Posted in Algebra

A challenging complex number problem with solution

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This complex number problem was selected on the basis of its uniqueness in terms of phrasing things within the Argand diagram/locus context. While my proposed solutions might be short, bear in mind this question truly demands/challenges the student to think unconventionally in order to formulate a viable solving approach.

 

Problem

A complex number z=x+iy is represented by the point P in an Argand diagram. If the complex number w where w = \frac{z-8i}{z+6}, (z\neq-6) has its real part zero, show that the locus of P in the Argand diagram is a circle and find the radius and the coordinates of the centre of this circle. If, however, w is real, find the locus of P in this case.

Solution

complex number problem

The author of this post is Mr. Frederick Koh. He is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com.

If you love this problem, I’m sure you will also enjoy the two other challenging problems shared by Mr. Koh in this site:

  1. Differentiation in parametric context
  2. Working with summation problems
Posted in Mathematics education

What is abstraction in mathematics?

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Abstraction is inherent to mathematics. It is a must for mathematics teachers  to know and understand what this process is and what its products are. Knowledge of it can enrich our reflection of our own practice as well us guide us and make us conscious of the type of learning activities we provide our students.

All the definitions below give emphasis about abstraction as a process. Note also that the direction of the abstraction is always from a set of contexts to an abstract concept. Abstraction is related to generalization which I discussed in another post.

Abstraction –
  • the omission of qualities from concrete experience – Aristotle
  • the process of separating a quality common to a number of objects/situations from other qualities – Davidov  (1972/1990, p. 13)
  • the act of detaching certain features from an object – Sierpinska (1991, p 61)
  • Abstracting is an activity by which we become aware of similarities … among our experiences. An abstraction is some kind of lasting change, the result of abstracting, which enables us to recognize new expereinces as having the similarities of an already formed class. … To distinguish between abstracting as an activity and abstraction as its end-product, we shall … call the latter a concept. – Skemp
Empirical vs reflective abstraction (Piaget et al)
  • Empirical abstraction is based on superficial similarities and is the type of abstraction involved in everyday concept formation.
  • Reflective abstraction is, according to Piaget, based on reflection one one’s actions. For example when one object and two objects are put together you always get three objects. This leads to recognition of invariance (later expressed as 1+2=3). These objects of invariance become concepts (the numbers 1, 2, and 3) and the invariant action becomes a relation between these concepts (addition). In reflective abstraction, concepts and relations are abstracted together.
Abstract-apart vs abstract-general (Mitchelmore et al)
  • abstract-apart:  concepts formed that exist apart from any contexts from which they might have been abstracted
  • abstract-general: concepts that have been abstracted through the recognition of similarities between contexts. These concepts derive their general meaning from the set of contexts from which it has been abstracted
Stages of abstraction
  • a cycle of interiorization-condensation-reification – by Sfard 1991
  • generalization-synthesis-abstraction cycle – Dreyfus (1991)

Reference: The Role of Abstraction and Generalization in the Development of Mathematical Knowledge by Michael Mitchelmore – paper presented during 2nd EARCOME.

You may want to read my post about assessing understanding of function in equation form for an example of abstracting based on Sfard’s interiorization-condensation-reification cycle.

Posted in Calculus

Differentiation problem in parametric context with solution

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This is hot off the press-a question taken from the recently concluded 2011 September Preliminary Examinations of a school in Singapore. It deals with applications  of differentiation in the parametric context. Extensive trigonometry is employed here together with the manipulation of surd forms. I have personally worked out everything for your (the student’s) reference.

If you want a real calculus challenge, the problem below should satisfy your appetite. Peace.

QUESTION :

The parametric equations of a curve  are

x = sin2t and y= a cos t

where is a positive constant \frac{-\pi}{2} \le t \le\frac{\pi}{2}

(i) Find the equation of the tangent to the curve at the point P  where t= \frac{\pi}{4}.

(ii) The normal to the curve at the point Q where t = \frac {\pi}{3} intersects the axis at R. Find the coordinates of R and hence show that the area enclosed by the normal at Q , the tangent P and the x-axis is

differentiation

Author

Frederick Koh is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com.

Mr Koh is also the author of the post Working with summation.

I have created a GeoGebra applet to visualize Question 1 above.

Posted in Algebra, Mathematics education

When is a math problem a problem?

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One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.