Unit Content
This unit is subdivided into three sections:
- A Foundational Understanding Approach to Place Value and Base Ten.
- How I Use a Number Line to Help My Students.
- Writing Expressions Using Scientific Notation and Comparing Quantities Using Scientific Notation.
Place Value and Base Ten
One of the most important conceptual understandings my students need is knowing how the system of place value and the base ten works. My students begin to learn about these essential mathematical concepts in the primary grades, but afterwards, they are no longer mentioned in our curriculum. Place value is generally understood as the way in which to determine the value of a number by the position of its digits: for the number 2,794, the 2 is in the thousands place, the 7 is in the hundreds place, the 9 is in the tens place, and the 4 is in the ones place. Although this way of describing place value is conventional, this is only the beginning. Students were taught to see numbers as one whole number: 2,794 comes after 2,793 and before 2,795. This unit can positively impact my students’ number sense by helping them to acquire a foundational understanding of the power of place value. This thought process begins by viewing the number 2,794 not as one number, but as representing a sum composed of special parts within the base ten place value system. Thus,
2,794 = 2,000 + 700 + 90 + 4
The parts, 2000, 700, 90 and 4, are called “place value pieces”1. We will also use this terminology in this unit. In article 2, the authors identify five sequential stages in interpreting a base ten number, and demonstrate a way of unpacking the numbers. The authors also describe a way to think about the number’s actual composition. Each stage presents a more advanced representation of the relationship between each digit and the value it represents in base ten notation. As a visual of processing a number in standard form, consider the number 2,794 written in standard notation:
Stage 1. The number is represented in standard form. Each number represents a specific positional value.
2,794
Stage 2. The number is written in expanded form. Numbers are broken apart with each piece identified by its place value, and assigned to the appropriate place value notation making a unique base ten piece meaning there is only one 2,000 piece, only one 700 piece, and so on. Base ten pieces are arranged in decreasing order, and then added together, resulting in the original standard form number.
2,794 = 2,000 + 700 + 90 + 4
Stage 3. The next step expands the number a second time by the multiplicative structure of each place value piece, to show that each individual digit is multiplied by a specific base ten unit (i.e., 1, 10, 100, 1000, etc.).
2 ×1000 + 7 ×100 + 9 ×10 + 4 ×1
Stage 4. The next step expands the number a third time by expressing each base ten unit as a product of certain number of factors of ten. This prepares my students for the concept of ten to the power of a (whole) number.
2 × (10 ×10 ×10) + 7 × (10 ×10) + 9 ×10 + 4 ×1
Stage 5. The final step uses exponential notation to put the expression in the form of a polynomial. This simplified way of representing powers of ten is a great steppingstone for writing in scientific notation, which comes later on in this unit, but is worth mentioning now. The stage five expression shows that whole numbers can be expressed as a sum of terms, each of which is a digit multiplied by a power of 10.
2 ×103 + 7 ×102 + 9 ×101 + 4 ×100
After practicing several problems using all five stages, I will assess student understanding by posting whole numbers written in standard form and asking them to write the five stages on their whiteboards. Students will have processing time to write their responses, which will then be used as part of a whole class discussion. Students will then be ready for more small group or individual practice with base ten and place value problems using manipulatives including base-ten blocks and a number line. Practice problems will also provide students the opportunity to brainstorm real-life situations and items that represent base ten to show place value. A number line with movable parts will be used as a visual to demonstrate how each place gets bigger or smaller using base ten and give students a more practical sense of what a million looks like. As an extension of this lesson, my students will learn how to interpret decimal fractions on a number line going from 0 to 1 and 0 to 10.
Teaching my students about the importance of each of these five stages will take more time than is usually allowed for this topic. Nevertheless, completely decomposing numbers written in standard form is necessary for them to grasp the power of the base ten place value structure. This may be the first opportunity students have to fully understand what each digit represents. This process will extend and refine the place value and base ten concept for them. It will build algebraic knowledge and prepare my students for future mathematical learning.
How Can Number Lines Help My Students?
A number line is a tool used to represent numbers and their relationship to each other, and can be instrumental in fostering numeracy and operational proficiency. A number line can be used as part of a strategy that helps us teach conceptual content in a concrete way. In this case a number line will be used to show how moving the decimal just one place affects a number’s size. It is important for students to be familiar with number lines prior to the eighth grade. In primary grades my students used number lines for counting, adding, and subtracting whole numbers. Number lines can be used to show positive integers increasing to the right, negative integers decreasing to the left, and showing the placement of fractions and decimals between whole numbers. Number lines are also helpful when learning how to interpret data on graphs. In the upper grades, number lines are typically used to display numerical data in plots, including dot plots, histograms, and box plots. Later on, number lines are used to teach scale, represent inequalities, make coordinate graphs, and help demystify irrational numbers when students are able to locate and graph them. Number lines play an important role in a variety of mathematical stages of learning, as they model a major way we naturally think about all numerical relationships and operations. In this unit, I use the number line to help students visualize place value and base ten.
My students have been overly reliant on symbolic procedures when comparing number size or numbers that contain decimals, which leads to self-doubt and misunderstandings. A common challenge for students when working with decimals is to know when and how to line up the decimals, add or subtract them, or count them. I want students to have some intuitive tool that will help them understand the number by looking at its important features first, rather than being concerned with the placement of decimals. In this case, I want them to visualize where each number belongs on the number line once the decimal is moved either to right. (The number get bigger or smaller? How much does it get bigger?) I notice that my students have several misconceptions about decimals that lead to erroneous thinking. One common misconception is the thought that a longer decimal is a bigger number than a shorter decimal. For instance, some students may think 2.4567 is bigger because it has five numbers, and 4.1 is smaller because it has two numbers. Another misconception, occurs when the decimal point is viewed as separating two distinct whole numbers. For example, 5.9 is seen as a 5 and a 9 instead of 5.9. I want my students to visualize on a number line to understand that 5.9 is 5, and 9 tenths more, between 5 and 6, but closer to 6. A number line is accessible to all students and offers them a way to intuitively know how to arrive at the correct answer. This unit uses the number line to help students develop place value and base ten proficiency by thinking in scale (finding where a certain a number lies on the number line).
I will begin discussing the number line by posting textbook problems and asking students to sketch the number lines in their notebooks. Figure 1 is an example of how I will have my students practice with the number line inside the classroom. Note that there are three number lines, one from 0 to 10, then one from 0 to 100, and a third from 0 to 1,000. I will begin by discussing the number lines, what the ten equal subintervals mean on each line, and what ten equal subdivisions of one of these intervals mean, on each line. I will then ask my students to study the first number line, and then the second one, and then asking guiding questions:
Can someone state where 7.68 lies on the first number line?
What do you notice about the numbers when you compare the two lines?
What do you notice about the intervals?
How do you say each number?
What would the next line look like?
As students move along the number line, they are able to share their thinking with their classmates. The number lines illustrate what is happening to the number when the decimal moves to the right (the number gets larger; precisely, ten times larger). Each number line begins at zero and is segmented by hatch marks evenly spaced apart, then ending at a specific point (i.e., 10, 100, and 1000). I use several differently spaced number lines (0 to 10, 0 to 100, and 0 to 1000) to illustrate the different sized numbers. The goal is for my students to see that a number changes in a big way when the decimal point is moved right: it gets multiplied by 10. For example, I ask my students to notice how multiplying by 10 moves the decimal point to the right, therefore increasing the original number as in Figure 1.
Figure 1.
I will design number lines to fit my students’ needs. For example, students use number lines when working with scaling and proportional relationships. Teacher-generated number lines become a tool I use to assess specific skills as a whole class or individual basis. Asking my students to draw number lines provides a good way to assess students’ comprehension as they demonstrate their thought processes. I include a “human number line,” a hands-on activity, in the Classroom Activities section below as way to help my students visualize and understand the relative size of numbers and to reinforce what is practiced inside the classroom. In addition to writing numbers on the number line, my students also learn to locate and stand in the appropriate position on the number line to demonstrate how numbers get bigger or smaller.
How Can I Make Scientific Notation Relevant to My Students?
The guiding questions for the lesson on scientific notation may also serve as a review, or help introduce some students to the concept for the first time:
What is scientific notation and why do we use it?
What are the components of an expression written in scientific notation?
What do they tell you about the number?
Throughout this lesson, I will pose these guiding questions to students as a means of formative assessment to determine if students are ready to move forward with their learning or if there needs to be some re-teaching prior to moving forward.
Scientific notation is more than just a convenient way of expressing very large or very small numbers. Textbooks typically show this expression for representing scientific notation:
a × 10n
Numbers written in scientific notation reveal a great deal about a number: writing a number in scientific notation highlights its size and how accurately the value is represented. Numbers expressed in scientific notation are presented as a decimal fraction a with non-zero single-digit whole number part, multiplied by some power of 10. The digits in a to the right of the decimal point tell how accurately the number is known.
Numbers greater than 1 are considered large numbers, and are written with a positive power of ten (e.g., 103). Numbers less than 1 are considered small numbers and are written with a negative power of ten (e.g., 10-4). The specific power of 10 indicates just how big or how small the number is. We will only work with positive powers in this unit. We deal with negative powers later in the year.
I want to make sure that my students understand the components of an expression written in scientific notation. Taking the number 2.3 × 102 as an example, the decimal term 2.3 is the called the coefficient a. The absolute value of a is restricted: it should be at least 1, but less than 10.
1≤|2.3|<10
The second term is a power of 10 (which is 102 in this example). I find it helpful to look at the exponential term first because it tells me just how big or small the number is. The exponent indicates how many times to multiply ten by itself. In this case, it tells me that the number will be in the hundreds, and less than 1,000.
The main purpose of the exponent is to tell us the order of magnitude, which is a big deal! The order of magnitude is a way of describing the size of quantities in terms of powers of base ten. For example, an American ant can lift its body thousands of time over, which means it can lift anywhere from 1 to 10 thousand, in units of its weight.
Explanation: 10n = a product of n 10s.
104 = 10 × 10 × 10 × 10 = 4 tens
103 = 10 × 10 × 10 = 3 tens
102 = 10 × 10 = 2 tens
101 = 10 × 1 = 1 ten
100 = 1
Although the thousands value is correct, we need more information to be precise. We need the coefficient to bring us even closer. For the expression 2.3 × 102, I know that I should multiply 10 times 10, which is 100. The coefficient tells me how many hundreds I have, 2.3 times one hundred, which is 2 hundreds and 30. Here are some more examples:
8 ×100 = 8 ×1 = 8
3 ×101 = 3 ×10 = 30
5.1 ×102 = 5.1 ×10 ×10 = 5.1 ×100 = 510
1.9 ×103 = 1.9 ×10 ×10 ×10 = 1.9 ×1000 = 1,900
2.6 × 104 = 2.6 ×10 ×10 ×10 10 = 2.6 ×10000 = 26,000
My students should understand the reasoning in these examples, from our discussion, described above, that moving the decimal point right multiplies a number by 10, and vice versa.
My students need to know automatically that there is implicitly a decimal point just to the right of the ones digit in whole numbers written in standard form. When a number is whole, it has no other parts: 39 is 39 wholes. On the other hand, a decimal fraction such as 39.5 is 39 wholes and 5 tenths.
Here is a practical procedure for converting numbers written in standard form to scientific notation.
1. Insert a decimal point just to the right of the ones place, then move it so that it is just to the right of the first non-zero digit.
Thus, for example, 123,000,000,000 = 123,000,000,000. -> 1.23,000,000,000.
2. Remove the trailing zeroes (if there are any).
Continuing, 1.23,000,000,000 -> 1.23.
3. Count how many places you moved the decimal point in step 1, and make that number the exponent for your base 10. In this, case we moved the decimal 11 places to the left.
So, 1.23 -> 1.23 × 1011.
4. Now write the coefficient 1.23 times 10 to the eleventh power.
Thus, we have found that 123,000,000,000 = 1.23 ×1011.
Examples:
678 = 6.78 ×102 1043 = 1.043 ×103.
These are some of my students’ common errors in performing this process:
- The final answer is not written in correct form because the coefficient is not between 1 and 10.
They write: 34 ×101
Instead of: 3.4 ×102.
- When comparing numbers, they think the larger coefficient signifies the larger number, because it has more digits. They are thinking of standard form for whole numbers, and ignore the power of ten.
Thus, they might claim that 1.2876 ×103 > 1.2 ×105.
Note that the example on the left side of this inequality is not a whole number. I want my students to understand that, for any two numbers written in scientific notation, if one has a larger exponent, it is larger (whatever the coefficients may be).
- They leave out the decimal or place it in the wrong position.
For example, 678 ×106 for 6.78 ×106 , or 105. ×102 for 1.05 ×102.
How Can I Help My Students Compare Expressions Written in Scientific Notation?
Once my students fully understand exponential notation and have practiced using exponents to denote powers of ten, they begin to evaluate and compare exponential expressions.
The guiding questions for writing numbers in standard notation form are meant to promote conversation about background knowledge and serve as a review for students:
What does it mean to write numbers in standard form?
What kind of terms do numbers in standard form and numbers in expanded form contain? When is it best to write numbers in standard form?
As previously mentioned, textbook problems usually refer to the field of science as a focus for lessons on scientific notation, but I want my students to know that scientific notation can also serve practical purposes for everyday life. At this point in the unit, I will have my students start to build upon their foundation with scientific notation while making conjectures about the relative size of numbers. For example, several of my students have video game competitions. We will write those numbers in scientific notation to compare scores on a number line to find out who has the highest score. My students might also compare city populations, movie blockbuster revenues, and the total annual cost of health care in the US, how much that is per person.
Comparing Magnitude
To compare the size and magnitude of numbers written in scientific notion, we first look at the exponents.
Which is larger, a) 2 ×103 or b) 2 ×105?
The answer is b because 105 = 10 ×10 ×10 ×10 ×10 = 100,000
103 = 10 ×10 ×10 = 1,000.
Thus, 2 ×105 > 2 ×103.
We will also compare numbers with different coefficients. For example,
1 ×106 > 9 ×105 , etc.
I will make sure that my students understand that, if two numbers written in scientific notation use different exponents, the one with the larger exponent is larger, no matter what the coefficients are. Only when the exponents are the same, do we look at the coefficients to compare. This may require students to process the number further, since students will have already practiced writing numbers in standard notation and can use this to determine which value is larger. Ultimately students should be able to recognize that when the exponents are the same, the number with the larger coefficient is larger, and also, able to compare coefficients efficiently.
Example 1:
a) 37 ×106 or b) 1.34 ×106
The answer is a because 1.37 is larger than 1.34.
Thus, 1.37 ×106 > 1.34 ×106.
Example 2:
a) 326 × 103 or b) 5.9 × 103
The answer is b because 5.9 is larger than 5.326.
Thus, 5.326 × 103 < 5.9 × 103.
These examples illustrate for my students the rule: to compare two base ten numbers between 1 and 10, compare their digits one place at a time, starting at the ones place, and moving right. The one with the larger digit in the first (leftmost) place where they differ, is the larger number.
We will use the problems below to help us visualize what happens to a number when we multiply it by different powers of 10. We make a conjecture and then simplify each expression and discuss how the numbers change. Our process:
- Notice the exponents.
- 97 ×10
- 97 ×102
- 6.97 × 103
- 6.97 ×104
- Write each number in standard notation form.
- 97 ×10 = 69.7
- 97 ×102 = 697
- 6.97 ×103 = 6,970
- 6.97 ×104 = 69,700
- Discuss what you notice with a partner and share with the class.
- Why would you want to write a number in scientific notation?
- What is your process for ordering numbers written in scientific notation?
- How do you convert a number in standard form to a number in scientific notation?
When the class is able to respond competently to these and similar questions, we will follow up with number talks helps to enrich our mathematical understanding and academic vocabulary.
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