Anyone who has ever ridden a bike uphill knows what a “Gradient” or “Slope” is all about !

But in Norway there is a clever way to save the hardship of big hills.

It’s called a “Bicycle Lift”, and is in Trondheim Norway.

Image Source: http://weburbanist.com

It is the world’s only functional bicycle lift, designed to help bike riders ascend the hill.

Image Source: http://weburbanist.com

The device is called a “Trampe” and opened in 1993. The hill it is on goes from the town center to the University. The “Trampe” has also become a major tourist attraction.

Basically you purchase tokens, then strap your foot into the escalator, and it pulls you up the hill.

Image Source: http://weburbanist.com

For the smoothest ascent the advice is to keep the right leg extended behind the body, shift the body weight to the footplate, and lean slightly forward on the bike.

So there is some weight distribution and balancing mathematics involved with the entire process.

Image Source: http://weburbanist.com

As can be seen in the diagram a continuous cable loop powers the lift, and it is driven along using electric motors with gears.

Image Copyright 2013 by Passy’s World of Mathematics

The Mathematical value of a Gradient or Slope involves comparing how far up we have gone, against how far across.

Original Image Purchased from Photozone.com

There are four types of Gradient: Uphill, Downhill, Flat Horizontal, and Straight Up Vertical.

Mathematically we name these : Positive, Negative, Zero, and Infinite.

POSITIVE Gradients and Slopes go UPHILL from left to right.

Original Image Purchased from Photozone.com

NEGATIVE Gradients and Slopes go DOWNHILL from left to right.

Original Image Purchased from Photozone.com

ZERO Gradients and Slopes are FLAT or horizontal and do not go up or down.

Original Image Purchased from Photozone.com

INFINITE or UNDEFINED Gradient or Slope goes straight up, and is neither uphill or downhill.

This final type of Gradient or Slope is one which goes straight up.

A bike cannot be ridden vertically, but it can be stored in this position.

Because it cannot be ridden vertically, we call the slope or gradient “UNDEFINED”.

original image from http://www.thehappybike.com

The following video gives a really good overview of what Gradient is, and then does a basic straight line gradient question.

We highly recommend watching this video.

Numeric Values for Gradients are usually worked out using measurements between points that are located on a reference grid.

Usually this Reference Grid is an X-Y “Cartesian Plane”, and the points are (x,y) coordinates on this grid.

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For our previous example, the Steepness or Gradient or Slope of the line connecting points A and B is 8/4 = 2.

We say the size of the uphill slope between the two points is “2”.

Because the direction of the line is uphill, the gradient is positive 2.

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To calculate Gradient or Slope, we always draw a right angled triangle around our two points.

We then count “up” squares, and “across” squares.

There are examples on how to do this later in this lesson.

Note that there is also a “Formula” way of working out gradient or slope from a pair of (x,y) points, but this is covered in a separate lesson.

In this lesson we are only looking at finding gradient or slope by manually counting squares on an X-Y Grid.

Here is a quick thirty second video with a smiley face emoticon which shows the four different types of Gradients and Slopes.

This next video gives really good explanations, and covers word problems, but is not embeddable.

Click the image of the video, or the link below to go to youtube and watch the video.

This next video shows how to calculate the Gradient Slope between two (x,y) points.

Here is another video about plotting two points, and then finding the Rise and Run and thereby calculating the Gradient.

This final video shows a real life example of how to calculate the Slope of a Building Block:

The first step is to find two points to use.

If using a grid with squares, make sure the points are on the corners of squares so their coordinates are both whole numbers.

We then work out the Vertical and Horizontal Distances between the points. We can draw a right angled triangle connecting the points to do this.

Finally we put the Vertical and Horizontal Distances into the Gradient Formula m = rise / run and then simplify our final answer.

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In this first example, we have a line sloping uphill from left to right, and so our answer will be a Positive Gradient Slope value.

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The next step is to work out the Rise and Run values.

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We now substitute the Rise and Run values into the Gradient Formula, and obtain our final answer.

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In this second example, we have a Downhill Gradient Slope, and so we need to make our final answer Negative.

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The next step is to work out the Rise and Run values.

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We now substitute the Rise and Run values into the Gradient Formula, and obtain our final answer.

Because we have a Downhill line, we need to make the final answer Negative.

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In this third example, we have a Flat Horizontal Gradient Slope, and so our final answer will be Zero.

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When doing questions, look for these four types of Gradient Slopes.

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Parallel Lines always have identical Gradient Slopes.

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This means that if we are told two lines have the same Gradient Slope, then they must be Parallel.

Perpendicular Lines are Lines which cross over each other at an angle of 90 degrees.

The Gradient Slope of two Perpendicular Lines are Negative Reciprocals of each other.

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This means that if we have one gradient, we can work out the other one, by flipping it over and changing its sign.

Eg. The following Gradient Slopes are negative reciprocals, and would indicate that we have Perpendicular Lines.

2/3 and -3/2

2 = 2/1 and -1/2

3 and – 1/3

1/4 and -4

-4/5 and 5/4

8/7 and -7/8

Blank X-Y Grid

Here is a blank X-Y Grid you can print out and use for plotting question Points, and working out Gradient Slopes.

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This first worksheet has some right triangles set up with measurements, and it is simply a matter of calculating the Rise / Run and simplifying the final answer.

This second worksheet gives line segments drawn on an X-Y Grid, and we need to add in right triangles to calculate the Rise and the Run.

This third worksheet is on Parallel and Perpendicular Lines.

This final worksheet is also on Parallel and Perpendicular Lines.

This first Online Activity is an interactive quiz where we can work up through a range of levels of difficulty.

Click the following link to play this Game

This next activity provides a series of Quiz Questions to answer online.

Click the following link to do the Quiz.

In this online activity we can carefull slide around the points to create any line we like, and its gradient will be calculated for us.

Click the following link to use this interactive.

(The interactive is about halfway down the web page)

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.

http://www.mathsisfun.com/straight-line-graph-calculate.html

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