Estimation

CONTENTS OF CURRICULUM UNIT 08.05.08

  1. Unit Guide
  1. Overview
  2. Objectives
  3. Mathematical Background
  4. Strategies
  5. Activities
  6. Teacher's Bibliography
  7. Student Bibliography and Resources
  8. Endnotes

Estimation in Ecology - the Horseshoe Crab Census

Brian Patrick Bell

Published September 2008

Tools for this Unit:

Mathematical Background

Now that you have a better understanding as to why estimating the horseshoe crab population in the Delaware Bay is important to many different groups of people, let's teach the students estimation using the place value system, expanded notation, very round numbers, order of magnitude, and the powers of 10.

Each of the math concepts listed above is directly connected to the others when teaching estimation. If you don't teach your students place value, how can you expect them to understand the relative sizes of different numbers. To understand that one number is 10, 100, or 1000 times larger than the other, the student must have a familiarity with the names within the place value system. How do you teach place value without a discussion leading into expanded notation? Expanded notation often helps students make sense of the place value system by showing them how the large numbers are broken down into smaller numbers, based on the place value system. It is almost impossible to teach expanded notation without a discussion on the concept of base ten. All of the topics are intertwined and each of them must be taught. However, the order in which they are taught may be more of an individual decision. Each school district, and perhaps even schools within the district have their own way, and order, of teaching these concepts, so do not feel obligated to teach them in the order that I have set forth in this unit. As long as the topics are taught and the students are able to use those topics with little teacher assistance, then the students will be able to accurately answer estimation questions.

For this unit, we will begin with a look at the place value system, also called the decimal system. It is important to get your students to realize the decimal system is the place value system because too often when they think of decimals, they only think of the numbers that are to the right of the decimal place. What I find to be the easiest way to begin is to simply ask the students what they remember about the place value system. In each of my classes there have always been a few students who were able to lead a discussion on the organization of the place value system. If you don't have any students volunteer for this, then a good place to start may be asking the students if they know why we write commas in our numbers. This is a good way to introduce the concept of periods in the place value system. You can also use money as a way of explaining the place value system. They always seem to have a good grasp of money related conversations. I like to begin by putting a decimal point on the board and then asking them what place value is in the first place to the left of the decimal. Once they give me the correct answer of ones, I like to organize their notebooks for this lesson. I have them turn their books sideways so the lines are now vertical instead of horizontal. I tell them to write a decimal point somewhere between two of the lines near the top of the page, but to make certain they have at least ten places to the left of the decimal, so we can go out to the billions place. If you want your students to go out further than the billions place, then have them organize their notebooks accordingly. At this time, I have them write the word "ones" in the first column to the left of the decimal. Then I ask what is the place value of the second column, after an answer of tens, I have them write "tens" in the proper column in their notebooks. After we get the correct answer of "hundreds" for the third column, and they write it in their notebooks, I tell them that they have just completed the first "period" of the decimal system. This is when I also explain to them that we write the commas in our numbers to separate the different periods. In their notebooks above the words, hundreds, tens, and ones, I have them write the word "units" to label the first period. We continue in this way out to the billions place. We also label each of the periods, units, thousands, millions, billions. Once we have made the place value system, we practice saying the numbers correctly, starting with the smaller, more familiar numbers, like those in the thousands. When teaching my students to say a large number correctly, we refer back to the periods. I tell them to say the number they see within a period, for example for 523,000 they would say five hundred twenty-three, then they just say the name of the period, thousand. We practice this quite often at this point and we continue it throughout the school year. Once I teach this, I don't want them to forget it. Teaching them to say the number correctly will assist them in writing the number correctly. This will also help them with decimals and fractions as well. I find it to be of great assistance when we begin converting decimals to fractions. At this time, I like to read numbers to them aloud to see if they can write the digits of the number in the correct place values in the chart we made in the notebook. It may be a good idea to make certain your students are using pencil as there may be a bit of erasing as they learn to write the numbers early on. If you don't want them to use the charts they made in their notebooks, create a handout and let them do it on the handout. Once they have are able to write the numbers without too much trouble, or assistance from me, I like to play a game with them. I write the digits 0 through 9 on large index cards, one number per card. I make three sets, using a different color for each set. These are something that you will use quite often in the future, so you may choose to laminate your sets after you create them. The way you play the game will depend on the total number of kids in your class. Here's how I play it with my students. I divide the class into three groups of equal numbers. Then each team is given a pack of the index cards. I call the groups by whatever color they were given, for example the red team, the green team, the purple team. Each student takes one card from the team's pile. Once they have that number, they remain that number unless I tell them otherwise. The game is simple. I call out a number to the class and the first team that gets in the correct order first wins. This is great for getting them out of their seats, working as a team, and practicing a math skill. I play this way for a few rounds, and then I change the rules. For the next round, the teams are not permitted to talk. This helps me determine whether everyone on the team understands the concept, or if there has been a team leader who has surfaced and is getting his teammates into place quickly for the win. The idea is not for one person to lead the team to victory, it's for everyone on the team to actively participate and the no talking rule helps ensure that this is happening. After I teach the right side of the decimal system, I add a decimal to each team. This takes the game to the next level. The decimal person may think he/she has it easy because he/she is just listening for the word "and." Remind the class that the whole number that you are calling out is not simply a matter of listening for "and," but rather the whole number is a combination of the sum from all of the places. You may even choose to leave out the word and to see if they are truly understanding the placement of the decimal. If you want to add yet another level of complexity to the game, once they are playing it well, purposely leave out digits from the numbers that you are calling out. See if they noticed that the person holding the four is not used on this number and therefore should be sitting down. This may cause some confusion on the first round, however, in the subsequent rounds it will actually cause the groups to increase their attention and as a result they will begin to play the game better than before.

Once they have a basic familiarity with the number names, I like to break the numbers down for them into the expanded form of the number. This is critical for getting them to understand that the leading digit of a number is the largest and therefore the most important digit in the number. As we go through this, they learn that the digits to the right of the leading digit are smaller and therefore a less significant part of the number. This is an important concept when we discuss rounding. Because the digits to the right of the leading digit are smaller, when we round a number, we are only changing the overall number by a small amount. If I give them a number, let's say, 4,235,681, I like to break the number down into its base ten expansion (expanded notation):

4,235,681 = 4,000,000 - millions

200,000 - hundred thousands

30,000 - ten thousands

5,000 - thousands

600 - hundreds

80 - tens

1 - ones

When a number is written in expanded notation, I call the individual components very round numbers. By writing the numbers out in their expanded notation, it is easier for the kids to see that place value is simply a way of writing the sum of a group of numbers of a special sort. For example, 4,235,681 is really the sum

4,000,000 + 200,000 + 30,000 + 5,000 + 600 + 80 + 1.

If your students are ready for it, you can also use this opportunity to talk about how these special numbers are written in terms of powers of ten. For example, 4,000,000 is written as 4 x 10 6. I find that they quickly understand the concept of the powers of ten. I start by teaching them 10 can also be written as 10 1. The one tells them how many zeros to write after the one, or the number they are using for a particular problem. For 10 2 I tell them that it simply tells us how many times we multiply 10 to itself. Make certain they understand that they are multiplying 10 by itself, not multiplying 10 times 2. This is a common mistake they make as they are learning the powers of ten. To practice base ten expansion (expanded notation), give them different numbers and ask them to write the numbers out in their base ten expansion. I find that they catch on to this quickly with relatively little assistance.

As the students improve upon their skills with base ten expansion and with converting those numbers to the powers of ten, it is a good time to discuss relative place value. What is meant by relative place value? It actually goes hand-in-hand with the powers of ten. Relative place value is simply the way of explaining how much larger one very round number is than another one. If we go back to place value, any given digit represents units that are ten times larger than the units of the digit to its right. So, as we start in the ones place and work our way to the left (towards the millions) each place value is ten times larger than the digit before it. The tens are ten times larger than the ones place, the hundreds place is ten times larger than the tens place, the thousands place is ten times larger than the hundreds place, and so forth. If we start at the billions place and work our way back toward the decimal, then we can say that each place value to the right of another is 1/10 its value. For instance, the hundred millions are to the right of the billions place, therefore the hundred millions are 1/10 the value of the billions. This holds true for all of the place values going from left to right through the decimal system, including the place values to the right of the decimal point. Make certain your students understand this relationship and that they don't just memorize the place values without understanding the relative place value. It might be useful to illustrate this point with your students in various concrete and pictorial ways, including using the number line to place numbers by means of their decimal expansions. They should also be able to deduce that a movement two places to the left on the place value system would be one hundred times larger than the other digit. Three places would be one thousand times larger, and so forth. To help them understand this concept a little better, I like to use a number such as 8,888,888. By using a large number with all digits the same, I can make certain they understand the relative place value of the numbers as opposed to the absolute value of the number of each digit. Then I can ask them questions such as "Would you rather have a summer vacation that lasts for eight days, or one that lasts for 80 days? Why?" or "Would you rather share a pizza with eight of your friends, or with 80 of your friends? Why?" and with that one I can also include, "What happens to the size of your slice if you share your pizza with 80 friends as opposed to 8 friends?" I am looking for answers that tell me the students understand that 80 is ten times larger than eight, or that the slices would be 1/10 as large with 80 friends. Money is also a good way to discuss this relative size comparison. Starting with a penny, you can ask them how much larger is a dime compared to a penny. They are all well aware that a dime is 10 cents, and therefore, ten times larger than the penny. Then go out to the $1 bill and ask them how much larger is it than the dime. Again, they will quickly realize that it is ten times larger than the dime. Continue in this manner to the $10, $100, $1000 bills. This is a real world example they can relate to in order to understand relative place value. Dr. Howe mentioned that he once read in a newspaper that "$100.00 to a billionaire, is like a dime to a millionaire." That may be something you can use with your students to help them visualize the size difference between these large numbers. As you continue with this exercise, remember to also go from larger to smaller so they can see that when it is getting smaller, the place value is 1/10 the size of the one to its' left. You should skip around to make certain they grasp the idea that a jump of two spaces is one hundred times larger (or smaller) number, and a jump of three place values would be one thousand times larger (or smaller) number no matter which place they are currently at.

This is also the time to discuss order of magnitude. As we increase by 10 we are increasing by one order of magnitude. Remember that to begin with order of magnitude, we must have a certain unit to work with for our problems. In this case, let's say a penny is our unit. Therefore, a penny has an order of magnitude of zero. Your base unit will always have zero magnitude, by definition. And remember that we only increase our order of magnitude when we multiply by 10. Therefore, any pennies from one penny to nine pennies will have an order of magnitude of zero. Ten pennies, or a dime, will have an order of magnitude of one, because a dime is 10x's larger than the penny. So, to get to an order of magnitude of three, we will have to multiply the dime by 10 to get 100 pennies, or one dollar. That means pennies 10 - 99 will have an order of magnitude of one. It is not an order of magnitude two until we reach 100. Again, this is just the definition. It allows us to talk about size in a qualitative and still precise way. As you see here, and hopefully your students will see, that the order of magnitude are bounded by successive powers of ten. An easy way to remember order of magnitude is to take the number of digits in your number and subtract one. 12 For example, 8,421, has an order of magnitude of three. There are four digits, so four minus one is three. Don't just give your students this shortcut without explaining the meaning, otherwise the shortcut will be useless because they won't know why they are getting the answer, or what it means.

OK, so we have covered place value, expanded notation, relative place value, base ten and order of magnitude. Let's put it all together now to complete some estimation. Now you can see that estimation is not a guess, but rather a culmination of all of the other math concepts described thus far. One point you will want to stress with your students and that is that estimation is not a precise answer. When we estimate, we want to try to get an answer that is between powers of 10, or increments of 10, 100, etc. 13 In other words, we first try to find out the order of magnitude of the number. If we can do that, the next job is to try to find out the leading digit of the number or which very round numbers it is between. According to Dr. Howe, a good analogy for the students is to tell them to think of goal posts. A kicker puts the ball between the field goal posts, not on them. We want the students to do the same thing with estimation, they need to get an answer that is between two acceptable powers of 10, or between the right two very round numbers. Sometimes if they are lucky, they may be able to find the next digit, but usually they should not try to expect to do better than knowing the first one or two decimal places. They should not be worried about exact answers or that their answer is different than someone else's in the class. As long as they are both within the same "posts," or they are within the same order of magnitude, then they are essentially in agreement. Again, I stress that this is not about exact answers. The reason for this is what we talked about earlier and that is the importance of the leading digit of a number. We won't take the space to justify it here, but it can be shown that the leading digit of a number alone gives us at least 50% of the whole number, the leading digit, and the next digit combined provide at least 90% of the whole number. 14 The first three digits provide over 99% of the whole number. So what does this mean? Let's take a large number like 12,435,694,392. The 1, the 2, and the 4 provide more than 99% of the size of that number, which means that the 3 must not be too important when we are discussing the accuracy of this number. And the 5 is even less important. In general, the importance of the numbers, as well as the likelihood that they are accurate, continues to decrease as we continue to the right. Why does this matter? Many times when your students are working on estimation problems, they may be dealing with rather large numbers. Don't let them worry about anything past the first two, or perhaps first three, digits of their answers. Let's look at our example number above once more. If a reporter gave us that number about, let's say, the number of dollars spent in the malls in the United States in a year. If the first three digits of a number provide us with 99% accuracy of a number, then can we even be sure about the accuracy of the rest of that large number? So, in truth, do we really need the rest of it? No, and to be as accurate as possible, we should probably leave it off and simply say that, "Malls in the United States made $12,400,000,000 last year." Even "made over $12 billion last year" would convey enough information for most people.

In terms of horseshoe crabs on the beach, this means that, for the estimate on Slaughter Beach (discussed below), we should be more than happy to replace the reported figure of 23,800 with 24,000. For many purposes, it might be satisfactory to report just the first digit: 20,000. We lose some horseshoe crabs here, but if on another beach there are 26,000 and we report that at 30,000, we get them back. The figure 24,000 is so close to the reported estimate (less than 1% more), that we can't be sure that the actual number might not be 24,000, or even a little more.

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