Scientific Notation


 

Unit Overview

In this unit, students will be able to:

·        Identify numbers in scientific notation

·        Convert numbers between scientific notation and standard form

·        Perform operations with scientific notation, including word problems

Key Concepts

·        Scientific notation

·        Standard form

·        Calculator form

 

Ohio’s Learning Standards

·        8.EE.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 Χ 108; and the population of the world as 7 Χ 109; and determine that the world population is more than 20 times larger.

Calculators

·        Here is a link to the Desmos scientific calculator that is provided for the Ohio State Test for 8th Grade Mathematics in the spring.

 

·        You are strongly encouraged to use the Texas Instruments TI-30XIIS handheld scientific calculator.  It is extremely user friendly.

o   We will be referring to several functions of this specific

calculator at times in this and other units.

 

 

Scientific Notation

·        A way of expressing a number, particularly extremely large or small numbers commonly seen with science topics

·        A number written in scientific notation must follow this form:

 

            

Example A Set →  Real life science numbers

Mass of Earth:   5,980,000,000,000,000,000,000,000 kg = 5.98 • 1024 kg

                             

Mass of an oxygen atom: 0.00000000000000000000000002657 kg = 2.657 • 10-26 kg

     

Example B Set → numbers not in scientific notation

 

  

 

Converting a number from standard (decimal) form to scientific notation

Example C Set:

1st:  make the coefficient between 1 and 10 by placing the decimal after the first number to the left.  Below, a carrot symbol   is used to represent the new decimal location.

 

 

2nd: deterimine the exponent by counting how many place values the decimal moves from the original decimal point to the new decimal point.

·        Big numbers greater than 1 (where the decimal is moved left) have positive exponents

·        Small numbers less than 1 (where the decimal is moved right) have negative exponents

 

·        Here’s why moving decimal point determines the exponent → you are basically multiplying or dividing by a power of 10

 

 

Converting a number from scientific notation to standard (decimal) form

Move the decimal as many places as the exponent value

·        Positive exponents represent big numbers greater than 1 → move decimal to the right

·        Negative exponents represent small numbers less than 1 → move decimal to the left

 

Example D Set → re-write in standard form (normal #s)

 

8 • 103 → move right 3 places →                 → 8 • 103 = 8,000                 

4.321• 105 → move right 5 places→  → 4.321 • 105 = 432,100       

1.5 x 10-2 → move left 2 places →   → 1.5 x 10-2 = 0.015  

8.321 x 10-4ΰ move left 4 places → → 8.321 x 10-4 = 0.0008321

 

 

TI-30XIIS handheld scientific calculator has a great tool for scientific notation.

 

 

Click on the words SCIENTIFIC NOTATION for further explanation:

 

 

Let’s Practice:

 

 

 

Operations with Scientific Notation

·        Carefully use your scientific calculator to work out. 

·        Use parentheses as needed. 

·        Be sure to give the answer in the number form that is requested (scientific notation or standard form)

 

 

Example E Set →

·        2.3 • 105 + 2.5 • 102   = 2.3025 • 105 = 230,250           

·        8.4 • 105 – 5.56 • 104 = 7.844 • 105 = 784,400                         

·        (4 • 107)(8 • 10-2)        = 3.2 • 106 = 3,200,000  

·        (3.6 • 10-2)3                       = 4.6656 • 10-5 = 0.000046656

 

Example F Set → Fractions and division can be tricky

·                     

·        A mistake is to enter this problem in your calculator as 6•4/2•3. 

o   The calculator follows the order of operations (multiply or divide from left to right):

 

§  6•4/2•3 = 24/2•3=12•3=36

 

·        To do enter this fraction correctly in your calculator, you either need to

o   work out the numerator and denominator separately, then divide those answers (see original problem above)

o   OR put both the numerator and denominator in separate parentheses

 

§  (6•4) / (2•3) =24/6 = 4

 

·        The same principle of using parentheses applies when scientific notation numbers are divided using the calculator.

 

o     (2.8 x 10^6) / (2 x 10^2) = 1.4 x 104

o   If you don’t use ( ) with this problem, you will get an incorrect answer: 1.4 x 108

 

Calculator Notation

·        This is an optional short-cut for typing numbers in scientific notation

o   There are less buttons to push

o   You do not ever need to use parentheses for numbers type in calculator form

2.5 • 102 = 2.5E2   

4.321 x 10-7= 4.321E-7

 

Example E and F Set re-written in calculator form

·        2.3 • 105 + 2.5 • 102 → 2.3E5+2.5E2                             

·        (4 • 107)(8 • 10-2)  → 4E5 • 8E-2   

·        (3.6 • 10-2)3  →  3.6E2^3       

·       → 2.8E6/2E2               

 

 

Scientific Notation Word Problems

 Example G Set → make the numbers simpler to see the operation

·       Jon sold 10 frogs.  Mike sold 2 frogs.

o   How many total frogs did they sell?           → 10 + 2 = 12 total frogs

o   How many more frogs did Jon sell?            → 10 – 2 = 8 more frogs

o   How many times more frogs did Jon sell?  → 10 / 2 = 5 times more

 

·       Jon sold 3.6 • 105 frogs.  Mike sold 2 • 103 frogs.

o   How many total frogs did they sell?     

→  3.6 • 105 + 2 • 103 = 362,000 total frogs = 3.62 • 105 total frogs

o   How many more frogs did Jon sell?      

→  3.6 • 105 - 2 • 103 = 358,000 more frogs = 3.58 • 105 more frogs

o   How many times more frogs did Jon sell?

→  3.6 • 105 / 2 • 103 = 180 times more = 1.8 • 102 times more

  

 

For further explanation on scientific notation word problems:

 

 

Let’s Practice: