Estimation

Objectives

This unit is about estimation and the benefits of being able to understand large numbers. Most of my students have only a vague idea of large numbers and if asked about one million, they would only think of the number of zeros that are after the one. They have heard of six figures in terms of money but would have a hard time figuring out how far one million inches is or how much time one million seconds is. It is not a common practice to think of distance in inches or to think of time in seconds because we are taught to use the largest unit. This system of using the largest unit is beneficial, but it makes students confused when they are asked to think in different units. It also puts them at a disadvantage when it comes to thinking of large numbers because the larger the unit, the smaller the number. This is one of the reasons that students do not think about large numbers.

I also have a fuzzy idea of large numbers, and I am looking forward to utilizing various strategies to help my students see the magnitude of these numbers and at the same time, gaining a clearer understanding for myself. I do not think of large numbers in terms of powers of ten, and I am pretty sure my do not either students. The idea of these numbers just being approximate digits times power of ten should be easy to understand once the students think about them, but I do not think that they have ever been taught to think in this way. I hope that this unit opens their eyes to the idea and simplifies the whole idea for them

This unit will be taught to students who need one more math class to graduate and are unwilling, or do not have the math background, to take either Calculus or Elementary Functions. They have below level math skills. They have taken Algebra 2, but they have not been recommended by their previous teacher to go on to the higher level class. These students have not had much luck with math in the past and are not very enthusiastic about math. I know that this is a common problem that teachers face, but the advantage is that these students need this class to graduate and they are mature enough to understand this. This does not mean that they are able to do the math; it just means that this is the last opportunity I have to change their ideas about math. I look at this as my responsibility to at least have them rethink math and have them realize that math does serve an important role in their lives.

Most students are willing and even excited to do the work if they are able, and when students are given the tools and know how to use them, the results are usually good. I usually begin the year with a quick review of common ideas and ask students what concepts are still fuzzy to them. The first thing that I hear is that the students cannot do fractions. Their lack of fraction knowledge never fails, so every year I begin with a quick review of the basic rules for adding, subtracting, multiplying, and dividing fractions. I have no problem going over very basic ideas to make my students comfortable. This sets the tone for the students to feel free to say that they do not remember how to change fractions into decimals or they never learned how to convert feet to yards. I have students who do not know how many feet are in a yard, and it is reassuring for the students when I just explain without judging.

I want us all to be on the same page when we begin our lesson on large numbers, and I want the problems to be interesting to the students. This is why I am creating problems that connect to their lives. I hope to engage my students and eventually have the students create some problems themselves. My objective is twofold. First, I want to change their attitudes and perceptions of math. Secondly, I want to show them that they are quite capable of answering interesting questions using math and really big numbers

I work in a Performing Arts High School and my students study various majors. The majors are dance, musical theater, technical theater, instrumental, vocal and art. I hope to have an even distribution of all majors so I can group the students and have them work on problems that pertain directly to their major. For example, I would have the dancers estimate the number of steps in a dance and then find the number for the entire company. Here are several examples of problems designed specifically with these students in mind. How many performances would the company have to perform before they have done one million steps and then one billion steps? I expect the students will be surprised by the difference between one million and one billion. The vocal majors will have to figure the length of a song and find out how long they would have to sing the song before they have sung for one million words and then one billion words. The art majors will have to estimate the number of hairs in a paint brush and then determine the size of a paint brush that has one million hairs and one billion hairs. This will be fun to see because now we are talking about area and the numbers will be different. I will have the instrumental majors estimate the number of hairs in a violin bow and then have them find the length of a bow that has one million hairs and then one billion. The theater majors will choose a play and find the number of times the play would have to be performed consecutively to have spoken one million words and then one billion words. Finally the technical theater majors will be asked to determine the size of a theater that holds one million people and one billion people. In each of these cases, I think the answers will surprise the students.

To help them think about large numbers, we will discuss the concept of grouping. For example, one million is ten groups of one hundred thousand and one billion is ten groups of one hundred million. This is quite a difference. If the students are having a hard time visualizing that, I will break it down further for them. I will ask them to picture one thousand, which should be easy to do and then one thousand more and one thousand more and so on. It will be hard to visualize one thousand groups of one thousand, but this visualization will certainly help them understand how large one million is. I wanted to have a visual and originally planned on using one million inches of yarn and running it through the building. Once I did the math, I realized that this was a very costly and dangerous idea. The conversion of one million inches into feet, then yards, then miles comes out to 15 miles, much to my surprise! Lesson learned! I hope the students have some surprise moments too. I will still have the students do the calculations and come to the same conclusion as I did. That is a lot of yarn, and wrapping it around the building might not be such a good idea, but asking the question who big the yarn would be might be a good question to ask here. Yarn is usually packaged with either weight or length, and either measurement could be used to ask this interesting questions.