Gradients and area under non-linear graphs
The resources in this list introduce students to the principles of how to calculate or estimate gradients of graphs and areas under graphs. The activities also provide students with the opportunity to interpret the results.
Visit the secondary mathematics webpage to access all lists.
Exploring area linked to graphs
This resource contains ten linked problems which are intended to lead to students developing a rule for integrating functions of the form y=axn where n not equal to -1.
Problem 1 begins by exploring the area enclosed under a straight line y=2x+3 from x=0 to x=a for varying values of a, leading to generalisation. This involves finding the area of trapezia.
Problem 2 extends to generalising the area under the line y=mx+c between x=a and x=b.
Problem 3 asks students to explore what happens when one of the limits is a negative value.
Problem 4 requires students to generalise the area under a straight line for different values of m and c, and covering x=0 to x=a as a varies.
The next section explores the area under a curve. Beginning by finding upper and lower bounds for the area under the curve y=x2 between x=1 and x=2 by finding areas of rectangles.
Problem 5 asks students to find upper and lower bounds for the area between x=1 and x=3, using 3 and 6 rectangles.
Problem 6 extends the task to finding upper and lower bounds for the area under the curve y =1/x between x=1 and x=4.
The following section explores the idea of splitting the area into a number of trapezoidal strips with problem 7 asking students to find the area under the curve y=x2 between limits using a number of strips.
Problems 8 and 9 require students to generalise using n strips whilst problem 10 explores the area under the curve y=axn.
The teacher notes provided could be adapted to lesson plans and presentations. They include detailed explanations as well as solutions to each of the problems. There are a range of problems that progress in difficult to include examples to challenge more able students.
The Mathematics Curriculum: from Graphs to Calculus
This book is about graphs, their drawing, their interpretation, their development and their use. It discusses the teaching of graphs from their early introduction and as far as the beginnings of integral and differential calculus. The book also places the teaching of graphs in an historical context as it reviews the curriculum at the time and the textbooks available.
The first five chapters set the scene by discussing graphical work and functions. Two separate themes develop from this: area leading to integration, and gradient leading to differentiation. These themes are developed throughout the book in parallel with developments based on students' experiences of distance, time and speed. The chapters on area and integration always precede the corresponding chapters on gradient and differentiation, because the authors believed that area is conceptually simpler than gradient. Chapters 9 and 10 on numbers and limits take their place because they provide essential background and Chapter 6 covers ‘Some Special Graphs’. A detailed contents page is included.
Whilst quiet complex this book gives a very good over view of the progression in learning from early understanding of functions and graphs towards area under the graph and gradient leading to integration and differentiation. It also makes strong likes to other related strands of mathematics.
Nuffield National Curriculum Mathematics 5
This resource is split into three sections; each section divided into chapters giving relevant information and presenting activities to do and questions to answer.
The 13 chapters on Number and algebra are suitable for those candidates following Higher GCSE courses. Each chapter is subdivided into sections showing the mathematics covered. Chapter 9 looks at gradients starting from what pupils should already know about gradients of a straight line moving on to gradient under a curve in part B. Part C focuses on interpretation including distance time graphs and velocity time graphs.
Chapter 10 introduces the area under a curve. This includes using the formula for the area of a trapezium to calculate the area under non-linear graphs. The material includes rigorous explanations and illustrations followed by a range of examples including real life examples.
Gradient
This excel program contains a series of interactive spread sheets designed to explore the concept of a gradient of a curve.
The first activity illustrates that the gradient of a function, at a point P, is the gradient of a chord PQ as the length of PQ tends to zero. The equation of the quadratic can be adjusted. The summary sheet summaries the findings for a particular example encouraging the student to deduce the formula for the gradient function.
The Function>Gradient activity provides the formula of 8 function, a mix of linear and quadratic, for students to differentiate to find the gradient function. Clicking on the empty cell reveals the answer.
The Gradient > Function activity requires the student to work backwards to find the original function given the gradient function.
The final activity provides a further opportunity to explore differentiation from first principles.
The workbook contains a further eight worksheets providing further questions for use in the classroom.
