Estimation and Brainstorming Lesson 1

Estimation is a technique that allows you get a reasonable idea of whether a project will work or not without having to calculate exact numbers. It involves using approximate numbers of the correct order of magnitiude. What do we mean by correct order of magnitude?

For example, imagine you are trying figure out how many pumps to install for a village. One pump can provide water for about 100-150 people. Would it matter if whether there were 200 people or 204 people? What about whether there were 200 or 240 people? No – whether there were 200, 204 or 240 people we'd still install 2 pumps.

But does it matter if there are 200 people or 2000 people? Yes! That means we would have to install a lot more pumps. That's because 200 and 240 are the same order of magnitude: for our purpose of building the pumps, they are similar numbers. 200 and 2000 are not similar – they require very different numbers of pumps!

An important aspect of estimation is there there is no “right answer”. For the water pump problem, imagine two students were asked to estimate the number of people in their village – one guesses 204, the other 240. Both are legitimate answers, because they will both give us the right number of pumps to install – so even though the students have different answers, they are both “correct”.

Say that a third student decides to take a survey of the village and finds that there are exactly 235 people living there. Even though he has the exactly 'right' answer, this student will also decide he needs to install two pumps – just like the two students that estimated the number in a few minutes! The third student could have saved a lot of time by estimating the number instead.

If a fourth student guesses 1000 people, however, that would not be a useful answer!

Estimation is a hugely useful tool in engineering, but often schools will not teach it – students are trained to find exactly the right numerical answer to a problem. So it might be difficult for students to grasp at first. While preparing for a lesson, come up with a list of simple estimation exercises that will be relevant to the students in the place you are teaching. The water pump example, for instance, might make sense to students in a rural, not urban area.

How many people living in your village/town?

How many children in an average high school class?

- How long would it take you to run around the school 5 times (it takes me this amount of time to run this distance, 5 times around the school is X times this distance)

What is the combined weight of all of the students in your school? (average student weight this much – gets a bit more complicated if there's a wide age range!)

These are exercises that require a bit more thought an a few more assumptions. The classic example is the “how many beans can fit in a litre bottle?” problem.

This problem can be worked through in several ways: Solution 1

1. What is the approximate size a bean? We can approximate a bean as a similar regular shape – for example, a small cylinder that measures about 2 cm long by about 1.5 cm in diameter.

2. Do the beans “completely fill the liter bottle”? The irregular shape of jelly beans result in them not being tightly packet; approximately 80% of the volume of the bottle is filled

3. The number of jelly beans is the occupied volume of the jar divided by the volume of a single bean Number of beans = (Occupied Volume of Jar)/(Volume of 1 Bean)

4. The volume of one jelly bean is approximated by the volume of a small cylincer 2 cm long and 1.5 cm in diameter Volume of 1 Jelly Bean = h(pi)(d/2)^2 = 2cm x 3 (1.5cm/2)^2 = 27/8 cubic centimers

5. Thus the approximate number of beans in the jar is Number of beans = (.80 x 1000 cubic centimeters)/(27/8 cubic centimeters) = approx 240 jelly beans

Solution 2

1. Imagine a paper cube that measures 1 cubic inch.

2. How many beans will fit in the cube?

3. Approximately 4

4. How many cubic inches are there in 1 liter?

5. 1 inch = approx 2.54 centimeters. Therefore 1 cubic inch = approx. 16 cubic centimeters

6. 1000 cubic centimeters/16 cubic centimeters = approx 62 cubic inches in one liter.

7. How many beans are there in the one liter container? 62 x 4 = approximately 248 beans

A few other good examples of Fermi problems:

How many litres of petrol/gasoline are using in your town/country every year? If you stacked up coins or notes of a certain value (e.g. one dollar bill), how high would be the stack if you had a certain amount (e.g. a billion dollars) How many revolutions will a wheel on school bus make on your trip to school?

Try to make up a few fermi problems before the lesson that are relevant to the students that you are going to teach – mention their own city or village, or things that they do in daily life. (e.g. rural Ghana specific problem – what total weight of fufu do the students in your school pound per year?)

Introduce lesson by talking about “so, you've made solar panels to power this device – now we need to power something larger”. You can use solar panels to power lighting in a building, or machinery, etc etc.

Designing a solar system is complicated. How do we figure out to decide what kind of system we want? How can we be reasonably sure that the system we have chosen will work? We could randomly guess – this would probably be wrong We could calculate exactly the amount of power we'd need – this would take a very long time, and we'd never get it exactly right anyway! Instead, we're going to use 'estimation' – this is half way between the two. It lets us quickly and easily get a reasonable number – not exact, but enough to tell us if we can do the project or not. Have an introduction to estimation, emphasising the idea that two people with different answers can be “right” if they have the same order of magnitude: e.g. water pump example in background notes

Run through some example Fermi problems with the class – either standard ones or ones that you've prepared to be relavant to the students. Start with simple ones (e.g. how much do all of us weigh?) and then, depending on how fast students understand it, work up to more complicated ones.

Can you estimate how much power your solar lamp from the previous lesson uses? There are lots of ways you can do this – have the class discuss how they might do this. Some example approaches (try to let the students figure out these themselves as much as possible!) Starting with.. how much power your panel generates, and how long your lamp can run How much energy the battery holds and again, how long the lamp can run Voltage and current through the LED when the lamp is on

Remind students to use simple numbers – we want the ORDER OF MAGNITUDE – 1 significant digit. As a challenge (and to emphasise this) tell them to do all the math in their head, and not to write anything down unti they have an answer.

Once students/groups have answers, they should share with the class how they got them.

EMPHASIS: There are different ways that are all correct and even though numbers might be different they should all be similar (20 and 40 are similar, 2 and 400 are not). If there is only one way students have used to find the answer, get the class to come up with other ways now. If students have very different answers, as a class look at each of them and see if any incorrect assumptions were made. This is the approach that should be used throughout the lesson.

What is the most important thing about estimation (correct assumptions, NOT precision!)

Next time we're going to use estimation to compare the two systems you came up with earlier – what kind of assumptions should we make?

## Discussion

Lesson Structure, 3. Examples:

Have students find answers in groups, then talk through how they did it with the class. You'll get different answers and be able to see how there are lots of ways to go about something, and get different but valid answers

Some general comments, upon re-reading this:

-Better define estimation in 'Background Info' - we have that it's not about getting an exact right answer, but not what it IS. #2 in lesson schedule covers this some, but should probly be before that…“Estimation is using information that we know (or is relatively easy to find out) and a few very simple calculations to come up with educated guesses on more complex problems”…or something. I just made that up right now

- More concrete definition of “order of magnitude” - what power of tens the number is in (ones/tens/hundreds/etc…)? # of digits? Not sure what the best way of phrasing it is. We get the idea across that 20 and 40 are similar, while 2 and 2000 are not, but that's kind of a fuzzy-edged definition. It's useful, but not necc. what “order of magnitude” will mean in other places. Did that make sense?

-Double time on examples - in DLab this took quite a bit of time, especially when you get to the slightly more complex problems