Unit 5. Equations and Inequalities.
Summative assessment
MYP : 4 Grade : 9
Criterion D: Applying mathematics in real-life contexts
Level | Level descriptor | Task – specific clarification |
0 | The student does not reach a standard described by any of the descriptors below. | The student have not submitted the project. |
1-2 | The student is able to: i. identify someof the elements of the authentic real-life situation ii.apply mathematical strategies to find a solution to the authentic real-life situation, with limited success. | The student: i. identifies a problem in which linear programming can be used and shows its personal importance. ii. clearly states their objective of the project. |
3-4 | The student is able to: i. identify the relevantelements of the authentic real-life situation ii. select, with some success, adequate mathematical strategies to model the authentic real-life situation iii. apply mathematical strategies to reach a solution to the authentic real-life situation iv.describewhether the solution makes sense in the context of the authentic real-life situation | The student: i. identifies the variables and constraints of the chosen scenario. ii. models the problem with equations and inequalities but makes mistakes. iii. solves the model but makes mistakes. iv. describes the constraints. |
5-6 | The student is able to: i. identify the relevantelements of the authentic real-life situation ii. select adequatemathematical strategies to model the authentic real-life situation iii. apply the selected mathematical strategies to reach a valid solution to the authentic real-life situation iv.explainthe degree of accuracy of the solution v.explainwhether the solution makes sense in the context of the authentic real-life situation. | The student: i. provides a valid solution with the correct answer. ii. provides a graph of the model. iii. describes the solution and its accuracy. iv. explains the application of the solution. |
7-8 | The student is able to: i. identify the relevantelements of the authentic real-life situation ii. select appropriatemathematical strategies to model the authentic real-life situation iii. apply the selected mathematical strategies to reach a correct solution iv.justifythe degree of accuracy of the solution v.justifywhether the solution makes sense in the context of the authentic real-life situation | The student: i. justifies the degree of accuracy of their calculations and answer through error analysis. ii. justifies the application of the solution through providing proof of the application. |
This project is an opportunity for you to apply mathematical concepts to real-world situations, enhancing your problem-solving and critical thinking skills. Linear programming is a powerful tool that mathematicians, engineers, economists, and various other professionals use to make optimal decisions within certain limitations.You will embark on a journey to explore and solve a problem from everyday life using linear programming. This task will require you to think creatively and apply what you've learned in class to identify a scenario where decisions need to be made, considering various constraints and objectives. This is the guideline to this project work.
This guideline will outlook the requirements to your project work, and there will be an example for each section to facilitate your understanding.
You must submit a project report. It will consist of 5 sections:
Introduction
Methodology
Results
Discussion
References
This is the IMRaD (Introduction, Methodology, Results and Discussion) structure and it is widely used in organizing research in social, natural, computer sciences and engineering. You can look through it deeper here.
The project report must be submitted to Google Classroom as a .docx (Microsoft Word) file. Please use Times New Roman 12 or Calibri 12, single-space interval.
You are given two weeksto submit this work. Your deadline is [DEADLINE].
Below are the requirements:
Introduction – choose a real-life scenario in which you want to apply the linear programming principles. Your introduction must consist of:
The description of the backgroundof your problem. This has to include all numerical background specific to the real-life scenario you’ve chosen.
Theobjectivethat you want to achieve.
The explanation of the personal importanceof this project.
Methodology – describe the specific problem you have chosen, including the context and any assumptions you've made. In your methodology, you must:
List and define all the variablesinvolved in your problem.
Present the objective functionyou are trying to maximize or minimize.
Explain what each part of the function represents in your context.
Detail the constraints that apply to your problem. Explain how each constraint affects the variables and the overall situation.
Results – here your main goal is to present the solution. Your results must consist of:
Thesystemof equations and inequalities that model your scenario.
Thegraphthat represents your equations and inequalities.
Theanswer to your problem.
Discussion – in this section, you reflect on your solution and present conclusions. Your discussion must consist of:
What your solution means in the context of your problem. How does it address the issue you set out to solve?
Accuracy justification – if your result is integer, you have to explain and justify why is it integer, and if it is rational, you have to also comment on your number of significant figures.
Reflection. You have to explain which part was the most challenging one, and how did you overcome any problems faced. Provide proofs that you are applying your solutions in real life.
Let’s look at the requirements to your project based on an example:
Suppose Arman is a school student who has a part-time job as promoter. During his free time, he can go to work, but he wants to work as little as possible. He wants to save up money, and his daily expenses are food and transportation. His aim is to maximise his savings.
Introduction:This project explores how linear programming can be used to optimize personal finances, specifically saving towards a goal. I want to save up money to buy a videogame The Witcher 3, which costs 22000 tg on Meloman.kz.To achieve this, I can economise from my daily allowance (1500 tg per day), and work as a promoter (they pay me 1000 tenge per hour). However, to reach to my work, I need to use either taxi (around 800 to 1000 tg) or bus (90 tg). Additionally, I have to spend some money in the school canteen (1000 to 1300 tg). In terms of time, I have 4 hours of free time per day if we exclude doing homework. Traveling by bus to work takes 1 hour each day, and taxi takes only 30 minutes.
In this introduction you can clearly see the objective,personal importance, and the backgroundof the problem with ALL important numerical values.
Methodology:The aim is to maximise daily savings – I will label it as S for “savings”. The numbers that change every day are taxi fares (T changes from 800 to 1000 tg), lunch prices in canteen (L in range -from 1000 to 1300 tg). I can travel by taxi, which will give me the most earnings, but that will mean that I spend a lot of money on taxi. Or I can work less if I travel by bus, and it will cost me very little. I will earn some money from work depending on how much I work (W – hours of work).
In this methodology you can clearly see the variablesandtime constraints.
Results:
My daily savings is the difference between daily incomes and spendings. I have to go to my work and get back from it. I can go by bus or taxi, so: either both times by taxi, or both times by bus, or one time with taxi and one time by bus. The total amount of travels is 2, so if I travel x times by taxi, I will travel times by bus.
My constraints are:
Additionally, I know that total amount of time is limited: I spend 0.5 hours on taxi, 1 hour on bus, and the rest for work. Total free time is at most 4 hours.
Simplifying this inequality will give us:
In this results section, you can see that the logic is provided: the variables are united into inequalities and equations. Full solutionwith all steps must be included. If you use any software including graphical software such as the GDC or Desmos/Geogebra, mathematical software such as the GDC functions or Symbolab/PhotoMath, you have to clearly state so and insert all screenshots with your solutions. The answer has to be clearly stated. This results section is not finished, so you can try to solve this problem yourself.
The discussion section for this example will not be revealed. You can ask guidance on the discussion section during the correction hours.
The references should include any resources you used in this project (apart from this guideline): names of any textbooks, websites, journal articlesorother materials you used to complete the project, including graphical software such as the GDC or Desmos/Geogebra, mathematical software such as the GDC functions or Symbolab/PhotoMath. All references must be completed in APA Reference style (look it up here).
Below are some ideas for you if you struggle to find a relevant scenario in which you can apply linear programming:
1. Personal Budget Optimization
Students can create a linear programming model to optimize their monthly personal budget. They would consider constraints like income (from allowances, part-time jobs, etc.), essential expenses (food, transport, savings), and discretionary spending (entertainment, hobbies). The objective could be to maximize savings or discretionary spending while meeting all essential needs.
2. Nutritional Meal Planning
Students can use linear programming to develop a weekly meal plan that meets nutritional requirements at minimum cost. Constraints include daily calorie intake, minimum requirements of macronutrients (proteins, fats, carbohydrates), and vitamins, while considering budget constraints. The objective function could be to minimize cost or maximize nutritional value.
3. Time Management for Activities
Students can optimize their weekly schedule to balance school work, extracurricular activities, personal hobbies, and relaxation. Constraints could include fixed time commitments (school hours, family time), maximum and minimum hours allocated to each activity, and personal goals (e.g., hours of study, physical activity). The objective might be to maximize personal or academic productivity.
4. Sports Team Management
For students interested in sports, they could use linear programming to manage a sports team effectively. This could involve optimizing the team lineup for the season based on player performance, rest requirements, and injury prevention, with constraints on team composition rules, budget for player acquisition, and player availability.
5. Travel Planning
Students could plan an optimal travel itinerary within a given budget. Constraints include travel and accommodation costs, the maximum number of cities they can visit, minimum and maximum days in each city, and must-see attractions. The objective could be to maximize the number of attractions visited or overall trip enjoyment.
6. School Event Planning
Organizing a school event (like a fair, sports day, or concert) using linear programming to optimize resource allocation. Constraints include budget, available volunteers, equipment, and space, while goals might include maximizing attendance, participant satisfaction, or fundraising.
7. Garden Layout Optimization
For those interested in gardening or sustainability, students could design an optimal garden layout. Constraints might include space, sunlight, water availability, and companion planting principles, with objectives to maximize yield, aesthetic appeal, or biodiversity.
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