Arithmetic for Praxis II ParaPro Test Prep Study Guide (page 4)

The practice quiz for this study guide can be found at:

Math for Praxis II ParaPro Test Prep Practice Problems

This section covers the basics of mathematical operations and their sequence. It also reviews whole numbers, integers, fractions, decimals, percents, and estimation.

The Number Line

The number line is a graphical representation of the order of numbers. As you move to the right, the value increases. As you move to the left, the value decreases.

A number line can show the approximate values of certain numbers. For example, the following number line shows that –4.5 is halfway between –5 and –4 and that 2.2 is closer to 2 than it is to 3.

Comparison Symbols

The following table will illustrate some comparison symbols:

Addition Symbols

In addition, the numbers being added are called addends. The result is called a sum. The symbol for addition is called a plus sign. In the following example, 4 and 5 are addends and 9 is the sum:

4 + 5 = 9

Subtraction Symbols

In subtraction, the number being subtracted is called the subtrahend. The number being subtracted from is called the minuend. The answer to a subtraction problem is called a difference. The symbol for subtraction is called a minus sign. In the following example, 15 is the minuend, 4 is the subtrahend, and 11 is the difference:

15 – 4 = 11

Multiplication Symbols

When two or more numbers are being multiplied, they are called factors. The answer that results is called the product. In the following example, 5 and 6 are factors and 30 is their product:

5 ×6 = 30

There are several ways to represent multiplication in this mathematical statement.

A dot between factors indicates multiplication:

5 · 6 = 30

Parentheses around any one or more factors indicate multiplication:

(5)6 = 30, 5(6) = 30, and (5)(6) = 30.

Multiplication is also indicated when a number is placed next to a variable: 5a = 30. In this equation, 5 is being multiplied by a.

Division Symbols

In division, the number being divided by is called the divisor. The number being divided into is called the dividend. The answer to a division problem is called the quotient.

There are a few different ways to represent division with symbols. In each of the following equivalent expressions, 3 is the divisor and 8 is the dividend:

Prime and Composite Numbers

A positive integer that is greater than the number 1 is either prime or composite, but not both.

• A prime number is a number that has exactly two factors: 1 and itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, and 23 are all prime numbers.
• A composite number is a number that has more than two factors. For example, 4, 6, 8, 9, 10, 12, 14, 15, and 16 are all composite numbers.
• The number 1 is neither prime nor composite since it has only one factor.

Whole Numbers

Whole numbers are the set of nonnegative numbers that are not expressed as a fraction or a decimal. For example, 0, 4, 39, and 3,318 are all whole numbers. For the ParaPro Assessment, you will be expected to know how to add, subtract, multiply, divide, compare, and order them.

Comparing and Ordering Whole Numbers

To compare and order whole numbers, it is essential that you are familiar with the place value system. The following table shows the place values for a very large number: 3,294,107.

To compare or order whole numbers, you need to look at the digits in the largest place value of a number first.

Example

Compare 3,419 and 3,491.

Begin by comparing the two numbers in their largest place value. They both have the digit 3 in the thousands place. Therefore, you do not know which number is larger. Move to the smaller place values (to the right) of each number and continue comparing. The digit in the hundreds place for each number is 4.You still do not know which number is larger. However, when you compare the digits in the tens places, you see that the 9 is greater than the 1. That means 3,491 is greater than 3,419. This can be represented with the greater than symbol: 3,491 > 4,419.

Example

Put the following numbers in order from greatest to least: 307, 319, 139, 301.

To order these numbers, the digits in their place values must be compared. Three of the numbers have a 3 in the hundreds place, but one number has a 1 in the hundreds place. Therefore, 139 is the smallest number. Next the digits in the tens places must be compared with the remaining numbers. The tens digit in 319 is 1, and the tens digit in 307 and 301 is 0. Therefore, 319 is the largest number. To order 307 and 301, compare the digits in the ones place: 7 is greater than 1, so 307 is greater than 301.

The correct order of the numbers, from greatest to least, is 319, 307, 301, and 139.

Adding Whole Numbers

Addition is used when it is necessary to combine amounts. It is easiest to add when the addends are stacked in a column with the place values aligned. Work from right to left, starting with the ones column.

Example

Add 40 + 129 + 24.

1. Align the addends in the ones column. Because it is necessary to work from right to left, begin to add starting with the ones column. The ones column totals 13, and 13 equals 1 ten and 3 ones, so write the 3 in the ones column of the answer, and regroup, or "carry" the 1 ten to the next column as a 1 over the tens column, so that it gets added with the other tens:



2. Add the tens column, including the regrouped 1.



3. Then add the hundreds column. Because there is only one value, write the 1 in the answer.

Subtracting Whole Numbers

Subtraction is used to find the difference between amounts. It is easiest to subtract when the minuend and subtrahend are in a column with the place values aligned. Again, just as in addition, work from right to left. It may be necessary to regroup.

Example

If Becky has 52 clients and Claire has 36, how many more clients does Becky have?

1. Find the difference between their client numbers by subtracting. Start with the ones column. Because 2 is less than the number being subtracted (6), regroup, or "borrow," a ten from the tens column. Add the regrouped amount to the ones column. Now subtract 12 – 6 in the ones column.



 

2. Regrouping 1 ten from the tens column left 4 tens. Subtract 4 – 3 and write the result in the tens column of the answer. Becky has 16 more clients than Claire. Check by addition: 16 + 36 = 52.

Multiplying Whole Numbers

In multiplication, the same amount is combined multiple times. For example, instead of adding 30 three times, 30 + 30 + 30, it is easier to simply multiply 30 by 3. If a problem asks for the product of two or more numbers, the numbers should be multiplied to arrive at the answer.

Example

A school auditorium contains 54 rows, each containing 34 seats. How many seats are there in total?

1. In order to solve this problem, you could add 34 to itself 54 times, but we can solve this problem more easily with multiplication. Line up the place values vertically, writing the problem in columns. Multiply the number in the ones place of the top factor (4) by the number in the ones place of the bottom factor (4): 4 ×4 = 16. Because 16 = 1 ten and 6 ones, write the 6 in the ones place in the first partial product.



2. Multiply the number in the tens place in the top factor (3) by the number in the ones place of the bottom factor (4): 4 ×3 = 12. Then add the regrouped amount: 12 + 1 = 13.Write the 3 in the tens column and the 1 in the hundreds column of the partial product.



3. The last calculations to be done require multiplying by the tens place of the bottom factor. Multiply 5 (tens from bottom factor) by 4 (ones from top factor); 5 ×4 = 20, but because the 5 really represents a number of tens, the actual value of the answer is 200 (50 ×4 = 200). Therefore, write the two zeros under the ones and tens columns of the second partial product and regroup or carry the 2 hundreds by writing a 2 above the tens place of the top factor.



4. Multiply 5 (tens from bottom factor) by 3 (tens from top factor); 5 ×3 = 15, but because the 5 and the 3 each represent a number of tens, the actual value of the answer is 1,500 (50 × 30 = 1,500). Add the two additional hundreds carried over from the last multiplication: 15 + 2 = 17 (hundreds).Write the 17 in front of the zeros in the second partial product.



5. Add the partial products to find the total product:

Note: It is easier to perform multiplication if you write the factor with the greater number of digits in the top row. In this example, both factors have an equal number of digits, so it does not matter which is written on top.

Dividing Whole Numbers

In division, the same amount is subtracted multiple times. For example, instead of subtracting 5 from 25 as many times as possible, 25 – 5 – 5 – 5 – 5 – 5, it is easier to simply divide, asking how many 5s are in 25: 25 ÷ 5.

Example

At a road show, three artists sold their beads for a total of $54. If they share the money equally, how much money should each artist receive?

1. Divide the total amount ($54) by the number of ways the money is to be split (3).Work from left to right. How many times does 3 divide into 5? Write the answer, 1, directly above the 5 in the dividend, because both the 5 and the 1 represent a number of tens. Now multiply: since 1(ten) ÷ 3(ones) = 3(tens), write the 3 under the 5, and subtract; 5(tens) – 3(tens) = 2(tens).
   
2. Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 divide into 24? Write the answer, 8, directly above the 4 in the dividend. Because 3 ÷ 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0.
   

Remainders

If you get a number other than zero after your last subtraction, this number is your remainder.

Example

What is 9 divided by 4?

1 is the remainder.

The answer is 2 R1. This answer can also be written as 2 , because there was one part left over out of the four parts needed to make a whole.

Estimating with Whole Numbers

Some questions on the ParaPro Assessment will ask you for an estimate. That means you will not need to find the actual answer, but should instead find an answer that is close to the actual answer. One way to solve estimation problems with whole numbers is to use numbers that are easy to work with, and that are close to the actual numbers.

Example

A television set weighs 21 pounds. About how much will a case weigh if it carries 46 television sets?

The number 21 is close to 20, and 20 is much easier to work with than 21. The number 46 is close to 50, and 50 is much easier to work with than 46. To find the approximate weight of the 46 television sets, you can just multiply 20 by 50. A proper estimate would be 1,000 pounds.

Integers

An integer is a whole number or its opposite. Here are some rules for performing operations with integers:

Adding Integers

Adding numbers with the same sign results in a sum of the same sign:

(positive) + (positive) = positive

(negative) + (negative) = negative

When adding numbers of different signs, follow this two-step process:

1. Subtract the positive values of the numbers. Positive values are the values of the numbers without any signs.
2. Keep the sign of the number with the larger positive value.

Example

–2 + 3 =

1. Subtract the positive values of the numbers: 3 – 2 = 1.
2. The number 3 is the larger of the two positive values. Its sign in the original example was positive, so the sign of the answer is positive. The answer is positive 1.

Example

8 + –11 =

1. Subtract the positive values of the numbers: 11 – 8 = 3.
2. The number 11 is the larger of the two positive values. Its sign in the original example was negative, so the sign of the answer is negative. The answer is negative 3.

Subtracting Integers

When subtracting integers, change the subtraction sign to an addition sign, and change the sign of the number being subtracted to its opposite. Then follow the rules for addition.

Examples

(+10) – (+12) = (+10) + (–12) = –2
(–5) – (–7) = (–5) + (+7) = +2

Multiplying and Dividing Integers

A simple method for remembering the rules of multiplying and dividing is that if the signs are the same when multiplying or dividing two quantities, the answer will be positive. If the signs are different, the answer will be negative.

(positive) × (positive) = positive            

(positive) × (negative) = negative            

(negative) × (negative) = positive            

Examples
(10)(–12) = –120
–5 × –7 = 35
12 ÷ –3 = –4
15 ÷ 3 = 5

Exponents

An exponent indicates the number of times a base is used as a factor to attain a product.

Example

Evaluate 2⁵.

In this example, 2 is the base and 5 is the exponent. Therefore, 2 should be used as a factor 5 times to attain a product:

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Zero Exponent

Any nonzero number raised to the zero power equals 1.

Examples

5⁰ = 1         70⁰ = 1                 29,874⁰ = 1

Perfect Squares

The number 5² is read "5 to the second power," or, more commonly,"5 squared." Perfect squares are numbers that are second powers of other numbers. Perfect squares are always zero or positive, because when you multiply a positive or a negative by itself, the result is always positive. The perfect squares are 0², 1², 2², 3² … Therefore, the perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …

Perfect Cubes

The number 53 is read as "5 to the third power," or, more commonly, "5 cubed." (Powers higher than three have no special name.) Perfect cubes are numbers that are third powers of other numbers. Perfect cubes, unlike perfect squares, can be either positive or negative. This is because when a negative is multiplied by itself three times, the result is negative. The perfect cubes are 0³, 1³, 2³, 3³ … Therefore, the perfect cubes are 0, 1, 8, 27, 64, 125 …

The Order of Operations

There is an order in which a sequence of mathematical operations must be performed, known as PEMDAS:

P: Parentheses/Grouping Symbols. Perform all operations within parentheses first. If there is more than one set of parentheses, begin to work with the innermost set and work toward the outside. If more than one operation is present within the parentheses, use the remaining rules of order to determine which operation to perform first.

E: Exponents. Evaluate exponents.

M/D: Multiply/Divide. Work from left to right in the expression.

A/S: Add/Subtract. Work from left to right in the expression.

This order and the acronym PEMDAS, which can be remembered by using the first letter of each of the words in the phrase: Please Excuse My Dear Aunt Sally.

Example