TEAS Measurement and Data

Units of Measurement and Conversions

Metric System

The metric system is a decimal-based system of measurement used by most countries worldwide, including units like meters, liters, and grams.

Each unit is 10 times larger or smaller than the one next to it (or 1 decimal place movement to the right or left).

All you need to do is memorize the prefixes. The prefixes are the same for all of the various types of measurements.

To convert metric units, follow these steps:

• Identify the units
• Determine the direction of conversion:
  • Larger unit (move the decimal point to the left)
  • Smaller unit (move the decimal point to the right)
• Count the steps
• Move the decimal point

Example: Convert 55.6 meters to millimeters.

Step 1: Identify the units.

We are working with the base unit (meters) and the milli- prefix.

Step 2: Determine the direction of the conversion.

We are converting to a smaller unit. This means we must move the decimal place to the right.

Step 3: Count the steps to move the decimal place.

From the base unit to the milli- prefix is 3 decimal places. Use the table above if you need a refresher.

Step 4: Move the decimal place.

Since we are going to a smaller unit, we need to move the decimal place 3 places to the right.

55.6 meters = 55600 millimeters

Conversion Math

Conversion math can be used to convert between units. As long as we have a conversion factor, we can set up a proportion and cross-multiply to solve. Here are the basic steps to use when working on conversion math:

1. Identify the units
2. Find the conversion factor
3. Set up the proportion
4. Cross-multiply and solve

Example: Convert 150 pounds to kilograms.

Step 1: Set up a proportion. Use the conversion factor 2.2 pounds = 1 kilogram.

\frac{2.2}{1} = \frac{150}{x}

Step 2: Solve the proportion by cross-multiplying and solving for x:

\frac{2.2}{1} = \frac{150}{x}
2.2 * x = 150 *1
2.2x = 150
x = 68.18 kg

150 pounds is equal to 68.18 kilograms.

Common Conversion Factors

Here are some of the common conversion factors within the standard system.

Here are some of the common conversion factors when going from the standard system to the metric system (or vice versa).

Temperature Conversions

To convert between Celsius and Fahrenheit, use these formulas:

C = (F – 32) × \frac{5}{9}

F = C × \frac{9}{5} + 32

Example: Convert 81 degrees Fahrenheit to Celsius.

Step 1: Plug our numbers into the formula and solve.

C = (81 – 32) × \frac{5}{9}
C = 49 × \frac{5}{9}
C = 27.22 degrees

81 degrees Fahrenheit is equal to 27.22 degrees Celsius.

Geometric Quantities

Perimeter and Area

Two of the most important geometric measurements are perimeter and area.

Perimeter is the distance around a two-dimensional shape. Some common formulas you should know are:

• Square: p = 4s
• Rectangle: p = 2l + 2w
• Triangle: p = s1 + s2 + s3
• Circumference of a Circle: 2πr OR πd

Area is the amount of space inside the boundary of a two-dimensional shape. Some common area formulas you should know are:

• Square: A = s²
• Rectangle: A = lw
• Parallelogram: A = bh
• Triangle: A = \frac{1}{2} bh
• Trapezoid: A = \frac{1}{2} h(b1 + b2)
• Circle: A = \pi r^2

Example: Find the perimeter and area of the rectangle below.

Step 1: Find the perimeter. We can use the formula for perimeter of a rectangle (2l + 2w), or we can just add up all of the sides.

11 + 11 + 9 + 9 = 40 cm

Step 2: Find the area. We can use the formula for area of a rectangle (length × width).

11 × 9 = 99 cm²

Volume and Surface Area

Three-dimensional shapes have length, width, and height, forming objects with volume and surface area.

Volume is a measure of the amount of space something occupies. Each shape has its own formula to calculate volume. Some common formulas are:

• Volume of a cube: V = side length³
• Volume of a rectangular prism (or rectangular solid): V = length × width × height
• Volume of a cylinder: V = \pi r^2 × h
• Volume of a pyramid: V = \frac{1}{3} × base × height

Example: Find the volume of an aquarium that measures 3 feet by 2 feet by 2 feet tall.

Step 1: Use the formula V = length × width × height and solve.

V = 3 × 2 × 2
V = 12 ft³

Surface area is a measure of the total area covering the surface of a 3D figure. Each shape has its own formula to calculate surface area. Some common formulas are:

• Rectangular Prism: SA = 2lw + 2lh + 2wh
• Cylinder: SA = 2\pi r^2+2\pi rh
• Sphere: SA = 4\pi r^2

Example: Find the surface area of a rectangular prism that has a length of 5 inches, a width of 4 inches, and a height of 3 inches.

Step 1: Use the formula SA = 2lw + 2lh + 2wh and solve.

SA = 2(5 × 4) + 2(5 × 3)+ 2(4 × 3)
SA = 2(20) + 2(15) + 2(12)
SA = 40 + 30 + 24
SA = 94 in²

Pythagorean Theorem

The Pythagorean theorem formula may be used to calculate a missing side for any triangle with a right angle, which is 90 degrees. The formula is:

a² + b² = c²

A and B are the two sides or legs on either side of the right angle, and C is the hypotenuse (slanted side).

Example: Find the hypotenuse of a right triangle with sides of 30 feet and 40 feet.

Step 1: Substitute the sides for a and b in the formula.

30² + 40² = c²

Step 2: Solve.

30² + 40² = c²
900 + 1600 = c²
2500 = c²
\sqrt{2500} = \sqrt{c^2}
50 = c

Graphs

Tables, Charts, and Graphs

We can use tables, charts, and graphs to visualize data. That visualized data can then be used to solve various problems.

Line Graph: A chart that is typically used to depict changes in data over time. The horizontal x-axis shows time units and the vertical y-axis shows the data being measured.

Bar Graph: A chart that uses rectangular bars to represent data values, where the length of each bar corresponds to the data quantity. Bar graphs are used to compare different items or categories.

Pie Chart: A circular chart divided into sectors where each sector represents a proportion of the whole. Since the whole circle is 100%, you can get a sense of the relative size of each sector.

Scatterplot: A graph that shows individual data points plotted based on two variables, often used to identify correlations.

Cartesian Coordinate Graph: A two-dimensional graph that represents data using an x-axis (horizontal) and a y-axis (vertical) to plot points. Each point on the graph is defined by a pair of numbers, known as coordinates (x, y).

In the graph below, point Q would be (4, 2) and point P would be (-2, 4).

Table: A set of data arranged in rows and columns for easy reference and analysis.

Example: Using the bar graph below, what percentage of its product does Organization B produce on Tuesday?

Step 1: To solve this, you need to realize that Organization B is red. You need to determine that Tuesday has 10,000 units by reading across to the left on the y-axis.

Step 2: Since percentage is part divided by whole, you also need to add all of the units for Monday through Friday, including Tuesday.

20,000 + 10,000 + 18,000 + 20,000 + 16,000 = 84,000.

Notice that you had to estimate the values for Wednesday and Friday based on the scale being used.

Step 3: Divide the number produced on Tuesday by the total number produced throughout the week, and multiply by 100 to get the percentage.

10,000 / 84,000 × 100 = 11.90%

Therefore, Organization B produces 11.90% of their product on Tuesday.

Data

Mean, Median, and Mode

Mean is the average of a set of numbers.

Example: Find the mean of the following set of numbers: 8, 12, 10, 14, 6

Step 1: Add all of the numbers together.

8 + 12 + 10 + 14 + 6 = 50

Step 2: Divide by the amount of numbers there are.

We were given 5 total numbers.

50 / 5 = 10

Therefore, the mean is 10.

Median is the middle number in a set of data when the numbers are arranged in order.

Example: Find the median of the following set of numbers: 8, 12, 6, 14, 10

Step 1: List the numbers in ascending order.

6, 8, 10, 12, 14

Step 2: Locate the middle number.

6, 8, 10, 12, 14

Therefore, the median is 10.

Mode is the number(s) that occur most frequently in a data set.

Example: Find the mode of the following set of numbers: 5, 7, 3, 7, 9, 5, 7, 8, 12

Step 1: Locate the number that appears the most often.

• 5, 7, 3, 7, 9, 5, 7, 8, 12

The mode is 7 because it appears three times (more than any other number).

Range is the difference between the largest and smallest values in a data set.

Example: Find the range of the following set of numbers: 5, 7, 3, 7, 9, 5, 7, 8, 12

Step 1: Locate the largest and smallest numbers.

• Largest = 12
• Smallest = 3

Step 2: Subtract the smallest number from the largest number.

• 12 – 3 = 9

Therefore, the range is 9.

Relationship Between Two Variables

There are two types of variables:

• Independent Variable: A variable that isn’t changed by other variables you are trying to measure. It is located on the x-axis.
• Dependent Variable: A variable that depends on other factors. The dependent variable is typically what you are measuring and studying in an experiment. It is located on the y-axis.

Covariance and correlation measure how variables vary together. There are three types:

• Positive Covariance/Correlation: Where variables tend to show similar behavior.
• Negative Covariance/Correlation: Where the variables tend to show opposite behavior.
• Zero Covariance/Correlation: Where there appears to be no relationship between the variables.

Example:

In a study conducted from 2015 to 2020, researchers found that students who spent more hours studying tended to achieve higher scores on their exams. This finding aligns with the understanding that increased study time can enhance comprehension and retention of the material, leading to better performance on assessments.

Which of the following sentences best describes this relationship?

a. The two variables show a positive correlation.
b. The two variables represent unrelated phenomena.
c. Exam scores are the independent variable.
d. The relationship between study time and exam scores is a negative correlation.

A: A is correct because the passage describes a relationship where an increase in study hours correlates with higher exam scores, indicating that as one variable (study hours) increases, so does the other variable (exam scores).

The passage does not support the idea of unrelated phenomena (option b). It also does not suggest that exam scores are the independent variable (option c). The relationship is not a negative correlation (option d), as there is no indication that more study time would lead to lower exam scores.