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Gradient of a line

Adapted a Don Steward diagram for a Year 8 lesson on the Gradient. Started with this, and asked students to put the staircases in order of steepness.



Students were talking in pairs initially (Think-Pair-Share) before coming back to a whole class discussion. We discussed how they were deciding on what steepness is and what the key features were. We came to the conclusion that more information was needed. 

 I asked students what measurements would they like to have. 

Student: "Can you tell us the angles?"
Me: "Sorry guys, I don't have my protractor on me!"

Instead I added these values and allowed students to discuss further. 



The variation in the numbers allowed for comparisons and a 'discovery' of how we can calculate the gradient. This was a really nice 'hook' into the lesson and helped students understand the concept of the gradient better. 

Following this we formalised our method for finding the gradient as rise over run. I went through one further example before setting students off on a task to work through independently. I chose the following questions from Corbettmaths.com. 



You may have noticed that all of the examples and staircases have been with a positive gradient, this is on purpose. Questions (f) and (g) gave my students that shock moment in a lesson that I was able to talk about further and link back to going down a set of stairs. This ensured that for future questions involving a negative gradient, students would remember the shock moment.

We also spoke about the differences in methods between students, e.g. some using 2/1 for (a) while some chose 4/2 or even 8/4, noting that they would all give the same value. I also displayed a students answers to (a) and (d) where they had, incorrectly, labelled both of them as a gradient of 2. Before I even mentioned whether he was right or wrong another student had said "but sir they are at different angles, how can they be the same?" at which point the student whose work it was suddenly realised his error!

Following this period of deliberate practice the class were happy to calculate the gradient from a graph given the axis. I then progressed onto the following set of questions, where the values on the x and y axis are removed or even when the graph itself is removed to leave just a set of coordinates. I modelled a question similar to question 6 and let students work independently from there. 




I questioned a number of students about how they were finding the gradient for Q7 where they couldn't even see the graph itself. Some students spoke about how to go from the first coordinate to the second, some were subtracting the first from the second, some were even sketching it to help visualise where the coordinates were. 

I finished and left students to ponder the following challenge (we will talk about it further next lesson):



  1. The aim of the lesson was for students to be able to calculate the gradient. 
  2. I wanted students to have a strong conceptual understanding built from a context that they could easily understand. 
  3. I wanted to show students lots of variations of calculating the gradient, progressing from on a graph with axis, to on a graph with no axis, to being given two coordinates, and finally to working backwards when given the gradient.
There are still a few tweaks I would make and obviously the work that the students produced is just a performance of what I had showed them. Learning would have to have been evidenced over time and I will make use of further retrieval practice to support long term learning. However this lesson has given these students a really strong understanding of what the gradient is and a number of different variations in questions to calculate the gradient.



**Massive thank you to the great Don Steward and the fantastic textbook exercises on Corbett Maths**

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