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Riemann sums to definite integral conversion

In the most recent tutorial there is a question about converting a Riemann sum to a definite integral, and it seems to be tripping up quite a few students. I wanted to run through one of the calculations in detail so you can see how to answer such a question.

Let’s look at the example:

$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(9\left(4+(i-1)\frac{6}{n}\right)^2-8\left(4+(i-1)\frac{6}{n}\right)+7\right)\frac{13}{n}$

There are many ways to tackle such a question but let’s take one particular path. Let’s start by the fact that when the limit is defined, the limit of a sum is the sum of the limits. We can split up our expression into 3, which looks like:

$\lim_{n\rightarrow\infty}\sum_{i=1}^n9\left(4+(i-1)\frac{6}{n}\right)^2\frac{13}{n}-\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(8\left(4+(i-1)\frac{6}{n}\right)\right)\frac{13}{n}+\lim_{n\rightarrow\infty}\sum_{i=1}^n7\frac{13}{n}$

Let’s tackle each of these separately. Let’s look at the first term:

$\lim_{n\rightarrow\infty}\sum_{i=1}^n9\left(4+(i-1)\frac{6}{n}\right)^2\frac{13}{n}$

Well, we can take the factor of 13 outside the front of the whole thing to start with, along with the factor of 9, and this will give

$13\times 9\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(4+(i-1)\frac{6}{n}\right)^2\frac{1}{n}$

We see here that we have a sum of terms, and a factor which looks like $\frac{1}{n}$ in each term.…

By Jonathan Shock| August 23rd, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|8 Comments

Some sum identities

During tutorials last week, a number of students asked how to understand identities that are used in the calculation of various Riemann sums and their limits.

These identities are:

$\sum_{i=1}^n 1=n$

$\sum_{i=1}^n i=\frac{n(n+1)}{2}$

$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}$

$\sum_{i=1}^n i^3=\left(\frac{n(n+1)}{2}\right)^2$

Let’s go through these one by one. We must first remember what the sigma notation means. If we have:

$\sum_{i=1}^n f(i)$

It means the sum of terms of the forms f(i) for i starting with 1 and going up to i=n. Sometimes n will actually be an integer, and sometimes it will be left arbitrary. So, the above sum can be written as:

$\sum_{i=1}^n f(i)=f(1)+f(2)+f(3)+f(4)+....+f(n-2)+f(n-1)+f(n)$

We haven’t specified what f is, but that’s because this statement is general and applies for any time of function of i. In the first of the identities above, the function is simply f(i)=1, which isn’t a very interesting function, but it still is one. It says, whatever i we put in, output 1. So this sum can be written as:

$\sum_{i=1}^n 1=1+1+1+1+....+1$

Where there are n terms.…

By Jonathan Shock| August 20th, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|3 Comments

MAM1000W 2017 semester 2, lecture 1 (part ii)

The distance problem

If I want to know how far I walked during an hour, I can ask how far I walked in the first five minutes, and how far I walked in the second five minutes, and how far I walked in the third five minutes, etc. and add them all together. ie. I could write:

$d=d_1+d_2+d_3+d_4+...d_{12}$

Where $d_i$ is the distance walked in the $i^{th}$ five minutes. To calculate a distance, we need to know how fast we are going, and for how long. In fact:

$distance=velocity \times time$

where you can think of velocity as the same thing as speed (though there are subtle differences which you will find out about later). This formula works if the velocity is constant, but what if it is changing. Well, if we have a graph of velocity against time, then we can think about splitting the graph into intervals (like the five minute intervals above), and approximating that during a small interval of time, the velocity is roughly constant.…

By Jonathan Shock| August 13th, 2017|Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate|1 Comment

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MAM1000W 2017 semester 2, lecture 1 (part i)

Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate

MAM1000W 2017 semester 2, lecture 1 (part i)

I wanted to put up a little summary of some of the most important things to remember from the end of last semester. There was a sudden input of new concepts, so let’s put some of them down here to get a clear reminder of what we need to know. A few things in this post:

The antiderivative
Sigma notation
Areas under curves

Antiderivatives

An antiderivative of a function $f$ on an open interval $I$ is a function $F$ such that:

$F'(x)=f(x)$ for every $x\in I$

Note that we say an antiderivative, not the antiderivative. There can be many functions whose derivatives give the same thing. While we know that:

$\frac{d}{dx}\sin x=\cos x$

and therefore $\sin x$ is an antiderivative of $\cos x$ , we can also say that:

$\frac{d}{dx}(\sin x+3)=\cos x$

So $\sin x+3$ is also an antiderivative of $\cos x$ . In fact for any constant $c$ it is true that $\sin x+c$ is an antiderivative of $\cos x$ . We will come up with some clever notation for the antiderivative soon.…

By Jonathan Shock| August 13th, 2017|Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate|1 Comment

I have decided to share something which I found interesting while reading up some Mathematics this holiday.

The idea I am going to talk about is that of the cardinality of a set. In simple terms,

Definition 1: Cardinality is a “measure of the size” of a set.

Example:
Suppose that we have a set, $A$ , such that $A=\{a,b,c,d,e\}$ . The cardinality of the set, denoted by $|A|$ , is $5$ , because there are $5$ elements.
It is indeed worth noting that unlike ‘lists’, in Mathematics, order and number of elements doesn’t determine much concerning the identity of a set. This means that

$\{a,b,c\}=\{a,a,a,b,c\}=\{a,b,b,b,c,c\}=\{a,...,b,...,c, ...\}$

as long as you use the same elements, they are all equal. Of course, cardinalities would be the same, because all of them are a representation of the same mathematical entity.

Let’s talk about something slightly more interesting. Suppose that you have another set with infinitely many elements, like the set of natural numbers, $N$ , or real numbers $\mathbb{R}$ .…

By Siphelele Danisa| July 28th, 2017|Uncategorized|0 Comments

Getting ideas into action: Statway and Quantway networked improvement communities

Siyaphumelela Conference 2017, The Wanders Club, Johannesburg 28 June 2017

Getting ideas into action: Statway and Quantway networked improvement communities

Bernadine Chuck Fong, Carnegie Math Pathways

Andre Freedman, Capital Community College

It’s an exciting time to be tackling problems that appear to be worldwide. The Carnegie Foundation is in California, home to many disruptive changes / technologies, e.g. Uber, Google.

An example of the problem: When asked which is bigger, a/5 or a/8 many students respond saying 8a = 5a so 8 = 5. For many, mathematics is about following algorithms and although they may pass algebra courses they can’t apply the knowledge.

Diagnostic tests can leave students feeling depressed, like they do not belong in higher education. A principle in the development of a disruptive transformation of mathematics in American colleges was to meet students where they are.

15% of students needing developmental maths complete the required college maths or stats course after 2 semesters.…

By Anita Campbell| June 30th, 2017|Conference, Uncategorized|0 Comments

Radically transforming mathematics learning experiences: Lessons from the Carnegie Math Pathways

Siyaphumelela Conference 2017, The Wanders Club, Johannesburg

http://www.siyaphumelela.org.za/conf/2017/#!

Carnegie Foundation Math Pathways Workshop 27 June 2017

Andre Freedman, Capital Community College, Connecticut

Bernadine Chuck Fong, Carnegie Math Pathways

Workshop goals:

Learn about the design, goals, implementations of Carnegie Math Pathways
Experience Pathways lessons
Engage in design tasks to improve student success in maths and college
Engage in conversations about professional learning to address issues and concerns that are specific to local contexts

Staff had to learn new ways to teaching maths, there had to be ‘buy in’ for it to be successful. This is a great challenge.

How to radically transform outcomes for all mathematics students?

aim to get more student to complete maths courses and degrees
persistence (students don’t easily give up when challenged)
quality of learning (e.g. students can explain what a function is years after taking a maths course)
identities of learning (students see themselves as someone who can do maths).

…

By Anita Campbell| June 28th, 2017|Uncategorized|0 Comments

On the place of struggle within Mathematics – how to truly get rid of distractions

I have had a lot of conversations with students over the last couple of weeks which made me want to write this post. I apologise in advance that it will be rather long.

It’s also important to state that this message doesn’t hold for everyone, but it is worth seriously thinking about.

Many students recently have asked me about how to go through tutorials, and how to revise. They are finding that while they are sitting down for a long time with their tutorials, the tests still feel really hard.

To an important extent, technologies have changed the way we think and act over the last couple of decades. In many ways, things were easier in my day when we didn’t have so many technological distractions. Cellphones were rare when I was a student and smartphones were still over a decade away! There was no Facebook, or Youtube, or Instagram.…

By Jonathan Shock| June 16th, 2017|Uncategorized|14 Comments

Modelling learning, unlearning and relearning in large classes

Claire Blackman, Department of Mathematics and Applied Mathematics, UCT

Presentation at the SANRC First Year Experience (FYE) conference, Johannesburg 24- – 26 May 2017

Context: Claire teachers first year commerce students who do not necessarily want to do maths.

Alvin Toffler quote: “The illiterate of the 21st century will not be those who cannot read and write, but those who cannot learn, unlearn, and relearn.”

Change is a way of life but adapting to change is hard and people are not very good at it. We have to learn how to learn and relearn and unlearn.

How I teach is more important than the content.

The world needs people who own and learn from their mistakes and think before responding.

Two useful models:

Krathwohl’s taxonomy (published by Bloom) for thinking about the individual
Group therapy to get students in an emotional comfortable space

Krathwohl’s taxonomy of the affective domain can help to see if students are learning

https://za.pinterest.com/source/dynamicflight.com

Living (Integrating values into life)
Organising
Valuing
Responding
Receiving

A group analytic framework

Environment (psychological structure, class boundaries)
- Students and teachers need to feel safe
Process (how is the class run)
Content
- Maths
- Tools for dealing with fear and uncertainty e.g.

…

By Anita Campbell| May 30th, 2017|Uncategorized|1 Comment

Finding Fibonacci, by Keith Devlin – a review

This book was sent to me by the publisher as a review copy.

I have a terrible admission to make. I came to this book with a paltry knowledge of Fibonacci (Leonardo of Pisa). The knowledge that I thought that I had was quickly shown in fact to be incorrect, so I was largely starting with a blank slate (Fibonacci did not discover the Fibonacci sequence, nor would he be terribly happy to know that in the popular psyche, this is what he is famous for).

In fact, this book is not really about Fibonacci (Devlin has another book about him). This is a book about the writing of a book, and about Devlin’s process of uncovering the history and importance of what Fibonacci had accomplished. It is a book about the research of the history of mathematics, and as such, it is a lovely tale: one of fortuitous moments of discovery, and of frustrations of searching for manuscripts.…

By Jonathan Shock| April 28th, 2017|Uncategorized|1 Comment

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