Estimation

Strategies

My first strategy will be to name and write the numbers, starting with one and doing powers of ten until we reach ten billion. I want the students to have a solid understanding of grouping because it is important for the understanding of large numbers. We will create a chart that will show this in words, numbers, expanded form and exponents. This will be a great time to discuss the importance of digits and their place value. Placing a large number at the end of a very large digit really won't have much impact on that number. This will be displayed and referred to throughout the unit. This chart will be an introduction or reminder, that the number one thousand is ten groups of one hundred and one million is ten groups of one hundred thousand and so on. This chart will also be a great reference when we use exponents in the future because students can look at the chart to see that one million can be written as ten raised to the power of six and surprise, that is the number of places after the digit! I am going to spend a lot of time talking about the power of ten and making sure that the students can explain large numbers in terms of powers of ten.

I believe that it is important to start small and work my way up, so I will ask questions that will force my students to think of the numbers as powers, and think of each successive power as ten times the previous one. For example, if I ask them how many students can fit in the theater, I will have them start with small numbers. Can ten people fit into the theater? Can ten more groups of ten, or one hundred people fit into the theater? This progression of multiples will be easier for students to visualize than just randomly picking a number. This way, the students will have a point of reference that they can use for all numbers once they begin to think of numbers as powers of ten, one hundred, one thousand and so on. I will continue to ask questions based on the idea of multiples. What if we take the seats out of the theater and we are all standing, then how many students can fit into the theater? How many students can fit in this classroom? Could I put more desks in to fit more students? If I take out all the furniture, how many will fit in now? Where in our school could you put one hundred, one thousand, ten thousand, and one hundred thousand all the way up to one million people? How long a row would one thousand people make? Could they fit inside our school in a straight line? How much area would one million people, in one thousand rows of one thousand ,, fill up? Could they fit inside the school grounds? I am going to reinforce the idea that one hundred is ten groups of ten and one thousand is then ten groups of one hundred and so on. I hope to have the students visualize each multiple of groups. This will result in my students developing a solid grasp of very large numbers.

It will be important to talk about notation to make sure that all the students understand the power rules. I am anticipating many diversions during this lesson because of the level of the students, but I would much rather spend the time up front to clarify things than have to go back, or worse yet, have the students just not understand. Preparation and clarification is always time well spent for any lesson.

I know that all of my students have been introduced to scientific notation but may not be comfortable with the operation of powers. This is when I will regress to teach addition, subtraction, multiplication, and division of exponents. Since Scientific Notation only concentrates on the significant digits and add the place values it should be a easy lesson. Once again, this short refresher lesson will serve my students well because it will remind them of what they already know or show them something that they may not be comfortable with. This is when I will refer back to our chart and using both the powers and expanded form, I will reintroduce the rules. This lesson will assure that we are all on the same page once again.

I will now begin to familiarize my students with the order of magnitude. The term will be unfamiliar to them but the idea is quite simple. The order of magnitude of a number is its largest non zero number single place component, or put simply the number of digits minus one. In the number 5279, the order of magnitude is 3 because 5279 would be 5x10 ³ since the five is the leading digit. The two is less important than the five and the seven is less important than the two and the nine is less important that the seven. This is a great opportunity to talk about the importance of place value because this concept will become very important when we begin estimation of large numbers. If you have one hundred dollars and lose ten, you may notice, but if you have one million dollars and lose ten, the difference is insignificant.

Since there will be many conversions during this lesson, I plan on spending some time going over simple conversions for the same reason as stated above. I had to look up exactly how many feet are in a mile, so I cannot expect my students to know this off the top of their heads. I will introduce them to conversion web sites that can help them, and we will do some problems to check for understanding. I want to make sure that my students have all the tools they need, and it is important to make certain the math is within their capabilities. It is also important to make sure that our calculations are correct and since we will be using a lot of rounded numbers, I will instruct my students on the percent of error. For example, the difference between using Π or 3.14 is about a 4% change in the answer, as is the difference between using 3 instead of 3.14. While 3.14 is accepted for most calculations, it is important to note that there is a difference in answers and to decide if that difference is acceptable.