson # 32 Aim: AIM: How can we find line reflections and test for line symmetry?
Students will be able to:
1. explain what is meant by a transformation and a line reflection
2. find the coordinates of a point under a reflection in the y-axis, the x- axis and the line y = x
3. graph the image of a plane figure under a reflection about these lines
4. test for line symmetry in the coordinate plane
Do Now: Draw a rectangle using the points A(1,1) B( 5,1) C( 1,4) D( 5,4) What is the rectangles Perimeter and Area?
Homework:
Properties of a Reflection:
· A reflection reverses orientation.
· A reflection is an isometry
Reflection in a Line
Remember, a reflection is often called a flip.Under a reflection, the figure does not change size. It is simply flipped over the line of reflection.
Do the following by folding a piece of paper along the x axis
Reflecting over the x-axis:
When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite.
The reflection of the point (x,y) across the x-axis is the point (x,-y).
Hint: If you forget this "rule", simply fold your graph paper along the x-axis to see where your new figure will be located. You can also measure how far your points are away from the axis as indicated in the picture above.
Reflecting over the y-axis:
When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite.
The reflection of the point (x,y) across the y-axis is the point (-x,y).
Hint: If you forget this "rule", simply fold your graph paper along the y-axis to see where your new figure will be located. You can also measure how far your points are away from the axis as indicated in the picture above.
Reflecting over the line y=x or y=-x:
When you reflect a point across the line y = x, the x-coordinate and the y-coordinate change places. When you reflect a point across the line y = -x, the x-coordinate and the y-coordinate change places and are negated (the signs are changed).
The reflection of the point (x,y) across the line y = x is the point (y,x).The reflection of the point (x,y) across the line y = -x is the point (-y,-x).
Reflecting over any line:
Each point of a reflected image is the same distance from the line of reflection as the corresponding point of the original figure. In other words, the line of reflection lies directly in the middle between the figure and its image. Keep this idea in mind when working with lines of reflections that are neither the x-axis nor the y-axis.
Notice how each point of the original figure and its image are the same distance away from the line of reflection.
1. The image of the point (4,-3) under a reflection across the x-axis is (-4,-3).
False. The answer is (4,3)
2. The image of the point (-5,4) under a reflection across the y-axis is (5,4).
(True)
3. The image of the point (-1, 8) under a reflection across the line y = x is
(8,-1). (True)
. / / a. Which point is a reflection of point A over the x-axis?Top of Form
Bottom of Form
b. Which point is a reflection of point A over the y-axis?
Top of Form
Bottom of Form
c. Which point is a reflection of point A over the line
y = x?
Top of Form
Bottom of Form
Lesson #33 AIM: How are images and pre-images related under point reflections and translations?
A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction.
Translations are SLIDES!!!Let's examine some translations related to coordinate geometry.
In the example below, notice how each vertex moves the same distance
in the same direction.
In this next example, the "slide" moves the figure 7 units to the left and 3 units down.
There are several ways that mathematicians indicate that a translation such as this is to occur.
1. / description: / 7 units to the left and 3 units down.2. / mapping: /
(This is read: "the x and y coordinates will become x-7 and y-3". Notice that movement left and down is negative, while movement right and up is positive - just as it is on coordinate axes.)
3. / symbol: /
(The -7 tells you to subtract 7 from all of your x-coordinates, while the -3 tells you to subtract 3 from all of your y-coordinates.)
This may also be seen as T-7,-3(x,y) = (x-7,y-3).
Ø Under the translation
, the point (2,5) will become (5,7). (True)
Ø Under the translation the point (-2,3) will become (-3,1). (False)
Ø This graph illustrates a translation of
. / / Top of FormTrue
False
Bottom of Form
Ø
080211aIf x = –2 and y = –1, which point on the accompanying set of axes represents the translation ?
(1) Q (2) R (3) S (4) T / (2)
060402a
What is the image of (x,y) after a translation of 3 units right and 7 units down?
(1) (x + 3,y – 7) (3) (x – 3,y – 7)
(2) (x + 3,y + 7) (4) (x – 3,y + 7) / (1)
069903a
What is the image of point (2,5) under the translation that shifts (x,y) to
(1) (0,3) (3) (5,3)
(2) (0,8) (4) (5,8) / (3)
080409a
What are the coordinates of P′, the image of P (–4, 0) under the translation
(1) (–7,6) (3) (1,6)
(2) (7,–6) (4) (2,–3) / (1)
010509a
The image of point (3,-5) under the translation that shifts (x,y) to (x–1,y–3) is
(1) (–4,8) (3) (2,8)
(2) (–3,15) (4) (2,–8) / (4)
080609a
What is the image of point (-3, 4) under the translation that shifts (x,y) to
(1) (0,6) (3) (-6,8)
(2) (6,6) (4) (-6,6) / (4)
060309a
A translation moves P(3,5) to (6,1). What are the coordinates of the image of point (–3,–5) under the same translation?
(1) (0,–9) (3) (–6,–1)
(2) (–5,–3) (4) (–6,–9) / (1)
The translation is
010614a
The image of point (-2,3) under translation T is (3,-l). What is the image of point (4,2) under the same translation?
(1) (-1,6) (3) (5,4)
(2) (0,7) (4) (9,-2) / (4)
The translation is
080508b
The image of the origin under a certain translation is the point (2,-6). The image of point (-3,-2) under the same translation is the point
(1) (-6,12) (3)
(2) (-5,4) (4) (-1,-8) / (4)
The translation is
AIM: How can we find point reflections and test for point symmetry?
Students will be able to:
1. explain what is meant by a point reflection and by point symmetry
2. find the coordinates of a point under a point reflection through the origin
3. graph the image of a plane figure under a point reflection through the origin
Writing Exercise: Explain the difference between point symmetry and a point reflection.
Point symmetry exists when a figure is built around a single point called the center of the figure. For every point in the figure, there is another point found directly opposite it on the other side of the center.
Study the diagrams below:
This figure shows two points and their reflections through a line. This is line symmetry. / This figure shows two points and their reflections through a point. This is point symmetry.In a point symmetry, the center point is a midpoint to every segment formed by joining a point to its image.
A simple test to determine whether a figure has point symmetry is to turn it upside-down and see if it looks the same. A figure that has point symmetry is unchanged in appearance by a 180 degree rotation.
Answer the following questions relating to point symmetry.
/ 1. A sign by a swimming hole displays the message shown at the left. The message is saying that there is no swimming on Mondays. What is special about the way the message is written?Top of Form
Bottom of Form
2. Does the word MATH possess point symmetry?
Top of Form
Bottom of Form
3. Does the word NOON possess point symmetry?
Which of the following lettered items possesses point symmetry?
a. the letter D
b. a square
c. the letter S
d. the word MOW
e. the letter B
f. the word DAD
g. the letter Z
/ Top of Form
a
b
c
d
e
f
g
Bottom of Form
/ 5. A kaleidoscope is a simple optical instrument that uses two mirrors to produce symmetrical patterns. Does the kaleidoscope image at the left possess point symmetry?
Top of Form
Bottom of Form
/ 6. Which of these cards has point symmetry?
Top of Form
Bottom of Form
Lesson #35 AIM: How are images and pre-images related under dilations?