Modelling Water

This lesson was based on an idea shared with me by Ron Lancaster at SEAMC 2015. It looks at the equation used to model the change in volume of 1kg of water as temperature changes. I like this task for the real life aspect and for the amount of maths hidden inside one idea.
[1] Students start by looking at the equation used. What do they know about it just by looking at the algebra? Here you can start by talking about the y intercept, cubic graphs, increasing or decreasing functions, etc. [2] The worksheet also asks students what degree of accuracy they will choose to work with. This is interexting to discuss as most shy away from "nasty" numbers, but the context mens that you can talk about implications of over simplifying. We agreed that 2dp would be good as this works with the intercept. However, if we choose this then the T cubed coefficient means that the curve becomes a parabola! 
[3] and [4] I used this part to talk about what a graph should look like and go over some fundamental ideas like dependent variables, scale, labelling etc. Then we began talking about the shape and reinforcing the idea that a cubic became a quadratic because of rounding and the extremely small nature of the t cubed coefficient. We discussed whether or not we thought the curve was symmetrical about its minimum (4 degrees). If required, you could go into tangents and gradients at this point.
[5] and [6] Next, we look at the graph showing water density and discussed how it was the reciprocal of our volume graph as we used 1kg as our mass of water. We looked at the point of discontinuity at 0, and discussed how we draw strange graphs in maths but often struggle to put them into a real life context. We looked at the "definition" of a point of discontinuity then wrote our own because its hard to read if you're not a mathematician.
[7] Finally, we looked at the real life repercussions of water's strange properties and the impact of ice being less dense than water on pond life. Then I referred my class to the science department to ask more advanced questions, because I love the graph and the maths but definitely can't answer why!
[5] and [6] Next, we look at the graph showing water density and discussed how it was the reciprocal of our volume graph as we used 1kg as our mass of water. We looked at the point of discontinuity at 0, and discussed how we draw strange graphs in maths but often struggle to put them into a real life context. We looked at the "definition" of a point of discontinuity then wrote our own because its hard to read if you're not a mathematician.
[7] Finally, we looked at the real life repercussions of water's strange properties and the impact of ice being less dense than water on pond life. Then I referred my class to the science department to ask more advanced questions, because I love the graph and the maths but definitely can't answer why!
water.pptx  
File Size:  1272 kb 
File Type:  pptx 
water_and_ice.docx  
File Size:  45 kb 
File Type:  docx 
chart.xlsx  
File Size:  13 kb 
File Type:  xlsx 