Mathematical Investigation: Print design investigation

For the following Mathematical Investigation task, students use Mathematics creatively to design an aesthetically pleasing pattern that can be used in real life by textile industry to produce fabric prints.

This activity is not covered in class creating a greater challenge for students, as one must utilize appropriately, concepts covered in calculus, including differentiation and integration techniques of various functions studied in our course, then apply their independent learning to completing the task.

Note:    Use  of  Desmos,  GeoGebra  or  any  other  maths  software  programs,  including  the  graphic  calculator,  is indicated as an exploration tool  in order to develop deeper and more meaningful connections between each question, sections of the investigation and consequently the entire Mathematical Investigation.

Mathematical Investigation Task must include:

•    An  introduction  that  demonstrates  an  understanding  of  the  features  of  the  problem  or  situation   being investigated

•    Appropriate presentation of information used, calculations and results

•    Analysis and Conclusion

For this investigation there will be minimal teacher direction (initial guideline questions are provided with no

background and students are expected to research relevant material)

Students must extend the investigation in an open-ended context.

This investigation exploreshow complex functions, differentiation and integration can be used to create a graphic design for a fabric.

Introductory Task:

Part 1

First pattern has been designed for you. You are required to find the functions and the circle used for this design, based on the description below.

The following functions/relations have been used to design this pattern:

-      one logarithmic function and its inverse

-      two cubic functions

-      one circle

Recommended function types:

y1  = f(x) = a ln(bx + c)

y2  =  f −1(x)

g(x) = ax [(x − ℎ)2  + k]

ℎ(x) = bx [(x − ℎ)2  + k]

Mathematical rules used for this design:

(i)   the four functions start at the origin.

(ii)  logarithmic function and its exponential inverse end at (4,4)

(iii) logarithmic function has  a  = 0.5

(iv) the point of inflection of the top cubic (green function) is located atx  =  1.6

(v)  the maximum value of the top cubic (green function) aligns vertically with the stationary point of inflection of the bottom cubic (blue function)

(vi) the minimum point of the top cubic (green function) is located at (2, 2)

(vii)  the area of the circle is one third of the area between the two cubic functions and its centre aligns with the minimum point of the top cubic.

(viii) the two cubic functions always meet on the logarithmic function.

Based on the above information find:

1.    The four functions and the equation of the circle.

2.    Calculate the areas between each curve combination (use of technology indicated).

Refer to the cubic functions found from above rules. Assume that theirrespective found hand k values do not change.

By varying only their leading coefficients, a and b respectively, investigate:

3.   The change in the area between the two cubic functions with respect to the totalarea (area between the logarithm and the exponential function).

4.    How a and b vary as the area between the two cubic functions approaches a maximum or a minimum.

Hence,

5.    Find two cubic functions such that the area between them is ¼ of the totalarea(area between the logarithm and the exponential function).

PART 2

Create your own print design stating the type of composition structure used and based on the following requirements:

•   In its final form your design should not exceed a A4 piece of paper

• Use at least three different functions out of which:

o one complex function

o one derivative

•   At least two areas should be in a ratio of your choice (you may consider the Golden ratio as a feature in your design when choosing colours for various areas).

•   Connections between functions must be considered (e.g. inflection points and turning points could be common or intersecting other functions, etc..).

•   For the final print, preferably 4 identical panels of the design should be joined together by using and

manipulating functions’ transformation properties (e.g. vertical & horizontal dilation, reflection, rotation, etc…)

•   At most three colours can be used and a statement of colour usage should be also included (area usage related).

•   Calculations, interpretation of results and analysis should be comprehensive throughout the investigation.

Your investigation will be assessed using the following assessment design criteria

Concepts and Techniques

CT1     Knowledge and understanding of concepts and relationships.

CT2    Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts.

CT3    Application of mathematical models.

CT4     Use of electronic technology to find solutions to mathematical problems.

Reasoning and Communication

RC1    Interpretation of mathematical results.

RC2    Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations.

RC3    Use of appropriate mathematical notation, representations, and terminology.

RC4    Communication of mathematical ideas and reasoning to develop logical arguments.

RC5    Development, testing, and proof of valid conjectures.

Performance Standards