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 y-intercept 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 y-intercept.
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 y-intercept.
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.