How to Measure Slope Using Graphs & Coordinate Geometry
SKILL: Thinking | Problem-Solving
In today’s episode of the Ask Diane podcast, I’m going to share a homeschool Logic Lesson Plan (skill # 2 of the classical trivium) for measuring slope using coordinate geometry. We’re going to capitalize on what your child is already learning (or has learned) in his packaged math curriculum by adding some real-life math lessons to make the theoretical concepts stick with actual applications and lots of practice.
This lesson is rich…measuring slope teaches lots of critical thinking skills like (1) memorizing math operations, (2) reducing fractions, and (3) solving word problems. I’ve given you a sample lesson plan, but you’ll soon see that you can come up with many creative variations of my lesson.
In This Homeschool Lesson Plan You’ll Learn:
- things you can do with coordinate geometry
- why mathematicians use the lowercase letter m to define slope
- the mathematical term for the steepness of a line
- the difference between rise, run, and pitch
- what Pythagoras knew about angles and degrees
- how to calculate a gradient
- why ski resorts need elevation data like altitude, vertical drop, and % grade
- how to make your own clinometer for measuring slope
What is Coordinate Geometry?
But before we start measuring slope, let’s review a little bit about geometry. In algebra, your homeschool student was introduced to Rene Descartes’ coordinate system where ordered pairs are plotted on the horizontal x-axis and the vertical y-axis. These 2 axes form right angles where they intersect; we call this the origin (0, 0).
We divide this flat surface (the coordinate plane) into 4 quadrants. Since most kids learn about the horizontal number line first (with zero being in the middle and positive numbers ascending on the right of the zero and negative numbers on the left), we start plotting our ordered pair on the horizontal x-axis.
Ordered pairs are shown in parentheses like (-2, 8) with the first number referring to the position on the x-axis and the 2nd number telling where it is on the y-axis. On the x-axis, values to the right are positive and those to the left are negative; on the y-axis, values above the origin are positive and those below are negative.
Together, the two numbers in the Cartesian ordered pair define a single unique position on the plane. When you add a 2nd ordered pair to the graph, you now have two points on the plane, and you can join them by a straight line. Depending on where you plot your points, you could have steep, moderate, or gentle sloping lines in all four quadrants of the coordinate plane.
What is Slope?
The slope of a line is a number that measures its steepness; we use the lowercase letter m to represent slope. Why m? Descartes was French, and the word for “to climb or rise” is monter which begins with the letter m. Slope is the the ratio of the line’s vertical change (on the y-axis…also called the “rise”) to the horizontal change (on the x-axis…known as the “run”).
So the simple formula for calculating slope is m=rise/run (“rise over run”) with rise in the numerator and run in the denominator. Slopes can be positive (line slopes upward to the right – think of climbing a ramp), negative (line slopes downward to the right – think of going down a playground slide), zero (horizontal lines – think balance beam), or undefined (vertical lines – think of a cliff face).
The “rise over run” fraction on the 2-dimensional coordinate plane is not the only way to measure slope; you can also express slope as an angle, measured in degrees. This is where real-life critical thinking skills get sharpened because your child has to take what he’s practiced on paper and translate it to 3-dimensional problem-solving.
On paper, slope refers to the steepness of a line; in real-life, slope refers to the angle, grade, or pitch of an incline. If you’ve ever driven through mountains, you’ve probably seen road signs like “6% Grade” and gravel safety ramps for runaway trucks. The percentage grade is simply the rise over run fraction. Roofers like fractions so they call slope pitch; architects like degrees so they call it gradient…basically same thing. (Don’t worry if you’re having trouble visualizing slope…I’ve got some examples for you in the show notes.)
What Did Pythagoras Prove?
In order to calculate slope in real-life, we need to use the Pythagorean theorem which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
To illustrate, let’s say you’re in the Colorado mountains and want to figure out how steep that ski run is that you’re planning to conquer this afternoon. With the click of a mouse, you can find the highest and lowest elevations (this gives you the vertical drop or leg a of the right triangle. Next you need to figure out the ground distance or length of the ski slope (like pulling a tape measure down the hill); this yardage is your hypotenuse.
You’ve got all you need to sketch out the right triangle and figure out the degrees of the acute angle (time to pull out your handy dandy travel protractor). And voila! You’ve got the facts to make a personal decision about whether you want to risk your pretty neck on that steep ski run or not!
How Do You Make Math Meaningful?
You show your kids how the abstract math works in real-life applications! Wheelchair ramps, staircases, roofs, cable cars, roller coasters, flag poles, skate ramps…hey, maybe even your own driveway…provide opportunities for applying mental math concepts to meaningful needs. All you need to do is imagine a coordinate plan or visually draw a right triangle, then solve your problem using the tools that Descartes and Pythagoras gave us.
By the way, I’ve included a link in the show notes for making your own clinometer (an optical measuring device used by architect and firefighters) so that you can take this lesson outside and continue learning.
Okay, so you can see the possibilities for measuring slope are all around you, right? It’s time to tackle that lesson plan, and I’ve laid out all six steps for you so that all you need to do is download and print a copy for your Homeschool Planning notebook.
Measuring slope teaches lots of critical thinking skills like (1) memorizing math operations, (2) reducing fractions, and (3) solving word problems. Plus, you can even practice using the scientific method to document your observations, predictions, and conclusions about real-life inclines.
Set aside 3 weeks to practice two skill sets: problem-solving and scientific method. By the end of this learning unit, your kid will have mastered the problem of measuring slope using Coordinate Geometry, the Pythagorean theorem, and a clinometer. Not only will he be prepared for the next step in his homeschool math curriculum, but he’ll also have some really practical knowledge that will help him in real-life for years to come.
Teaching Resources for this Lesson Plan:
- Problem-solving tool using Pythagorean formula
- Contractor tool for calculating pitch
- Firefighter tutorial on measuring slope % (hillside graphic)
- Architect explanation: calc gradient, %, and degrees
- Make your own Clinometer (plus don’t miss: how to use it)
- Refresher on Scientific Method (fact # 10)
- How to make a lap book
- Here are the docs mentioned in the podcast: CHANGE!
If you’d like type your own LOGIC Lesson Plan like mine, DOWNLOAD A BLANK PDF of the Logic Lesson Plan. Don’t discount your own ability to write your own DIY homeschool curriculum…knowing what skills you need to teach frees you up to be creative and make learning fun!
P.S. If you liked this homeschool lesson plan, try a science lesson plan about molecules.