Estimation

Objectives/Rationale

The hustle and bustle of our high tech world is leaving very little room for our children to estimate. Children are constantly being entertained and stimulated by both active and interactive shows and games. They come to school expecting the same type of programming. When it is not received, the children become bored, have unfinished assignments, and suffer academically. If this attitude starts in the lower grades it will be much worse in the upper grades. It is our job as teachers to find activities that will stimulate their academic needs and get them to become active learners.

Children of all academic levels need to be given time to be wrong, time to think, and time to struggle, with the thought that it is ok to be wrong the first time. If they are to experience mathematics in depth, they must have enough time to become engaged in real mathematical problems. The ideal situation for a teacher is to be able to step back and be come the facilitator. The teacher should learn by listening to what the children have to say. Reading, journal writing and class discussion should be a part of every class.

There is evidence from research that elementary school children do not understand our numeration system, especially place value.(Kamii and Joseph, 1988;Ross,1986, 1989; Smith 1973) According to the research, only 33% of third graders knew that the numeral 16 means 1 ten and 6 ones. But with carefully planned activities of counting and grouping by hundreds, tens and ones, a third grader can be expected to understand and work with three digit and larger numbers. ²

It should be obvious that it is often easier to calculate using estimates rather than using exact figures. It is less obvious that then, even with calculators and computers taking the work out of computation, estimating may make things a lot easier without an important loss in the quality of the answers. In fact, answers derived using shrewd estimates may be more reasonable and more realistic than those that attempt to be exact. ³

In order for a child to be a successful mathematician we as teachers must build a strong foundation in not only the basic skills of adding, subtracting, multiplication and division but also a strong foundation in the use of zero as a place holder in place value, the zero in rounding, and the zero in order of magnitude, thus, using estimation as a tool. Elementary teachers need to give the child all tools needed to be successful. Many children think that a math tool is a calculator, ruler, base 10 blocks, or compass. The children don't realize that tools can be algorithms, formulas or just a firm knowledge of the subject.

The lessons will be created so as to invite all students into mathematics- boys and girls, members of diverse cultural, ethnic, and language groups and students with different strengths and interests. The unit will help the children with their growth in the enjoyment and appreciation of mathematics, develop flexibility and confidence in approaching problems, fluency in using mathematical skills and tools to solve problems and proficiency in evaluating their mathematical thinking. Because third grade is typically a self contained classroom reading and writing skills will also be an emphasis.

This unit will use several stories and songs to reinforce the concepts. Children need an opportunity to reflect upon and explain or justify their ideas and solutions, both orally and in writing. Children's literature provides a context through which mathematical concepts, patterns, problem solving and real-world contexts may be explored. The children's books, focusing on math, makes mathematics readily accessible to all children. Therefore, I will be using a variety of math related books throughout the unit. The class will have written or verbal reflections or responses. In order to present the solution and write mathematically they need to compile their information in an orderly way.

It is important that the children find their own ways of organizing and recording their work. They need to learn how to explain their thinking with both drawings and written words, and how to organize their results so someone else can understand them. For this reason it is suggested that a teacher does not always provide a student recording sheet, for example; if a tally sheet is provided a child will automatically figure out the problem using tallies rather than finding their own strategy. ⁴

The idea of nothingness and emptiness has inspired and puzzled mathematicians, physicists, and even philosophers. What does empty space mean? If the space is empty, does it have any physical meaning or purpose? The use of zero in estimation can be a hard concept to understand. A young child needs constant repetition and a complete understanding of what it means to estimate. The process is more important than an exact answer. Using real-life situations that require estimations will give the children a chance to become comfortable with estimating. An estimate is created by using prior knowledge. Everyone gets better at estimation with practice. From the schoolroom to the supermarket and lots of places in between we must perfect estimation in order to survive.

Estimation starts and ends with zero.

The zero has many responsibilities in math. First and foremost zero will always be the digit before one when counting. It may not be stated when counting but everyone knows that before one of something there was nothing, or zero. The zero has the ability to turn a two, into twenty and then two hundred. Each time a zero is added to make a number bigger it multiplies that number by 10.

Place value enables a person to distinguish between the face value of a digit and its value because of its particular position in a numeral. In teaching place value vocabulary words such as ones, tens, hundreds, thousands, ten-thousands, hundred-thousands etc are required. However, the key point that students need in estimating, especially with front end estimation, is that each place is ten times bigger than the next smaller place.

The order of magnitude is a concept that is used in class but is not made explicit as such in the Elementary and Middle grades. The idea of order of magnitude is based on powers of ten. A power of ten is ten multiplied by itself a certain number of times. A power of ten is written as a 1 followed by some zeros, and the number of zeros tells the power. Thus, the powers of 10 are 1 = 100, 10 = 101, 100 = 102, 1000 = 103, and so forth. The order of magnitude of a number is based on the largest power of ten less than the number. For example: 1000 7, 452 10,000, so the order of magnitude of 7,452 is 3. To simplify this example: the order of magnitude is 3 because there are 3 digits that follow the 7 in 7,452. To make it clear: 100 256 1000, so the order of magnitude of 256 is 2. Simplified: the order of magnitude is 2 because there are 2 digits following the leading 2 in 256.

Using the exponent a ten can be raised to whatever power is required to represent a size. The exponent can be a positive to represent a number larger than one or a negative number to represent a number less than one. Now you might be thinking "Well what does this have to do with the zero?" That is easy! The exponent represents the number of zeros that must be after the 1 in the ten therefore 10^5 is 1 with 5 zeros or 100,000. This concept leads into scientific notation. For example: My savings account has $1.20 x 10^5 ( one dollar and twenty cents times ten to the fifth power). Ten to the fifth power would be a one and five zeros or 100,000. multiplied by $1.20 and I have $120,000 in my savings. (I wish!). If the order of magnitude of my savings account were six, I would have over a million dollars. The order of magnitude can have an infinite amount of zeros with each one representing a multiple of 10. Wow! That zero has a lot of power.

Reys (1986) describes five strategies for estimation: front-end, rounding, clustering, compatible numbers and special numbers. This unit will utilize the front end and leading digit strategies but first we will begin with the zero and place value. The leading digit rounding lesson will coincide with the development of the order of magnitude concept. Base ten blocks will be used to discover how easily one can add and subtract using the expanded form. Base 10 blocks consist of unit cubes (1/2" x 1/2" x 1/2"), ten rods (1/2" x 1/2" x 5"), hundred flats (5" x 5" x 1/2") and thousand cubes (5" x 5" x 5"). These blocks can be used to represent the expanded form to give a visual representation of a number.

The Rule for front-end estimation is: For any number in the calculations: between 1 and 10, round off to the nearest whole number; between 10 and 100, round off to the nearest multiple of 10; for example: choose from 10, 20,30..... 100 , between 100 and 1000, round off to the nearest multiple of 100, 200, 300.….etc. An example of this would be as follows: To add 347 + 129 we first add the hundreds (300 + 100) then the tens (40 + 20) then the ones (7 + 9) to obtain 476. This method is using the expanded form of each number and adding the hundreds, tens, and ones separately. Vertically this problem would look like this:

347

+ 129

300 + 100 = 400

40 + 20 = 60

7 + 9 = 16

476

Mathematicians at Rutgers state that the precise definition of Leading digit estimation is "A reasonable approximation, then, of a multi-digit sum or difference can always be made by considering only the leftmost places and ignoring the others. This strategy is referred to as front end estimation and is the main estimation strategy that many adults use. In third and fourth grades, it should accompany the traditional rounding strategies." ⁵

The leading digit captures most of the number, the next digit captures most of what is left, and after that, the digits are less important, as far as size is concerned. Within the lessons we will discuss the fact that when you add very large numbers the relative size of the number really does not change much when you look at each digit. For example: 1,354,596 + 2,485,439 relatively could be rounded to the nearest million and still be in a ball park figure of estimation. Here we would add 1,000,000 + 2,000,000 » 3,000,000. The importance of zero is very prominent here and should be dwelled upon.