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TEKSING TOWARD STAAR
MATHEMATICS
TEKS/STAAR-BASED
LESSONS
GRADE 8
Six Weeks 4
TEKSING TOWARD STAAR © 2014
®
TEKSING TOWARD STAAR SCOPE AND SEQUENCE
Grade 8 Mathematics
SIX WEEKS 4
Lesson
Lesson 1
Lesson 2
TEKS-BASED LESSON
8.9A/identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y  mx  b from the
intersections of the graphed equations
8.10A/generalize the properties of orientation and congruence of ….reflections… of two-dimensional shapes on a coordinate
plane
8.10C/explain the effects of …reflections over the x-and y-axis,…, as applied to two-dimensional shapes on a coordinate plane
using an algebraic representation
Lesson 3
8.10A/generalize the properties of orientation and congruence of rotations… of two-dimensional shapes on a coordinate plane
8.10B/differentiate between transformations that preserve congruence and those that do not
8.10C/explain the effects of …rotations limited to 90°, 180°, 270°, and 360° as applied to two-dimensional shapes on a
coordinate plane using an algebraic representation
Lesson 4
8.10D/model the effect on linear and area measurements of dilated two-dimensional shapes
Lesson 5
8.8B/write a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides
of the equal sign using rational number coefficients and constants
8.7B/use previous knowledge of surface area to make connections to the formulas for lateral and total surface area
and determine solutions for problems involving …cylinders
8.12G/estimate the cost of a two-year and four-year college education, including family contribution, and devise a periodic
savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college
8.12B/calculate the total cost of repaying a loan, including credit cards and easy access loans, under various rates of interest
and over different periods using an online calculator
Lesson 6
Lesson 7
Lesson 8
8.12E/identify and explain the advantages and disadvantages of different payment plans
TEKSING TOWARD STAAR
 2014
Page 1
STAAR Category 2
Grade 8 Mathematics
TEKS 8.9A
Parent Guide
Six Weeks 4 Lesson 1
For this lesson, students should be able to demonstrate an understanding of how to perform
operations and represent algebraic relationships. Students are expected to apply
mathematical process standards to use multiple representations to develop foundational
concepts of simultaneous linear equations.
Students are expected to identify and verify the values of x and y that simultaneously satisfy
two linear equations in the form y  mx  b from the intersections of the graphed equations.
The process standards incorporated in this lesson include:
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation and number sense as
appropriate to solve problems.
8.1D Communicate mathematical ideas, reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs, and language as appropriate.
Math Background- Identifying and Verifying the Values of x and y that Satisfy Two
Linear Equations Using the Graphs of the Two Equations
Make sure every student has a graphing calculator for this lesson. If your students need
help using the y= , graph, setting a window, or table features of the calculator, spend some
time demonstrating those features to the class.
Several previous lessons involved graphing linear equations of the form y  mx  b . The graph will be a
line, and m represents the slope of the line. The b represents the y-intercept of the line.
If two equations are graphed on the same grid, then there are three possible situations for the graphs
of the two equations. These situations are:
1. The two lines are intersecting lines. They share one point. The value of x and y for that
intersection point will satisfy both equations simultaneously (at the same time).
2. The two lines are parallel. The lines do NOT share any point. There is no solution for the equations
in this situation. If the two lines have the same slope (m) and different y-intercepts (b), then the
lines will be parallel.
3. The two lines coincide. This means they graph as only one line as they share all the points on the
line. If the two lines have the same slope (m) and the same y-intercept (b), then they will coincide
or be the same line when graphed.
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 2
Grade 8 Mathematics
TEKS 8.9A
Situation 1: Graph y  3 x  1 and y  2 x  6 . What point(s) do they have in common?
The point they have in
common is the point where
they intersect. This point
is (1, 4). To verify this
point satisfies both
equations, substitute 1 for
x and 4 for y in both
equations. Analyze the
resulting equations for
their truth values. Both
equations must be true.
y
9
8
y = 3x + 1
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
1
-1
2
3 4
-2
5
6 7
8
9
x
y = −2x + 6
-3
-4
4 = 3(1) + 1 =3 + 1 Yes
-5
-6
4 = −2(1) + 6 =−2(1) + 6
= −2 + 6 = 4 Yes
-7
-8
-9
Situation 2: Graph y  2 x  1 and y  2 x  3 . What point(s) do they have in common?
y
9
8 + 1
y = 2x
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
-2
-3
1
2
3 4
5
6 7
y = 2x − 3
8
9
x
The point they have in
common is the point where
they intersect. Since the
lines are parallel, they will
not intersect. They do not
share any points. There
are no values of x and y
that satisfy both equations
simultaneously (at the
same time.)
-4
-5
-6
-7
-8
-9
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 2
Situation 3: Graph y 
Grade 8 Mathematics
TEKS 8.9A
1
x  2 and y  0.5x  2 . What point(s) do they lines share?
2
y
9
8
7
6
1 5
y = x4 + 2
2
3
y = 0.5x +2
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
1
2
3 4
5
6 7
8
9
x
The point they have in
common is the point where
they intersect. Since the
lines are the same line,
they intersect at every
point on the line. They do
share every point on the
line. There are infinite
values of x and y that
satisfy both equations
simultaneously (at the
same time).
-5
-6
-7
-8
-9
To determine which situation exists for a pair of equations, check the slope and y-intercepts. If the
slopes are different, it is situation 1. If the slopes are equal but the y-intercepts are different, then it is
situation 2. If the slopes are the same and the y-intercepts are the same, then it is situation 3.
Identifying and Verifying the Values of x and y that Satisfy Two Linear Equations
This part of the lesson will focus on intersecting lines and the ordered pair that belongs to the
intersection point.
When using your calculator as an aid to finding the intersection point of two linear equations, enter the
two equations in the y= screen. Hit the graph button. Be sure your window is set so that you can see
the intersection point. The table feature will be helpful once you have your equations entered in the
calculator. By looking at the graphing screen you can approximate the ordered pair for the intersection
point. When you open the table feature, look for the x value you approximated belonged to the
ordered pair of the intersection point. Compare the y- columns of the table, until you find a row where
the y values are the same. If you have your table set on whole values and the answer is (3, 1.5) then
you will be able to see the 1.5 in both y-columns. However, if the answer is (1.5, 3), then you will not
see a value for x that is 1.5 as it will only show whole values for x. Go to your table set and change the
Tbl to 0.1. Practice using your calculator to graph two equations, and find the intersection point if it
exists.

You need to be able to graph your equations on a grid also. Using your calculator as a check of your
work is very beneficial. It will help you decide if your work is correct or if you need to redo it.
TEKSING TOWARD STAAR © 2014
Page 3
STAAR Category 2
Grade 8 Mathematics
TEKS 8.9A
Example: Determine the intersection point of y  6 x  3 and y  2 x  7 . Verify your solution is
correct.
The slopes of the lines are 6 and 2. The lines are intersecting lines.
Graph the lines to locate the intersection point.
Line 1 has a slope of 6 and a y-intercept of 3. Line 2 has a slope of 2 and a y-intercept of 7.
y
9
8
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
-7
-8
-9
The lines intersect at (1, 9).
satisfy both equations.
y = 6x + 3
?
9 = 6(1) + 3
?
9=6+3
√
9=9
To verify this is the correct point, check to make sure it will
y = 2x + 7
?
9 = 2(1) + 7
?
9=2+7
√
9=9
Our solution is correct.
Enter both equations in the y = screen. Check the table to see if when x = 1 do both y values = 9.
TEKSING TOWARD STAAR © 2014
Page 4
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10C
Parent Guide
Six Weeks 4 Lesson 2
For this lesson, students should be able to demonstrate an understanding of how to represent
and apply geometry and measurement concepts. Students are expected to apply
mathematical process standards to develop transformational geometry concepts.
Students are expected to generalize the properties of orientation and congruence of
…reflections… of two dimensional shapes on a coordinate plane. Students are also expected
to explain the effects of …reflections…..as applied to two-dimensional shapes on a coordinate
plane using an algebraic representation.
The process standards incorporated in this lesson include:
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1D Communicate mathematical ideas, reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs, and language as appropriate.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas
Math Background- Understanding Orientation and Congruency with Reflections
We have already studied two of the transformations. The two we have already studied are
translations and dilations. This lesson will focus on reflections. We will review translations
and dilations in this lesson also.
A reflection of a figure is its mirror image. A figure is reflected across a line called the line of
reflection. The line of reflection serves as the mirror on which the figure is reflected. A figure and its
reflected image are always congruent.
Each point of the reflected image is the same distance from the line of reflection as the corresponding
point of the original figure, but it is on the opposite side of the line of reflection. Any line can be a line
of reflection, but we will focus on the x and y-axis as the lines of reflection.
Example: If triangle ABC is reflected across the x-axis, what will be the coordinates for the reflection
y
of point A?
9
8
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
B
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
A
-7
-8
C
-9
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10C
y
9
A′(−8, 7)
8
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
1
-1
B
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
-7
A
C
-8
-9
 The coordinates of point A are (−8, −7).
 The x-coordinate of point A is −8. When point A is reflected across the x-axis, its new
x-coordinate will be −8.
 The y-coordinate of point A is −7. When point A is reflected across the x-axis, its new
y-coordinate will be 7.
The coordinates for the reflection of point A will be (−8, 7). This point is labeled on the grid as
A′(−8, 7).
Point A is 7 units below the x-axis and Point A′ is 7 above the x-axis. They are the same distance from
the axis of reflection.
When reflecting a point across the x-axis, the x value will remain the same in the ordered pair and the
y-value will change signs for the reflection.
Example: If triangle ABC is reflected across y-axis, what will be the coordinates for the reflection of
point A?
y
9
8
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
B
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
A
-7
-8
C
-9
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10C
y
9
8
7
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
B
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
-7
A
-8
C
-9
A′(8, −7)
 The coordinates of point A are (−8, −7).
 The x-coordinate of point A is −8. When point A is reflected across the y-axis, its new
x-coordinate will be 8.
 The y-coordinate of point A is −7. When point A is reflected across the y-axis, its new
y-coordinate will be −7.
The coordinates for the reflection of point A will be (8, −7). This point is labeled on the grid as
A′(8, −7).
Point A is 8 units from the y-axis to the left and Point A′ is 8 units from the y-axis to the right.
Notice part of the triangle is in Quadrant III and part is in Quadrant IV.
When reflecting a point across the y-axis, the x value will change signs in the ordered pair and the
y-value will remain the same for the reflection.
Mathematically, this is stated as:
If (x, y) is reflected across the x-axis, then the reflection will be (x, −y).
If (x, y) is reflected across the y-axis, then the reflection will be (−x, y).
In both examples, the triangle did not change size when it was reflected. The original (pre-image) and
the reflection (image) are congruent. The change that occurred was in the orientation. The reflection
is not in the same position (orientation) as it was before.
Reflections and Other Transformations
A reflection can be indicated algebraically by using the symbolism:
Line of reflection: x-axis
Line of reflection: y-axis
TEKSING TOWARD STAAR © 2014
(x, y)
(x, y)
(x, −y)
(−x, y)
Page 3
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10C
In a reflection, the image will have coordinates that are the coordinates of the preimage with one value
changing to its opposite. We must remember which value changes to its opposite for each line of
reflection.
Sometimes two transformations can occur to a figure. For example, a quadrilateral might be translated
and then reflected. We complete the first transformation and then the second transformation occurs to
the image of the first transformation. Translations and reflections are two transformations that do not
change the size of the figure. If both transformations are used, the final image should be congruent to
the original preimage.
Example: Determine the coordinates of the final image if quadrilateral ABCD below is translated using
the representation (x, y)
(x + 2, y + 1) and then reflected across the x –axis.
y
y
9
8
8
7
7
B′
6
B
6
5
5
4
4
3
3
2
2
1
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
A′
-3
2
3 4
5
6 7
8
9
x
-3
-4
-4
-5
-5
-6
1
-2
-2
A
C′
9
C
D
D′
-6
-7
-7
-8
-8
-9
-9
Original Preimage
First Image (Translation)
y
9
8
7
6
5
D′′
4
3
A′′
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
B′′
-5
-6
-7
-8
-9
C′′
Final Image (Reflection of the Translation)
TEKSING TOWARD STAAR © 2014
Page 4
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10C
The table below shows the coordinates of the vertices for the original, the first image (translation), and
the final image (reflection of the translation).
Vertex
A
B
C
D
TEKSING TOWARD STAAR © 2014
Original
(−6, −3)
(−7, 5)
(4, 8)
(2, −5)
Translation
(−4, −2)
(−5, 6)
(6, 9)
(4, −4)
Reflection
(−4, 2)
(−5, −6)
(6, −9)
(4, 4)
Page 5
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10B/8.10C
Parent Guide
Six Weeks 4 Lesson 3
For this lesson, students should be able to demonstrate an understanding of how to represent
and apply geometry and measurement concepts. Students are expected to apply
mathematical process standards to develop transformational geometry concepts.
Students are expected to generalize the properties of orientation and congruence of
…rotations… of two dimensional shapes on a coordinate plane. Students are also expected to
explain the effects of …rotations…..as applied to two-dimensional shapes on a coordinate
plane using an algebraic representation. Students are also expected to differentiate between
transformations that preserve congruence and those that do not.
The process standards incorporated in this lesson include:
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1D Communicate mathematical ideas, reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs, and language as appropriate.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas
Math Background- Understanding Orientation and Congruency with Rotations
We have already studied three of the transformations. The three we have already studied
are translations, reflections, and dilations. This lesson will focus on rotations. This will
finish TEKS 8.10A and TEKS 8.10C.
A rotation is a transformation that turns a figure about a set point. The set point is called the point or
center of rotation. Any point on the coordinate plane can be the center of rotation, but we will only
consider the center of rotation to be the origin.
The rotation or turn can be clockwise or counterclockwise. The amount of the turn or rotation is
given in degrees and can be any amount of degrees of a circle. We will only consider 90° clockwise or
counterclockwise, 180° (clockwise or counterclockwise are the same result), 270° clockwise or
counterclockwise, and 360°. Turning an object 270° clockwise has the same result as turning the
object 90° counterclockwise. Likewise, turning an object 270° counterclockwise has the same result
as turning the object 90° clockwise.
Think of the numbers on a clock. If you start at 12 and go clockwise around to the 9, you have rotated
the hand of the clock 270° clockwise. If you move the same hand from the 12 and rotate 90°
counterclockwise, you will also be at the 9. The results are the same.
Rotating an object 360° turns the object in a complete circle and the figure is back where it started. It
doesn’t matter if you go clockwise or counterclockwise.
The rotation of an object about the origin only changes the figures position on the grid. It does not
change the size of the figure.
The transformations that preserve congruency of the figure are translations, reflections, and rotations.
The transformations that preserve orientation are translations and dilations.
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10B/8.10C
To draw a rotation on a grid, you will need a protractor and a ruler. First we will just rotate one point
90° clockwise.
1. Plot the point to be rotated.
2. Draw a ray from the origin to the point, the preimage.
3. Use your protractor. Place the center of the protractor on the origin, and the ray you’ve drawn on
the 0 of the protractor so that the measures of the protractor are to the right (clockwise).
4. Mark where the 90 is on the protractor on your grid. Connect that point to the origin.
5. The rotation point is on this ray. If using a ruler: To determine where it is located, measure the
distance the preimage is from the origin using your ruler. Mark an equivalent distance from the
origin on the other ray. That is the rotation point, the image.
If using a compass, draw a circle or arc of the circle whose center is the origin with a radius the
length equivalent to the distances between the origin and the preimage.
Example: Rotate point C (4, 6) 90° clockwise.
1. Plot the point.
2. Draw a ray from the C to the origin.
y
9
8
7
C
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
1
-1
2
3 4
5
6 7
8
x
9
-2
-3
-4
-5
-6
-7
-8
-9
y
3. Using a protractor, measure a
90° clockwise from C, with the
vertex of the angle being the
origin.
9
8
7
C
6
5
4
3
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
-7
-8
-9
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10B/8.10C
y
Either use a ruler or a compass and
mark a point on the new ray that is
the same distance from the origin
as C.
9
8
7
C
6
5
4
3
2
The point marked is the image, C′.
The coordinates of C′ are (6, −4).
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
C′
-5
-6
-7
-8
-9
Note the preimage was (4, 6) and the image is (6, −4). The two points have their coordinates reversed
and the sign on y changes. This will be true for any point rotated 90° clockwise.
To rotate a figure, such as a triangle, 90° clockwise, rotate the vertices of the figure and then connect
them to form the image. Rotate each vertex individually following the steps above.
Example: Triangle ABC is shown below. Rotate the triangle 90° clockwise and give the coordinates of
the vertices of the preimage and image on the table below.
y
9
8
7
B
6
5
4
3
2
A
C
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
-6
-7
-8
-9
TEKSING TOWARD STAAR © 2014
Page 3
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10A/8.10B/8.10C
The rays and circle for A are shown in orange. The rays and circle for B are shown in blue. The rays
and circle for C are shown in green.
y
9
8
7
B
6
A′
5
4
B′
3
2
A
C
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
8
9
x
-2
-3
-4
-5
C′
-6
-7
-8
-9
The grid below shows the preimage and image of the rotation.
y
9
8
7
B
6
A′
5
4
3
2
A
C
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
B′
6 7
8
9
x
-2
-3
-4
-5
-6
C′
-7
-8
-9
Vertex
A
B
C
(−5, 2)
(−2, 6)
(5, 2)
Vertex
A′
B′
C′
(2, 5)
(6, 2)
(2, −5)
A rotation using the origin as the center of rotation can be indicated algebraically by using the
symbolism:
90° clockwise: (x, y)
(y, −x)
90° counterclockwise: (x, y)
180° clockwise or counterclockwise: (x, y)
TEKSING TOWARD STAAR © 2014
(−y, x)
(−x, −y)
Page 4
STAAR Category 3
Grade 8 Mathematics
270° clockwise: (x, y)
TEKS 8.10A/8.10B/8.10C
(−y, x)
270° counter clockwise: (x, y)
(y, −x)
360° clockwise or counterclockwise: (x, y)
(x, y)
Look at the patterns in the relationships in the algebraic representations shown above.
When the rotation is 180°, the origin, the preimage and the image will lie on a line.
Example: Give the image of the following rotations using the algebraic representation given. Identify
the rotation defined by the representation.
(6, 2) (x, y)
(−x, −y)
The rotation indicated is 180°. The image is (−6, −2).
(5, −2) (x, y)
(−y, x)
The rotation indicated is 90° counterclockwise or 270° clockwise. The image is (2, 5).
Example: Triangle ABC has vertices A(1, 7), B(−4, 5), and C(8, −2). The triangle is to be rotated
180°. Complete the table of values for the coordinates of the vertices of both the preimage and the
image. Graph both triangles on a coordinate grid.
Vertex
A
B
C
(x, y)
(1, 7)
(−4, 5)
(8, −2)
(1, 7)
(−4, 5)
(8, −2)
(−x, −y)
(−(1), −(7))
(−(−4),−(5))
(−(8), −(−2))
Vertex
A′
B′
C′
(−1, −7)
(4, −5)
(−8, 2)
y
9
8
7
A
6
B
5
4
3
C′
2
1
-9 -8 –7 -6 -5 -4 -3 -2 -1
-1
1
2
3 4
5
6 7
-2
8
9
x
C
-3
-4
-5
-6
A′
B′
-7
-8
-9
Preimage and Image
Do the two triangles appear to be congruent?
Should they be congruent?
How could you determine the length of each side of the triangles?
TEKSING TOWARD STAAR © 2014
Page 5
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10D
Parent Guide
Six Weeks 4 Lesson 4
For this lesson, students should be able to demonstrate an understanding of how to represent
and apply geometry and measurement concepts. Students are expected to apply
mathematical process standards to develop transformational geometry concepts.
Students are expected to model the effect on linear and area measurements of dilated twodimensional shapes.
The process standards incorporated in this lesson include:
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1D Communicate mathematical ideas, reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs, and language as appropriate.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas
Math Background- Understanding the Effect on Linear and Area Measurements of
Dilated Two-Dimensional Figures
A dilated figure is similar to the original figure. The ratios of the lengths of the corresponding sides are
equivalent. The ratio of the length of a side of the image to the ratio of the corresponding side of the
preimage is the scale factor of the dilation.
If the dilation of the original figure is an enlargement, the scale factor is greater than 1. If the dilation
of the original figure is a reduction, the scale factor is less than 1.
Example: The rectangle on the left has been dilated by a scale factor of
1
to form the rectangle on
2
the right.
Scale factor of
1
2
The perimeter of the dilated figure will change by the same scale factor as the dilation. The scale factor
will be less than 1 because the dilation is a reduction.
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 3
Grade 8 Mathematics
TEKS 8.10D
Example: The triangle on the left has been dilated by a scale factor of 2 to form the triangle on the
right.
Scale factor of 2
Scale factor of 2
The perimeter of the dilated figure will change by the same scale factor as the dilation. The scale
factor will be more than 1 because the dilation is an enlargement.
Example: A triangle with a perimeter of 36 inches is dilated with a scale factor of 3. What is the
perimeter of the dilation?
The perimeter of the original triangle will be changed by a scale factor of 3 to give the
perimeter of the dilation. The new perimeter will be 36(3) or 108 inches.
The new perimeter will be the original perimeter multiplied by the scale factor.
The original perimeter will be the new perimeter divided by the scale factor.
The Effect on Area Measurements of Dilated Two-Dimensional Figures
We have learned that the perimeters of dilated figures will change by the scale factor. What change
happens to the area?
Example: Look at triangle 1 below. It is dilated by a factor of 2 to create triangle 2.
1
2
It takes 4 of the smaller triangles to cover the triangle 2. the area of triangle 2 is 4 times the area of
triangle 1. How is 4 related to the scale factor 2? 4 = 2 2 . The area of the dilation is the square of
the scale factor times the area of the original triangle.
The area of any dilated figure will be (scale factor) 2 (area of the original figure).
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 3
Grade 8 Mathematics
Example: Rectangle 1 is dilated with a scale factor of
TEKS 8.10D
1
to create rectangle 2.
2
1
2
Scale factor of
The area of rectangle 2 is
1
2
1
1
1
the area of rectangle 1. How is
related to the scale factor, ?
4
4
2
2
1
1
=   .
4
2
The area of the dilated figure is (scale factor) 2 (area of original figure).
The area of the original figure is area of original figure divided by (scale factor) 2
Example: The area of a triangle is 42 square centimeters. It is dilated by a scale factor of 3. What is
the area of the new triangle?
The area of the dilated figure is (scale factor) 2 (area of original figure).
The area of the dilated figure is (3) 2 (42) = 9 (42) = 378 square centimeters.
Example: A figure has an area of 100 square inches. It is dilated and the new figure has an area of
900 square inches. What was the scale factor of the dilation?
2
Area of dilation
  scale factor 
Area of Original
(scale factor) 2 = 9
900
= 9 = (scale factor)
100
2
scale factor = 3 since 3 2 = 9
TEKSING TOWARD STAAR © 2014
Page 3
STAAR Category 3
Grade 8 Mathematics
TEKS 8.8B
Parent Guide
Six Weeks 4 Lesson 5
For this lesson, students should be able to demonstrate an understanding of how to represent
and apply geometry and measurement concepts. Students are expected to apply
mathematical process standards to use one-variable equations or inequalities in problem
situations.
Students are expected to write a corresponding real-world problem when given a one-variable
equation of inequality with variables on both sides of the equal sign using rational number
coefficients and constants.
The process standards incorporated in this lesson include:
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as
appropriate to solve problems.
8.1E Analyze mathematical relationships to organize, record, and communicate
mathematical ideas
Math Background- Write Real–World Problem When Given a One-Variable Equation
or Inequality with Variables on Both Sides of the Equal Sign
Equations with variables on both sides of the equation are two-step equations. These equations are of
the form ax + b = cx + d. To create a problem from an equation with variables on both sides, describe
a quantity using x in two different ways.
Example: Write a problem that corresponds to the equation 2x + 6 = 3x − 8.
Let x represent the number of blue beads in a bag.
Twice the number of blue beads in a bag increased by 6 is the same as three times the
number of blue beads decreased by 8. How many blue beads are in the bag?
Example: Write a problem that corresponds to the equation x + 10 = 4x − 18.
Let x represent Betty’s age.
Betty’s age increased by 10 is the same as four times her age decreased by 18. What is
Betty’s age?
Remember:
+ can be interpreted as increased by, added to, sum of, etc.
− can be interpreted as decreased by, subtracted from, difference of, etc.
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 3
Grade 8 Mathematics
TEKS 8.8B
Subtraction is not a commutative operation so the order you write the two quantities is important.
x − b can be interpreted as x decreased by b, b subtracted from x, the difference of x and b, etc. It is
not the difference of b and x. That would be b − x. Another interpretation for x − b is x less b.
(not less)
For example, if x represents the number of students on a bus, x − 10 could be interpreted as the
number of students on the bus decreased by 10, the number of students on the bus less 10, the
difference of the number of students on the bus and 10.
Writing a Problem to Correspond to an Inequality with Variables on Both Sides
When working with inequalities, we must be able to recall the meaning of the inequality symbols.
<

>

means is less than
means is less than or equal to
means is greater than (more than)
means is greater than or equal to
 also means a maximum of
 also means a minimum of
Example: Write a problem to correspond with the inequality 3 x  5  2 x  10 .
Let x represent the number of girls at Beth’s birthday party.
Beth made the statement that three times the number of girls at her party increased by 5 is
more than 10 increased by twice the number girls at her party. What inequality describes
the number of girls at her party?
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 3
Grade 8 Mathematics
TEKS 8.7B
Parent Guide
Six Weeks 4 Lesson 6
For this lesson, students should be able to demonstrate an understanding of how to represent
and apply geometry and measurement concepts. Students apply mathematical process
standards to use geometry to solve problems.
The students are also expected to use previous knowledge of surface area to make
connections to the formulas for lateral and total surface area and determine solutions for
problems involving …, cylinders.
The process standards incorporated in this lesson include:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as
appropriate, to solve problems
8.1E Create and use representations to connect and communicate mathematical ideas
Math Background-Surface Areas of Cylinders
You can use models or formulas to find the lateral and total surface area of a cylinder.
 A cylinder is a three-dimensional figure with two bases. The bases are congruent circles. The other
surface is a curved surface.
Base
Curved Surface
Base
Like the area of a plane figure, the surface area of a three-dimensional figure is measured in square
units.
 The total surface area of a three-dimensional figure is equal to the sum of the areas of all its
surfaces.
 The lateral surface area of a three-dimensional figure is equal to the sum of the areas of all its
faces or curved surfaces. It does not include the areas of the figure’s base(s).
One way to compute the surface area of a three-dimensional figure is to use a net of the figure. A net
of a three-dimensional figure is a two-dimensional drawing that shows what the figure would look like if
it were unfolded with all its surfaces laid flat. The net is used to find the area of each surface.
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 3
Grade 8 Mathematics
Cylinder
TEKS 8.7B
Net for the Cylinder
The net for a cylinder consists of two congruent circles and one rectangle. The dimensions of the
rectangle are the height of the cylinder and the circumference of the circular base.
Circumference
of the base
Height of the
cylinder
The area of the rectangle is the product of the dimensions. Therefore, for a cylinder the lateral surface
area is 2 rh , where r is the radius of the circular base and h is the height of the cylinder. This formula
is given on your STAAR Reference Materials. The total surface area is the sum of the lateral surface
area and the two bases. This formula is 2 rh  2 r 2 . Using the Distributive Property, this formula can
also be written as 2 r (h  r ) . Answers for lateral or total surface areas of cylinders can be given in
terms of  or you can use your  key on your calculator to substitute a rational approximation for  .
The calculator will use an approximation for  with many more digits than 3.14. If you are answering
a multiple choice question such as the STAAR test, look at your answer choices to decide if you need to
substitute for  .
Example: What is the lateral surface area of a cylinder with a radius of 8 units and a height of 10
units?
8u
10 u
The lateral surface area is the area of the curved surface which is
a rectangle in the net of the figure. Its dimensions are 10 u and
16  u. The lateral surface area is (10)(16  ) square units or
160  square units. The approximate lateral surface area to the
nearest tenth of a square unit is 502.7 square units.
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 3
Grade 8 Mathematics
TEKS 8.7B
Solve Problems Involving Cylinders
The formula for the lateral surface area can be used to determine the height of the cylinder if the radius
and the lateral surface area are known.
LA  2 rh can be rewritten to isolate the variable, h. Using the Division Property of Equality, divide
LA
2 rh
LA
LA
both sides of the formula by 2 r .

Simplifying, this formula becomes
 h or h 
.
2 r
2 r
2 r
2 r
Example: A cylinder has a lateral surface area of 36  square inches. The radius is 4 inches. What is
the height of the cylinder?
Using the formula for h, h 
LA
36 in.2
,h=
. h = 4.5 inches (Note: Square inches divided
2 r
8 in.
by inches gives inches.)
By the same procedure, the radius can be determined if the lateral surface area and the height are
given.
Using LA  2 rh , divide both sides by 2 h.
r 
LA
2 rh

2 h 2 h
Simplifying this formula becomes
LA
 r or
2 h
LA
.
2 h
Example: A cylinder has a lateral surface area of 40  square inches. The height is 8 inches. What is
the radius of the cylinder?
Using the formula for h, r 
LA
40 in.2
,h=
. r = 2.5 inches
2 h
16 in.
Example: A cylinder has a total surface area of about 326.7 square inches. The lateral surface area is
about 226.2 square inches. What is the radius of the cylinder?
The total surface area decreased by the lateral surface area will give the area of the two
bases. 326.7−226.2 = 110.5 square inches. Divide this by two to give the area of one base.
100.5/2 = 50.25 square inches.  r 2  50.25 . Divide both sides by  . (use your calculator)
r 2  15.99 so r  15.99 or about 4 . The radius is about 4 inches.
TEKSING TOWARD STAAR © 2014
Page 3
STAAR Category 4
Grade 8 Mathematics
TEKS 8.12G
Parent Guide
Six Weeks 4 Lesson 7
For this lesson, students should be able to demonstrate an understanding of how to describe
and apply personal financial concepts. Students are expected to apply mathematical process
standards to develop an economic way of thinking and problem solving useful in one’s life as a
knowledgeable consumer and investor.
Students are expected to estimate the cost of a two-year and four-year college education,
including family contribution, and devise a periodic savings plan for accumulating the money
needed to contribute to the total cost of attendance for at least the first year of college
The process standards incorporated in this lesson include:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as
appropriate to solve problems.
Math Background-Cost of a Two-Year or Four-Year College Education
Planning for the cost of a college education should start early in a child’s life. The cost of a 2-year or 4year college education is rising every year. If a plan is started early in a child’s life, the savings have
time to earn interest and increase over the years.
The cost of a college education can be affected by many things. If you decide to attend a local college
or university and live at home, the cost can be reduced significantly. Housing and food are a large part
of the cost of a college education. Regardless of where you live, you will have tuition, fees, textbooks,
and transportation costs.
One thing you can do to aid in the expense of a college education is to start saving part of any money
you earn at a summer job or after school job. Even if it is just $50 a month to begin with, open a
savings account and start saving. As you get older, you will earn more in the summer so save more.
Many parents will not be able to pay for the entire college education for their child or children.
Sometimes grandparents may be willing and be able to aid in the cost. However, you should plan on
helping pay for the cost also.
There are grants that are available for those who financially qualify. If you are one of those, be sure
you get your application filed. The federal Pell grant is one of the most popular education grants.
Grants do not have to be repaid so take advantage of the opportunity if you qualify. You can research
Pell grants and it will give the qualifications for receiving this grant.
There are many scholarships available also. They are given based on need and qualifications. Many
local organizations such as PTAs and service organizations may give scholarships. The scholarships
available for your school should be listed with your counselors.
Most students can apply for a student loan. There are state and federal student loans available. Some
of these loans do not have to be paid back until you have graduated or leave school. Look for
information about these on the college of your choice’s website.
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 4
Grade 8 Mathematics
TEKS 8.12G
If you want a college education, then start planning for it now.
Example: Betty wants to attend Kilgore Junior College for 2 years. The yearly tuition for a state
resident, out-of-district is $2304. The fees are estimated to be $672 and $2303 for books and
supplies. Room and board for the year is $4,407. She estimates her personal expenses to be about
$1500 a year. What will be the total cost for Betty to attend Kilgore Junior College for 2 years?
Total all the costs for a year and then multiply by 2.
$2304 + $672 + $2303 + $4407 +$1500 = $11,186
$11,186 (2) = $22,372.
It will cost Betty about $22,372 to attend Kilgore Junior College if the costs remain the
same. It is very possible that tuition and room and board could rise.
While attending college you can apply for work study. This is a program that provides you a job on
campus doing various jobs to help assist in the cost. Jobs can be at the library, the cafeteria, or any
office on campus.
Example: Starting the beginning of her freshman year and the remaining 3 years of high school, Alice
deposited $2500 in a savings account from her summer job at the local city park where she was a life
guard. Her money accrued interest at 2.5% compounded annually. How much money should the
account have at the beginning of her freshman year of college?
Year
1
2
3
4
Beginning
Balance
$0
$2562.50
$5188.56
$7880.77
Deposit
Amount
$2500
$2500
$2500
$2500
New
Balance
$2500
$5062.50
$7688.56
$10,380.77
Amount of Interest Earned
$2500(0.025)=$62.50
$5062.5(0.025)=$126.56
$7688.56(0.025)=$192.21
$10,380.77(0.025)=$259.52
Ending
Balance
$2562.50
$5188.56
$7880.77
$10,640.29
The interest in the table above was found by using the simple interest formula since it was being
calculated for one year at a time, the compound interest or simple interest formula for 1 year are
equivalent.
If Alice is planning to attend Kilgore Junior College like Betty, she has saved almost enough for the
entire first year.
TEKSING TOWARD STAAR © 2014
Page 2
STAAR Category 4
Grade 8 Mathematics
TEKS 8.12B/8.12E
Parent Guide
Six Weeks 4 Lesson 8
For this lesson, students should be able to demonstrate an understanding of how to describe
and apply personal financial concepts. Students are expected to apply mathematical process
standards to develop an economic way of thinking and problem solving useful in one’s life as a
knowledgeable consumer and investor.
Students are expected to calculate the total cost of repaying a loan, including credit cards and
easy access loans, under various rates of interest an over different periods using an online
calculator. Students are also expected to identify and explain the advantages and
disadvantages of different payment plans.
The process standards incorporated in this lesson include:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution
8.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as
appropriate to solve problems.
Math Background-Calculate Total Cost of Repaying a Loan and Identify Different
Advantages and Disadvantages of Different Repayment Plans
When you repay a loan you repay the principal (amount of the loan) and the interest for the loan. An
amortization table can show you the amount of your monthly payment that is principal and what is
interest. The institution giving you the loan should provide you with one if you request it. If you look
an amortization table, you can see that the amount of principal increases each payment and the
amount of interest decreases. When repaying a mortgage on a home, the early payments are almost
entirely interest.
One way to avoid paying so much interest is to shop around for the institution that has the lowest
annual interest rate. Even a 0.5% less interest rate can save you lots of money in interest if the loan is
for very many years.
Another way to pay less interest is to have the term of the loan fewer months or years. While a shorter
loan time will increase your monthly payment more of the payment will be principal. When deciding
what amount of time to get the loan for, you want the least number of months or years that will give
you a monthly payment you can financially handle for that amount of time. Most home loans are for 15
years or 30 years. A 30-year loan will give you a smaller monthly payment, but you will pay little
principal the first 10 years. The less principal you pay then the less escrow you are accumulating on
the house.
An online calculator can be used to determine the amount of your monthly payment for a given loan
amount and a given interest rate. You can find an online calculator at www.bankrate.com
TEKSING TOWARD STAAR © 2014
Page 1
STAAR Category 4
Grade 8 Mathematics
TEKS 8.12B/8.12E
Example: What is the monthly payment and total repayment for a $5,000 loan at 6% interest for 36
months?
Using the online calculator, input 5000 for the loan amount, 36 for the months and 6 for the
interest rate. The calculator gives a monthly payment of $152.11. Multiplying $152.11 by
36 gives the total repayment of $5475.96. The interest paid on the loan is $475.06.
If the person with the loan had paid an extra $20 each month, you will pay the loan off 5 months early.
That is one advantage of paying a “little extra” each month. You don’t even have to pay the same
amount extra each month.
Credit cards charge a much larger interest rate than bank loans. They can charge as high as 25%
annual interest rate. It is important to pay off credit card balances every month. If the total balance is
paid each month and on time, then you do not have to pay any interest or late charges. For large
purchases that you cannot pay for in one payment, it is better to get a bank loan than use a credit
card.
TEKSING TOWARD STAAR © 2014
Page 2