A bar graph is a type of graph in which the lengths of the bars are used to represent and
compare data. A numerical scale is used to determine the lengths of the bars. It is important
to note that the scale used on the y-axis can show any type of numerical value. Bar graphs are
useful for displaying data collected from distinct experimental conditions or from distinct time
points during an experiment. They should not be used to attempt to demonstrate a connection
from one condition to another or among time points.
TRANSPARENCY, PAGE E11
Page E11 contains a grid that has vertical and horizontal axes already drawn onto it. You can
use the transparency to model for students how to make a bar graph using your own data, data
that students may have already collected, or data in the examples below.
Example 1
Example 2
Use a bar graph to show the acceleration of
blocks of ice across a smooth, flat surface.
Use a bar graph to compare the respiration
rates of smokers and nonsmokers during
different physical activities.
TABLE 2. RESPIRATION RATE
Acceleration
(m/s2)
Block
Mass (kg)
1
1.0
4.0
2
2.0
3
3.0
Activity
Smokers
Nonsmokers
At rest
22
15
2.0
Walking
26
23
1.3
Running
36
33
Figure 2. Respiration Rates
Figure 1. Acceleration of Ice
5.0
Breaths per minute
40
4.0
(Acceleration m/s2)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
TABLE 1. ACCELERATION OF ICE
3.0
2.0
Nonsmokers
Smokers
30
25
20
15
10
5
1.0
0.0
35
0
At rest
1.0
2.0
Walking
Running
3.0
Mass of block (kg)
SCIENCE TOOLKIT E1
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Math in Science Tools
Bar Graph
Math in Science Tools
Data Table
A data table is useful for organizing, recording, and presenting scientific data. Typically, the
first column of a data table contains the independent variable, and additional columns are
filled in with the experimental observations. Often, data collected from an experiment are
transferred from a data table to a graph. In some cases, a better representation of the data is
to simply leave the information in a table. For example, if the data are all related to a single
part of an experiment, but the numerical values for different conditions have a large range, it
may not be practical to use a graph.
TRANSPARENCY, PAGE E12
Page E12 contains a grid of empty cells that can be used as the starting point for a data table.
You can use the transparency to model for students how to make a data table using your own
data, data that students may have already collected, or data in the examples below.
Example 1
Example 2
Organize the following information about
electrons, protons, and neutrons in a data table.
Organize the following information about
four mineral samples in a data table.
Electrons: relative mass  1; relative
charge  1
Sample E: density  3.7 g/cm3; hardness
 8.5 on the Mohs scale; is not magnetic
Protons: relative mass  2000; relative
charge  1
Sample F: density  5.2 g/cm3; hardness
 5.5 on the Mohs scale; is magnetic
Neutrons: relative mass  2000; relative
charge  0
Sample G: density  2.7 g/cm3; hardness
 7.0 on the Mohs scale; is not magnetic
Sample H: density  2.7 g/cm3; hardness
 3.0 on the Mohs scale; is not magnetic
TABLE 1. PARTICLE CHARGES AND MASS
Relative
Mass
Particle
Relative
Charge
TABLE 2. MINERALS
Sample
Density
(g/cm3)
Hardness
in Mohs
scale
Magnetic
Electron
1
1
Proton
2000
1
E
3.7
8.5
no
Neutron
2000
0
F
5.2
5.5
yes
G
2.7
7.0
no
H
2.7
3.0
no
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A circle graph is a type of graph in which the sizes of sectors of a circle are used to represent
and compare data. A degree measurement is used to determine the size of each sector. In
order to calculate the degree measurement, it is necessary to find the sum of all data points,
which will be used as the denominator of a fraction. The numerator of this fraction is the
data point for an individual category. Multiplying the fraction by 360° provides the degree
measurement for that sector of the graph. It is important to note that it is only appropriate to
use a circle graph when the data from the experiment are in the form of a percentage; 100
percent is represented by the entire 360° of the circle.
TRANSPARENCY, PAGE E13
Page E13 contains a blank circle that can be filled to demonstrate the steps involved in
making a circle graph. You can use the transparency to model for students how to make a
circle graph using your own data, data that students may have already collected, or data in the
examples below.
Example 1
Example 2
Use a circle graph to compare the proportion
of sedimentary rock to that of igneous and
metamorphic rock in Earth’s crust.
Use a circle graph to compare the amounts of
different metals in the alloy called pewter.
TABLE 1. ROCK IN EARTH’S CRUST
Type
Percentage
Sedimentary
5.0
Igneous and
metamorphic
95.0
Degree Measurements for Graph
Sedimentary:
.05 p 360°  18°
Igneous and Metamorphic:
.95 p 360°  342°
Figure 1. Rock in Earth’s crust
TABLE 2. METALS IN PEWTER
Metal
Percentage by Mass
Antimony
2.0
Bismuth
6.0
Copper
7.0
Tin
85.0
Degree Measurements for GraphAntimony:
.02 p 360°  7.2°
Bismuth: .06 p 360°  21.6°
Copper: .07 p 360°  25.2°
Tin: .85 p 360°  306°
Figure 2. Metals in Pewter
Antimony 2%
Bismuth 6%
Copper 7%
Sedimentary
rock 95%
Igneous and
metamorphic rock
95%
Tin 85%
SCIENCE TOOLKIT E3
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Math in Science Tools
Circle Graph
Math in Science Tools
Line Graph
A line graph is a type of graph in which a line represents a continuous relationship between
an independent variable and a dependent variable. Often, a line graph is used to demonstrate
a change in a variable over time. A numerical scale is used to determine the magnitude of the
change. Line graphs are useful for displaying data collected from experimental conditions or
from time points that are linked. It is important to note that the greater the distance between
points on the x-axis, the greater is the probability that a straight line does not actually connect
the data points. To decrease the likelihood of overlooking such variations, as many data points
as possible should be collected.
TRANSPARENCY, PAGE E14
Page E14 contains a grid that has vertical and horizontal axes already drawn onto it. You can
use the transparency to model for students how to make a line graph using your own data,
data that students may have already collected, or data in the examples below.
Example 1
Example 2
Use a line graph to show the amount of
ragweed pollen in the air during a sixweek period.
Use a double line graph to compare the
amounts of two pollutants in the air between
1950 and 1990.
Ragweed (grains/m3)
Week
1
10
2
250
3
130
4
240
5
140
6
25
Year
Pollutant 1
(millions of tons)
Pollutant 2
(millions of tons)
1950
22
10
1960
20
15
1970
30
20
1980
25
25
1990
20
27
250
200
150
100
50
0
Figure 2. Pollution Levels
Millions of tons
of pollutant per year
Grains per cubic meter
Figure 1. Weekly Pollen Counts
300
1
2
3
4
Week
5
6
50
40
30
20
10
0
1950
1960
Year
1970
1980
Pollutant 1
Pollutant 2
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1990
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
TABLE 2. POLLUTION LEVELS
TABLE 1. WEEKLY POLLEN COUNTS
Math in Science Tools
Percent Grid
A percent grid is a type of graph in which the sizes of portions of 10 by 10 grid are used to
represent and compare data. In order to use a percent grid, it is necessary to find the sum of
all data points, which will be used as the denominator of a fraction. The numerator of this
fraction is the data point for an individual category. Using cross-products to multiply the
x
fraction by 100
converts the fraction into a percent that can be plotted on the graph by shading
the appropriate number of squares. A percent grid should only be used when the data from the
experiment are in the form of a percentage; 100 percent is represented by the 100 squares in
the grid.
TRANSPARENCY, PAGE E15
Page E15 contains a blank 10 by 10 grid that can be used to demonstrate the steps involved
in making a percent grid. You can use the transparency to model for students how to make a
percent grid using your own data, data that students may have already collected, or data in the
examples below.
Example 1
Example 2
Use a percent grid to represent the percentages
of frozen and free flowing fresh water on Earth.
Use a percent grid to represent a mixture that
is 51 sand, 52 water, and 52 salt.
TABLE 1. FORMS OF FRESH WATER
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Form
TABLE 2. PARTS OF MIXTURE
Percentage
Frozen
70
Free flowing
30
Part
1
5
2
5
2
5
Sand
Water
Salt
Figure 1. Forms of Fresh Water
Free flowing
Frozen
Fraction of Mixture
Percentages
x
Sand: 51  100
 0.2  20%
x
2
Water: 5  100  0.4  40%
x
Salt: 52  100
 0.4  40%
Figure 2. Parts of Mixture
Sand
Water
Salt
SCIENCE TOOLKIT E5
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Math in Science Tools
X, Y Coordinate Grid
An x, y coordinate grid can be used to plot many different types of data that are related in
some way. Data points are plotted on an x, y coordinate grid using ordered pairs of values.
The origin of the grid, or the point at which the axes intersect one another, is represented by
the ordered pair (0, 0). Data points are plotted in relation to the origin. It is important to note
that an x, y coordinate grid is a general tool for the plotting of data. The coordinate system
can be used with continuous data to produce a line graph, or with discrete data points that
represent the relationship between the variables on the axes.
TRANSPARENCY, PAGE E16
Page E16 contains a grid that has vertical and horizontal axes already drawn onto it. You can
use the transparency to model for students how to use the grid by plotting your own data,
data that students may have already collected, or data in the examples below.
Example 1
Example 2
Use an x, y coordinate grid to plot the following
ordered pairs.
Use an x, y coordinate grid draw a cross
section of a region of mountains. The data
in the table can be used as ordered pairs.
Ordered Pairs:
(1, 5)
(1, 1)
(4, 1)
TABLE 2. MOUNTAIN HEIGHTS
6
5
A
Height (m)
1
293
2
381
3
593
4
517
5
204
4
Figure 2. Mountain Heights
3
600
500
1
0
B
Height (m)
2
C
400
300
200
100
1
2
3
4
5
1
2
3
4
Position
E6 SCIENCE TOOLKIT
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Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Position
The calculations to determine area and volume use two basic mathematical formulas
important in science. These formulas, as well as all other formulas used in scientific
investigations, are expressions of facts, rules, or principles written in terms of an equation.
The formulas for area and volume can be stated in both word form and in symbolic form;
the symbolic form substitutes mathematical variables for the words in the word form of the
expression. These formulas, and any others, can be solved if the values for all but one of the
variables are known. It is important to note that the formulas and examples given for area and
volume are only appropriate for simple geometric figures and rectangular solids, respectively.
TRANSPARENCY, PAGE E17
Page E17 contains formulas for calculating the area of two-dimensional figures, and the
formula for calculating the volume of a regularly shaped solid. You can use the transparency
to demonstrate for students how to calculate area and volume using your own numerical
values or the values in the examples below.
Example 1
Example 2
Calculate the area of a square that has
sides that are 38 cm long.
Calculate the width of a rectangular tank
if the tank’s volume is 190,000 cm3, its
length is 250 cm, and its height is 19 cm.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
h  19 cm
s  38 cm
w?
l  250 cm
s  38 cm
Area  (side length)2
Volume  length  width  height
V  lwh
A  s2
190,000  250  w  19
A  382
Area  1444 cm2
190,000  4750  w
40 cm  width
SCIENCE TOOLKIT E7
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Math in Science Tools
Area and Volume
Math in Science Tools
Significant Figures
Significant figures in measurement and calculation are important for determining scientific
accuracy. There are several rules to be considered when dealing with a question about the
number of significant figures in a measurement or calculation.
• Nonzero integers always count as significant figures.
• Zeros that come before all nonzero digits (called leading zeros) never count as
significant figures.
• Zeros that fall between nonzero digits (called captive zeros) always count as
significant figures.
• Zeros that come after all nonzero digits (called trailing zeros) count as significant
figures when the number contains a decimal point.
• When multiplying or dividing, the number of significant figures in the result is equal
to that in the measurement with the smallest number of significant figures.
• When adding or subtracting, the number of significant figures in the result is equal to
that in the measurement with the fewest decimal places.
TRANSPARENCY, PAGE E18
Example 1
Example 2
Identify the number of significant figures in the
half-lives of radioactive isotopes.
Express the answer of the following calculation
using the correct number of significant figures.
TABLE 1. HALF-LIVES OF ISOTOPES
Isotope
Half-Life
Uranium-238
4,510,000,000 years
Carbon-14
5730 years
Lead-214
27 minutes
Polonium-214
0.00016 seconds
8.315
 0.0279027
298
Smallest number of significant figures is 3,
from the number 298.
Answer: 0.0279 has three significant figures.
Significant Figures
U-238: 3
C-14: 3
Pb-214: 2
Po-214: 2
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Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Page E18 contains information about using significant figures, along with an example. You
can add the examples below as well.
Scientific notation is used to express very large or very small numbers as a number between
1 and 10 multiplied by a power of 10. Scientific notation is used because frequently in
science the numbers associated with measurements contain many zeros. Such measurements
are inconvenient to use, and scientific notation is a way to simplify these numbers. Any
number can be represented using scientific notation. When converting a number from
standard form into scientific notation, it is necessary to take into account the rules for
significant figures (see p. E8).
TRANSPARENCY, PAGE E19
Page E19 contains instructions for using scientific notation and converting between standard
form and scientific notation. You can use the transparency to demonstrate for students how to
convert values between scientific notation and standard form using your own data, data the
students may have already collected, or data in the examples below.
Example 1
Example 2
Use scientific notation to express the height
of a redwood tree in millimeters if the tree is
100 meters tall.
Use both standard notation and scientific
notation to express the size of an oxygen
atom if the oxygen atom is 0.00000000014
meters across.
1 m  1000 mm
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Standard notation
100 m  100000 mm
0.00000000014 m
Move the decimal point five places to the left.
Scientific notation
The exponent is 5.
Move the decimal point 10 places to the right.
1.0  105 mm is the height of the redwood tree in
millimeters
The exponent is –10
1.4  10–10 m is the distance across an oxygen
atom in meters
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Math in Science Tools
Scientific Notation
Mean, median, mode and range are called descriptive statistics. Descriptive statistics are used
to provide an overview of a set of data without looking at every measurement. Mean, median,
and mode are measures of central tendency, which means that they fall near the middle of the
data. Range is a measure of variation in a set of data. Depending on the data being collected,
one of these statistics will be more useful than the others.
• The mean is the arithmetic average of a set of data, and is the most commonly used
measure of central tendency. Mean should be used when the data involve continuous,
well-defined measurement scales.
• The median is the value represented by the data point in the middle of the entire set
of data. The median separates the lower half of the data from the upper half, and can
be thought of as the 50th percentile. Median should be used when the data involve a
ranking or relative degree of a characteristic.
• The mode is the most common value in a set of data. Mode should be used when the
data involve the presence or absence of a characteristic. A set of data can have more
than one mode; if two values occur most frequently, the set of data is said to be bimodal.
• The range is the difference between the largest data point and the smallest data
point. The range of a set of data is important because it indicates how consistent
measurements are. If the set of data has a large range, it means the measurements
are widely scattered and the mean may not represent the data with a great degree
of accuracy.
TRANSPARENCY, PAGE E20
Page E20 contains instructions and formulas for calculating the mean, median, mode,
and range for a data set. You can use the transparency to demonstrate for students how to
calculate these statistics using your own data, data the students may have already collected,
or the data in the examples below.
Example 1
Example 2
Find the mean, median, mode, and range
for five weeks of wave height data.
Find the mean, median, mode, and range for the
number of lynx observed during a six-year period.
TABLE 1. WAVE HEIGHT
TABLE 2. LYNX IN HABITAT
Week
Wave Height (m)
Year
1
1.2
1
2
2
1.6
2
15
3
1.4
3
65
4
1.7
4
75
5
1.7
5
100
6
95
Mean  1.5 m
Median  1.6 m
Mode  1.7 m
Range  0.5 m
Number of Lynx
Mean  59
Median  70
Mode  none
Range  98
E10 SCIENCE TOOLKIT
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Math in Science Tools
Mean, Median, Mode, and Range
A bar graph is a type of graph in which the lengths of the bars
are used to represent and compare data. A numerical scale is
used to determine the lengths of the bars.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Title: _____________________________________________________
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Math in Science Tools
Bar Graph
You can use a data table to organize and record the
measurements that you make. Some examples of information
that might be recorded in data tables are frequencies, times,
and amounts.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Table _____________________________________________________
SCIENCE TOOLKIT E12
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Math in Science Tools
Data Table
Math in Science Tools
Circle Graph
You can use a circle graph, sometimes called a pie chart,
to represent data as parts of a circle. Circle graphs are used
only when the data can be expressed as percentages of a
whole. The entire circle shown in a circle graph is equal to
100 percent of the data.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Title: _____________________________________________________
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A line graph is a type of graph that is used to show a
relationship between variables. Line graphs are particularly
useful for showing changes in variables over time.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Title: _____________________________________________________
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Math in Science Tools
Line Graph
You can use a percent grid to graphically represent data as
making up a portion of a 10 square  10 square grid. A grid
of this type is used only when the data can be expressed as
percentages of a whole. The entire 10  10 grid used is equal
to 100 percent of the data.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Title: _____________________________________________________
SCIENCE TOOLKIT E15
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Math in Science Tools
Percent Grid
An x, y coordinate grid is useful for plotting distinct points
that show a relationship between variables.
Title: _____________________________________________________
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
y
x
SCIENCE TOOLKIT E16
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Math in Science Tools
X, Y Coordinate Grid
Math in Science Tools
Area and Volume Formulas
Area is the amount of surface a geometric figure covers. Area
is measured in square units, such as square meters (m2) or
square centimeters (cm2).
w
s
h
l
s
b
Area  (side length)2
A  s2
1
Area  2 (base
A  21 bh
Area  length  width
A  lw
 height)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
The volume of a solid object is the amount of space contained
by a solid. Volume is measured in cubic units, such as cubic
meters (m3) or cubic centimeters (cm3).
h
l
w
Volume  length  width  height
V  lwh
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The significant figures in a decimal number are the digits that
are warranted by the accuracy of a measuring device. When
you perform a calculation with measurements, the number of
significant figures to include in the result depends in part on
the number of significant figures in the measurements.
Example
A marble has a mass of 8.0 grams and a volume of 3.5 cubic
centimeters. Calculate the marble’s density to the correct
number of significant figures.
Density 
mass
Volume

8.0 g
3.5 cm3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
 2.285714286 g/cm3
Both the mass and volume have two significant figures, so the
answer is expressed with two significant figures.
Density  2.3 g/cm3
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Math in Science Tools
Significant Figures
Scientific notation is used as a short-hand method for writing
very large or very small numbers that have a large number of
zeros on either side of a decimal point. Numbers expressed
in scientific notation can be converted to standard form, and
numbers in standard form can be converted into scientific
notation. Exponents are used to express numbers in scientific
notation.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
You can convert from standard form to scientific notation.
Standard Form
Scientific Notation
720,000
7.2  105
5 decimal places left
Exponent is 5.
0.000291
2.91  10–4
4 decimal places right
Exponent is –4.
You can convert from scientific notation to standard form.
Scientific Notation
Standard Form
4.63  107
46,300,000
Exponent is 7.
7 decimal places right
1.08  10–6
0.00000108
Exponent is –6.
6 decimal places left
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Math in Science Tools
Scientific Notation
The mean of a set of data is the sum of the values divided by
the number of values.
Instructions for calculating mean:
1. Find the sum of all measurement values.
2. Divide the sum of all values by the number of values
included in the sum.
The median of a set of data is the middle value when the
values are written in numerical order.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company
Instructions for calculating median:
1. Arrange your data in order from smallest to largest.
2. Count the number of data points.
3. If you have an odd number of data points, the median
is the middle value. If you have an even number of data
points, the median is the mean of the two middle values.
The mode of a set of data is the measurement that occurs
the most.
Instructions for calculating mode:
1. Arrange your data in order from smallest to largest.
2. Examine the data set to determine the value that occurs
most frequently.
The range of a set of data is the difference between the
largest and smallest measurements.
Instructions for calculating range:
1. Arrange your data in order from smallest to largest.
2. Subtract the smallest value from the largest.
SCIENCE TOOLKIT E20
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Math in Science Tools
Mean, Median, Mode, and Range