Stress-Free Math: A Visual Guide to Acing Math in Grades 4-9

Introduction

There are many keys to being a proficient math problem solver, and two of the most important elements to successful problem solving are knowing:

•what the problem is asking (understanding the vocabulary and being able to determine which strategy to employ), and

•how to perform the operation(s) (being able to quickly and automatically perform the necessary operations).

Besides providing information that will help math students in these two important areas of problem solving, this visual guide also offers a wealth of other information, compiled in an easy-to-use format.

Stress-Free Math is much more than a compilation of words and definitions. This book has been organized to reflect the different areas of mathematics taught in elementary and junior high schools. Each category includes the terms commonly used in this field of study, concise definitions, and many examples and illustrations. In addition, the book provides quick reference guides for basic operations and tables of commonly used facts and equivalents.

Once you use a reference like this book, you’ll agree that it truly is absolutely essential. It will be the reference material you will use again and again to supplement and reinforce topics throughout your math classes.

Visual Definitions by Topic

WHOLE NUMBERS AND OPERATIONS

Addends

Numbers in addition problems that are added together to form a sum.

Example:

3 + 7 = 10

3 and 7 are addends.

Addition

The process of uniting two or more numbers into a sum; counting the total.

Key word: altogether

Examples:

Arabic Numbers

A Base-10 place-value number system that uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Also called Hindu-Arabic numbers.

Arithmetic Progression

A series of numbers in which each number differs from the preceding number by a fixed amount. A series of numbers that follows a pattern.

Example:

1, 5, 9, 13, 17, 21, 25, 29, . . .

Each number differs by 4.

Array

An organized arrangement of objects using rows and columns. These can be very helpful in building multiplication problems and division problems.

Example 1:

6 rows of 7 columns = 42 squares altogether

6 x 7 = 42

Example 2:

1. Build the problem 20 ÷ 4 in an array using as many of the 20 tiles as you can.

2. Count the number of rows that have been built, and count the remainder.

3. 20 ÷ 4 = 5. There is no remainder in this problem.

Example 3:

1. Count out 21 tiles.

2. Build the problem 21 ÷ 4 in an array using as many of the 20 tiles as you can.

3. In this case, 20 tiles can be used to build the array with 1 tile left over.

4. Count the number of rows that have been built, and count the remainder.

5. 20 ÷ 4 = 5. There is 1 remainder. Write the remainder as a fraction. 21 ÷ 4 = 5¼

Ascend

To increase in number, value, or amount.

Ascending Order

Increasing from least to greatest, but not necessarily according to a fixed pattern. To count upward from smallest to largest; counting up.

Examples:

35, 37, 39, 41, 43, 45, 47, 49, . . .

An example of an ascending arithmetic progression.

1, 5, 13, 40, 53

A series of numbers in ascending order.

Binary Numbers

A numerical system that is based on the number 2. Each place has a value equal to a power of 2, as indicated or shown by the symbols 0 or 1.

Cardinal Numbers

Numbers used for counting or answering the question how many?; they show quantity.

Key words: counting numbers

Example:

Numbers such as 1, 2, 3, 47, and 104 are cardinal numbers.

Common Factor/Common Divisor

A factor that two or more numbers have in common. A number that divides two or more numbers evenly (without a remainder).

Example 1:

When using tiles to look at two numbers, the common factors are the factors that both numbers share.

Example 2:

The factors that 6 and 8 have in common, or share, are 1 and 2.

Common Multiple

Any number that is a multiple of two or more numbers; a multiple that two or more numbers have in common, or share.

Key words: common, shared

Example:

12 is a common multiple of 2, 3, 4, and 6.

Composing Numbers

The process of creating a larger number through addition.

Example:

1,000 + 300 + 10 + 7 is composed as 1,317.

Composite Number

A number that has factors other than 1 and itself. A number that can be built in more than one way using tiles. Composite numbers can be written as the product of prime numbers.

Example:

Counting Numbers

The positive whole numbers. Also called natural numbers.

Example:

1, 2, 3, 4, 5, . . .

Decline

To decrease in number, value, or amount.

Decomposing Numbers

The process of separating a number into its component parts.

Example:

3,487 can be decomposed as 3,000 + 400 + 80 + 7.

Decrease

To make less (smaller).

Descending Order

Decreasing from greatest to least, but not necessarily in a fixed pattern. To count downward from largest to smallest.

Example:

101, 90, 87, 72, 56

A series of numbers in descending order.

Difference

The answer to a subtraction problem.

Example:

7 – 5 = 2

2 is the difference between the numbers.

Dividend

The number in division that is to be divided, or broken, into equal parts.

Example:

30 is the dividend.

Divisibility Rules

2A number is divisible by 2 if it is even or if the last digit is divisible by 2.

3A number is divisible by 3 if the sum of its digits is divisible by 3.

4A number is divisible by 4 if the number formed by the last two digits is divisible by 4 or if the last two digits are two zeros.

5A number is divisible by 5 if its last digit is 5 or zero.

6A number is divisible by 6 if the number is even and is divisible by 3.

9A number is divisible by 9 if the sum of its digits is divisible by 9.

10 A number is divisible by 10 if its last digit is a zero.

Divisible

Capable of being divided evenly without leaving a remainder.

Example:

20 ÷ 4 = 5

20 is divisible by 4.

Division

The process of division, meaning:

1. Breaking a number into smaller groups of equal quantities.

2. Repeated subtraction; subtracting the same number again and again.

3. Breaking a number into an equal amount of same-sized pieces.

Key words: evenly divided, evenly split, evenly shared between or among, fair shares, even groups of, how many would each get . . ., divisor, dividend, quotient, remainder, left over, fraction

Example:

Division Strategies

Larger Numbers: Multiples Table

1. Make a multiples table (see p. 11) for the number you are dividing by, the divisor.

2. Subtract the largest multiple of the divisor that does not exceed the dividend.

3. When you can’t subtract any more multiples of 100s, begin subtracting multiples of 10s, and then multiples of 1 through 9.

4. When you can’t subtract any more multiples, add the number of multiples that have been subtracted. This final number is the quotient.

Repeated Subtraction

23 was subtracted a total of four times with 6 remaining, so the answer is 4 r 6 or 4 .

Use Manipulatives

Use beans, counters, or other objects, or draw a picture. Count out the number of beans that need to be divided. Divide them equally into the number of groups that are you are dividing by.

Use tiles or paper squares. Count out the number of tiles that need to be divided. Put them into the number of rows you are dividing by. Count the number of columns you make, and then count the remainder.

Divisor

The number in a division problem by which the dividend is divided. The number used to divide by.

Example:

Double

To count something twice.

Twice as much.

Example:

Double 3 means the same thing as

3 + 3 = 6 or 3 x 2 = 6.

Equal

Having the same value in quantity, size, or amount.

Example:

Equality

The property of being equal. The following are true of equal numbers:

•a = a

•If a = b, then b = a

•If a = b and b = c, then a = c

Even Number

A whole number that has 0, 2, 4, 6, or 8 in the ones place. It can be divided evenly into two equal groups with no remainder. Even numbers are divisible by 2.

Example:

14 is an even number; when put into two equal groups, there is no remainder.

Expanded Form

Numbers broken up into their individual place values.

Example:

3,422 = 3,000 + 400 + 20 + 2

          = (3 x 1,000) + (4 x 100) + (2 x 10) + 2

          = (3 x 10³) + (4 x 10²) + (2 x 10¹) + (2 x 10⁰)

Fact Family

A group of addition and subtraction or multiplication and division facts made from the same three numbers.

Examples:

Addition/Subtraction (3, 7, 10)

7 + 3 = 10        10 – 3 = 7