# MU123 Discovering Mathematics Assignment Sample UK

MU123 is a course offered by The Open University and is part of their undergraduate program. The course is called “Discovering Mathematics” and is designed to help students develop their mathematical skills and understanding. The course covers a range of topics including algebra, geometry, trigonometry, calculus, and statistics. It aims to provide students with a solid foundation in mathematics and to help them develop their problem-solving skills.

Throughout the course, students will engage in a range of activities including online tutorials, interactive computer sessions, and written assignments. There will also be opportunities for students to work in groups and to receive feedback from their tutors. The course is designed to be flexible and is suitable for students with a range of mathematical backgrounds. It is also designed to be self-contained, so students do not need any prior knowledge of mathematics to take the course.

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### Assignment Task 1: Key ideas in mathematics, including some statistics, algebra, geometry and trigonometry.

Here are some key ideas in mathematics, including some statistics, algebra, geometry, and trigonometry:

1. Statistics: Statistics is the branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It includes topics such as probability, distributions, hypothesis testing, and regression analysis.
2. Algebra: Algebra is a branch of mathematics that deals with mathematical operations and their rules. It includes topics such as equations, inequalities, exponents, polynomials, and functions.
3. Geometry: Geometry is the branch of mathematics that deals with shapes, sizes, and positions of objects in space. It includes topics such as angles, lines, circles, polygons, and three-dimensional objects.
4. Trigonometry: Trigonometry is the branch of mathematics that deals with the relationships between angles and sides of triangles. It includes topics such as trigonometric functions, identities, and equations.
5. Calculus: Calculus is a branch of mathematics that deals with rates of change and accumulation. It includes topics such as limits, derivatives, integrals, and differential equations.
6. Number theory: Number theory is the branch of mathematics that deals with the properties of numbers. It includes topics such as prime numbers, factorization, and divisibility.
7. Combinatorics: Combinatorics is the branch of mathematics that deals with counting and arranging objects. It includes topics such as permutations, combinations, and graph theory.
8. Logic: Logic is the branch of mathematics that deals with reasoning and argumentation. It includes topics such as propositional logic, predicate logic, and Boolean algebra.
9. Probability theory: Probability theory is the branch of mathematics that deals with random events and their probabilities. It includes topics such as probability distributions, Bayes’ theorem, and stochastic processes.
10. Linear algebra: Linear algebra is the branch of mathematics that deals with linear equations and their properties. It includes topics such as matrices, vectors, and systems of linear equations.

These are just some of the key ideas in mathematics, and each topic can be explored in greater depth and complexity.

### Assignment Task 2: Mathematical vocabulary and notation introduced and developed in the module.

Without knowing the specific module you are referring to, it is difficult to provide a comprehensive list of all mathematical vocabulary and notation that may be introduced and developed. However, here are some commonly used mathematical terms and symbols:

1. Function: A mathematical object that takes an input and produces an output.
2. Domain: The set of all possible inputs for a function.
3. Range: The set of all possible outputs for a function.
4. Graph: A visual representation of a function.
5. Equation: A statement that asserts the equality of two mathematical expressions.
6. Variable: A symbol that represents a quantity that can vary.
7. Constant: A symbol that represents a fixed value.
8. Derivative: A measure of how much a function changes as its input changes.
9. Integral: A measure of the area under a curve.
10. Summation: A shorthand notation for adding a sequence of numbers.
11. Matrix: A rectangular array of numbers or variables.
12. Vector: A mathematical object with both magnitude and direction.
13. Set: A collection of objects.
14. Subset: A set that contains only elements that are also in another set.
15. Union: The set of all elements that are in at least one of two or more sets.
16. Intersection: The set of all elements that are in both of two or more sets.
17. Complement: The set of all elements that are not in a given set.
18. Cartesian product: The set of all ordered pairs of elements from two or more sets.
19. Congruence: A relation between two objects that are the same size and shape.
20. Similarity: A relation between two objects that have the same shape but not necessarily the same size.

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### Assignment Task 3: Selection and use of mathematical techniques for solving problems.

Mathematical techniques are used in various fields, from engineering to finance, from computer science to physics, and beyond. Here are some steps to follow when selecting and using mathematical techniques for solving problems:

1. Define the problem: Start by clearly defining the problem you want to solve. This will help you determine what kind of mathematical technique you need to apply.
2. Choose the appropriate technique: Once you have defined the problem, choose the appropriate mathematical technique to solve it. Some common mathematical techniques include algebra, calculus, statistics, and probability.
3. Check assumptions: Before applying the mathematical technique, make sure to check any assumptions that are involved. These assumptions could affect the accuracy of the results you obtain.
4. Apply the technique: Apply the chosen mathematical technique to the problem. Make sure to show all the steps involved in your calculations and explain your reasoning.
5. Check your results: Once you have obtained your results, check them to make sure they are reasonable and make sense. Check if they are consistent with any known laws, theories, or empirical data.
6. Communicate your findings: Finally, communicate your findings in a clear and concise way, using appropriate mathematical notation and terminology. Explain the implications of your results and how they can be used to inform decision-making.

Remember that the process of selecting and using mathematical techniques for problem-solving is iterative. You may need to revise your assumptions, choose a different technique, or check your results again as you work through the problem.

### Assignment Task 4: Interpretation of results in the context of real life situations.

Interpreting results in the context of real-life situations involves understanding how the findings of a study or analysis can be applied to the world outside of the research setting. This requires taking into account the specific context of the study, the limitations of the research design, and the potential impact of the results on individuals and society.

For example, if a study finds that a new medication is effective in treating a particular condition, the interpretation of the results in the context of real-life situations would involve considering factors such as the availability and affordability of the medication, the potential side effects and risks associated with its use, and how it could improve the lives of patients with the condition.

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### Assignment Task 5: Simple mathematical arguments.

1. The sum of two even numbers is always an even number.

Proof: Let a and b even number, then a = 2x and b = 2y, where x and y are integers. The sum of a and b is a + b = 2x + 2y = 2(x + y), which is clearly an even number. Therefore, the sum of two even numbers is always an even number.

2. The square of any even number is always an even number.

Proof: Let a be an even number, then a = 2x, where x is an integer. The square of a is a^2 = (2x)^2 = 4x^2 = 2(2x^2), which is clearly an even number. Therefore, the square of any even number is always an even number.

3. The product of any two odd numbers is always an odd number.

Proof: Let a and b be odd numbers, then a = 2x + 1 and b = 2y + 1, where x and y are integers. The product of a and b is ab = (2x + 1)(2y + 1) = 4xy + 2x + 2y + 1 = 2(2xy + x + y) + 1, which is clearly an odd number. Therefore, the product of any two odd numbers is always an odd number.

4. The sum of the first n odd numbers is n^2.

Proof: Let S be the sum of the first n odd numbers. Then the first odd number is 1, the second odd number is 3, the third odd number is 5, and so on. Therefore, the nth odd number is 2n – 1. Thus, S = 1 + 3 + 5 + … + (2n – 1).

Now, if we add the first and last terms, second and second-to-last terms, and so on, we get (2n) for each pair. There are n pairs in total. Therefore, S = n(2n)/2 = n^2. Therefore, the sum of the first n odd numbers is n^2.

5. The sum of the first n even numbers is n(n+1).

Proof: Let S be the sum of the first n even numbers. Then the first even number is 2, the second even number is 4, the third even number is 6, and so on. Therefore, the nth even number is 2n. Thus, S = 2 + 4 + 6 + … + 2n.

Now, if we divide each term by 2, we get 1 + 2 + 3 + … + n, which is the sum of the first n natural numbers. Using the formula for the sum of the first n natural numbers, we get S/2 = n(n+1)/2. Therefore, S = n(n+1). Therefore, the sum of the first n even numbers is n(n+1).

### Assignment Task 6: How to explain mathematical ideas from the module in writing.

Explaining mathematical ideas from a module in writing can be a challenging task, but it can also be a very rewarding one. Here are some steps you can follow to help you explain mathematical ideas effectively in writing:

1. Start by identifying the key concepts or ideas you want to explain. This will help you organize your thoughts and ensure that you cover all the important points.
2. Use clear and concise language to explain mathematical ideas. Avoid using overly technical jargon or complicated terminology, unless it is necessary to convey a specific concept.
3. Provide examples or illustrations to help clarify your explanations. This can be particularly helpful for abstract or complex concepts.
4. Use diagrams, charts, or graphs to illustrate your points. These can be very effective in conveying complex ideas or relationships.
5. Check your work for accuracy and clarity. Make sure that your explanations are logically sound and that you have provided all the necessary information.
6. Finally, consider your audience when writing your explanations. If you are writing for a general audience, avoid using too much technical language or assuming too much prior knowledge. If you are writing for a more specialized audience, make sure that you are using the appropriate terminology and providing enough detail to be useful.

By following these steps, you should be able to explain mathematical ideas from a module in writing in a clear and concise manner that is accessible to your intended audience.

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