This study was conducted in an online course environment via Zoom.

Subject: Numbers and Operations

Topic: Ratio and Proportion
Age: 12-13 years

Time: 1 week

Real World Motivation: Mathematics and real life are closely related. It is important to see mathematics in life and to be able to transfer mathematical knowledge to daily events. In this study, students will realize this relationship.

Goals of the lesson: By the end of the lesson the students will be able to

  • understand the link between mathematics and real life
  • make a bicycle by using waste/recycle materials
Mathematicsunderstand the inverse proportionality of two multiples in real lifefind and interpret one of two inversely proportional multiplicities when the other is given solve problems related to inverse proportionalityformulate and solve problems
Scienceunderstand the idea behind a simple machinewill notice the interdisciplinary link between science and  mathcreate a simple machine (bicycle) on their own
Technologysearch the internet for the productuse technology as learning material and assessment material
Engineeringuse materials, tools and ingredients in order to produce a certain product in a correct waydevelop a strategy to make a design
21st Century SkillsInformation literacy: In order to make well-founded judgements about a problem, students find, use and evaluate the necessary mathematical knowledge.   Creativity: Students will develop a bicycle with  wheels of different sizes.   Productivity: In order to develop the product, students will create a plan, implement this plan and use time efficiently.
Inverse proportion Proportionality constant Gear system Pulley system      Problem solving Direct instruction Class discussion Inquiry-based instruction    Computer Zoom application Web tools
Household waste / recycling materials (e.g. wooden stick, plastic bottle, rope) Ruler Scissors Band Glue
Students were asked to carry out project studies in a home environment under the supervision of their parents.


Lemonade Activity (15 minutes)

Based on the recipe given below, you need to prepare lemonade for the whole class.  

(For 4 persons )
Ingredients for Lemonade:
10 lemons
2 glasses of sugar
5 glasses of water
1/2 bunch of mint    

In this case, How is the amount of material you will use?If you wanted the lemonade to taste more sour, how would you make changes in which ingredients?
These question is intended to remind the students’ prior knowledge of direct proportion.It will also make them realize that there is no one way to get a more sour taste, such as reducing the amount of sugar / water or increasing the lemon


Introduction: Guessing Part (15 minutes)

The teacher shows the picture below and asks the students “Why wheels are shaped as circular?”.

Students think, share and discuss the ideas.


The following videos are watched:


Teacher explain:

In order to minimize the friction between road surface and wheel surface, wheels are designed as circular. If wheels are designed as other shape, for example: square, rectangular then the motion of the vehicle will not be smooth and more disturbance will be experienced by the passengers. Because more resistance will be created between surface of road and wheel.

Tractor problem (20 minutes)

The sizes of the front and rear wheels of the tractor are different. Find out how many turns the front wheel and rear wheel will make when this tractor travels 600 meters?

The given information is not sufficient to solve the question. Therefore, students will not be able to solve the question. The teacher guides the students with additional questions.

  • Review the information provided. Is it enough for a solution? If not enough, what information is needed?
  • Is the path taken the same for both wheels?
  • Guess! Which wheel turns less and which one turns more? Why?

These questions are used for brainstorming. Students think, share and discuss their ideas.


Lecture (30 minutes)

Then, the teacher explains the subject and solves the tractor problem in task and more problems.

Inverse proportionality is the relationship between two variables when their product is equal to a constant value. When the value of one variable increases, the other decreases, so their product is unchanged.

For example: Speed and travel time are inversely proportional. The faster you go, the less time it takes to complete your trip.

y is inversely proportional to x when the equation takes the form:

x.y = k      where k is an inverse proportionality constant.

Q: It takes 8 days for 35 laborers to harvest coffee on a plantation. How long will 20 laborers take to harvest coffee on the same plantation?

S: 35 laborers harvest coffee in 8 days. So, duration taken by one worker = (35 × 8) days.

The duration taken by 20 workers = (35 × 8)/20 = 14 days
Therefore, 20 laborers will take 14 days.

Tractor Problem: Both wheels take the same path, but the smaller one makes more turns.

In order to determine the number of turns precisely, the radius or diameter information of the wheels is required.

Circumference of the front wheel X The number of turns of the front wheel = Circumference of the rear wheel X The number of turns of the rear wheel.


This part is planned in two sections.

First section: Drill and Practice (30 minutes)

After the lecture, the teacher projects the questions one by one on the screen, gives time to the students to solve and mark, and  then solves it by explaining.

Depending on the level of students, questions can be taken from the following sources:


Second section: Real Life Examples (30 minutes)

Gear wheel and pulley systems are related to the inverse proportion. In order to better understand the two issues and the relationship between them, the following videos are watched, examples are discussed and questions are solved.




Questions were taken from https://www.eba.gov.tr/


(The presentation phase of the project was done in the online course environment. – 30 minutes)


The video is watched.   Today, bicycles with the same wheel sizes are produced. When we watch the video showing the historical development of the bicycle, it is seen that bicycles with different sizes of wheels have not been successful. You are expected to solve this problem and to make a model bicycle using wheels of different sizes.  

The points to take into account:
* Prepare your design as a model. Don’t forget! It must look like a bicycle.
* You must use two wheels of different sizes.
* Do not forget the fact that there is an inverse proportion between the wheels. The smaller wheel should turn more the biggger wheel should turn less.
* The wheels should turn.
* Try to use the waste materials from your home while preparing the design  


At this stage, students are assisted in the design process. Teacher encourages them to follow these steps:

  • Find ideas
  • Make a plan
  • Create a model
  • Evaluate the model
  • Improve the model


Each student presents the product. Students have the opportunity to examine other designed products. The aim is to make students evaluate their products by comparing them with other products.


Each project is evaluated in two stages: peer evaluation and teacher evaluation.

Peer evaluation done via google form: (The evaluation form has been translated into English.)


 Should be improvedAcceptableVery good
Does the model look like a bicycle?   
Are the wheels in different sizes?   
Are the wheels movable?   
Was waste material used?   
Was the project presentation clear and understandable?   


Students are given a summative assessment to demonstrate what they know and can do.

The inverse proportion exercise containing 10 questions in the eba.gov.tr system is given to the students as homework:



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