All about gradient

When we teach linear graphs at Key Stage 3, we often miss the opportunity to explore useful real-life mathematics. In this post I share practical ideas and resources for teaching the concept of gradient.

A sense of steepness
At school we usually express gradients as integers or fractions but road signs in the UK show gradients as percentages.  Older signs showed ratios (rise:run) - arguably these were easier to interpret.

The road sign pictured indicates a gradient of 25% ie one quarter. We say the gradient of this road is 'one in four'. Think about this as 'rise over run' and therefore interpret this as 'we go up one unit for every 4 units we go across'. Wouldn't it be great if we could jump in the school minibus with our class and drive to a road with a 25% gradient to see how steep it is? Unfortunately this is probably impractical (though it could be set as a homework, depending on how hilly your local area is!).

There are some impressively steep roads in San Francisco but New Zealand boasts the 'world's steepest road' according to the Guiness Book of Records - Baldwin Street in Dunedin has a gradient of 35%. So how can we interpret this gradient? It means we go up 35 units for every 100 units we go across. We could write this as the ratio 35:100, simplified to 7:20. But we usually prefer to state gradient ratios in the form 1:n, so in this case the gradient is 1 in 2.86. Incidentally, 25,000 balls of chocolate are rolled down this 350m-long street in an annual charity Cadbury Jaffa Race.

Give your students a sense of 'steepness' by showing this short video of someone cycling up a road with 38% gradient. Looks like hard work! As a matter of interest, this rather technical article attempts to calculate the steepest gradient that one can cycle up.  This article says that anything over 16% is considered very challenging for cyclists of all abilities.

Ripley Street Ridge, San Francisco

Practical activities using trigonometry
In real life it is normally impossible to measure the rise and run of a slope so we use trigonometry to calculate these lengths. Picture a slope as a right-angled triangle - we can use a clinometer to measure the angle of elevation and perhaps a trundle wheel to measure the length of the hypotenuse. Trigonometry can then be used to calculate the rise and run in order to find the gradient. The table below shows gradients and their equivalent angles of elevation.  I think it's important that students understand the connection between a 100% gradient and a 45 degree angle.

Source: Wikipedia

Gradient can also be calculated using Pythagoras' Theorem. Say a man cycles 5km on a slope and knows (from his altimeter watch) that he has climbed 250m vertically.  Again, picture a right-angled triangle - the hypotenuse is 5000m and the height is 250m. Use Pythagoras to calculate the base (4994m).  Now use 'rise over run' to calculate the gradient - in this case we get 5%.

These real-life applications of mathematics can easily be made into practical class activities. For example you could use home-made clinometers and trigonometry to calculate the gradients of slopes in and around school, like staircases and ramps.

Resources for teaching gradient

• My worksheet on real-life gradients on TES
• A wide range of exercises in this pack. Note that this is a American resource so uses the terms slope and grade instead of gradient.
• A measure of slope - lesson and activities from Mathematics Assessment Project
• A worksheet of contextual gradient problems
• A gradient activity from Nrich
• Mixed gradient questions by linzbfc on TES
• PowerPoint on gradients by Andrea Frame on TES
• What is gradient? worksheet by MathsPad
• Alphabet slopes activity

Possible homework investigation questions
What is a funicular? Why can funiculars travel at steeper than 90 degrees?
What’s the maximum grade allowed for a railway without cable or cogs?
Why do you think road builders have to worry about the grade of the road?
Why do you think we have regulations for building wheelchair ramps?
At what steepness do avalanches occur?
Is it possible to ski at 90 degrees steepness?
(source)

I hope this post has inspired you to try out some new ideas when teaching gradient!  Let me know how you get on (@mathsjem).

Unusual gradient sign in Hertfordshire, UK