Estimation

Mathematical Background

Assessing Prior Knowledge

Before starting this unit, I would assess to see if students could identify and name place value positions and whole numbers both orally and in written form up through 9999. The purpose of this is twofold; first, in order to evaluate numbers up to the millions, students need a firm grasp of the ordering of the periods. If they understand the "ones, tens, and hundreds" places of the hundreds period, they will be able to apply this knowledge toward understanding the place value names in the subsequent periods. Secondly, students also need to be able to write these numbers correctly in order to work with them accurately. A common error in third and fourth grade numeracy is for students to translate numbers literally instead of using place value notation. For example:

Teacher says: "Six thousand one hundred fifty two"

Student A writes: 6000100502

Conversely, another misconception with this number could present itself as such:

Teacher writes: 6,152

Student B says: "Six one five two"

Student A obviously understands the value of the individual place value components of the number, but needs assistance in writing it correctly. Student B is only understanding the number as a grouping of individual digits without place value. Much of the intent of this unit lies in an attempt to "undo" these misconceptions and redirect those personal understandings towards better fluency with the conventional notations and algorithms that students will encounter on tests and in later grades.

The practice of composing and decomposing numbers, or breaking the standard form of a number into its expanded form is one of the fundamental principles of the base ten system. An error such as one made by Student A provides an excellent opportunity to introduce the expanded form of a base ten number which will form the basis for our work on estimation. Rewriting Student A's response as a sum of four place value amounts shows the relationship between the standard form of a number and its place value components:

6152 = 6000 + 100 + 50 + 2

Each addend is a digit times a unit or a denomination of a given order of magnitude.

6000 = 6 x 1000 100 = 1 x 100

50 = 5 x 10 2 = 2 x 1

These numbers that have only one non-zero digit are the building blocks of the decimal system. We will call them very round numbers. They will form the basis of our approach to estimation. ^a4a

I am assuming that most beginning fourth graders have been introduced to addition, subtraction, and multiplication with multi-digit numbers. However, student mastery of the conventional algorithms will certainly vary widely, and it should be expected that these will need to be reviewed or perhaps discarded in favor of other methods that are more readily understood by the individual student. Although explicit teaching methods of these fall outside the scope of this unit, the concepts addressed here are designed to assist students in strengthening their number concepts in order to develop and assess their own techniques. Again, understanding of expanded form gives students greater flexibility and individual control over their approach to computation.

Comparing Large Numbers

One of the overall goals of this unit is that students grasp the meaning and value of large numbers in order to evaluate their purpose in the context of text and the author's intent in using that particular number. I also hope that students will begin to use mathematical arguments and expressions to strengthen their own research and writing. Exposure to a wide variety of exact and estimated large numbers in context will provide multiple opportunities to evaluate their meaning and purpose. For example, we might question why the author of the EPA statement mentioned earlier decided to use 990,000. What does this number represent? Is there another number close to 990,000 that would be easier to understand? What about one million? What's the difference between 990,000 and one million?

One of the principles of estimation is that the relative error between the actual number and the estimated number is less than ten percent. At the fourth grade level, I would not teach this explicitly, but would nevertheless examine the idea. For example, the difference 10,000 in the above example may seem like a big number by itself, but it is only 1/100 of a million and not very significant when compared to a million. This could be demonstrated visually so that students could see that it would indeed be reasonable to use one million as a replacement for 990,000 in that context.

Place Value and Order of Magnitude

A complete understanding of the fundamentals of place value and order of magnitude is essential to alleviate misconceptions in numeracy, errors in computation and notation, and understanding the measurements represented by numerical expressions. We will make manipulatives with which students may work with order of magnitude concepts visually and kinesthetically. (See Appendix 1). These manipulatives help students solidify their understanding of the above principle; that is, that all numbers are the sum of very round numbers and will improve students' fluency through composing and decomposing numbers. The terms "standard notation" and "expanded notation" will be applied to these exercises as they consistently appear on standardized tests and in many textbooks. These exercises also reveal the gradual increase in the number of digits numbers contain as the order of magnitude increases. Discussion of the role of zero will segue into preparation for estimation activities using very round numbers. To address the ten percent principle mentioned earlier, we will sometimes use pretty round numbers (i.e., numbers with a whole number other than zero in the first two places such as 25,000).

Importance of the Leading Digit and Very Round Numbers

When a number is expressed in its expanded form, i.e.,

7,543 = 7000 + 400 + 50 + 3

the addends or very round numbers in the sum are called very round components or single place components of the number. The largest one is called the leading single place component. ^a5a In this case it is 7000.

Examination of the value of the leading single place component in both arbitrary and very round numbers will help students understand that the leading single place component gives one a great deal of information about the value of the number; in fact, it is always more than half of the number and often quite a bit more than half. This information is crucial in estimating the approximate value of large numbers and the relative size of other numbers in comparison.

Application of Concepts to Computation and Self-Evaluation

At this point it would be appropriate to demonstrate the application of these strategies towards computation. Mastery of multiplication strategies is an essential component of the fourth grade curriculum, and is necessary to compute area and volume. Although mastery of addition is required in earlier grades, its use with large numbers is applied in fourth grade, and continued practice will strengthen those skills for application to division. Focusing on very round components of a number will help me demonstrate that when adding, we always add the very round components of the same order of magnitude. When multiplying, we multiply the very round components of one factor with the very round components of the other and add the products. This process lends itself well to estimating sums and products.

The Four Square model of problem solving requires students to evaluate their solutions for reasonableness, a strategy that is certainly helpful in solving any problem. However, students without explicit instruction will simply say "Yes, it's reasonable because I checked it." if they are not taught specific techniques. For the purpose of this unit, I will not address specific algorithms, as this varies from teacher to teacher. My intent is for students to develop a greater conceptual understanding of operations rather than adherence to a specific procedure. These activities demonstrate the use of estimation strategies as they relate to student self-assessment of their addition and multiplication computation skills.

Measurement

Using multiple units of measurement based on the metric system as well as non-standard units will provide continued practice with different units of measurement, as well as reinforce the idea that any object that can be replicated can be used as a unit of measure. Using conventional metric measurements will reinforce the base ten structure of our place value system. Computing length may require adding numbers with different numbers of digits, giving students the opportunity to practice using order of magnitude and place value knowledge to add correctly. Students computing area problems will demonstrate understanding of multiplication strategies and use order of magnitude with leading digits to evaluate whether solutions are reasonable. Volume problems may be solved by using repeated addition or multiplication. The relationship between area, length and volume will be explored to introduce students to conservation concepts and proportional space. For example, 1000 centimeter cubes in a line will stretch for 10 meters, but if arranged in a plane array, they will fit inside a square with sides of less than 1/3 meter. If they are stacked in a cube formation, they will fit in a cube with sides that are only 10 cm long.

Application of Knowledge to Experimentation and Reporting Techniques

The final portion of the unit demonstrates application of all of these strategies as applied to the "real-world" issue of waste management in the school cafeteria. Students will design their own investigations, experiment, compute, evaluate, and report their findings using both comparative statements and visual aids.