Strategies
In each lesson, I will ask my students to write and read numbers, to do calculations, to make comparisons, to do approximations and to make meaningful conclusions. As we work with these numbers, I will visit and revisit the basic facts of base ten notation, and the meaning of scientific notation, and with issues of approximation until my students are comfortable with them, and tell me that I don't need to explain any more.
It would be important to start with a short review the basic structure of base ten notation,
for example that
732 = 700 + 30 + 2
so that 732 is really shorthand for the sum of certain special numbers, like 700, and 30
and 2. Each of these numbers is a digit, meaning 1,2,3, 4,5,6,7,8,9 or 0, times a base ten unit. The base ten units are 1, or 10, or 100 = 10x10, or 1000 = 10x10x10, and so forth.
Each one is the product of a certain number of 10s. The number of zeros in the unit
tell how many tens have been multiplied together to make it. I will call a digit times a base ten unit a single place number. 3
Writing a number explicitly as a sum of single place numbers is called expanded form.
If a number has zero as a digit, then we don't have to explicitly write the term corresponding to the zero. For example
4093 = 4000 + 90 + 2.
No confusion is caused by leaving out 000, because each single place number in this sum tells us how large it is. But when we recompress the number, we need to put in the zero in the hundreds place, or the 4 won't correctly signal that it is representing thousands.
After a brief review like this, I will write a fairly large number on the board, for example
1,988,929,000,000,000,000,000,000,000,000,
which tells the mass of the Sun in kilograms. I will ask my students to write this number in expanded form. I hope they think it is a lot of work! When they complain, I will tell them about a shorter way – exponential notation.
In exponential notation, instead of writing out all the zeros in a base ten unit, you just count how many there are, and put that number at the upper left of 10. For example,
1000 = 10x10x10 = 10 3, and 1,000,000 = 10x10x10x10x10x10x10 = 10 6 and so on.
After this discussion, I will ask them to write the first number I gave them using exponential notation for the base ten units. For the example above, this would be
1x10 3 0 + 9x10 2 9 + 8x10 2 8 + 8 x10 2 7 + 9x10 2 6+2x10 2 5+9x10 2 4 .
This is still some work, but a lot less than writing all the zeros! After doing this, we would work with some other similar numbers. For this particular lesson, I would also give them the mass of Earth and the Moon in kilograms, and their volumes in cubic kilometers, and perhaps the comparable figures for some of the planets.
All lessons will begin with an introductory questions to prepare the students for what they learn that day. This will warm up their brains and get them in the right train of thought. Accordingly, the following questions will be used to introduce two of the lessons: "How long would it take you to count to 1 million? or "How long is a million seconds? 1 billion? 1 trillion?" or "How much would 10 American dollars be worth in Zimbabwe ( 3.33 x 10 - 1 7 = 1 Zimbabwean cent)". These questions and ones like them, encourage students to think about size and how it can be measured with scientific notation. Additionally, they will recognize how significant order of magnitude is when comparing numbers. This will be most evident when the students observe the extreme differences in the answers they got for the introductory question. The teacher will discuss these concepts aloud with the class following a group discussion at their assigned table. The time needed to successfully complete these questions range from 7-10 minutes. Time to expound on the concepts will be given during the lesson through built in supplementary task. In the time given, students should have prepared themselves to achieve the objectives of the day. This should be accomplished by successfully completing each of the given tasks.
In general, all lessons included in this unit will contain a specific theme or objective that guides the entire lesson. As seen above, there are particular themes to be covered in each lesson given to my students. Specifically, I plan to include, order of magnitude, accuracy, rounding, and the notation of numbers as common themes interweaving the lessons together. Overall, my students will witness how these ideas make their lives a whole lot easier when attempting to get a handle on various concepts. These intentions will be made clear in the introductory lesson. They will be asked to write out huge numbers such as the mass of the Sun in kilograms and a light year measured in kilometers. These two numbers will be used because they allow the students to see how these numbers can easily be easily understood through the use of proper notation. This will be a reoccurring theme that will persist throughout the entire unit. It will help to reinforce the importance of scientific notation and how it simplifies their life. These concepts can also serve as a springboard into other topics like accuracy and absolute relative error, enhancing their understanding.
As mentioned before, I would sell each idea on the basis that it simplifies things. To start this discussion, the introductory lesson will tell them that there is an even simpler way to write the mass of the Sun. This will lead to an introduction on the idea of the law of exponents, and negative exponents, where we would factor out 10 3 0, and then write the quotient in standard base ten notation, and there we would have scientific notation. Next I could say that, although scientific notation is really wonderful compared with the regular base ten notation, we would still would like not to have to write out all those digits, so we would approximate our number with a simpler one – like 2x10 3 0. A discussion on the relative sizes of the places will follow, making sure they understand that the leftmost place is largest, and worth more than all the others combined. My students can then begin their introduction into relative error, by replacing the number above by 2x10 3 0, and observing how much error it makes. After computing it, they will be asked if it a big number – they will probably say yes. Then I would say, but we should compare the error to the number we are approximating – is it big compared with the total mass? When they divide the error by 2x10 3 0, they will find that, relatively speaking, the error is quite small. Many problems like this one will be given to the students before they actually receive a formal definition for relative error.
Once they become more familiar with writing these types of numbers, they should have the skills to compare the ratio of the mass of the Sun to the mass of Earth, and the mass of the Moon to the mass of Earth. Next, they would do the same calculations with the approximations they have found. Then, they should verify that the results are good approximations to the computed ratios. Once they have computed the ratios of volume and of masses, they should compare these two, again by division, and conclude that Earth is a lot denser than the Sun, and the Earth is denser than the moon. Then we could have a discussion of why this might be. The eventual understanding of these ideas should lead my students to provide the explanations instead of myself.
Eventually my students should be able to understand relative error since it a highlights a major concept of scientific notation. When using scientific notation numbers are usually expressed within using 1 or more of the first 3 digits. Those digits are then multiplied by a power of 10. To understand this better my students will learn how we determine the leftmost digits to be so accurate. They will start with a number like, 2 x 10 4. In the case of the example above, knowing the 1 st digit of the base 10 number always tells you at least 50% accuracy of the actual value. Therefore, using 1000 to represent a number like 1,484 puts you at least 50% accuracy of knowing the actual value of 1,484. Students should see that by computing the subsequent expression 1000 1,484 2,000 and so 1000/1,484 > 1000/2000 = ½ = .5 x 100= 50%. It ought to be clear that as the first digit of the base 10 number increases so does the accuracy. This will be done multiple comparisons of numbers. This could be followed, by asking the students what happens if more digits are added to the estimation. They should see the more digits added the more dramatic the increase of accuracy. Using the 1 st first two digits of a base 10 number will always put you within at least 90% accuracy of the value (3.4 x 10 5). Following that same pattern, using the 1 st 3 digits of a base 10 number will always tell you 99% of the number. This is illustrated in the following expression:
149,000 149,623 150,000 so 149,000/149,623 > 149,000/150,000 = 149/150 > 99/100 = 99%.
All of this is possible because of the expression (x/x+1) with x representing the digits used to get the estimation. Mastering this concept can become very powerful in proving how practical applying scientific notation can be.
The next technique implemented will provide the students with an introductory question connecting scientific notation to temperature, luminosity, and size in astronomical terms. This will be done with introductory questions that stimulate their curiosity. The type of question asked will encourage them to think about the significance of big and small numbers. For example, the question used to introduce this lesson on temperature, luminosity, and size will have them simply identify the difference in temperature between our sun and another familiar star. This should prepare the students by leading them to think about how size is related to other qualities of a star. Because the qualities of a star can be measured with such huge numbers scientific notation can be used to express these values. This will assist the students with achieving the objectives of the unit.
Once they become comfortable with the concepts they will begin with the application of scientific notation to temperature and its correlation to star color. Students will be provided a Hertzprung-Russell Diagram. A Hertzprung-Russell or HR Diagram is a chart that displays the absolute magnitude (luminosity/brightness) of a star and its connection to its temperature and color. The chart does an excellent job of illustrating the connection that exists between all three characteristics of a star. It also brilliantly demonstrates the vast differences amongst the stars by using scientific notation to represent numerical value. The HR diagram itself will help the students to better understand scientific notation by comparing the different magnitudes to each other. The difference in magnitudes can also be compared to other qualities of star represented by the diagram. Therefore, one example might allow the students to observe how the greater magnitude of a star's luminosity strongly correlates with the size of star. This helps to prove how scientific notation can be used to compare several qualities of a star illuminating another purpose of this specific representation. An example of the HR Diagram can found in the appendices to supplement classroom activities.
Although all the strategies mentioned can serve the purpose of realizing the objectives, the strategy my students will rely on most involves repetition and practice of many problem sets. Provided in the appendices is an example of the types of problems I will give my students. Ultimately, I feel what best serves the students is the consistent practice of the theories discussed throughout this unit. As with any thing in life, the more you work with concepts, the more comfortable they will become to you. Consequently, the students will spend more time practicing the ideas as opposed to the more time spent listening to lectures. In effect, the students will "learn by doing" gaining knowledge from their experiences with success and failure in the unit. After all, practice makes perfect.
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