Gabrielle Kotelnicki
Title: Exploring and Identifying Properties of Lines on The Cartesian Plane
NJ Core Curriculum Content Standards:
4.5.A. Problem Solving: Students will draw on their knowledge to formulate the distance
formula from The Pythagorean Theorem and also be able to construct the equation of a
line from the information given.
4.5.E Representation: Students will see graphical representations of how to calculate
slope, draw straight lines based on given information, and visually identify The
Pythagorean Theorem and Distance Formula.
4.5.B Communication: Students will participate in an interactive lecture where they will
answer questions and be asked to reason their findings in an effort to be clear and
convincing to the class.
4.5.C Connections: Students will form connections between The Pythagorean Theorem
and The Distance Formula, slope and finding the equation of a line.
4.5.D Reasoning and Proof: Students will see the proof of The Distance Formula and be
able to justify it in their own words.
Length of unit: The unit will consist of one week, or five integrated lessons.
Resources needed: Students will need to have a graphing calculator and grid paper.
Teacher will need an overhead projector.
Prior Knowledge: Students should:
a) Be able to simplify algebraic expressions involving exponents
b) Know the basic properties of a right triangle
c) Be familiar with the Cartesian Plane and be able to plot any given points on it
Objectives: After completing this unit, students should:
a) Be able to state the formula for slope, distance, and midpoint as well as the
Pythagorean Theorem and equation of a line in slope intercept form.
b) Be able to apply the above formula’s/theorem’s to find missing values
c) Understand and interpret graphical representations of a line.
d) Be able to construct a graphical representation of the Distance Formula by
forming a right triangle from any given two points and using this representation to
find the length between the points.
Relationship between this unit and other units: In previous units, students learned how
to interpret a graph and be able to plot and connect points on the Cartesian Plane. This
expands the previous lessons in that students will be able to identify the properties of
lines and shapes and learn how to manipulate them using what they will learn in this
lesson (e.g., before they could draw lines on a graph, whereas after this unit, they can find
the slope, midpoint and distance of the line and between the points.
Identify important ideas in terms of the subject area: The Pythagorean Theorem is
one of the most widely used mathematical relationships. It is an important introduction
to Trigonometry where it is used to find angle degree. Graphical topics such as slope and
line equations are also important introductions to Calculus where students will have to
find the intersection of two lines and also form functions.
In terms of real life, this lesson can be used to figure out what size ladder is needed to
reach the top roof of a building, or in measuring the size of a television where the length
of its diagonal is needed.
Sequence of lessons:
Lesson #1: Calculating Slope
Objectives: Students will:
a) Be able to determine slope by counting, using the “rise over run” method.
b) Be able to state the formula for slope
c) Be able to calculate the slope of a line on the Cartesian Plane
Lesson #2: Formulating and graphing the equation of a line
Objectives: Students will:
a) Be able to state and identify the equation of a line if slope-intercept form.
b) Be able to identify and solve for the slope and y-intercept of a line and apply this
information to create the equation of a line
c) Understand what it means for a point to be on a line or not.
d) Interpret a graph and find it equations
Lesson #3: The Pythagorean Theorem
Objectives: Students will:
a) Be able to state The Pythagorean Theorem
b) Understand when it is possible to apply this theorem
c) Be able to find the missing side of a right triangle given the lengths of the other
two sides
Lesson #4: Distance Formula
Objectives: Students will:
a) Be able to state the distance formula
b) Be able to apply this formula to find the distance between any two points on a
graph
c) Be able to form a graphical representation of the distance formula by forming a
right triangle from any given two points and using this representation to find the
length between the points
d) Understand the connection between the distance formula and the Pythagorean
theorem, and use this knowledge to prove the distance formula
Lesson #5: Midpoints of a line
Objectives: Students will:
a) Be able to state the formula for the midpoint of a line
b) Be able to find the midpoint between two points of a line
c) Understand that the midpoint of a line on a graph is a point
Relationship to students’ lives: The relationship to students’ lives is primarily the
everyday application of these formulas. For example, students would use the Distance
Formula, Pythagorean Theorem, and even midpoint to help measure and hang a television
on a wall, or for home decorating use in general. Students can also use what they have
learned in this lesson to determine how steep stairs or ski slopes are.
Potential student difficulties:
Students may have difficulties forming equations based off of graphical line
representations when they are given little information, and asked to interpret the graph.
They may also find it difficult to use the connections between lessons to prove theorems
and formulas from this unit.
Modifications:
Students will be put into heterogeneous groups during activities so that students having
difficulty can work with more sophisticated students. Weaker students, who are learning
disabled, will be given more organized worksheets so that they are able to approach the
material in a step-by-step manner. All lessons (1-5) will incorporate technology and
substantial drawing, which will be beneficial for students who learn better visually.
Assessments:
The final assessment will ask students to state various formulas and theorems and apply
these to find the missing lengths or values of a line or triangle. Students will also be
asked to graph and also interpret graphs while then being able to formulate equations
based on the information shown. Such questions might include:
What is the formula for slope and how could you use it to find the slope between the
points (2,4) and (1,2)? (OBJECTIVES A & B)
Find the equation of the straight line that has slope m=4 and passes through the point (-1,
-6). (OBJECTIVES A & B & C)
How do you determine what side of a triangle is the hypotenuse? Use this information to
find the length of the hypotenuse of the right triangle with side lengths 8 and 6.
(OBJECTIVES A & B)
Given the points (-7,1) and (-1, 9), on a Cartesian Plane, construct a right triangle
connecting them and find the distance between the two points using:
a) The Pythagorean Theorem
b) The Distance Formula
(OBJECTIVES B & D)
State the midpoint formula and use it to find the midpoint between the points (-1,2) and
(3, -6). (OBJECTIVES A & B)