How to Use Your Scientific Calculator
Objectives
After completing this activity, the student will be able to:
1. Convert numbers between scientific notation and standard form (decimal notation).
2. Enter scientific notation properly using their exponent key.
3. Understand the mathematical order of operations and properly perform calculations using it.
4. Understand basic algebra concepts.
Background
One of the most important tools in the arsenal of any scientist or engineer is their scientific calculator. Knowing
how to properly use it not only ensures accurate results, but also saves time. This activity will cover a few of
the basic calculator functions and underlying mathematical principles.
Scientific Notation vs. Standard Form
Scientists frequently deal with numbers that are both very large and very small. For example, the distance
between two oxygen atoms in O2 is 0.000000000121 meters, while the distance between Earth and the Moon
is 384,400,000 meters. Scientific notation is a more compact form that retains significant digits while
removing placeholder zeroes. For example, 0.000000000121 meters can also be written as 1.21×10−10
meters, while 384,400,000 meters can be written as 3.844×108 meters.
Order of Operations
When performing a series of calculations, there is a set of rules that covers the precedence for which operation
to perform first. In other words, we don’t simply enter calculations from left to right; rather we prioritize them
according to the order of Parentheses followed by Exponents, then Multiplication & Division, and finally
Addition & Subtraction. This order is often referred to as PEMDAS. (One way to remember the order is with
the mnemonic device, “Please Excuse My Dear Aunt Sally.”) Consider the following example:
94 – (72 + 3 × 8)
First, we perform the portion in Parentheses, (72 + 3 × 8). Of the three operations inside the parentheses, we
perform the Exponent (72) then the Multiplication (3 × 8) and finally Add them together.
94 – (49 + 3 × 8)
94 – (49 + 24)
94 – (73)
Lastly, we perform the Subtraction operation outside the parentheses:
94 – 73 = 21
Distributing in Algebra
Distributing means spreading items out equally. Algebraic distribution means to multiply each of the terms
within the parentheses by another term that is outside the parentheses. Each term gets multiplied by the same
amount. For instance, if we wanted to multiply all of the terms within the following parentheses (b + c + d + e)
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by the term a, where ‘a’ is any real number: positive, negative, integer, or fraction, we would multiply each of
the terms inside the parentheses by the term a.
a(b + c + d + e) = ab + ac + ad + ae
If there is a positive (+) or negative sign (–) they are also distributed. Keep in mind that if a negative is
multiplied by a negative the answer will be positive and if a negative is multiplied by a positive the answer will
be negative.
EXAMPLE 1: Distribute 2 over the terms 4x + 3y – 6
1.
Multiply each term by the number(s) and/or variable(s) outside the parentheses
2(4x + 3y – 6)
2(4x) + 2(3y) – 2(6)
2. Perform the multiplication operation in each term.
8x + 6y – 12
EXAMPLE 2: Simplify the expression by distributing and combining like terms: 4x(x – 2) – (5x + 3)
1. Distribute the 4x over the x and the –2 by multiplying both terms by 4x:
4x(x – 2) = 4x(x) – 4x(2) = 4x2 – 8x
2.
Distribute the negative sign over the 5x and the 3 by changing the sign of each term:
– (5x + 3) = – (+5x) – (+3) = – 5x – 3
3. Combine the like terms:
4x2 – 8x – 5x – 3 = 4x2 – 8x – 5x – 3 = 4x2 – 13x – 3
Solving Equations using Algebra
Solving linear equations involves using combinations of multiplication, division, addition and subtraction. Many
formulas and equations include a coefficient (multiplier) with the variable. To get rid of the coefficient and solve
the equation, you divide.
EXAMPLE 1: Solve for x in 20x = 170
1.
Determine the coefficient (in this example it is 20) of the variable and divide both sides by it:
20x
170
=
20
20
2. Reduce both sides of the equal sign:
x = 8.5
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Multiplication is used where a number already divides the variable. Remember that the opposite operation of
multiplication is division.
EXAMPLE 2: Solve for y in
y
11
= −2
1. Determine the value that divides the variable and multiply both sides by it. In this example, 11 is
dividing the y, so that’s what you multiply by:
y
11 ( ) = (−2)(11)
11
2. Reduce on the left side and multiply on the right:
y = –22
The standard form of a linear equation is ax + b = c. In the previous two examples the value of b was 0. If
instead b is a value you will solve the equation through addition/subtraction and then multiplication/division.
EXAMPLE 3: Solve for z in 3z – 11 = 19
1. To isolate the z term, you add 11 to each side of the equation. The number 11 is chosen, because it is
the opposite of –11 and the sum of –11 and 11 is 0.
3y – 11 = 19
+ 11 +11
3y
= 30
2. Now you have a linear equation (3y = 30) which can be solved by dividing each side of the equation by
3:
3y
30
=
3
3
y = 10
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ACTIVITY PROCEDURE
Part I –Scientific Notation
A. Entering Scientific Notation
First, locate the exponent key on your scientific calculator. It will most likely either be EXP or EE. If you
cannot find it on your own, ask a classmate or your professor for help. If your calculator’s exponent key looks
like ×10y or similar then you may run in to difficulties below so pay close attention. (Do NOT use the 10x
located above the log key as this is antilog and will cause problems in some calculations below.)
This exponent key is a shorthand way of entering “× 10 ^”. For example, we can enter 3.4×108 by typing “3 . 4
EXP 8” or “3 . 4 EE 8”. Notice, this saved us 3 keystrokes! But in addition to saving us time, it also prevents
errors in the order of operations. Let’s try the following calculation both with and without using the exponent
key.
5.12×108
6.5×10−4
Option #1: Without using the exponent key, you might enter:
Button*
Keystroke
5
1
.
2
1
3
2
4
×
5
1
6
0
7
^
8
8
9
÷
10
6
11
.
12
5
13
×
14
1
15
0
16
+/−
17
4
18
* Some calculators have yx instead of ^.
This gives an answer of 7876.92308.
Option #2: Now let’s try it using the exponent key.
Button*
Keystroke
5
1
.
2
1
3
2
4
EXP
5
8
6
÷
7
6
8
.
9
5
10
EXP +/−
11
12
4
13
* Some calculators have EE instead of EXP.
This gives an answer of 7.87692×1011.
Why the difference? Because your calculator strictly follows the order of operations. It multiplies 5.12 by 108,
divides it by 6.5, and then multiplies that by 10−4. In other words, it thinks you meant:
5.12×108
6.5
× 10−4
The exponent key solves this by entering 6.5×10−4 as a single number rather than a series of operations. Not
only that, it also saved us 5 keystrokes!
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Practice:
Solve the following calculations on your scientific calculator, being sure to use the exponent key (EXP or EE)
as necessary.
1) 7.1×103 + 4.2×103 =
2) 9.2×10-6 × 3.11×10-3 =
3) 8.27×105 ÷ 5.1×10-4 =
4)
5)
6)
3.2×1032 × 5.1×10−29
7.4×1012
4.17×1014 × 6.8×1025
8.3×10−6 × 3.22×1030
7.3×10−8 × 5.1×1014
6.1×1012 × 2.89×1021
=
=
=
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When performing metric conversions, you will frequently find yourself using numbers such as 10−2, 10−6, and
103. Some calculators allow you enter this simply as EXP −2, EXP −6, and EXP 3, while others require the
number 1 before the EXP or EE key like so: 1 EXP −2, 1 EXP −6, and 1 EXP 6.
Practice
Solve the following calculations on your scientific calculator, being sure to use the exponent key (EXP or EE)
and a 1 as necessary.
103 m
7)
5.11 km ×
8)
7.229 × 10−2 pm ×
9)
14 dL ×
1 km
10−1 L
1 dL
×
×
1 cm
10−2 m
=
10−12 m
1 pm
1 GL
109 L
×
1 μm
10−6 m
=
=
B. Switching Between Scientific and Standard Notation
Another useful skill on your scientific calculator is learning to switch numbers between standard notation
(decimal notation) and scientific notation. Most scientific calculators make this easy with a simple keystroke or
two. If you want your calculator to display answers in scientific notation, you will need to enter SCI mode. To
display answers in standard notation, you will need to enter FLO mode. (FLO stands for “floating decimal”
which causes the decimal point to appear in the normal position rather than always after the first digit.)
Work with a classmate to find either the SCI and FLO modes (or keys) on your calculator. Once you’ve found
it, try the following calculation:
6.41×108 × 0.037
After obtaining the answer, switch back and forth between SCI and FLO several times. When in SCI you
should see 2.3717E7 or 2.3717×107. When in FLO you should see 23,717,000.
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Practice:
Solve the following calculations on your scientific calculator, being sure to use the exponent key as necessary.
Record your answers in both standard notation (FLO) and scientific notation (SCI).
10)
6.38×103 + 1.9×104 =
FLO: ____________________
11)
1.442×10−2 ÷ 7.6×103 =
FLO: ____________________
12)
SCI: ____________________
0.00073 × 0.00499
FLO: ____________________
13)
SCI: ____________________
SCI: ____________________
8,400,000 ÷ 2.77×10−3
FLO: ____________________
SCI: ____________________
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Part II –Order of Operations
Practice
Solve the following calculations, being sure to follow the order of operations (PEMDAS).
14)
2 + 3 × 7 – 4 =
15)
5 + (12 − 3) × 2 =
16)
3.1 × 4.2
5.7 × 2.8
=
17)
4 ÷ 22 + 8 × 32 =
18)
27 − 3 × (5 − 3)3 + 5 =
19)
42 + 53 ÷ 24 − (6 + 23)2 =
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Part III –Algebra
Practice
Distribute the following through the terms in the expression given:
20) -1 over the terms in the expression (4x + 2y – 3x +7)
21) a over the terms in the expression(a4 + 2a2 + 3)
Solve for x in the following equations:
22) 12x = 300
23)
24)
𝑥
9
4𝑥
5
= 500,000
= 12
3
25) 𝑥 = 4
26) 3x + 10 = 31
27) 2(x + 4) = 8
1
28) 3 (𝑥 + 3) = −9
29) 3(x + 7) = 7(x + 2)
30) 2 – (2x + 1) = 4(x + 2)
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Introduction to Measurements
EXPERIMENTAL TASK
To learn to use a variety of common laboratory equipment while recording measurements with proper
significant digits and units.
Objectives
After completing this experiment, the student will be able to:
1. Identify and use common laboratory equipment.
2. Determine the correct number of significant digits given by a device.
3. Record measurements with proper significant digits and units.
Background
This lab will give you an introduction to laboratory equipment, performing measurements, and recording data
with correct units and significant digits.
For any measurement, there is always a degree of uncertainty. Typically,
we record all of the digits that we know with certainty plus a single uncertain
digit. For example, let’s consider the volume of the liquid in the graduated
cylinder to the right. First, note how the top of the liquid is curved. This is
called a meniscus and the volume is always recorded at the bottom of the
curve. Now, let’s determine which increments we can be certain of. In this
case, both the 10’s place (40 mL, 50 mL) and 1’s place (41 mL, 42 mL, 43
mL, etc.) are marked. Therefore, we can be certain that the answer is 43 mL
but we must estimate one more uncertain digit. In our case, the bottom of
the meniscus appears to be directly on the 43 mark so we could record the
volume as 43.0 mL. However, as the last digit is uncertain any reasonably
close measurement (within +0.1 mL) would be acceptable (42.9 mL, 43.1
mL).
Keep in the mind that you must always estimate one uncertain digit for any
measurement that does not provide an exact reading – thermometers,
rulers, analog clocks, etc. However, for devices with digital readouts such as analytical balances, digital
clocks, or digital thermometers all digits displayed should be recorded. The last digit is still uncertain but the
instrument did the estimating for you.
Accuracy and Precision
Precision is a measure of how consistent a series of measurements are. The more significant digits a device
generates, the more precise it is considered because the measurements it produces will be consistently closer
to one another. For example, a measurement of 43.0 mL (3 significant digits) is more precise than 43 mL (2
significant digits). This is different from accuracy which is how correct the answer is, or how close to the true
value the measurement is. Although the two are frequently confused, they are completely different! For
example, a measurement can be very precise without being accurate and vice-versa. (You can be consistently
wrong – precise but inaccurate!)
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Density
Density is the ratio of a substance’s mass to its volume.
The density of liquids and solids is typically recorded in units such as g/mL or g/cm3, while the density of gases
generally has units of g/L. Remember that 1 mL = 1 cm3.
EXPERIMENT PROCEDURE
Part I –Recording Measurements
A. Mass Measurements
1. Determine the mass of a 250-mL beaker on an analytical balance. Be sure to “zero” or tare the balance
before each measurement.
2. Determine the mass of a watch glass on an analytical balance.
3. Determine the mass of a 125-mL Erlenmeyer flask on an analytical balance.
B. Volume Measurements I
1. Fill a 100-mL graduated cylinder with water until the bottom of the meniscus (curve in the upper part of
the water) is exactly on the 100 mL mark. Record the volume.
2. Fill a 13×100 mm test tube with water from the graduated cylinder. Record the new volume in the
graduated cylinder.
3. Fill a second test tube with water from the graduated cylinder. Record the volume in the graduated
cylinder.
C. Volume Measurements II
1. Fill a burette with water until the bottom of the meniscus is exactly on the 0 mL mark. Record the initial
reading.
2. Dispense approximately 10 mL into a 125-mL Erlenmeyer flask. Record the new burette reading.
3. Dispense approximately 5 mL into the same 125-mL Erlenmeyer flask. Record the new burette
reading.
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D. Temperature Measurements
1. Half fill a 250-mL beaker with deionized water. Record the temperature of the water in Celsius using a
thermometer.
2. Add ice to the beaker until the beaker is approximately two-thirds full. Hold the thermometer in the ice
water and record the coldest observed temperature.
3. Empty the beaker and half-fill it again with deionized water. Place the beaker on a ring stand with an
iron ring and wire gauze. The beaker should be approximately six inches above the Bunsen burner.
Heat the water using a Bunsen burner to boiling then shut off the burner. Hold the thermometer in the
water and record the hottest temperature.
Part II –Density
For this part of the experiment, we will record the volume by two different methods. First, we will take
measurements using a ruler and calculate the volume using the appropriate formula (Direct Measurement).
Then we will take the unknown and place it in a graduated cylinder containing a known amount of water
(Displacement).
1. Obtain an unknown metal solid of any regular shape (rectangle, cylinder, or sphere).
2. Find the mass of the unknown metal solid on an analytical balance.
3. Using a metric ruler, take appropriate measurements to determine the volume of the unknown metal
solid. Record all measurements in centimeters but still with proper significant digits.
4. Calculate the volume by direct measurement using the appropriate formula.
a. Volume of a rectangular solid = length × width × height
b. Volume of a cylinder = πr2h
4
c. Volume of a sphere = 3 𝜋r 3
5. Next, we will measure the volume of the unknown metal solid by displacement. Half-fill a 100-mL
graduated cylinder and record the initial volume.
6. Tilt the graduated cylinder to a 45° angle and slowly add the unknown metal solid to the graduated
cylinder. Record the final volume.
7. Calculate the volume by displacement by subtracting the initial volume from the final volume.
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Name: ______________________________
Date: _________________
Lab Partner: _________________________
Section: _______________
DATA, OBSERVATIONS, and RESULTS
Record all measurements with proper significant digits and units.
Part I –Recording Measurements
A. Mass Measurements
Mass of 250-mL beaker
__________________
Mass of watch glass
__________________
Mass of 125-mL Erlenmeyer flask
__________________
B. Volume Measurements I
Initial volume of water in graduated cylinder
__________________
Volume minus one test tube of water
__________________
Volume minus two test tubes of water
__________________
C. Volume Measurements II
Initial burette reading
__________________
Burette reading after dispensing ~10 mL of water
__________________
Burette reading after dispensing ~5 mL of water
__________________
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D. Temperature Measurements
Temperature of deionized water
__________________
Temperature of deionized water and ice
__________________
Temperature of boiling water
__________________
Part II –Density
Mass of unknown metal solid
__________________
Measurements of unknown metal solid (in centimeters)
(May include length, height, diameter, etc. as necessary)
__________________
__________________
__________________
Calculate the volume of your unknown solid
(Show work below)
__________________
Initial volume of graduated cylinder
__________________
Final volume of graduated cylinder plus unknown metal solid
__________________
Volume of unknown metal solid (final – initial)
__________________
Temperature of water
__________________
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Name:
Date:
Lab Partner:
Section:
POST−LABORATORY QUESTIONS
Show all work. Give all answers with proper significant digits and units.
1. Calculate the density of your unknown metal solid using the volume found by direct measurement.
_______________ g/cm3
2. Calculate the density your unknown metal solid using the volume found by displacement.
_______________ g/mL
3. Which method do you think gave a more precise (not necessarily accurate) result – direct measurement or
displacement? Explain your reasoning.
4. State the number of significant digits in each of the following measurements.
o
5.00 ºC
__________
o
0.03 cm
__________
o
10.000 g
__________
o
0.450 mL
__________
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Name:
Lab Partner:
Date:
Section:
PRE−LABORATORY ASSIGNMENT
Answer the following questions BEFORE coming to lab. You will not be allowed to participate in lab
unless ALL questions are complete. Show all work. Give all results with proper significant digits.
1. Sketch a beaker, an Erlenmeyer flask, a watch glass, and a graduated cylinder. Label each drawing.
2. Record the length of the line below using proper significant digits and units.
Length of line:
_________________
3. A graduated cylinder contains 37.5 mL of water. A lump of iron with a mass of 45.38 g is dropped into
the graduated cylinder and the water level rises to 43.3mL. What is the density of the iron sample?
(Show work below.)
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