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MODELING LINES WITH SLOPE AND A POINT
INTRODUCTION
The objective for this lesson on Modeling Lines with Slope and a Point is, the student will derive the equation of a line
using the slope and a point on the line in order to solve mathematical and real world problems.
The skills students should have in order to help them in this lesson include slope, slope intercept form and y equals mx
plus b the general form of an equation when written in slope intercept form.
We will have three essential questions that will be guiding our lesson. Number one, how is graphing a line given the
slope and y-intercept similar to graphing a line given the slope and a point? Number two, explain the steps for graphing
a line if you know the slope and a point. And number three, explain the steps for writing the equation of a line if you
know the slope and a point.
We will begin by completing the warm-up identifying the equation of a line from a given graph to prepare for Modeling
Line with Slope and a Point in this lesson.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Mrs. Rosen’s fifth grade class recently planted a tree. The type of tree planted
tends to grow at a rate of two inches per month. After three months, the tree measured twenty inches tall. What is the
equation of a line that represents the height of the tree, y, after x months?
We will begin by Studying the Problem. First we need to identify where the question is located within the problem and
underline the question. The question for this problem is, what is the equation of a line that represents the height of the
tree, y, after x months?
Now that we have identified the question we want to put this question in our own words in the form of a statement.
This problem is asking me to find the representation of the height of the tree, y, after x months as an equation.
During this lesson we will learn how to determine the equation of a line given the slope and a point. We will use this
knowledge to complete this SOLVE problem at the end of the lesson.
DERIVING THE EQUATION USING THE SLOPE AND A POINT WITH GRAPHING
Now we are going to apply our knowledge of graphing in order to find the equation of a line, given the slope and point
on the line.
In the previous lesson, what two pieces of information were necessary for graphing a line? We needed to know the
slope and the y-intercept.
What is the first step of graphing a line given the slope and y-intercept? We need to plot a point at the y-intercept first.
What is the next step of graphing a line given the slope and y-intercept? Next, we apply the slope to the y-intercept to
plot a second point.
And once we have done these two things, what is the final step? We can connect the two points with a straight line.
Why do we use the y-intercept as a starting point for graphing? Justify your thinking. We know that the y-intercept is a
definite point on the line.
Now let’s graph the line that passes through the point one, one and has a slope of two. What is different about using
this information and the graphing you have done in the previous lessons? This time we are given the slope and a point
on the line, rather than the y-intercept.
So where do you think we should start? We should graph the point that we are given. What point are we given? We’re
given the point one, one. Let’s graph it now using the coordinate grid.
Now what is the slope that we are given? We are told that the slope is two. So how can we write the slope so that we
can interpret the rise over the run? We can write it as a fraction with a rise of two over a run of one. The fraction is
written as two over one.
What does a rise of two over a run of one mean? It means that move vertically two units and one unit horizontally to
plot another point on the line. Let’s do this now. We will rise two units and then run vertically one unit to plot the
second point.
Why do we move up two units? The rise is a positive two and a positive rise means that we move up.
And why do we move to the right one unit? The run is a positive one and a positive run means that we move to the
right.
So what is the location of the new point after we apply the slope? The location of our new point is two, three.
Now what is the final step? Now that we have two points we can connect the two points and draw a line using the
straightedge.
So what is a y-intercept? It is a point on the graph where it crosses the y-axis.
What is the y-intercept of this line that we graphed? The line crosses the y-axis at negative one. The y-intercept is
negative one.
What is the general form of an equation written in slope-intercept form? This general form of the equation is y equals
mx plus b.
If we wanted to write the equation of this line in slope-intercept form what two pieces of information would we need?
Defend your thinking. We would need to know the slope and the y-intercept. These are the two pieces of information
that would be needed in order to put the equation in slope-intercept form, y equals mx plus b. The slope is represented
by m, and the y-intercept is represented by b in the equation.
Do we know the slope? Yes, it is two. And do we know the y-intercept? Yes, it is negative one. We learned that when
we graphed the line.
What is the equation of the line that passes through the point one, one with a slope of two? Remember that the slope is
two and the y-intercept is negative one. So the equation of the line is y equals two x minus one.
Now let’s take a look at Graph A. What is this problem asking us to do? It is asking us to graph the line that passes
through the point two, zero with negative one over two as the slope.
What point do we know is on this line? The point two, zero. Let’s graph this point now.
And what is the slope of this line? The slope we are told is negative one over two.
How do we apply the slope? Let’s apply the slope now. Since our rise is negative one, we will go down one unit and our
run is two, we will move to the right two units to plot the next point.
What does a rise of negative one over a run of two mean? Explain your thinking. It means that we move vertically one
unit and two units horizontally to plot another point on the line.
Why do we move down one unit? The rise is a negative one and a negative rise means that we move down.
Why do we move to the right two units? The run is a positive two and a positive run means that we move to the right.
So what is the location of the new point after we apply the slope? The new point that we plotted is located at the point
four, negative one.
Now that we have plotted two points on this line, what is the final step? We can connect the two points and draw a line
using the straightedge.
So what is the y-intercept of this line that we graphed? We can see that the line crosses the y-axis at one. The yintercept is one.
What is the general form of an equation written in slope-intercept form? It is y equals mx plus b.
If we wanted to write the equation of this line in slope-intercept form, what two pieces of information would we need?
We need to know the slope and y-intercept. Do we know the slope? Yes, it is negative one over two, which was given to
us at the beginning of the problem. Do we know the y-intercept? Yes, it is one. We learned that when we drew the
graph of the line on the coordinate grid.
So what is the equation of the line that passes through the point two, zero with a slope of negative one over two? Since
the slope is negative one over two and the y intercept is positive one, the equation is y equals negative one over two
times x plus one.
DERIVING THE EQUATION USING SLOPE AND A POINT
Let’s take a look at the graphic organizer seen here. What do you notice is provided in the top of the graphic organizer?
It is the general form of an equation in slope-intercept form. It is y equals mx plus b.
When we found the equation of the line by graphing, what steps did we follow? Explain your thinking. We plotted the
point that we know, applied the slope and drew the line. After the line was drawn, we went back and found the yintercept.
Let’s take a look at this problem. A line has a slope of two and passes through the point three, four. Write the equation
of the line in slope-intercept form.
Do we know the slope of the line? Yes, it is two. Let’s include this information in our graphic organizer. The slope of the
line that we are looking at is two.
Do we know an ordered pair, or a point, that is on the line? Yes, it is the point three, four. Let’s include this information
in our graphic organizer as an ordered pair that we know that is on the line.
And do we know the y-intercept? No, this information was not given to us in the problem.
What do you notice about all the variables in the second column of the graphic organizer? The variables are m, x, y and
b. They are all of the variables that are in the general equation for slope-intercept form.
What do x and y represent in the equation? They are the unknown input and output values that create specific ordered
pairs related to the line.
Do we know any specific input-output values that create an ordered pair for this equation? Yes, we know the ordered
pair, three, four, which is a point on the line.
If we know numerical values for x, y, and m, as we see in our graphic organizer, what can we do with them? Justify your
answer. We can substitute them into the general equation for the slope-intercept form. Let’s do this now. We know
that the general equation for slope-intercept form is y equals mx plus b. We know a point on this line is the point three,
four. So the value of y is four and the value of x is three. We also know the slope is two. We can plot these values into
the equation. Four equals two times three plus b.
So what variable is still unknown? It is b. What does b represent? It represents the y-intercept.
How can we find the missing value of b? The only missing variable is b, so we can solve the equation for b using simple
algebra. Let’s do this now. We need to isolate the variable that we are trying to find, which is b. First let’s multiply two
times three. Two times three equals six. So we can simplify this equation as four equals six plus b. Now to get b by
itself we need to subtract six from both sides of the equation. When we subtract six from the left we get negative two.
And when we subtract six from the right we get zero, which leaves us with b, our missing variable. We have found that b
is equal to negative two. Let’s include this information in the graphic organizer. The y-intercept for this line is negative
two.
Now what do we need to write the equation of the line in slope-intercept form? We need the slope and the y-intercept.
So what is the equation of the line that has a slope of two and passes through the point, three, four? Remember the
slope is two and the y-intercept is negative two. The equation of this line is y equals two x minus two.
Now let’s take a look at another problem together. This problem says to find the slope-intercept form of the equation
described. Then, graph the line to check your solution.
The problem describes the line that passes through the point four, one and has a slope of negative one. What is this
problem asking us to find? It’s asking us to find the slope-intercept form of the equation described. Then, graph the line
to check the solution.
So do we know an ordered pair, or point that is on the line? Yes, we know that there is a point four, one that is on the
line.
What is the value of the x-coordinate for this point? X equals four.
And what is the value of the y-coordinate for this point? Y equals one.
What is the value of m? Remember that m is the slope of the line. M equals negative one.
Do we know the y-intercept for this line? No we don’t know the y-intercept yet.
If we know numerical values for x, y and m, what can we do? Explain your thinking. We can substitute them into the
general equation for slope-intercept form. Let’s do this now. The general equation for slope-intercept form is y equals
mx plus b. We know that x equals four, y equals one and m equals negative one form the information that was given to
us in this problem. We can substitute these values into the general equation for slope-intercept form. One equals
negative one times four plus b. What is the only variable for which we do not have a value? It is the variable b. What
does b represent? It represents the y-intercept. We need to find b in order to find the y-intercept.
How can we find the missing value of b? The only missing variable is b, so we can solve the equation for b using simple
algebra. Let’s do this now. First, let’s simplify the right side of the equation. Negative one times four is negative four.
We can rewrite the equation as one equals negative four plus b. Now we want to get b by itself on the right side of the
equation. So we will add four to each side of the equation. When we add four to the left side of the equation, one plus
four equals five. And on the right side of the equation we are left with b. We have found that b is equal to five.
So what do we need to write the equation of the line in slope-intercept form? We need to know the slope and the yintercept.
So what is the equation of the line that has a slope of negative one and passes through the point four, one? We found
that b is equals to five, which is our y-intercept and the slope is negative one. So the equation for this line is y equals
negative one times x plus five, or we can simplify this equation to, y equals negative x plus five. Since negative one
times x equals negative x.
Now that we have found the equation of the line, let’s graph the line to be sure that we have the correct equation.
What point do we know? We know that there is the point four, one on the line. Let’s graph it now. We can graph the
point four, one the coordinate grid.
What is the slope of the line? The slope is negative one.
How do we apply the slope? If we change it to a fraction it is a rise of negative one over a run of one. The fraction is
negative one over one.
What does a rise of negative one over a run of one mean? Defend your thinking. It means that we move vertically down
one unit and one unit to the right to plot another point on the line.
So what is the location of the new point after we apply the slope? Our new point is located at five, zero.
So what is the final step, now that we have two points plotted on our coordinate grid? We can connect the two points
and draw a line using the straightedge.
What is the y-intercept of this line that we graphed together? The y-intercept is five.
Does the y-intercept of the graph match the y-intercept that we solved for? Justify your answer. Yes it does, they are
both five.
FOLDABLE
We are now going to add to our foldable to help us to organize the information we have learned in this lesson for future
reference.
On the third section of the foldable write, Modeling Lines with Slope and a Point. When you lift this flap, on the top
section of the flap we are going to include the graph of a line. You can draw this line using the points negative two,
negative four; negative one, negative one; zero, two; and one, five.
Now on the bottom flap for the Modeling Line with Slope and a Point section of the foldable, we are going to include
information that we are given in order to solve the problem. We are told that the slope of this line is three, and that a
point on the line is one, five. Knowing this point provides an x-value and a y-value that can help us when we are trying
to find the equation of the line.
Remember that the general formula of the equation in slope-intercept form is y equals mx plus b. We can plug in the
values for x, y and m and solve for b. We can find that b is equal to two for this problem. Now that we have found that
b equal two, we know the y-intercept and we can go back to the previous flap of the foldable in order to derive the
equation of a line.
Be sure to hold on to this foldable so that you can use it for future reference when working with these types of
problems.
IF TIME PREMITS
We are now going to take a look at the, if time permits section of this lesson. An additional method can be used to find
the equation of a line when given the slope and a point on the line. This method is called, point-slope form.
Point-slope form of an equation is y minus y one equals m times the quantity x minus x one. Why do you think it is
called point-slope form? We need a point on the line and the slope to find the equation of the line.
The most important part of this formula is to know that we are going to use it as our final equation! So what variables
are almost always in our final equation when we substitute the slope and y-intercept? The variables x and y.
Therefore, the first y and the first x you see in the formula represent permanent variables for the final equation. This
means we will not be substituting values in for those variables.
The y in this equation is the permanent y–variable for the final equation. And the x in the equation is the permanent xvariable for the final equation. What do you think x one and y one represent? They represent the coordinates of the
point that is given. Y one represents the y-coordinate of the given point, and x one represents the x-coordinate of the
given point. What do you think m represents in the equation. It represents the slope. Include this on your graphic
organizer.
Now let’s take a look at an example and see if we can use the information that we have just talked about in order to find
the equation of the line.
This example is the line passes through the point two, five and has a slope of negative three? Let’s substitute the value
that we know. Remember the x and y without a subscript are the final variables and they remain permanent, without
substitution.
Since the point that we know is the point two, five the x one will equal two and the y one will equal five when we
substitute in the values into the equation. The equation is y minus five equals negative three times the quantity x minus
two.
Looking at the equation, what do you think our goal is? Our goal is to get it into slope-intercept form. How can we get
this into slope-intercept form? We can begin by distributing the negative three to all of the terms in parentheses.
Begin by distributing the negative three to all of the terms in parentheses on the right side of the equation. Our
equation can now be written as y minus five equals negative three x plus six. Since negative three times x is negative
three x and negative three times negative two is positive six.
Now what is the last step to manipulate the equation so that it is in slope-intercept form? We need to add five to each
side of the equation so that we have the equation in the form y equals mx plus b. When we add five to each side of the
equation we get y equals negative three x plus eleven.
For this problem we have use point-slope form to find the equation of the line.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson. The SOLVE problem is, Mrs.
Rosen’s fifth grade class recently planted a tree. The type of tree planted tends to grow at a rate of two inches per
month. After three months, the tree measured twenty inches tall. What is the equation of a line that represents the
height of the tree, y, after x months?
At the beginning of the lesson we Studied the Problem. First we identified where the question is located within the
problem and underlined the question.
We then put this question in our own words in the form of a statement. This problem is asking me to find the
representation of the height of the three, y, after x months as an equation.
In Step O, we will Organize the Facts. First we need to identify the facts. Mrs. Rosen’s fifth grade class recently planted
a tree/fact. The type of tree planted tends to grow at a rate of two inches per month/fact. After three months, the tree
measured twenty inches tall/fact. What is the equation of a line that represents the height of the tree, y, after x
months? The question also contains facts in this problem.
Now that we have identified the facts we are ready to eliminate the unnecessary facts. These are the facts that will not
help us to find the representation of the height of the tree y after x months as an equation.
Mrs. Rosen’s fifth grade class recently planted a tree. Knowing that they planted a tree will not help us to find the
equation that represents the height of the tree after a certain number of months. So we will eliminate this fact. The
type of tree planted tends to grow at a rate of two inches per month. We want to know the rate at which the tree
grows, so that we can use it to help us to form an equation. We will keep this fact. After three months, the tree
measured twenty inches tall. We also want to know what happened after a certain number of months, to help us in
representing the height of the tree after a certain number of months as an equation. So we will keep this fact as well.
What is the equation of a line that represents the height of tree, y, after x months? Since we are writing an equation we
need to know what x and y will represent within the equation. So we will keep the facts located in the question as well.
Now that we have eliminated the unnecessary facts we are ready to list the necessary facts. They number of months
represented by x is our independent variable. And the height of the tree represented by the variable y is the dependent
variable. The rate at which the tree grows is two inches per month. And at three months the tree was twenty inches
height. This gives us a point on the line three, twenty.
Now in Step L, we will Line Up a Plan. First we need to write in words what your plan of action will be. We can
substitute the x-coordinate and y-coordinate, as well as the slope, into the general slope-intercept equation. Then,
solve for the y-intercept and use m and b to write the equation of a line.
What operation or operations will we use in our plan? We will use multiplication and subtraction as we solve to find the
y-intercept.
Now let’s Verify Your Plan with Action. First we can estimate your answer. We can estimate that our answer is going to
be an equation in slope-intercept form.
Now let’s carry out your plan. We said that we would substitute the x-coordinate and y-coordinate as well as the slope
into the general slope-intercept equation. Let’s talk about what each of the variables in our equation represent. Y is the
height of the tree, or the dependent variable. X is the number of months, or the independent variable. Two inches per
month is the rate, or the slope. And the point three, twenty is a point that would be on the line. Because we know that
at three months the tree was twenty inches tall.
Now we’re going to substitute the x-coordinate and y-coordinate as well as the slope into the general slope-intercept
equation. We know that y is equal to twenty, x is equal to three and the slope is two.
Our equation is twenty equals two times three plus b. Remember that we want to solve for the y-intercept. We can
simplify the equation as twenty equals six plus b. By multiplying two times three on the right side of the equation. Now
we need to subtract six from each side of the equation, which leaves us to b equaling fourteen. The y-intercept is
fourteen.
Now that we know the slope and y-intercept we can use them to write the equation of the line. The equation of the line
is the y equals two x plus fourteen. Because the slope is two and the y-intercept is fourteen.
Now let’s Examine Your Results.
Does your answer make sense? Here compare your answer to the question. Yes, because we found the equation that
represents the height of the tree.
Is your answer reasonable? Here compare your answer to the estimate. Yes, because it matches our estimate as an
equation in slope-intercept form.
And is your answer accurate? Here check your work. Yes, the answer is accurate.
We are now ready to write your answer in a complete sentence. The equation that represents the height of the tree
over time is y equals two x plus fourteen.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, how is graphing a line given the slope and y-intercept similar to graphing a line given the slope
and a point? They are similar because in both cases you start with a point that is known to be on the line, either the yintercept or the point given, and then apply the slope to find another point on the line. Connect the points to graph the
line.
Our second question was, explain the steps for graphing a line if you know the slope and a point. Begin by plotting the
point that is known. Identify the rise over the run for the slope provided and apply the slope until a second point can be
graphed. Then, connect the two points with a straight line.
And our third question was, explain the steps for writing the equation of a line if you know the slope and a point. Begin
by filling in the values for the x-coordinate and y-coordinate, as well as the slope in the general equation for slopeintercept form. The only variable unknown will be the y-intercept, represented by the variable b. Solve for b and then
substitute the value of the y-intercept and the value of the slope back into the general equation for slope-intercept
form.