Maths

1. Preliminary Investigation
Industrial Packaging Design Task
BY LOUISA MAY ALCOTT

3. Overarching Inquiry
Decision-making can be
improved by using a model
to represent relationships

4. Definition of keywords
 Decisions
 Modelling in mathematics
 Relationships and patterns

5. General Concept Map for Algebra

6. Worldviews
 Social Constructivist or pragmatism

7. Deductive vs Inductive Research
 The main difference between inductive and deductive reasoning is that inductive
reasoning aims at developing a theory while deductive reasoning aims at
testing an existing theory. Inductive reasoning moves from specific
observations to broad generalizations, and deductive reasoning the other way
around.
 In my research, I have taken a deductive stance because…

8. Deontological vs Consequential views
 Optional..
 Ethics
 Trolley problem

9. Application of learning and Progression of
Research
*Checklist of concepts that you are applying and showcasing
 Concept of quadratics
 Concept of vertex form
 Concept of circles
 Concept of cubics? Quartics? Conics? Integration (Calculus)?
 Concept of polynomials in degrees P(x)
 Concept of factor theorem
 Personal Inquiry based on curiosities after the parabolic research

10. Curiosities
 Factual
 How do we maximize packaging volume?
 Conceptual
 Will hyperbolas be involved in this investigation?
 Debatable
 3D modelling is the most effective way to present our hypothesis?

11. Background and Abstract
In this investigation, I am to use non-linear algebra in estimating packaging dimensions
that are cost effective and sustainable.
I will use 3 methods in this parabolic investigation. They are:
1) Arithmetic – Define Arithmetic approach in algebra
2) Procedural – Define Procedural approach in algebra
3) Conceptual – Define
Each of these three methods have its merits and is dependent on the availability of
resources. However, efficiency is maximized when we employ the conceptual method as it
minimizes wastage and keeps the process sustainable. XXXXXXXXXXXX* Provide more
insights on the three methods..

12. Progression of research
 Polynomials
 What are polynomials?
 What are not polynomials?
 How are polynomials relevant in this project?
 Xxxxxxxxxx
 xxxxxxxxxx

13. Arithmetic Approach (Concrete Model) Models)
If x = 6,
length of box =
Width of box =
If x = 10,
length of box =
Width of box =
If x = 6,
length of box = 2
Width of box = 2
If x = 6,
length of box = 2
Width of box = 2

14. Investigation 1: Representation of
Polynomials (Degree 1) - Perimeter
Net (Representation 1) 3D Tray (Representation 2)
2(x-4) + 2(y-4) = 80
Provides the same results

15. Investigation 2: Representation of
Polynomials (Degree 2) - Area
Area = (x-4) (y-4)
Since x + y = 40,
Substitute y = 40-x
Area = (x-4) (40-x-4)
Let P(x) be the area,
P(x) = (x-4)(36-x)

16. Maximum Area
Graphical Representation of Area – Relationship between x and the area of the base of
the packaging

17. Investigation 3: Representation of
Polynomials (Degree 3) - Volume
Solution 1:
Completing the square
Solution 2:
- b/2a

18. Procedural Approach

19. Generalization of patterns
 XXXXXXXXXX

23. Personal Inquiry
 Personal Inquiry: How are conic graphs applied in real world contexts?

24. Discoveries
 When a body falls under gravity when it's initial velocity is zero and acceleration is
due to gravity and g remains fairly constant when height h is not large, the
equation reduces to the form h= 1/2gt².

25. Future implications and further
explorations
 How can we maximize packaging to optimize capacity in production through
algebra?
 Cubic, quartic, conic? Future implications

26. Lesh’s Translation Model (Reflection)

27. Bibliography (links) or References (APA7)