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Closing Deficits Exponentially: Addressing Base Ten & Small Numbers Using Exponents
byTierra Ingram“In an age of exponential change, we need the power of diverse thinking, and we cannot afford to leave any talent untapped.”  Cathy Englebert
Introduction
Do your students become disengaged in class? Have deficits in standards taught in previous years? Are you always trying to find ways to remediate the skills that they struggle with during class time? I can assure you, I experience these problems every year. As summer comes to an end and a new school year is on our horizon, there are several hundred thousand students excited to return to their classrooms. Most students at this time are still excited about school and have deep love for Math Class. They haven’t begun to experience the anxiety that comes from the new concepts that they will soon encounter. Somewhere around third or fourth grade when they're introduced to fractions and decimals, mathematical anxiety is something that many of my students have expressed. For most of my students, this is when they said they either grew to appreciate and love math or cringe and do the bare minimum to get by. Students often fall through the cracks with no real understanding as to what fractions and decimals are and their place in our number system; nor do they truly understand the place value for a decimal fraction. They then continue their young careers as math students being taught to manipulate and apply operations to these numbers and are left clueless as to what the values of these numbers actually are. This curriculum unit will aim to address some of those misunderstandings while exposing them to a new standard of algebraic representation.
This unit will introduce an innovative way to decompose numbers, identify their characteristics, and use Base 10 to introduce the Law of Exponents. Throughout my experience as a teacher, I have seen that many students can often apply the various properties of exponents but, are lacking a solid understanding around what exponents define, especially negative exponents. I'm hoping to use the property of negative exponents to model that negative powers are the reciprocals of positive powers, and also to make connections with the Base 10 model and place value.
As the curriculum progresses, students are often asked to demonstrate their understanding of the properties of exponents by representing them in various forms. These properties of exponents can also be represented in graphical form, tabular form, and written as equations. This is also what students are expected to do after mastering the basic laws of exponents. These types of questions can often include openended and inquirybased questions which require students not only to show their mathematical skills but relate them to a realworld concept and/or situation. My students will be expected to answer a collection of questions that will be written on different levels and designed to assess multiple aspects of content knowledge.
By teaching my students to compare the process of applying the basic exponent rule combined with the negative exponent rule, I aim to broaden their understanding of positive and negative integers, inverse operations, and the value of tiny numbers/decimals on a number line. This will allow my students to solve problems more fluidly, and a number of the mathematical procedures will become innate. Through this process, my students will be able to compare the value of small numbers, which will not only enhance their abilities to make more accurate approximations and estimations as they arise, but also, they’ll be able to describe the relationship between one number and another with more accuracy and meaning. I will be encouraging students to invent new ways of accessing the material & encouraging them to share their thoughts with their peers. Through this level of constructivism in their math classroom, all my students will encounter increasing levels of success, by being able to persevere through outside distractions and make personal connections with the content. Through these experiences’ students will strengthen their knowledge concerning what one number means in relation to another.
Lastly, my unit will provide my students several different ways to show their understanding of what is taught. Especially, I will be requiring them to do a little more writing than their average math class. This writing will permit the unit to be presented through questioning and inquirybased problems. This will allow me to facilitate mathematical discourse around what is being covered. By creating this safe space, I will make my students feel more comfortable, addressing any misunderstandings that they may have or had and allow for new learning to happen. Because dealing with negative exponents becomes more and more important as you progress through high school, students must be given a fair chance to unlearn any bad practices and learn new and fluent ways of manipulating these small numbers.
Demographics
This past academic year I served as the 11th Grade Algebra II, Honors Algebra II, and Honors PreCalculus Math Teacher at Ballou Senior High School (BSHS), where 100% of the school’s population was economically disadvantaged. Before working at BSHS I served as a Middle School Math teacher for 4 years, familiarizing myself with the Common Core State Standards (CCSS) and the subsets of skills needed to promote mathematical progress. This year, I will be transitioning to one of the feeder schools to BSHS and will take on the role of the 8th grade Math/Algebra I teacher at Johnson Middle School (JMS). JMS is located in Ward 8 of the southeast quadrant of Washington, DC. Much like Ballou, Johnson Middle School also serves a 100% economically disadvantaged student population. Of the 252 students that attended last year, 80% of the student population have been reported as being 2 grade levels below in Math. Like many other inner city/urban schools, JMS is plagued by a number of violent crimes, generational trauma, and pockets of poverty due to heavy gentrification across the DC Metropolitan area. JMS has a 38% truancy rate, and when they do attend, most students carry the emotional/physical trauma into the classroom, often causing extreme behaviors in their academic setting. In my efforts to address these issues, I have found great benefits in allowing students to arrive at new mathematical ideas using the knowledge that they’ve gained through their own math and/or realworld objectives.
Objectives
This 4week instructional unit is designed to ground and solidify middle school students’ knowledge of the numerical makeup of decimals and decimal fractions (place value, order of magnitude, significant figures, exponential properties, and scientific notation). It will be introduced in parallel with the Module 1 of the 8th Grade Eureka/EngageNY curriculum. Through varied studentcentered activities such as problem sets, inquirybased activities, comparative analysis practices, using the Standards of Mathematical Practice, and through promoting constructivism in class, students will be able to access the content material from many different learning levels. All of the topics covered in my unit will be designed around the Common Core State Standards (CCSS) as outlined in the Appendices and will follow the pacing guidelines suggested in the DCPS Scope & Sequence. This unit is composed of 8 topics: the 5 stages of place value, place value of a decimal number, order of magnitude, the Law of Exponents (also known as the product rule), and its related rules the quotient rule for exponents, zero property of exponents, negative property of exponents, and the power rule of exponents. Three overall objectives will be used to monitor and measure student growth including:
 Know & apply the 5 stages of place value (which will be reviewed for whole numbers) to efficiently decompose a number written in decimal form.
 Understand and use the Law of Exponents for positive, zero, and negative whole number exponents.
 Apply the Law of Exponents to develop equivalent expressions using the powers of 10.
Unit Content
5 Stages of Place Value (Whole Numbers)
Mathematics is the science and study of numbers; their size, representations, relationships and their applications in the real world and much more. The digit placement and value determine how the numbers will be categorized and measured. All numbers are rooted in the Base 10 system and it is assumed that students are grounded in their ideas around the numerical makeup of whole multidigit numbers. However, they will be introduced to new ideas around that numerical makeup to make new connections with place value and small numbers. Through decomposing a whole number, students are familiarized with the idea of numbers written in base10 notation. This decomposition will allow students to recognize numbers in various ways, while defining the value of each digit in comparison to the others. These ideas can be found in Taking Place Value Seriously, Arithmetic, Estimation, and Algebra.^{1}
It is imperative for students to become acclimated to the process of numerical decomposition. We will consider a number of various examples that will ask students to decompose whole numbers to build fluency around Base10.
Decomposition of a 4digit whole number
5,632 = 5000 + 600 + 30 + 2 (a)
= 5 x 1000 + 6 x 100 + 3 x 10 + 2 x 1 (b)
= 5 x (10 x 10 x10) + 6 x (10 x 10) + 3 x (10) + 2 x (1) (c)
= 5 x 10^{3} + 6 x 10^{2} + 3 x 10^{1} + 2 x 10^{0 }(d)
Justification
This decomposition details the process in identifying the many mathematical structures used to specify the said number. The successive lines of decomposition allow students to see mathematical ideas that have been labeled elementary evolve into concepts that are grounded in algebraic thinking.
Line a: 5000 + 600 + 30 + 2 (Expanded Form/Sum of Numbers)
We will refer to each number in this expanded form as a place value piece.
Line b: 5 x 1000 + 6 x 100 + 3 x 10 + 2 x 1 (Powers of Ten)
Line c: 5 x (10 x 10 x 10) + 6 x (10 x 10) + 3 x (10) + 2 x (1)
These lines that detail how place value pieces have multiplicative structures: each one is the product of a digit with a base ten unit, which is a product of 10s. Line c presents powers of ten being written out explicitly as products.
Line d: 5 x 10^{3} + 6 x 10^{2} + 3 x 10^{1} + 2 x 10^{0 }(Order of Magnitude)
The final step in this decomposition introduces the most important concept in writing numbers using scientific notation and introducing the order of magnitude. For place value pieces, the order of magnitude is essentially the number of zeros used to write out the number, which is the same as the exponent of 10. If you reference Line b, 5 x 1000 = 5 x 10^{3} therefore, the order of magnitude is 3; and because 600 = 6 x 100 = 6 x 10^{2}, the order of magnitude is 2.
The general idea is to build a deeper understanding between the ideas and application of the identity property, multiplicative inverse property and the zero & negative property of exponents. We will begin to explore how these general ideas of decomposing numbers will be used by students to understand the basic Laws of Exponents. We will look at each property closely and define each for number base of 10. In a followup to this unit, I will discuss powers of any nonzero number.
Product Rule for Exponents (The Basic Rule of Exponents)
The powers of 10 stand in a multiplicative relationship to each other. If we multiply one of them by 10, we get the next one:
10^{1} x 10 = 10 x 10 = 10^{2} 
10^{2} x 10 = (10 x 10) x 10 = 10^{3}, 
In general, 10^{m }x 10 = 10^{m}^{+1}, will also be valid and demonstrated with any nonzero integer m. In fact, any product of powers of 10 is another power of 10. For example:
Example 1 (Product Rule with Positive Exponents)
Solution I 
Solution II 
10^{3} × 10^{5} = 10^{(3 + 5)} = 10^{8} 
10^{3} × 10^{5} = (10 × 10 × 10) × (10 × 10 × 10 × 10 × 10) = 10^{8} 
Solution I & II Rationale:
In both these examples, solution I is the formal application of the Product Rule, and solution II is the justification in terms of the definition of powers.
Common Misconceptions about the Product Rule: When traditionally presented as problems students will be tempted to multiply the base as well as the exponent. However, if they are asked to compute as in Solution II, they may see that the base remains the same, only the number of times it is used as a factor (i.e., the exponent) changes.
As this example shows, there is a very simple and pretty relationship between the exponents of the two factors and the exponent of the product. The total number of 10s in the product is just the sum of the number of 10s in the two factors. This is summarized by the Law of Exponents, also known as the
Product rule for exponents: 10^{m} x 10^{n }= 10^{m+n}, for any whole numbers m and n.
Quotient Rule for Exponents
Division, of a larger power by a smaller power, also results in a power, and the number of factors of 10 left in the quotient is the difference between the numbers of factors in the two powers. For example:
Example 2 (Quotient Rule with Positive Exponents)
Solution I 
Solution II 
10^{4} / 10^{2} = 10^{(42)} = 10^{2} 
10^{4} / 10^{2} = (10 x 10 x 10 x 10) / 10 x 10 = 10^{2} 
These relationships are summarized in the Quotient rule for exponents, for whole numbers m and n, with m³ n. The Quotient rule follows from the Product rule, in the form 10^{m} = 10^{n} x 10^{mn}, by dividing by 10^{n}.
Quotient Rule for Exponents 10^{m} ¸ 10^{n} = 10^{mn}
Defining Negative Powers
To understand decimal fractions, we need to work with tenths, hundredths, etc. We naturally wonder if exponents can also apply to these fractional quantities. One of the basic features of fractions is that they allow us to unite multiplication and division in a single operation. For our situation, the relevant fact is that division by 10 is the same thing as multiplication by 1 / 10. Since 10^{m} / 10 = 10^{m}^{1}, this suggests that we should define 1 / 10 = 10^{1}. If we do that, then the product rule for exponents remains true when n = 1. Even better, we can extend this strategy to the reciprocals of all powers of 10. Since we know that (1/m) x (1/n) = 1/mn, it follows that 1/10^{m }= (1/10)^{m}. In other words, all reciprocals of powers of 10 are powers of 1/10. This and the Quotient rule for exponents suggests that we should define
(1/10)^{ m} = 10^{m},
for any whole number m. If we do that, then the Product rule for exponents can be checked to remain valid for all nonzero integers m and n, and the Quotient rule (for positive exponents) gets absorbed into the Product rule for signed numbers. This idea can be seen in the above example of decimal number decomposition and can be used to solidify this basic law of exponents. This generates new ideas and builds connections between the numeric/arithmetic procedural way of thinking to more solid algebraic ideas. In upper level math, the notation of negative exponents is favored to prove the process of multiplicative inverses. Using the negative property of exponents will allow students to get familiar with writing extremely small numbers using scientific notation and will introduce them to new concepts in upper level math classes such as, Algebra I, Algebra II, Differential Calculus, Group Theory, etc. Unlike using multiplication by10 for positive exponents, each negative exponent demonstrates dividing by 10, perhaps several times. By applying division via the negative exponent property, we create, fractional base ten units, as illustrated in the table below.
Example 3
…10^{1}/10 = 10/10 = 1 = 10^{0} > 10^{0}/10 = 1/10 = 10^{1} > 10^{1}/10 = (1/10)/10 = (1/10)/10 = 10^{2} …
Base Ten Units (including fractional ones)
Ten Thousand 
Thousand 
Hundred 
Ten 
One 
Tenths 
Hundredths 
Thousandths 
Ten Thousandths 
10^{4} 
10^{3} 
10^{2} 
10^{1} 
10^{0} 
10^{1} 
10^{2} 
10^{3} 
10^{4} 
10,000/1 
1,000/1 
100/1 
10/1 
1/1 
1/10 
1/100 
1/1,000 
1/1,000 
10,000 
1,000 
100 
10 
1 
0.10 
0.01 
0.001 
0.0001 
(Note: Once you approach one if fractional base ten units need to be represented then a decimal is added to the right of one, so each decimal number can be represented.)
In the above table you can note that if you move to the right of the zero the numerical values are getting smaller and smaller by 10. Each place is only one tenth of the place to the left. In parallel, the exponent of 10 goes down by 1 with each step, moving from positive values, to 0 at 1, to negative values to the right of 1. This is true on both sides of the decimal point. What is special about the right side is that the place just left of the decimal point is the ones place and dividing it by 10 produces a fraction; this continues the further right you go.
Students will explore problems with negative powers and will be encouraged to discover that although the sign of the power/exponent is negative the general rule is still correct.
Example 4 (Product Rule with Negative Exponents)
Solution I 
Solution II 
10^{3} × 10^{5} = 10^{(3+5)} = 10^{8} 
10^{3} × 10^{5} = (1/10 ×1/10 ×1/10) ×(1/10 ×1/10 ×1/10 ×1/10 ×1/10) = ×1/10^{8} = 10^{8} 
We will look at a variety of examples while investigating the quotient rule for exponents, I want to guarantee that my students are introduced to this property that contain negative powers. The next example illustrates how such a problem may be presented.
Example 5 (Quotient Rule with Negative Exponents)
Solution I 
Solution II 
10^{4}/10^{2}= 10^{(4)  (2)} = 10^{2} 
10^{4}/10^{2} = (1/10 × 1/10 × 1/10 × 1/10)/(1/10 × 1/10) = (1/10 × 1/10)/1 = (1/100)/1 = 10^{2} 
Solution I & Solution II Rationale: By implementing the same strategies used in the above examples with positive exponents. There may be increased room for error as we are moving into a very uncomfortable territory with our students as we apply the basic addition operation with negative integers. Using expanded form along with the Base Ten Unit Table can aide in make connections with negative numbers and integer operations.
Defining the Power of Zero
What would happen if we multiplied the following:
10^{n} × (1/10)^{n} = 10^{n}/10^{n} = 1
The Laws of Exponents would want this to be , but on the other hand we know that it is 1 therefore, this tells us that we should define = 1.
Let n = 2; 10^{2} × (1/10)^{2} = 10^{2}/10^{2} = 100/100 = 1 or 10^{2} × (1/10)^{2} = 10^{2}/10^{2} = 10^{22 }= 10^{0} = 1
Over the years I’ve been bombarded with confused students who often wonder how this is true and makes mathematical sense in relation to zero’s normal multiplicative relationships with numbers. I’ve been asked, “How is it 1 for every single base, and not equal to zero? Why and how would it not be zero?” Most people, students included, associate the number zero with, well, zero. The misconception around this is that when you multiply any number by zero your product would simply be zero. But you are not multiplying by zero, you are not multiplying by anything, you are not multiplying at all, you are doing nothing. This is very different from multiplying by zero. You start with 1 and don’t change it, so of course you end up with 1.
So far, we have defined 10^{m} for all nonzero integers m. Could it be possible to define 10^{0} also? Since 0 = m – m, if we want to include zero as an exponent, and also preserve the Law of Exponents, we would have to conclude 10^{0} = 10^{mm }= 10^{m} ¸ 10^{m} = 1. Thus, we define 10^{0} = 1. This definition is forced on us by the desire to preserve the Law of Exponents, and we can check that it does! With the definitions that
10^{m }= (1/10)^{ m} and 10^{0} = 1,
it is easy to check that the Law of Exponents, and its consequence, the Quotient rule for exponents, remain true for all integers. These two equations are sometimes called, the “Negative property of exponents” and the “Zero property of exponents”.
The following visual models should provide a basis for student engagement and grounding, models as such should be used to provide a deeper understanding and begin to make connections between basic elementary and algebraic ideas.
Example 6
A. It is important to note that as the positive exponent of the base is increasing, we continue to multiply the base by itself to satisfy the exponent rule. Each time we multiply by the base, we increase the exponent by 1. Each time we divide by the base, we decrease the exponent by 1. This is true for all integer powers, positive, negative or zero.
B. By applying the rules of the inverse property of multiplication, or simply dividing students should approach applying the exponential rules in reverse to notice the pattern in arriving to the product. The table also supports the idea that 1/10 = 10^{1}.
Example 7: 10^{1 }÷ 10 = 10^{0 } = 10 ÷ 10 = 1
Sample Problem 1:
Select ALL that apply that show the definition on a^{0}, for a positive a.
A. 10^{0} = 1 
B. 10^{1} ÷ 1 = 0 
C. 10^{1} ÷ 10 = 1 
D. 1^{0} = 0 
E. 10^{0 }÷ 1 = 1 
Correct Responses: A, E
Distractors:
B. 10^{1} ÷ 1 = 0 Students who chose this answer may see it as visually appealing because of the exponent of 1 and 0 quotient; but did not apply the division carefully enough to see that the zero property is not being applied. The quotient of two nonzero numbers is never zero. 
C. 10^{1} ÷ 10 = 1 This is the most contentious distractor. Some educators may say that this answer choice is up for debate. However, this equation is true by definition of 10^{1} and division, it is not dependent on the definition of 10^{0}. 
D. 1^{0} = 0 This student simply applied the zero property incorrectly and calculated the incorrect quotient. 
Power Rule for Exponents
Before leaving this discussion of exponents, we will record one more property. We have been discussing the powers of 10, but we can raise any number to a whole number power. If we raise a power of 10 to a power, the result will still be a product of 10s, so it will still be a power of 10, but what power? Look at some examples:
Example 8
Raised to a Positive Power 
Raised to a Negative Power 
(10^{2})^{3} = (10^{2}) × (10^{2}) × (10^{2}) = 10^{6} 
(10^{2})^{2} = 10^{4} = 1/10^{4} 
(10^{2})^{3} = (10^{2}) × (10^{2}) × (10^{2}) = 10^{6} 
(10^{2})^{2} = 10^{4} = 10000 
In fact, if we take any power 10^{m} of 10, then by the Product formula,
(10^{m})^{2} = 10^{m} x 10^{m} = 10 ^{m}^{ + m} = 10^{2m}
From this, we can conclude by similar reasoning that
(10^{m})^{3} = 10^{m} x 10^{m} x 10^{m} = (10^{m} x 10^{m}) x 10^{m} = (10^{m})^{2} x 10^{m} = 10^{2m} x 10^{m} = 10^{2m + m }= 10^{3m}
If we continue like this, we can show the general “Power rule for exponents”: for any integers m and n, we have
(10^{m})^{ n} = 10^{mn}
The Order of Magnitude
Sometimes referred to as decimal fractions, decimals are often seen as a terrifying subject in Math Classes. They stand for small parts of a whole (fractions) and are persistently avoided by students. The term decimals can be simply defined as using 10s, but I think it's crucial to add to that definition that it is using powers of 10 as a fraction denominator. The denominator should signal to students that the numerator will be divided into that many parts (denominator) and will place the numerator in position by the proper decimal place value.
Example: 8/10 = Dividing 8 into 10 equal parts = .8 = eight tenths
One of the common misconceptions and hangups is failing to read 8/10 as eight parts of size one tenth. Some will interpret it as ten parts of size one eighth, giving them a quotient of 1.25. By the 8th grade it is anticipated that students will have mastered this skill and will need little to no remediation on this topic. It might help to allow students use a 4 function or scientific calculator to arrive at the proper calculations. Through the tedious process of decomposing decimal numbers, I hope that their foundational knowledge will be strengthened and will enable them to do these calculations without the use of a calculator.
When discussing decimals and rewriting them using exponential notation, one often needs to state the order of magnitude. It is essential that students understand and familiarize themselves with determining the magnitude of decimal numbers. We will start by reviewing this idea for whole numbers, which have nonnegative orders of magnitude. We will establish some important formal relationships in this situation. Then we will discuss how to extend these ideas to decimals and negative orders of magnitude.
We have
10^{1} = 10.
10^{2} = 10 x 10.
10^{3} = 10 x 10 x 10,
and so forth. The formal name for the superscripts 1, 2, 3 in the lines above is exponent, but we also refer to these numbers as the order of magnitude of the various powers of 10, because they reflect that these numbers have very different sizes: 10^{3} is 10 times bigger than 10^{2}, which is 10 times bigger than 10^{1}. I plan to show my students some number lines to help them see how fast these numbers grow as the exponent/order of magnitude increases., as illustrated below.
Next, we blow up the interval from 0 to 1, and show that the numbers that divide into equal tenths are .1, ,2, .3. . . , .9, and finally 1 at the far right. Then the numbers that divide the interval from 0 to .1 into equal tenths are .01, .02, .03. . . , .9, and then .1 at the right end.
We could blow up the little interval at the right on this number line, to produce a number line that only goes from 0 to .01. Then tic marks dividing this line into ten parts would be labeled .001, .002, up to .009, with .01 at the end. Thus, the relationship between 1, .1, .01 and .001 is parallel to the relationship between 1,000, 100, 10 and 1, or between one million, one hundred thousand, ten thousand and one thousand. Each number in each list is only 1/10 of the number that comes just before it.
The magnitude of a number is a pretty basic concept when determining it from a positive whole number; when you begin to identify the magnitude of a decimal fraction students will run into the obstacle of assuming that simply counting the number of zeros will suffice. However, the direction that the zeros are written in a decimal is quite important and should be taken into consideration when identifying it. In the below example you can observe that we’ve used the same base number however, because the exponent in b is negative therefore it takes on a negative magnitude. I will emphasize that the magnitude can be derived simply by noting the exponent.
Positive Exponent 
Negative Exponent 
a. 8 x 10^{2} = 8 x 100 = 800 Magnitude = 2 
b. 8 x 10^{2} = 8 x 1/100 = 8/100 = 0.08 or 8 x 10^{2} = 8 x 0.01 = 0.08 Magnitude = 2 
5 Stages of Place Value (Decimal Fractions)
For proper decimal fractions, you can use positive powers of (1/10). I would like to extend these ideas so that they also apply to decimal fractions & introduce negative powers. We will begin with the same decomposition practice with decimal fractions, with more emphasis on the relationship between fractions and decimal numbers. It takes on the same procedures as those in the example of whole numbers but offers a new view on the meaning of very small numbers.
.325 = .3 + .02 + .005 (e)
= 3/10 + 2/100 + 5/1000 (f)
= 3/ (10 x 1) + 2/(10 x 10) + 5 (10 x 10 x 10) (g)
= 3/10^{1} + 2/10^{2} + 5/10^{3} (h)
= 3 x10^{1} + 2 x 10^{2} + 5 x 10^{3 }(i)
Line e: .3 + .02 + .005 (Expanded Form/ Sum of Decimals)
Like the previous example this illustrates the expanded form of decimals, the obvious difference is that the decimal point indicates that the addends are all less than 1. This should be a clear indication that the next line should demonstrate how they will be represented using the multiplicative inverse property.
Line f: 3/10 + 2/100 + 5/1000 (Sum of Fractions)
The word decimal means “using tens” and this step presents the relationship between ten and the whole number numerator.
Line g: 3/(10 x 1) + 2/(10 x 10) + 5/(10 x 10 x 10) & Line h: 3/10^{1} + 2/10^{2} + 5/10^{3}
The denominators are powers of ten, which can be written as products or using exponents. These two lines begin to allow the student to see the how the denominators can be written with algebraic concepts.
Line i: 3 x 10^{1} + 2 x 10^{2} + 5 x 10^{3}
This final step is nearly identical to that in Line d. The main difference is the negative exponents. It is important to emphasize that the negative exponent is going to mathematically communicate that your product will be a decimal fraction. It also emphasizes the order of magnitude through counting how many times 10 is multiplied or by simply counting the number of digits in each number. With these decimal places to the right of the decimal point, we are dividing by 10, rather than multiplying.
In the example below, we include the skill of decomposing decimal fractions that include both whole and fractional pieces. Students will be encouraged to use the same procedural practices to decompose this number. This process is critical for allowing students of various learning levels to pick up on the patterns between the digit place value and the exponents/scientific notation.
632.45 = 600 + 30 + 2 +.4 + .05
= 6 x (100) + 3 x (10) + 2 x (1) + 4 x (.1) + 5 x (.01)
= 6 x (10 x 10) + 3 x (10) + 2 x (1) + 4 x (1/10) + 5 x (1/(10 x 10))
= 6 x 10^{2} + 3 x 10^{1} + 2 x 10^{0} + 4 x 10^{1} + 5 x 10^{2}
It is best to put everything in a seamless system that works for whole numbers and fractions. We will explore a few numerical models to solidify the basic rule of the Law of Exponents. We will use the definitions of decimal fractions as a way to solidify the ideas of The Product and Quotient Rule. We will first begin by working with the multiplicative inverse; I will make sure that my students understand the relationship between multiplication and division. Especially, they will need to recognize that multiplying a number by 1/10 is the same as dividing by 10. Below we will define this idea through numerical representations:
Example 9 (Defining Multiplicative Inverse)
4 1/10 = 4/10 = .4 
16 × 1/100 = 16/100 = .16 
125 × 1/1000 = 125/1000 = .125 
Teaching Strategies
Inquiry Based Learning & Activities
Offering class time for students to think, explore, and make connections with mathematics is key to promote active student learning. Students in urban school settings tend to fall off the radar when it comes to taking deep dives into content and many are written off as not having enough to contribute to real learning. Through experience I’ve found that students find more meaning, make stronger connections, and feel more involved in mathematics when using inquirybased strategies. It promotes student inquisitiveness, bolsters student intuition, and offers the opportunity for students to have a voice. Because this unit will require students to utilize some of their prior knowledge and previously learned standards (regardless of mastery or achievement levels), it is critical that when students are introduced to new content that they get the opportunity to experience how the content is associated with previous content.
This unit in particular will seek to strengthen students’ previous experiences with mathematical operations with whole numbers, fractions, decimals, and integers while moving from basic elementary ideas to solid algebraic foundational skills. From previous years of working at Ballou and having large classes with students on varied levels, I found that inquiry activities gave students the ability to experience success in all lessons. Inquiry will be used to introduce the unit as a way to promote student dexterity and give them the opportunity to reform their general ideas about the concepts being introduced. These exercises will vary in level of difficulty, duration, and types of inquiry activities.
Blended Learning Stations
These small stations will offer differentiated methods of gaining knowledge and practicing the Law of Exponents. Students will rotate throughout the class on different programs giving them multiple opportunities to solidify their conceptual understanding, practice applying strategies learned and multiple chances to show mastery. Using the learning stations, I have found great success in breaking up the monotony in traditional teacher led instruction and rote activities. Because the unit will focus on topics that normally cause much confusion in my students, the programs used in the rotations will target students with various learning styles (kinesthetic, auditory, visual). Students will be expected to use a tracker that will be used as a progress monitoring tool and ensure they are spending enough time working through the different unit objectives.
Practice PARCC Problem Sets
Students will be presented with questions that mirror the district selected standardized assessment, PARCC. The questions will be formulated using released questions and common databases to allow students to see taught objectives in ways that they will see them assessed in the future. Students will be asked to organize, analyze, and define the different properties of exponents, with varying degrees of rigor. We will use these questions at various points during the unit to collect formative data on each objective. The collected data will be used to adjust daily lesson plans and to measure overall student achievement. Students will be presented with problem sets that will allow them to dissect and create their own experience with the math.
Error Analysis/Math Talks
In order to challenge students, and promote conceptual knowledge and higher order thinking, students will have the opportunity to critique and analyze the work of other students. A number of problems will be collected and/or created that will serve as a talk piece for math discourse in class. Students will be required to analyze an array of problems that vary in difficulty and skill level. Each problem will be accessible at some point for all students, creating a safe space and environment where are students will feel they can contribute to the class discussion.
Classroom Activities
Find Your Place
Objective: Know & apply the 5 stages of place value reviewed with a whole number to efficiently decompose a number written in decimal form, using the content covered.
Days Covered: 10 Days (5 Days Whole Numbers / 5 Days Decimal Fractions)
Materials: WarmUp Sheet, 36 Digit Numbers
During the first 10 days of school students will be given or select various 36digit whole numbers and decimal fractions. With these numbers I will facilitate the process of decomposing these numbers through questioning. It is important to note that some students may not be able to demonstrate an understanding of the zero exponent in the last line of decomposition. As the unit progresses, and students begin to define the zero power, I will continue to offer examples and push them to use it in the decomposition.
Example: 352 = 300 + 50 + 2
= 3 x 100 + 5 x 10 + 2 x 1
= 3 x 10^{2} + 5 x 10^{1} + 2 x 10^{0}
 What does the 3 represent in context of this number? …5? …2?
 How can we decompose this number that represents what these digits mean using Base 10?
 How can we visually represent 3 Hundreds? 5 Tens? 2 Ones?
 In what way can we show the relationship between the digit and the Base 10 number? How can we use multiplication to help us do that?
 Is there a way we can write 100 using a shorthand algebraic method? What different ways can we represent 100? Can we use exponents?
 How can we now use that to multiply the digit by the power of ten?
Example 2: 424 = 400 + 20 + 4
= 4 x100 + 2 x 10 + 4 x 1
= 4 x 10^{2} + 2 x 10^{1} + 4 x 10^{0}
In this example using the above set of questions will allow students to follow the proper procedures of decomposing this whole number. However, once students are familiar with this process pushing them to compare the relationship of digits in different places will give you a good gauge of if students are learning the true value of each place. For example, asking a student to discuss the relationship between the two 4s in the above example 424, will push students to this off Base 10 multiplicity.
 What is the difference between the 4 in the hundreds place & the 4 in the ones place?
 How can I state that multiplicatively?
These decomposition exercises will allow students to make elementary connections with algebraic processes. This is critical in rooting a deep understanding around the numbers and their makeup. This activity can be facilitated with students of all learning levels and abilities.
The Negative Power of Zero
Objective: Understand and use the Law of Exponents for positive, zero, and negative whole number exponents.
Days covered: 23 Days
Materials: Calculator (if needed), Function Table Sheet (for zero & negative exponents)
I have found great success using this activity. The overall idea is to introduce students to a problem and/or problem set, giving them the answer, and asking them to find their way to the answer. This can be presented in a number of different ways and using a diverse group of problem sets. This also supports the idea around inquirybased learning, relinquishing responsibility, and students taking ownership in solving problems. Giving students the answer allows them to have something to not only work with but something to also work towards. There are a number of ways that you can push student thinking and bolster student discourse in class. The Negative Power of Zero gives the teacher the ability to get a pulse for the classes’ understanding overall and provides methods to collect individual informative student data. As it relates to this unit this would be an excellent way to use comparison model and identify differences in exponents, i.e. positive exponents vs negative exponents.
This activity will be used to allow students to define the zero and negative properties of exponents and formally demonstrate consistency with the definition of the product rule for exponents; how negative exponents are used to show numbers smaller than one, given that the base is larger than 1. This will be practiced through an inquirybased activity where students will be given little to no math context or direction in the beginning. After being separated into small groups of 45 students they will be given a function table sheet (Example 9ab), with exponents written in expanded form. To promote student engagement and reasoning students should be asked to fill in the missing components leaving the exponent with the zero/negative powers alone. After the first part of this task is complete they will then be given an allotted time to define/justify why any number to the power of zero is 1.
Example 9a
2^{0} 
? 
1 
2^{1} 
2 

2 × 2 
4 

2^{3} 
2 × 2 × 2 

2^{4} 
16 

2^{5} 
2 × 2 × 2 × 2 × 2 
Once students are able to define the zero power of exponents, the same activity and procedures will be used to define what happens when a number has a negative power.
Example 9b
2^{3} 
? 
.125 or 1/8 
2^{2} 
? 
.25 or 1/4 
2^{1} 
? 
.5 or 1/2 
2^{0} 
2 ÷ 2 
1 
2^{1} 
2 × 1 
2 
2^{2} 
2 × 2 
4 
What’s the Value? ( < , >, = )
This is a great tool for building rigor, higher order thinking, scaffolding, and promoting the students’ growth mindset. This activity would be an activity that would be best if taught after having discussions and solidifying student understanding on previously addressed standards and topics. It is suggested that students be grouped in either heterogeneous, homogeneous groups, or in groups that is up to the teachers’ discretion, and each group should be given some type of progress monitoring tool. This activity can vary in duration and adjustments can be made based on content objectives and overarching topics. For this particular unit students will be required to compare the quantities of numbers written in various representations. This activity will be scaffolded in a way that will allow students to work towards more difficult problem sets. There will some opportunities to incorporate topics learned from other courses, allowing collaboration cross curricula. It is important to note that these activities will be done using both positive and negative powers; introducing the activities with positive powers will allow students to become familiar with the activity allowing the transition into negative powers more fluid.
Example: Using 2^{10} = 1024, show that 2^{10 }> 10^{3}, (or 2^{20} > 10^{6})
By presenting problems written this way, it allows students the autonomy to reason and arrive at their own conclusions without limitations. It will be essential to allow students multiple opportunities to discuss strategies and collaborate in finding easier ways to compare quantities. I would monitor the progress and success of my students and increase the number of quantities being compared.
Example: Use <, >, = to define each quantity
1,100,000,000 _____1 billion
To efficiently wrap up the other groups shall provide feedback and ask questions while promoting positive classroom culture and foster the Standards of Mathematical Practice.
Real World, Real Costs
The following problems are rooted in students’ knowledge to collect data and being to develop rational relationships between multiple quantities. These examples offer students the opportunity to connect real world quantities with mathematical concepts and make use of procedures structures outlined in this unit. Although by finding the answers to such questions can seem basic and easily attainable, I will require students to answer the same questions multiple ways.
For example, number one can be answered by writing the number out in standard form as a prerequisite skill; after moving through the unit and introducing scientific notation students will be asked to write these figures in scientific notation. By providing my students with questions like these and aligning them to skills taught in this unit, they will begin to make personal connections with the math.
 What is the total population of your school?
 What would be the total cost to purchase each student a computer?
 What is the population of the DC jail?
 What is the total amount spent of healthcare for each prisoner?
 How does the amount in #2 compare to #4?
 What is the cost to purchase basic school supplies for one student ($ 50 avg.)?
 What would be the total cost of a pair of Jordan sneakers ($ 150)?
 What fraction of the cost of supplies is that?
 Paramecium reproduce into 2 parts through a process called binary fission. They split themselves in half creating identical copy of itself.
 Create a table displaying how many paramecia would be present after 8 hours of splitting every hour. Every 1/2 hour.
 Write an equation for the number of amoebas (p) after (h) hours?
 John decided to trace his family create a family tree. He started to wonder how many ancestors he has from his past 8 generations. As he begins to create his family tree, he found that it began to get difficult to figure out by drawing.
 Create a table and equation showing the number of ancestors in each of the 8 generations.
 Write an equation showing the number of ancestors in each of the 8 generations.
 How many ancestors are there in all, in 12 generations.
Sample Problems
1. Which expression is equivalent to 4^{3?}
 3 x 3 x 3 x 3
 4 x 4 x 4
 1 / (4 x 4 x 4)
 (4 x 3) (4 x 3) (4 x 3)
2. Which equation has both 3 and 3 as possible values for x?
 x^{2} = 6
 x^{3} = 12
 x^{2 }= 9
 x^{3} = 27
3. Create & write expressions that are equivalent 4^{9} / 4^{5}?
a. 
b. 
4. Gabe entered the mass in kilograms of 4 substances into a table. His table converted the numbers into scientific notation. List the substances in order from least to greatest.
A. 
1.8 E4 
B. 
3.2 E6 
C. 
7.28 E3 
D. 
6.0 E 2 
5. Milan used his scientific calculator to multiply 3,000, 000,000 x 3,000,000,000. The answer displayed on the screen was 9e+18. Is that correct? Why or why not?
6. Delilah wrote (4^{3})^{2} = 4^{5}. What is her mistake? Correct her mistake by writing it in exponential form.
7. Taylor types 2^{5} words per minute. How many words does she type in 3^{4} minutes?
Teacher & Student Resources
For additional support, there are a number of resources to aid in academic support.
Khan Academy
Khan Academy is a nonprofit educational organization, that serves as a portal for multiple content areas. The math portal offers lesson plans, practice problems, and a detailed scope and sequence to give the learners’ individual learning paths. The website offers personal access to content that can be controlled and managed after user's discretion, while offering the user multiple times and accessing the content. the lessons are scaffold it and our online to previously taught standards.
Clever (DCPS Portal)
Clever is an online portal that serves as a hub for various educational resources purchased by DCPS. Teachers have the option of creating their own content page that can be made accessible to parents, students, and other members of the educational community. Clever is home to some of the following websites/apps that will be used to support content throughout the unit: Edulastic & EdCite (online assessment tools), Office365, Discovery Education, etc.
Appendix
Standards
Common Core State Standards (CCSS)
This unit will include standards from the CCSS throughout this unit. These standards will focus on using the properties of exponents to represent large and small numbers using scientific notation. We will concentrate on understanding the influence of Base 10 and exponents to ensure students can use scientific notation to represent such numbers.
Essential Standards
 MATH.CONTENT.8.EE. A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^{2}× 3^{5} = 3^{3} = 1/3^{3} = 1/27.
Related Standards
 MATH.CONTENT.8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10^{8}and the population of the world as 7 times 10^{9}, and determine that the world population is more than 20 times larger.
 MATH.CONTENT.8.EE. A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology
 MATH.CONTENT.5.NBT. A.1 Recognize that in a multidigit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
 MATH.CONTENT.5.NBT. A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10 and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use wholenumber exponents to denote powers of 10.
 MATH.CONTENT.5.NBT. A.3.A Read and write decimals to thousandths using baseten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
 MATH.CONTENT.5.NBT. A.3.B Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Standards of Mathematical Practice
 Make sense of problems and persevere in solving them
 Reason abstractly and quantitatively
 Construct viable arguments and critique the reasoning of others
 Model with mathematics
 Use appropriate tools strategically
 Attend to precision
 Look for and make use of structure
 Look for and express regularity in repeated reasoning
Bibliography
 Epps, Susanna. Howe, Roger. Taking Place Value Seriously: Arithmetic, Estimation, & Algebra. (2004) :13, 2729.
 Ifrah, Georges. From One to Zero: A Universal History of Numbers. Penguin House, 1985.
 Jackson, Tom. Numbers: How Counting Changed the World. Shelter Harbor Press, 2017.
 Brickwedde, James. Developing Base Ten Understanding: Working with Tens, The Difference Between Numbers, Doubling, Tripling…, Splitting & Scaling Up” (2012,2008): 26.
 Ifrah, Georges. From One to Zero: A Universal History of Numbers. Penguin House, 1985.
 Dietz, Geoffrey. What is So Negative About Negative Exponents? Volume 1Issue 1 (2014): 125
 Battista, Michael. Clements, Douglas. Constructivist Learning and Learning. (2009): 611
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