Including key questions, key teaching points, models and resources Including Support and Extension Including key questions and homework 3 Recap finding the equation of a line between two points. Project a graph of y=x2. Find the equation of the line through the points with coordinates: (3, 9) and (3.5, 12.25)? (3,9) and (3.1,9.61)? (3,9) and (3.01,9.0601)? Students then repeat this for another point of their choosing (e.g. 2.5, 2.1, 2.01). Make sure as many different points as possible are chosen. Class discussion to find that all points end up with y=2x. Repeat the process as a class using x, x+h to derive differentiation formula. Use the ‘new’ formula to compare with answers from the previous lesson’s homework, and identify that this works for different initial functions. working more quickly can work with even finer intervals (3.001 etc.) Worksheet/Exercise to gain routine practice from old resources or integral. 4 Students recap understanding from GCSE that gradients of distance-time graphs give speed, and gradients of speed-time graphs give acceleration - use old GCSE Qs, such as parts of questions found Using calculus to find speed from displacement/time, and acceleration from velocity/time graphs. Students use this activity to match up displacement-time and velocity-time graphs (assuming that students know the difference between distance and displacement - if not then ) Students then work on , their understanding of the relationships between displacement/velocity/acceleration, to generate graphs of the other two given one. Some students may be able to work back from the later questions. Complete the sketched graphs. Starter Main teaching Including key questions, key teaching points, models and resources Notes Including Support and Extension Consolidation/Plenary Including key questions and homework
5 Recap the solutions from the homework. Ensure clarity around the concept that the gradient of displacement gives velocity, and gradient of velocity gives acceleration. Link this to the previous work of finding gradient functions (lesson 2), and identify that velocity is a gradient function of displacement, therefore to find velocity, we can differentiate displacement. Rates of change. Demonstrate example(s) of being given a function for displacement and finding a function for velocity and acceleration. Students then attempt a similar exercise, purely just using their differentiation skills, but under a new context. Support may be needed to understand that differentiating with t is no different to x Worksheet/Exercise to gain practice working with different variables (simple as using letters other than x & t) 6 Recap solutions from homework - Up to 3 starter questions to check understanding Students undergo their own to appreciate and understand the reality of how a function of x relating to t works. Students more in depth about modelling, including assumptions. Some students may need a reminder on the general shape of functions. Some students may be able to predict what type of function we will end up with. Students to list all possible assumptions of the experiment. Students to use their knowledge from previous lessons to find a function for velocity and acceleration for the train. Starter Main teaching Including key questions, key teaching points, models and resources Notes Including Support and Extension Consolidation/Plenary Including key questions and homework 7 Review train experiment, Provide two or more displacement-time graphs (they could be with or without context), such that they are Continue second investigation.
lesson/homework, going through problems as a whole class where necessary. differentiation to solve more practical problems regarding rates of change. Go through one or two that are teacher-led. Students can then work on one question individually/paired/grouped in depth, and present their entire thought process to someone else. Depending on class size/skill, it may be that the presenting is kept to smaller groups/pairs. Difficulty of questions could be targeted dependent on students.