Connecting Math to Our Lives


Karachi, Pakistan: Local report from academic coordinator Abeer Rizvi from Shahwilayat Public School.


Can we integrate "integration" in our daily lives?

Hi everyone,
I hope that you all are well and enjoying good times. Let me introduce myself. I'm an academic coordinator at Shahwilayat Public School in Karachi, Pakistan. I'm also teaching additional mathematics to o-levels. Nowadays I'm working in an on-line iearn course. One of my course coordinators has asked me to be part of this segment too, so I'm sending my project template for a start.  I welcome your feedback!
Abeer Rizvi

Designing a Problem-Base Unit Template

Teacher: Abeer Rizvi

Problem Name/Title:
       Can we integrate "integration" in our daily lives?

iEARN Project this will contribute to:
 As integration is a new concept for the students at this level and they are a bit reluctant to investigate the actual depth and breadth of the topic. However, the condition is reverse in this case as this project of mine will make it very clear that they have come across the integration in their daily life while learning many of the concepts in physics and advance physics. It will  enable students develop an awareness and understanding of the different huge mysteries of daily life are being resolved by the help of integration.

Teacher's Role:

As the project is going to uncover many different daily life phenomena  so I'll be a mentor throughout.

Student's Role:

Observers, Analyzers, Facilitators and Researchers, Designers.

Time Frame for the Unit:

Approximately  needs 4 -5 weeks.

Meet the Problem:

Well, the thing which is going to give my students a hook surely are my unit questions:
Do you think that "rate of change" is what life all about?
Where can we relate "integration" in our daily lives?
This will motivate them enough to start this project as they are already quite into to
Investigate the application of integration in daily life.

Know/Need to Know: 

 My comments for the KNK are:

Know (K)  Need to Know (NK)
Clear concept of "rate of change" i.e "differentiation".       
Summing up all the small changes gives the new concept of "integretion".
History of "integration".
Different concepts of physics for example, work, Kinetic energy, Potential energy, Thermal expansion, Rockets thrust, impulse and collisions, Rotational inertia, Fluid motion, Thermodynamics.
To relate and  investigate    all these scientific concepts already learnt in their previous grades and to learn how they have been derived using the "integration".
They all are frequent users of computers and internet resources.       
They will need guidance in the keywords for browsing the internet resources.
They are familiar with the softwares like word,power point and publisher.      
They will need mentoring in organizing and designing their projects using all these softwares.
Browsing around the sites they know to handle the data provided at these sites.
They need thorough monitoring and understanding of "what is copyrights?" and "what is fairuse?"

Problem Definition:

 Being the researchers and analyzers, how can we investigate the daily life examples in such a way so that we can relate them with the concept of integration and also how the queries of our previous years which were left unanswered by writing "beyond the scope of the book" can be answered by applying  integration.
Information Gathering & Sharing:

For  gathering the information the students will be utilizing:
Different reference books
Different CD's of physics
Browsing different sites (key words -- guided by the teacher)
Discussion sessions with their mathematics and physics teachers

For sharing the information they will:
Word documents for soft board presentations
Power point presentations
Class Presentations
Reflection forms

Generating Possible Solutions and Determining a Fit:

As I have mentioned earlier this project is not touching some particular problem, but it is uncovering many previous concepts on which the whole physics is based.
At the end the students will be able to relate the following physics concepts with integration:
Work, Kinetic energy, Potential energy, Thermal expansion, Rockets thrust, impulse and collisions, Rotational inertia, Fluid motion, Thermodynamics and some more depending upon the time frame.


1. Students will gain an understanding what is the area under a curve.
2. Be able to develop and understand the "rate of change".
3. Also be able to develop the understand that how these all rates of changes can all be summed up to determine the area under the curve.
4. Understand the integration as a reverse process of differentiation.
5. Investigate different types of integrals i.e indefinite and definite integrals.
6. Students will relate the integration procedures with different daily  life examples.
7. Students will use collaboration skills to convey ideas and create presentations.
8. .Students will use  research skills; basic and intermediate computer skills, navigate resources including, an internet browser, word processing software, CD ROMs, and presentation software.
9. Using right key words for browsing the internet.
10. Students will observe copyright rules.
  Students will develop and promote Higher-Level Thinking skills that is they will be able to recognize, comprehend, apply, analyze, schematize, and assess/ evaluate the different situations provided from time to time by the mentor.
12. Students will be able to design intelligent reflection forms to get the feed back from other group mates. 

Curriculum Compass:
Core Concept
The goal of the project is to enable students by systematic study to acquire the knowledge, skill, and judgment to continue to learn for themselves; to participate intelligently, justly, and responsibly in civic life and to avail themselves resources like internet sites, libraries, multimedia information sources wherever they may live or travel. It lets the students to learn to apply the mathematics in their daily lives.

Curriculum areas covered:
Additional Mathematics
1. Area under the curve.
2. Studying the area under the curve.
3. Rates of changes.
4. Integration
5. Integration as a reverse process of differentiation.
6. Relating integration with the daily life examples.
7. Different techniques of integration.
8. Definite integration and indefinite integration.

Working with the mathematical calculations of:
Work, Kinetic energy, Potential energy, Thermal expansion, Rockets thrust, impulse and collisions, Rotational inertia, Fluid motion, Thermodynamics and some more depending upon the time frame.

Computer Studies:
Working with the softwares
1. Power point
2. Publisher

Embedded Instruction Activities to Meet the Outcomes: 

The activities are necessary to give students the knowledge and skills they need to solve the problem and achieve the learning outcomes:

  What is:             
1. Area under the curve.
2. Studying the area under the curve.
3. Rates of changes.
4. Integration.
  How is:
1. Integration as a reverse of differentiation.
2. Integration related to mathematical calculations of different concepts in physics.

Different groups will work on different fields of physics like Mechanics, Waves,
Oscillations, fluids, Thermodynamics and Eletromagnetism, designing documents, forms, templates, soft board material and class presentations.

Rules for integration
Definite and indefinite integrals
Resources of internet, CD Roms etc.
Solutions to some unique problems of integration.

Class discussions
Penal discussions
Sessions with Physics teacher


The groups will be assessed on the basis of how:
1. Each group will use resources to present its research work i.e soft boards, class presentations.
2. They will also design forms to collect reflections from other groups of the class.
3. The students will develop a plan of action to determine how to get the research completed.
4. The group should be able to summarise the unit.
5. They should also be able to design a Tool kit which should include "terms to understand, Symbols to recognize, Limitations to keep in mind.
6. After completing the research,the way the students will create PowerPoint presentations with partners over their area of expertise.
7. Students will present PowerPoint presentations to the class.
8. They will also design rubrics.
9. Usage of publisher software will give them some extra credit points.

1. Students will be interviewed about their developed concepts.
2. Individual class presentation to be given.
3. MCQ's, quizzes, word problems, conceptual numericals will be given to assess them.
4. Students will also be required to design observation charts during the class presentation of other class mates.
5. Every day participation in the respective group.

Debriefing the Problem/Process:
Yes, as far as I am concerned, debriefing the unit again will help the students to have a contented feeling of completing a task. I am sure that with this project the approach of the students will totally be changed as that will be more motivated and investigating towards very complex learnings. The best thing about this project is that the student will always be ready to uncover the depth and the breadth of the topic and can relate different fields at all stages.
This project can easily be expanded as the concept of integration is also useful in different types of motion e.g Projectile, Vertical, Horizontal, angular, sliding and molecular motions etc.
The whole picture will come up as a summation of all the areas covered of different curriculums and thus developing much better higher level thinking skills.   

Other Concern and/or Notes:

Its my practice that I always train my students to make their time schedules for the whole week, their weekend and then their assessment too.
I'm going to keep a check in their project work that they do work keeping a written organizer with them and working accordingly.
Another thing which is also very important as I have mentioned above that they should be able to design a checklist for their project as they will be designing a Tool kit at the end of the project.