Big Numbers, Small Numbers

Content Objectives

Concept #1- Understanding the Relative Size of Numbers

To lay the foundation for this unit I will address the concepts of the relative size of numbers and orders of magnitude. Any number past the thousands place is difficult for my students to conceptually understand. When calculating large numbers, my students operate in a procedural sense without reasoning about the size of the numbers they are dealing with. When they arrive at their answers, the lack the tools and habits to test for reasonableness. It is my aim for students to more readily reason with the magnitude of quantities and think about the relative size before blindly starting a procedure. In order to achieve this, I will direct students to reconsider powers of 10 and the basic structure of our base 10 system. A useful vehicle to think about this is the number line. Consider the following task.

Graph the following populations on the given number line. Be as precise as possible.

Enrollment at Brentano Elementary School: 502

Population of Logan Square Neighborhood: 73,702

Population of Chicago: 2,695,598

Population of the United States: 327,996,618

Students will initially have a difficult time completing this task because of the wide range in magnitude of the given populations. Students must first be able to identify the intervals that divide 1 billion into tenths which would be 100,000,000. Then consider the intervals that split each tenth by another tenth which would be 10,000,000, or one hundredth of 1 billion. Just this act of labeling the intervals is an important opportunity for students to discuss place value and the number of digits. Every time we divide an interval by 10, we are losing a digit. Instead of writing out the full numbers it will be useful to remind students that these numbers can be represented as a power of 10.  One billion can be written as 1 × 10⁹, and 100 million as 1 × 10⁸.

Figure 2

Having students then plot these values on a number line provides the opportunity to think about the relative size of the numbers. The first number line labeled from 0 to 1 billion tells us that the difference between the population of Chicago and the U.S. is massive: the U.S. population is more than 100 times that of Chicago, which in turn is more than 40 times (but less than 100 times) the population of Logan Square. Most students wouldn’t understand before seeing this number line that, even though in the millions, Chicago’s population is almost negligible on a scale of 0 to 1 billion. In order to see where exactly the population of Chicago, Logan Square, and Brentano land on the number line, we need to expand the first of the hundred small intervals and break it up into another hundred intervals, as seen in the second number line in Figure 2. Using the context of a number line will allow students to visualize the degree of difference more concretely than if we were to just discuss the difference by just reading the numbers alone.

Concept #2- Numbers as powers of 10 and laws of exponents

Although my students have spent a great deal of time thinking about place value in earlier grade levels, these basic understandings of our number system are not drawn upon for several years by the time they reach 8^th grade and have considerable holes. Consider the question of how many times bigger is the population of the U.S. compared to Chicago. Students may want to take the two numbers and divide them, but if we consider the magnitude of each number in terms of a power of 10, we can reach a reasonable answer without going through the process of division. See the table below.

Chicago’s population is somewhat less than 3 million people (but more than 2 ½ million). The 3 is in the millions place can be seen as 3 multiplied by 10, 6 times. The U.S. is somewhat more than 300 million, which is 3 multiplied by 10, 8 times. Therefore, the U.S. population 10 x 10 or 100 times larger than Chicago’s. This is an important realization that students rarely make because they have not been exposed to seeing number in terms of powers of 10. A more refined point can be made by paying attention to how the populations were rounded. We know that we rounded up in the case of Chicago’s population and rounded down in the case of the population of the U.S. This means that our determined ratio of 100 is an underestimate. Another prompt to encourage students to pay attention to place value and powers of ten is to give students a prompt like the one below.

Consider the following large number: 1 , 1 1 1 , 1 1 1 , 1 1 1 , 1 1 1

How many times bigger is the 1 in the hundred millions place worth compared to   the 1 in the ten thousands place?

We can see that the 1 in the hundred millions place is worth 100,000,000 or (1 × 10⁸) and the 1 in the ten thousands place is worth 10,000 or (1 × 10⁴). Then we can see that the first 1 is multiplied by 10, 4 more times than the second 1. So it is worth 10⁴ or 10,000 times more! This will be a new way of thinking about numbers, and will be beneficial in our future study of large numbers and laws of exponents.

How many pennies would it take to make $10,000?

In order to solve this problem, students must complete the calculation of 10,000 × 100. To find the product, my students would line up the factors and work the standard algorithm or remember the shortcut of adding the number of total zeroes seen without being able to provide a mathematical argument as to why it works. To remedy this, I will prompt students to break the numbers down by powers of 10 into what I will call place value pieces.

10,000 × 100

= (1 × 10 × 10 × 10 × 10)  × ( 1 × 10 × 10)

= (1 × 10⁴)  × (1 × 10²)

Seeing the two factors as a product of a number of 10’s allows for a logical justification that the final product being the product of the total number of 10s being multiplied. We can then make the connection to the base ten system. Every time we multiply by ten we move to the next place value, therefore if we are multiplying by 10, 6 times, then the final product will be a 1 with 6 digits to the right, namely 1,000,000. Next, I will direct students to write numbers, starting with the numbers just calculated, as a power of 10 to make the numbers more compact and easier to work with. Although this might seem pedantic, it will allow students to notice some important patterns with powers of tens. After giving students enough iterations of multiplying different powers of ten together, I will prompt students to pay attention to the exponents. Essentially they should notice that when multiplying the powers of 10 we can simply add the exponents. Consider another example:

4,000,000 x 2,000

= (4 × 10 × 10 × 10 × 10 × 10 × 10)  × (2 × 10 × 10 × 10)

= (4 × 10⁶)  × (2 × 10³)

Here, by means of the commutative property we can take this in two parts by first seeing that 4 x 2 is 8 and 10⁶ × 10³ is essentially 10 multiplied by itself a total of 9 times or 10⁹. The key aspect of showing the numbers broken up by powers of 10 is that it provides a clear justification for adding the exponents. This is an important step in students understanding the rationale behind this rule that is many times reached by students memorizing a procedure rather than being understood conceptually. Students can then also see that this works inversely for division. See the example below.

4,000,000 ÷ 2,000

= (4 × 10 × 10 × 10 × 10 × 10 × 10) / (2 × 10 × 10 × 10)

= (4 × 10⁶) ÷ (2 × 10³)

= (4 ÷ 2) × 10^(6-3))

= 2,000

By considering the numbers as place value pieces, we can calculate separately 4 ÷ 2 = 2 and then easily see that 10 multiplied by itself 6 times is being divided by 10 multiplied by itself  3 times, leaving the quotient of 10³. After going through this process several times we will generalize and define these laws of exponents in the following terms.

For a positive whole numbers x and when n > m,

x^m ∙ xⁿ = x^m+n

x^m / xⁿ = x^m-n

The 8^th grade Common Core Standards calls for a broader coverage of the laws of exponents including fractional exponents and negative exponents. Although there are more rules, these are the only ones necessary for this unit where the culminating objective is for student to perform operations with scientific notation for large quantities. The other laws of exponents will be referred to later in the year after students are comfortable using the laws stated above with integer powers of 10.

Concept #3- Estimating

After students have a solid grasp of the relative size of numbers and a fresh understanding of seeing numbers in terms of powers of ten, we will cover the topic of estimation. There are two meaningful justifications for estimating numbers that I want students to understand by the end of the unit. The first is that when considering large numbers that we get from measurement, it is difficult to know numbers beyond a couple of digits, due to human error. The other utility of estimating is that it allows us to execute computations with high relative accuracy by rounding numbers to their first 2 or 3 digits.

To illustrate the point that we do not really know most quantities past the first few digits, I will task students with considering certain measurements. I want my students to understand that there is a certain level of error involved with all measurements. Take the population of Chicago for example. The 2010 census reported that there were 2,695,598 people living in Chicago. I will have my students consider the question: Is it possible to know the exact population of a city? My students will discuss factors such as counting errors, people moving in and out of the city, and people being born and dying. Students should reach the conclusion that it would be safer to say that around 2,700,000 people live in Chicago, because in a fairly short period of time the population could fluctuate by thousands of people. Measurements will always have a degree of error. I will be asking students some questions that will involve the surface area of Lake Michigan. A quick search on the internet will give you several different figures for this measurement. See table A.

Table A

There could be a variety of reasons why these figures are different ranging from the technology used to the year that the measurement was taken. At this point we will discuss measurement error and why we should only use digits that we are sure of. In this case we could safely round the number to hundreds place and use 22,400 as the surface area for Lake Michigan, because the tens and ones place varies in all three figures. We can not be sure that they are accurate and therefore are not significant.

I also want students to appreciate that approximating large numbers serves the purpose of making calculations easier. Students should eventually see rounding as a function of efficiency.

How much water in total do Chicagoans consume in a year? Assume that the   average American consumes 58 gallons of water every year.

It is much less daunting to take a calculation like 2,695,598 x 58 and instead multiply 2,700,000 x 60. I expect that this point will be easily received and will most likely be a review for most of my eighth graders. However, a point that I think they have not grasped yet is that these estimated answers are actually fairly accurate. The justification for why we are allowed to round numbers is not covered in the traditional curriculum. In order for students to realize the relatively small impact of rounding numbers, I will have them plot numbers and their respective approximate values on a number line.

Figure 3 illustrates the point that when the number 2,695,598 is rounded to the first two digits that the change in relative value is hardly noticeable when considering such a large number. Even though we changed the value by 4,402 the impact of its location on the number line is hardly detectable when considering a number in the millions. Further, Figure 4 shows the impact of rounding to one digit is still relatively minimal. It will also be important to show that approximating by rounding to the first 2 or 3 digits roughly the same effect on all sizes of numbers. Throughout the unit we will revisit the number line when rounding numbers to justify why we are able to approximate to the first few digits.

After students are convinced that rounding numbers to the first 2 or 3 digits does not cause a big change in the relative location on a number line, I will push the class to further prove that rounding numbers gives us a very good approximation to work with. To achieve this, we will address the concept of percent error. Consider the following problem:

565,000 is rounded to 570,000 and 565 is rounded to 570. Which rounded number is more off from its original value?

With a prompt such as this one, I expect students to have a discussion about which number changed more as a result of rounding. Most students will only consider the amount changed, and will not compare it to the size of the number being changed. Because 565,000 increased by 5,000 to 570,000 and 565 is only increased by 5 to 570, one could argue that 570,000 is more off. My hope is that some students will employ proportional reasoning and take into account how much was rounded in relation to the whole original number. It is important that students take the time to discuss this nuanced point because it is crucial to understanding the concept of rounding. To illustrate the point of how it makes sense to compare the error with the number being approximated I will ask students, if they lost $100, would they be upset? Then I will ask them to suppose they had a million dollars, and lost $100. Would they be as upset? The points raised from these discussions will allow me to provide the formal language and procedure of finding percent error. The formula to find percent can be defined as:

|(A-N)/N| × 100 = Percent Error

A = Approximated number

N = Actual Number

In the case above, when 565,000 is rounded to 570,000 the number is rounded up by 5,000 which is less than 1 % error. Students should then see that when 565 is rounded up by 5 to 570 that there is the exact same percent error. To solidify this point, I will have students execute many iterations of rounding numbers to varying place values to realize that if we keep three, or even two, digits when rounding numbers, we introduce a relatively low percent error. By noticing what happens when numbers are rounded to 1, 2, or 3 places we can state the following general rules.

When rounding to 1 digit there will always be less than 50% error and half of the time less than 10%.

When rounding to 2 digits there will always be less than 10% error and half of the time less than 1%.

When rounding to 3 digits there will always be less than 1% error and half of the time less than 0.1%.

By understanding rounding and the impact on percent error, my aim is that students will understand why rounding is an accurate and useful tool rather than an abstract task that is sometimes recommended by their math teacher. Students are often thrown off by large strings of numbers, and by proving that only the first few digits are important, I hope students will be less daunted by large numbers and approach them with greater confidence.  It also provides a meaningful framework for understanding scientific notation, the next key concept addressed in this unit.

Concept #4- Scientific notation

After students understand why we can round numbers to the first 2 or 3 digits with an acceptable degree of accuracy, I will introduce the concept of scientific notation. In the past I have taught this concept by simply showing students how to convert numbers into scientific notation and vice versa void of any mathematical rationale beyond scientific notation is easier way to write complicated numbers. Scientific notation is certainly a simple way to write and read numbers with many digits, but there is reason as to why we are allowed to use it. We will consider the following problem:

Chicago residents get their drinking water from Lake Michigan. Assuming that a typical American consumes 58 gallons per year, is there enough water in Lake Michigan for all Americans to drink for a year? How long would Lake Michigan last if everyone in the world drank its water?

Lake Michigan has a volume of 1,186 cubic miles. A useful fact that would be provided to students is that cubic mile can hold 1,101,000,000,000 gallons. This can be computed exactly, by computing the number of cubic inches in a cubic mile. This is a massive number that is confusing to read, with so many digits, and hard to reason with when read in standard form. Tasking students to reason with this number would provide the opportunity to consider an easier way to write this number. It will be important here to have students grapple with the questions: Which digit is most important in this number? How can we quickly know how big this number is? Students should come to the conclusion that the first digit tells us how many trillions of gallons there are in a cubic mile. The 12 digits to the right of the 1 tells us the magnitude of the number or how many powers of ten the 1 is multiplied by, in this case 12.  These descriptions lay the framework to understanding that the number can be written in scientific notation as 1.101 x 10¹². Scientific notation expresses a number as a decimal fraction between 1 and 10 multiplied by a power of 10. The factor 1.101 tells us how accurately we know the number, in this case we can be confident up to the billions place. The exponent of 12 indicates the size or magnitude of the number. It will be important to note that it would be sufficient to simplify this number even further and round to just 3 digits, 1.10 x 10¹². This rounded number would only introduce an error of less than 0.1%.

Concept #5- Operating with Scientific Notation

Once students understand the relative size of large numbers, can make approximations by rounding, and express them simply, using scientific notation, they will be ready to perform calculations involving big numbers. This will not be the first time they are attempting to calculate big numbers in this unit but rather the first time they will be expected to demonstrate all of the previously learned concepts to work in an efficient manner. In this section I will discuss the key mechanics of adding, subtracting, dividing and multiplying with scientific notation.

Adding and Subtracting

If you were to stack the 10 tallest buildings in Chicago on top of each other, how   high would they reach in inches?

In the task above we must simply add all of the heights. The sum could be reached in multiple ways. One could efficiently estimate the answer by rounding the heights of Trump International and Willis Tower to 2 × 10⁴ inches and the remaining 8  buildings to  1 × 10⁴  inches. The next step would be to then add all of the heights to get a sum of  12 × 10⁴ inches. Alternatively, students could reach a more exact sum by taking all of the given digits of each height and adding them. A common mistake that students could make here is to add the coefficients of the Water Tower Place and Aqua Building without paying attention to the power of 10. Here, we must first change the placement of the decimal to represent the heights using the same order of 10.  See this adjustment below.

The sum would come to 11.57 × 10⁴ inches. I will make sure that we take note that this is rather close to the sum reached in the first method which used rounded numbers. Overall, when adding or subtracting with scientific notation the main point that should be highlighted for student is that just as we line up numbers by place value when working with numbers in standard from we must ensure that we are adding digits that multiply the same power of 10. 

Multiplying and Dividing

Chicago Public Schools spent a total of 5,460,000,000 dollars in 2017. CPS reports that there were 371,382 student enrolled last school year. How much money did CPS pay per student?

This problem will allow students to see the utility of applying all of the previously learned concepts from this unit. First, we can reason with these numbers better if they are rounded, then converted into scientific notation. The amount of money CPS spent in a year can be represented as 5.46 × 10⁹. Then the number of students enrolled in CPS can be safely rounded to 3.72 × 10⁵ without introducing a significant amount of error. Finally we are left with the calculation (5.46 × 10⁹) ÷ (3.72 × 10⁵). Before calculating for a final answer, I will ask students to figure how big the number will be. Essentially, I will require them to consider powers of 10 before finding a precise number. Before they deal with the coefficients, they should know that the answer is going to be in the tens of thousands. Ultimate we can divide the coefficients 5.46 and 3.72 and then subtract the exponents for the power of 10 to arrive at an answer  of 1.47 × 10⁴. Another strategy that students might choose is to round the figures to (6 ´ 10⁹)  and (4 ´ 10⁵). The quotient of these two numbers would be 1.5 ´ 10⁴, extremely close to our previous answer.