Lesson Plan - Get It!
Get out your phone and open the stopwatch app — you're going to be racing against yourself!
You can do this in a large hallway, large room, in a driveway, or wherever there's a good amount of space. You're going to run in three paths from one side of your area to the other. The first will be a straight line, with the second and third having progressively more turns. Models of the paths you take are shown below! Time each one to see how long they each take. Which do you think will be the fastest path?
- So, which was the fastest path?
You should have found that the straight line was the fastest way to get from one point to another. This is because the shortest distance between any two points is exactly that, a straight line! So, what does this mean for us in the world of math? Well, it helps us graph a linear relationship so long as you know two points on the graph. The two points used in order to do this are called intercepts.
An intercept is the point at which a graphed equation intersects with an axis.
- The x-intercept is an ordered pair where the line crosses the x-axis. Its ordered pair is in the form (x,0).
- The y-intercept is an ordered pair where the line crosses the y-axis. Its ordered pair is in the form (0,y).
You may be wondering, "Since these intercepts make it so easy to graph linear relationships, how do I find them?" Well, it's good that you asked! There are two methods you can use: the substitution method and the cover up method. You're going to learn both so you can decide which you like best!
The substitution method is all about replacing a variable with something else, like another variable or a number value. When using this method to find x and y intercepts, you plug in zero. Here is a walk through on how this is done by finding the x and y intercepts of y= 3x + 6. Start with the x and then work on y.
- You know the x-intercept takes the form of (x,0), so our y-value is zero. Plug that into your equation: 0 = 3x + 6.
- Now, solve as you normally would, by subtracting over the 6. You will get 3x = -6.
- Finally, divide over the three, giving an answer of x = -2. Now that you have both the x and y values, you can see that the x-intercept is (-2,0).
Time to move on to your y-intercept!
- You know the y-intercept takes the form of (0,y), so just like our y-value was 0 last time, your x-value is zero this time.
- Putting that into your equation and solving, you get y = 3(0) + 6, or y = 6. And with that, you're done. Your y-value is 6, making your y-intercept (0,6).
Cover up method
To use the cover up method, all you need is your finger! When solving for one intercept, try covering up the other variable in the equation. It's that easy! Just pretend it isn't there! Like last time, first find the x-intercept, then move on to the y-intercept.
Now, try a different equation, 9x - 6y = 18. Since you're doing the x-intercept, cover up the y part with your finger and solve for x. Doing this with your finger should make the equation look like 9x = 18.
- Now, solve for x!
- Dividing out the 9 gives you x = 2, making the x-intercept (2,0).
Now, do the y-intercept.
- When you find the y-intercept, you want to cover up the x part and solve for the y! Covering up the x part with your finger makes our equation look like –6y = 18.
- Time to solve again! Dividing the –6 over gives us y = -3, and our y-intercept is (0,-3).
Now that you have your intercepts, it's time to graph! You've already done the hard work when you found the intercepts; this is the easy part! To graph your equation, simply plot both intercept points on a coordinate plane. You know your graph is linear, which means the graph will be a straight line.
- What is the shortest distance between two points?
A straight line! Use a ruler to draw a straight line connecting the two intercept points, and boom! You're all done — you've found the graph of the equation.
- How are the substitution and cover up methods similar? How are they different?
Continue on to the Got It? section to practice each method.