Estimation, Counting and Fermi Problems | | Instructions: | Suggested Lessons or Activities: | Special Features: | Other Related Modules and Useful Links:

Estimation, Counting and Fermi Problems

Be patient! It may take a few seconds to load large number of dots or bubbles in higher levels.

Choose your Level:

(Please be patient! It may take a bit of time to load large number of dots or bubbles)

Level 1 - Numbers (10-20)
Level 2 - Numbers (25-50)
Level 3 - Numbers (50-100)
Level 4 - Numbers (100-250)
Level 5 - Numbers (20-500)
Level 6 - Numbers (500-10,000)

© R. Mirshahi


  • Click the Start button to begin. Estimate how many bubbles there are on the screen. You can click the "eye" button to reveal or hide the actual number of bubbles on the screen at any one time. You can burst the bubbles to count them. As you burst the bubbles, you will see the number of bubbles you have burst right at the mouse pointer. For level 6 you will see a large number of white dots instead of bubbles.

Suggested Lessons or Activities:

  • In these activities, you are presented with a number of bubbles that you have to estimate. Since the bubbles move around, it is hard to count them. Students are encouraged to come up with estimation strategies to arrive at a number close to the actual count. Imagine you want to count a large colony of birds. Since there are thousands of birds and they are constantly moving, how would you estimate the size of the colony that is reasonably close to their actual number? This question can be considered a Fermi Problem. Estimations of these kinds where it is impossible to count or measure what is practically uncountable is very useful in science. How do scientists estimate the number of cells in the human body? How do we count the number of bacteria in a cup of drinking water? How do we estimate the amount of oxygen produced by the Amazon rainforest? How do we estimate the amount of carbon dioxide humans produce each year? These are difficult but important questions that we need to answer.

  • These exercises here are carefully designed so that they are beneficial for a wide variety of ages. Ask the students to come up with strategies for finding estimates that are close to the actual number. Students are encouraged to develop a strategy for estimation that is different from simple guessing (e.g., sampling, creating a grid, etc.).
  • This resource is also great for very young students when they are beginning to count. Encourage them to count out loud as they burst the bubbles. It is also appropriate for much older students who are trying to understand Fermi problems.
  • Since at any one time you can see the number of bubbles you have burst and the ones still remaining on the screen, this resource can be used to model addition and subtraction. For example if you start with 155 bubbles and burst 55, then there must be 100 bubbles left on the screen.

Special Features:

  • full screen
  • 5 different levels, suitable for different ages and abilities
  • Each time you launch the program, it looks different (number, colour, size and position of the bubbles vary each time)
  • attractive bubbles make them so irresistible that you just have to burst them!

Other Related Modules and Useful Links:

  • TBA