Geometry
Grade Eight
Standard 8-4: The student will demonstrate through the mathematical
processes an understanding of the Pythagorean theorem; the use of ordered pairs,
equations, intercepts, and intersections to locate points and lines in a coordinate
plane; and the effect of a dilation in a coordinate plane.
Indicator 8-4.1
Apply the Pythagorean Theorem.
Continuum of Knowledge:
In seventh grade the students learned about the inverse relationship between
perfect squares and square roots (7-2.10)
In eighth grade, students apply the Pythagorean Theorem (8-4.1).
This topic is also addressed in high school Geometry.
Taxonomy Level
Cognitive Dimension: Apply
Knowledge Dimension: Conceptual
Key Concepts
Vocabulary
Hypotenuse
Legs
Right triangle
Square root
Radical
Mathematical Formula
a2+b2=c2
Instructional Guidelines
For this indicator, it is essential for students to:
Recall the formula
Understand the relationship among the sides beyond memorization of the
formula
Know that the hypotenuse is always the longest side and that the legs are
connected to the right angle.
1
Geometry
Grade Eight
Know how to use the Pythagorean Theorem in situations involving both
perfect and non-perfect squares.
Recognize when the Pythagorean Theorem is being described in a story
problem
For this indicator, it is not essential for students to:
None noted
Student Misconceptions/Errors
The students often forget to take the square root of the sum to find the
length of the missing side.
Students will often misuse the formula thinking all values go in the place of a
and b because they do not have a sound understanding of the difference
between the legs and the hypotenuse.
Instructional Resources and Strategies
Although the focus of the indicator is procedural, students need to have an
understanding of the “concept” of the Pythagorean Theorem. Conceptual
knowledge is based on relationships; therefore, students should gain of an
understanding of how these values connect beyond the reciting the formula.
The Pythagorean relationship states that if a square is constructed on each side of a
right triangle, the sum of the areas of the two smaller squares equals the area of
the largest square. The Pythagorean Theorem is most frequently represented as a2
+ b2 = c2, with "a" and "b" being the legs (the two sides that create the right angle)
and "c" being the hypotenuse (the side directly across from the
right angle) of a right triangle.
It is extremely important that the students understand (not just memorize) the
different applications of the Pythagorean Theorem. The students should be given a
variety of story problems that can be solved by using the Pythagorean Theorem. An
example of this: A diver dove off of the dock and swam 50 feet to the where the
buoy was attached to the bottom of the lake. The buoy is 25 feet above his head;
how far is it from the dock to the buoy? This question requires the students to draw
2
Geometry
Grade Eight
a picture and to realize that they are looking for a missing leg; the information that
was given is the length of the hypotenuse and one leg.
Children’s book: “What’s your angle Pythagoras” By: Julie Ellis
Assessment Guidelines
The objective of this indicator is to apply which is in the “apply procedural”
knowledge cell of the Revised Taxonomy. Although the focus of the indicator is to
gain computational fluency with using the Pythagorean Theorem, the learning
progression should also build the student’s conceptual knowledge in order to
support retention. The learning progression to apply requires students to recall
explore the Pythagorean relationship using a variety of examples. Students use
their observations to generalize connections (8-1.7) among the areas of the
squares. They then generalize a mathematical statement (8-1.5) summarizing this
connection using correct and clearly written or spoken words (8.1.6). Students
translate this verbal description to mathematical notation understanding that both
are equivalent symbolic expressions of the same relationship (8-1.4). They use
their conceptual understanding of the Pythagorean Theorem as they engage in
problem solving and repeated practice to gain computational fluency.
3
Geometry
Grade Eight
Standard 8-4: The student will demonstrate through the mathematical
processes an understanding of the Pythagorean theorem; the use of ordered pairs,
equations, intercepts, and intersections to locate points and lines in a coordinate
plane; and the effect of a dilation in a coordinate plane.
Indicator 8-4.2
Use ordered pairs, equations, intercepts, and intersections to locate points and lines
in a coordinate plane.
Continuum of Knowledge:
In the sixth grade students represented with ordered pairs of integers the location
of a point in the coordinate grid (6-4.1). In seventh grade, students analyzed tables
and graphs to determine the rate of change between and among quantities (7-3.2).
Students gained an understanding of slope as a constant rate of change (7-3.3).
In eighth grade, students use ordered pairs, equations, intercepts, and
intersections to locate points and lines in a coordinate system (8-4.2). They also
the coordinates of the x- and y- intercepts of a linear equation from a graph,
equation, and/or graph (8-3.6)
Taxonomy Level
Cognitive Dimension: Apply
Knowledge Dimension: Procedural
Key Concepts
Vocabulary
y-intercept
x-intercept
slope
function table
function rule
coordinate system
ordered pair
y-axis
x-axis
linear function
4
Geometry
Grade Eight
Instructional Guidelines
For this indicator, it is essential for students to:
Use an equation to locate points by creating a function table
Use points from a table to graph a line
Use (plot) intercepts to graph a line
Determine the point of intersection after graphing two equations or from a
given graph.
Understand that the point of intersection is where both equations have the
same value for x and y.
Understand that a line consists of infinitely many points
Understand that the x-intercept is the result of the intersection between a
line and the x-axis line. It is in the form (x, 0)
Understand that the y-intercept is the result of the intersection between a
line and the y-axis line. It is in the form (0, y).
For this indicator, it is not essential for students to:
Know that the point of intersection is the solution to a system of equations.
Solve for the x- and y- intercept without the graph.
Student Misconceptions/Errors
None noted
Instructional Resources and Strategies
To clarify the intent of this indicator, replace the word locate with graph. For
example,
o Students use ordered pairs to graph points and lines.
o Students use equations to graph points and line.
o Students use intercepts to graph points and line.
o Students use intersections to graph points and line.
The intent of the indicator as it relates to intersections of lines is not to solve
a system of linear equations. Real world story problems can be used to
explore the concept of intersection. For example, at Kira’s Video, they
charge $12 a month and $0.50 for each video. At Arika’s video there is no
fee to join but charges $1 for each video. For how many movies will the cost
at each video store by the same?
5
Geometry
Grade Eight
Below is an example that may be used to illustrate how students use
points to determine lines.
o Which line contains the following points (4,4) and (0,0)? Answer: c
o What two lines intersect at (4,4)? Answer: c & d
o What is the line that has an x-intercept of 4? Answer: d
o What line has a y-intercept of 4? Answer: a
o What line has the point (0,-1) on it? Is there another way to name
this point? Answer b, yes it’s the y intercept of -1
Assessment Guidelines
The objective of this indicator is to use which is in the “apply procedural” cell of the
Revised Taxonomy. Procedural knowledge is not only knowledge of steps and
techniques but also knowing when to use appropriate those steps. The learning
progression to use requires students to recall how to plot points and how to write
points given the graph. Students use their understanding of ordered pairs,
equations, intercepts and intersections to create point and lines on the coordinate
plane. They generalize connections (7-1.7) among these concepts and understand
that each is a distinct symbolic form that represents the same linear relationship
(7-1.4). Students translate from equation to ordered pairs to graph, from
intercepts to graph and explore real world problems to gain a conceptual
understanding of intersection.
6
Geometry
Grade Eight
Standard 8-4: The student will demonstrate through the mathematical processes
an understanding of the Pythagorean theorem; the use of ordered
pairs, equations, intercepts, and intersections to locate points and
lines in a coordinate plane; and the effect of a dilation in a
coordinate plane.
Indicator 8-4.3
Apply a dilation to a square, rectangle, or a right triangle in a coordinate plane.
Continuum of Knowledge:
In sixth grade the students applied strategies to find the missing vertices of various
polygons (6-4.2).
In eighth grade, students apply a dilation to a square, rectangle, or a right triangle
in a coordinate plane (8-4.3)
Taxonomy Level
Cognitive Dimension: Apply
Knowledge Dimension: Procedural
Key Concepts
Vocabulary
Dilation
Scale Factor
Ordered Pairs
Image
Center of dilation
Notation
A’- A prime and denotes the image of the original or pre-image.
Instructional Guidelines
For this indicator, it is essential for students to:
Understand the meaning of dilation
Recall the characteristics of a square, rectangle and right triangle
Apply a dilation factor to enlarge or reduce squares, rectangles, and right
triangles.
Multiply the ordered pairs by the dilation factor.
Use the new coordinates to graph the image of the pre-image.
7
Geometry
Grade Eight
Understand that if the scale factor of the dilation is greater than 1, the image
will be bigger than the original.
Understand if the scale factor of the dilation is less than 1 but greater than 0,
the image will be smaller than the original.
Understand that the image and pre-mage may overlap or one may be inside
the other.
Use appropriate notation to denote the image and pre-image
For this indicator, it is not essential for students to:
Apply a dilation factor to enlarge or reduce irregular shapes.
Student Misconceptions/Errors
Students sometimes have difficulty plotting ordered pairs, by wanting to
reverse the x and y axis “moves.” If this is the case, location activities
should precede or be used in a differentiated lesson for those students
having difficulty.
Students often think that all dilation factors will always enlarge the pre-
image. They do not connect the idea that multiplying by a fraction or decimal
less than one but greater that zero reduces the size and that multiplying by a
whole number 1 and greater enlarges the image.
Instructional Resources and Strategies
Students should be given several opportunities to investigate the process of
applying dilation to various polygons. A dilation is defined as a transformation that
copies, enlarges, or reduces an image or polygon. One strategy would be to have
students construct a square, rectangle or a right triangle in a coordinate plane and
record the original coordinate points next to the vertices of the shape. Students
should then multiply these coordinate pairs by a number (this number is called the
scale factor) to create a similar figure. The new shape should then be constructed
using the new coordinate pairs and the new coordinates recorded next to the
vertices of the new shape. By doing so, students have performed a dilation on the
figure. This provides the opportunity to discuss similarities and differences between
the old and new ordered pairs and the shapes they produced to assist in the
development of mental models. Once students have a deeper understanding of the
effect of a dilation, the teacher may want to move to more abstract examples by
giving them only the coordinates of the vertices of a polygon and having them find
the coordinates of a dilation of the given polygon, given the scale factor.
Assessment Guidelines
The objective of this indicator is to apply which is in the “apply procedural
knowledge” cell of the Revised Taxonomy. To apply is to carry out or use a
8
Geometry
Grade Eight
procedure in various situations. The learning progression to apply requires
students to recall and understand the meaning of dilation. They also recall the
characteristics of squares, rectangles and right triangles. Students explore
examples where they multiply by several different scale factors. They generalize
connections (8-1.7) among examples where the factors are between zero and one
or greater than one. They generalize mathematical statements (8-1.5) about these
relationships using correct and clearly written or spoken words (8-1.6). They use
their understanding of the relationships to apply dilations to other examples and
explain and justify their answers to their classmates and teacher.
9
Geometry
Grade Eight
Standard 8-4: The student will demonstrate through the mathematical
processes an understanding of the Pythagorean theorem; the use of ordered pairs,
equations, intercepts, and intersections to locate points and lines in a coordinate
plane; and the effect of a dilation in a coordinate plane
Indicator 8-4.4
Analyze the effect of a dilation on a square, rectangle, or right triangle in a
coordinate plane.
Continuum of Knowledge:
In sixth grade the students applied strategies to find the missing vertices of various
polygons (6-4.2).
In eighth grade, students apply a dilation to a square, rectangle, or a right triangle
in a coordinate plane (8-4.3)
Taxonomy Level
Cognitive Dimension: Analyze
Knowledge Dimension: Conceptual
Key Concepts
Vocabulary
Dilation
Scale Factor
Ordered Pairs
Image
center of dilation
Proportional reasoning
Notation
A’- A prime and denotes the image of the original or pre-image.
Instructional Guidelines
For this indicator, it is essential for students to:
Name the new coordinates of a polygon when given the coordinates and the
scale factor.
10
Geometry
Grade Eight
Understand that a fractional scale factor will reduce the pre-image and a
scale factor greater that 1 will enlarge the pre-image.
Know that a scale factor of 1 will copy the image.
Analyze the impact the scale factor has on the area of the polygon.
Understand that the image and pre-image are similar shapes.
Use proportional reasoning to analyze relationships
For this indicator, it is not essential for students to:
Apply a dilation to irregular shapes.
Student Misconceptions/Errors
Students often think that dilations will always enlarge the pre- image. They do not
connect the idea that multiplying by a fraction or decimal less than one but greater
that zero reduces the size and that multiplying by a whole number 1 and greater
enlarges the image.
Student may not understand that if the anchor point or the center of dilation is a
point within the pre-image the dilation factor would not be applied to that point.
Instructional Resources and Strategies
This is an extension of 8-4.3. When students are provided with examples of
dilations, they should be able to identify them as either being a dilation or as not
being a dilation. As part of this indicator, students also explore the effect of a
dilation on the area of the figures.
http://www.mathopenref.com/dilate.html- interactive dilation and describes
another technique that can be used to create dilations.
Assessment Guidelines
The objective of this indicator is to analyze which is in the “analyze conceptual”
knowledge cell of the Revised Taxonomy. To analyze means to break material
down into its constituent parts and determine how the parts relate to each other
and the overall purpose; therefore, students examine the coordinates and area of
figures (parts) and determine how their relate to dilations (purpose). The learning
progression to analyze requires students to recall the meaning of dilations and
understand how to perform dilations. They recognize the relationships among scale
factor and proportional relationship between the pre-image and image. They
explore a variety of examples and generalize connections (8-1.7) among those
examples in order to generalize mathematical statements (8-1.5) related to the
effect of dilations. Students use deductive reasoning (8-1.3) to move from
generalized statements to specific relationships in order to describe the effects.
11