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The Gradient Slope of the Snowboard Jump Ramp in the photo above, would need to be created at the correct incline to give the Snowboarder sufficient height to complete his aerial acrobatics.
Obviously some basic mathematical calculations would have been involved with building this jump ramp.
In a previous lesson we looked at Gradient and Slope, and how to calculate them by plotting points on a Grid, and drawing and measuring right traingles.
If you have not done this previous lesson, then you need to go through the material at the link below, before doing the “Gradient Formula” lesson.
Intro to Gradient and Slope Lesson
In our “Gradient Formula” lesson, we will be looking at finding the Gradient Slope between two points using an Algebra Formula.
The nice thing about the formula method, is that we do not actually need to draw the points onto a Cartesian Plane Grid.
This saves us a lot of time and work…. Nice!
Definition of Gradient
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The Mathematical value of a Gradient or Slope involves comparing how far up we have gone, against how far across.
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Change in Y and Change in X
Previously we have found Gradient Slope by drawing our points onto a grid and counting squares.
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We can actually determine the UP and ACROSS squares counts by SUBTRACTING PAIRS OF X AND Y COORDINATES.
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This observation led mathematicians to develop a Gradient Slope Formula which does the coordinate pairs subtractions.
Gradient Slope Formula
The Gradient Slope Formula involves labelling the x and y coordinates, and then subtracting the y’s and subtracting the x’s.
This gives us the “Change in Y” and the “Change in X”.
To get the Gradient we divide these two subtractions.
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Applying this formula to our previous example points, we obtain the same Gradient Slope answer of “2”.
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Gradient Slope Example 1
In this first example, we have plotted the points and graphed them on a Cartesian Grid.
HOWEVER, Graphing the points is not actually necessary, as we shall see in Example 2 later.
Have a go at substituting the required values into the Gradient Formula, and working out the final answer.
(You can then scroll down to see the fully worked solution).
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Here is the fully worked solution to Example 1.
(It can be checked by counting squares on the Grid, and is Negative because from left to right it is a Downhill Line connecting the two points.
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Gradient Slope Example 2
In this example we have not bothered plotting the points, because this is not necessary.
This is the great thing about using the Gradient Slope Formula…. it is so much faster to get to the answer!
Have a go at substituting the required values into the Gradient Formula, and working out the final answer.
(You can then scroll down to see the fully worked solution).
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Here is the fully worked solution to Example 2.
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Gradient Slope Videos
The following is a Gradient by Formula video, (10 minutes long), that shows the line plotted and how the formula works.
This next video talks about the change in y versus the change in x, using the Gradient Slope Formula.
The following is an excellent, but quite long YayMath video, which uses the Gradient Slope Formula.
The video even shows the calculation of zero and undefined gradients using the formula.
Straight Line Between Two Points Calculator
All we have to do is enter the (x,y) coordinates of any two points and click “Go” and this online interactive will calculate the slope, find the rule for the line, and even plot the line for us on an X-Y Grid.
It also tells us about Parallel and Perpendicular Lines.
Click the following link to use this interactive calculator.
http://www.mathsisfun.com/straight-line-graph-calculate.html
Gradient Worksheet
This worksheet does not have any answers; however the answers can be checked using the Gradient Slope Calculator available at the following link:
http://www.mathsisfun.com/straight-line-graph-calculate.html
Click the following link to get to the worksheet.
Click here for Gradient Slope Formula Worksheet
Related Items
Gradient and Slope
Gradient and Slope in the Mountains
The Cartesian Plane
Plotting Graphs from Horizontal Values Tables
Plotting a Linear Graph using a Rule Equation
Plotting Graphs from T-Tables of Values
Finding Linear Rules
Distance Between Two Points
Mountain Gradients
Real World Straight Line Graphs I
Real World Straight Line Graphs II
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