## When a Distance-Time Graph lesson turns into Calculus.

We recently had some great fun learning about Distance-Time graphs in year 8 and ended up discussing a bit of advanced mathematics while we were at it. I started the lesson with a quick question about gradients of straight lines to make sure that they could use this in the task later on in the lesson. Once we´d discussed it, I decided that I wanted to go and get some fresh air so we left to go on a little journey around school. I borrowed some stop watches from the Science department and 2 students took the stop watches to record the times at the different points of the journey while two others wrote down the times on clip boards. Here´s the Distance-Time graph showing the journey from the goal post to the school gate and back again.

From A – B: We marched from the goal post to the big tree.

From B – C: We stopped to ponder the problem of me dropping my pen.

From C – D: We started off shuffling at an extremely slow rate and slowly increased our speed to a sprint until we reached the gate.

From D – E: We stopped to catch our breaths.

From E – F: We ran back to the goal post.

When we arrived back into class we managed to construct the graph using the times given by the students who recorded the information. At first the students wanted to draw a straight line from C to D but I mentioned that we only manged to get 5 metres in the first 20 seconds and then 15m in the next 20 seconds and carried on with this until we had a set of points that looked like a curve.

I then set the students off on a worksheet that I created specifically for the lesson. While making the worksheet, I decided to stick in a question (which was badly worded in the sheet, so I changed it before any of them had got onto it) about the speed between the points C and D:

Is it possible to find our speed at the 100 second point of the journey? Explain your answer.

Some of the students left this section blank and moved on but others wrote down some remarkable answers which touch on the very essence of calculus – the idea of finding the instanteous rate of change of something.

Here are some interesting answers from 2 students: Student 1 and Student 2. It´s amazing to think that 12 year olds can begin to understand the fundamentals of calculus by themselves! They essentially realised that it was difficult to find the gradient of the curve because it was constantly changing.

As a side note, they then started to do a few problems from the text book to apply what they´d learnt and I was quite impressed with this question in addressing a common misconception.

Jake cycles along a flat road, then up a hill, then down the other side. Which graph best describes jake´s cycle journey?

Two students made a common mistake on this one. Which graph do you think they went for?

(Also, see my Distance-Time graph Geogebra file)