During 2012 I wrote a letter to a niece explaining how to find areas of common shapes (plane figures). I later typed it up as a PDF file which you can download here. This free PDF file summarises most of the material in the first three videos below. I am confident that, whether you are a young student, a senior student, or even a mathematics teacher, you will find something new and useful here … because my approach makes very simple what a lot of textbooks make complicated.
Basically, every area is composed of a length multiplied by a breadth! That is, every area can be thought of in terms of the rectangles of which it is composed. I may upset some purists here but, even when we integrate (in calculus), we are essentially adding a vast number of microscopically thin rectangles and we write of the area of a shape as the sum (using the old fashioned S) of the y-value multiplied by the very thin dx-value. Hence A = ∫y.dx.
Although I acknowledge that triangles have bases and heights and that there is a special terminology associated with other shapes as well, in my booklet I advocate not using those terms when calculating areas. This is because such terms lead to more formulae than students need and they therefore fail to see the essential simplicity underlying calculations of area. Also, I reserve the notion of height for formulae concerned with volumes (we speak of prisms, pyramids, cylinders and cones as having height). This means that, to avoid confusion (with some volume formulae having two heights … one for the shape of the cross-section and one for the prism/pyramid itself), all my area formulae use length and breadth! In this way, my students only have to learn four formulae in order to calculate the areas of all the basic plane figures.
School textbooks unnecessarily complicate their descriptions of how to find areas of basic shapes. I extend this accusation even to 'area' topics in senior high school because they are often dealt with in an unimaginative way. I have helped many, many senior high school students, for example, who have encountered great difficulty in understanding Simpson's Rule (for approximating areas under curves).
What I will be sharing in this video is very simple. Young mathematics students should find that following the instructions here will make their calculating of areas very logical and very straight-forward. If you are a senior student, I encourage you to watch this series (including this first video) because it will help you understand Simpson's Rule and integration with greater clarity. It is a long video (25 minutes) but I decided not to shorten it because, if you listen to all the explanations, you will be much more likely to remember the principle and the four formulae!
So, please watch it right through. In my next video I will show how to use the four formulae to calculate the area of simple shapes. The emphasis then will be on clear setting out of work. For the moment, let's learn the basics!
In this video, I summarise the four formulae that we learned in the previous video and show how they are all strongly related to a common theme. This will help you think of them properly and remember them well.
Most of the video is devoted, however, to showing how to use these formulae to find simple areas. I show you how to set your work out clearly, and why. Many students insist on writing as little as possible, but that usually creates confusion and greatly increases their risk of making careless errors. The methods that I show here can help you work very quickly and accurately ... and therefore earn very good results with confidence!
I am convinced, after many years of teaching, that clear setting out of work will benefit you in three ways:
It will help you think more clearly. It is amazing the number of times students tell me that something looks so easy when I do it, but they find it hard to do afterwards. Almost without exception I find that, if the student sets work out in the way I do, they find the work easy to understand too! Please do not underestimate the power of what I just shared with you ... I have seen talented students perform poorly because their sloppy setting out reflects the fact that they are not thinking clearly about a topic ... and I have seen less talented students excel when they have adopted this philosophy and these techniques. Please try them for a while and don't take short-cuts. I am confident that you will be impressed with the results!
Anyone reading your work will find it much easier to understand ... and this includes examiners! When marking papers, I have often been confronted with a mass of numbers and numerical calculations extending from one side of a page to the other ... calculations everywhere ... and not a single word in sight to explain what is going on ... and, often, not even an '=' sign either. After a brief search, trying to find out what the student was thinking, it is very easy to give up! The answers from such students are rarely at the bottom right hand side of the mess! I will say more. The answers (if I can fnd them) from such students are often wrong as well. Confused minds produce confused work ... but it works the other way as well ... clear work can clarify thinking. I encourage you to try it out. Remember ... no short cuts (not for a while, anyway).
You will almost certainly make fewer careless mistakes ... and that means higher marks. I have seen students' marks rise significantly as they begin to set their work out neatly. I am sufficiently realistic to realise that, for most students, their results are their main motivation for studying. However, if you are one of the very rare individuals who wishes to master mathematics for the sheer joy of it ... you, also, will benefit from clear thinking and the greater accuracy that comes with this kind of setting out.
I think I have said enough. Please watch this video, and the one following it (about finding areas of composite figures). I am confident that, if you follow what I suggest, you will not be disappointed.
Many students get confused when finding areas of composite shapes. They lose track of what they are doing and (as I did at school) sometimes forget to add the 'bits' together at the end.
In this video I show you how to use the four formulae summary (from the first videos in the series) to calculate the area of composite shapes using an extremely simple and efficient method. It is quite important that all the steps be followed as I show them. Every line is there for a reason (to train your mind to present your work in a logical way). As you master the method you may feel free to modify it to suit yourself, but please practise setting it out this way first.
Being able to find the area of a trapezium (or trapezoid) is a skill that many students fail to master in junior high school. Consequently, they often struggle with using the Trapezoidal Rule to approximate an area under a curve in senior high school as part of their 'pre-calculus' studies. Some students don't even recognise the extremely simple connection between the two skills.
Here, I present, explain and demonstrate a profoundly simple method for using the Trapezoidal Rule. I also show how the formulae using the function notation and the y-notation are actually the same formulae that you used years earlier when calculating the area of trapezia (trapezoids).
There is really nothing mysterious about Simpson's Rule. If you struggle with using this rule to estimate areas under curves, please watch this video and allow me to explain how to understand it as a 'rectangle' with a width and average length!
I will explain the basic concept very clearly and then show how to understand the notation used in most textbooks. I then show how to use Simpson's Rule as a 'one off' and also when it is used multiple times.
Watching this video could mean that you never struggle with this aspect of mathematics again!
In it, I summarise the basic concepts underlying the finding of areas of plane figures and how these concepts relate to the Trapezoidal Rule and Simpson's Rule. In this way, you can see all these aspects as a complete, comprehensive whole. I then show how the same principles underly integration.
This is not an introduction to calculus and is certainly not a rigorous explanation or description of it. It is simply offering you a very clear understanding of what integration achieves and what the terminology means. It would be worth your while viewing this video AFTER you have learned the basics of integration. It may help put it in perspective for you.
Graeme’s approach to explaining maths formulas made it easy for my children to grasp. Graeme had a number of methods by which he could explain each problem, giving the students a clear understanding of how to approach each area of maths. My students came away feeling confident of when, and how to apply each formula to solve the maths problems.
Sarah G (parent, 2011)