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Math 8/7 A

Lesson 1

Arithmetic with Whole Numbers

and Money –

Variables and Evaluation

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Basic Information

Counting numbers or natural numbers:

{1, 2, 3, 4, 5, …}

Whole numbers:

{0, 1, 2, 3, 4, …}

Amounts of money are indicated with a dollar sign ($) or with

a cent sign (¢) but not both. $0.50¢ is incorrect!

Operations of arithmetic:

Addition

Subtraction

Multiplication

Division

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Addition

Addition is bringing two or more numbers (or things) together to make a new total.

Numbers that are added together are called addends. The result of addition is the sum.

addend + addend = sum

Practice Problems:

Add:

a. 36 + 472 + 3614

b. $1.45 + $6 + 8¢

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Subtraction

Subtraction is taking one number away from another.

In subtraction the subtrahend is taken from the minuend. The result is the difference.

Other names used in subtraction are Minus, Less, Difference, Decrease, Take Away, Deduct

Practice Problems:

Subtract:

a. 5207 – 948

b. $5 - 25¢

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Multiplication

Numbers that are multiplied together are called factors. The result of their

multiplication is the product.

factor * factor = product

Multiplication can be indicated in several ways:

4 · 5 4 x 5 4 * 5 4(5) ab

Multiplication Words:

Groups of - 3 groups of 2 make 6.

Sets of - 2 sets of 4 make 8.

Lots of - 2 lots of 2 make 4.

Multiplied -3 multiplied by 4 is 12

Product - the product of 5 and 3 is 15.

Times - 3 times 2 is 6.

Practice Problems:

Multiply: (a) 164 · 23 (b) $4.68 x 20 (c) 5(29¢)

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Division

Division is splitting into equal parts or groups.

In division the dividend is divided by the divisor. The result is the quotient.

Division can be indicated in several ways:

dividend ÷ divisor = quotient dividend/divisor = quotient

Divide: (a) 1234 ÷ 56 (b) $12.60/5

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Valuations and EvaluationA variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A constant is a number we know. A constant has a fixed value.

We evaluate an expression by calculating its value when the variables are

assigned specific numbers.

Evaluate each expression for x = 10 and y = 5:

(a) x + y (b) x – y (c) xy (d) x/y

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Math 8/7 A

Lesson 2

Properties of Operations

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Properties of OperationsAddition and subtraction are inverse operations. They undo each other.

2 + 3 = 5 5 – 3 = 2

Together, the numbers 2, 3, and 5 form an addition and subtraction fact family. We

can write two addition facts and two subtraction facts:

2 + 3 = 5 5 – 3 = 2

3 + 2 = 5 5 – 2 = 3

The order of addends do not matter. This is called the commutative property of

addition. a + b = b + a

The order of the minuend and subtrahend do matter. Therefore subtraction is not

commutative! 5 – 3 ≠ 3 – 5

The identity property of addition states when zero is added to any number, the

sum is that number. x + 0 = x Zero is the additive identity.

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Properties of Operations (cont)Multiplication and division are inverse operations. They undo each other.

4 * 5 = 20 20 ÷ 5 = 4

Together, the numbers 4, 5, and 20 form a multiplication and division fact family.

We can write two multiplication facts and two division facts:

4 * 5 = 20 20 ÷ 5 = 4

5 * 4 = 20 20 ÷ 4 = 5

The order of factors do not matter. This is called the commutative property of

multiplication. a * b = b * a

The order of the dividend and divisor do matter. Therefore division is not

commutative! 20 ÷ 5 ≠ 5 ÷ 20

The identity property of multiplication states when any number is multiplied by

1, the product is that number. x * 1 = x One is the multiplicative identity.

The property of zero for multiplication states that when a number is multiplied

by zero, the product is zero.

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More Properties of OperationsArithmetic is binary, which means we work with only two numbers in one step.

2 + 3 + 4 = 9

2 + (3 + 4) = 9

(2 + 3) + 4 = 9

The associative property of addition states that the grouping of addends do not

change the sum. (a + b) + c = a + (b + c)

The associative property of multiplication states that the grouping of factors does

not change the product. (a * b) * c = a * (b * c)

Neither subtraction nor division are associative!

(8 – 4) – 2 ≠ 8 – (4 – 2)

(8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2)

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Practice Problems

Example 1: Name each property:

a) 5 * 3 = 3 * 5

b) (3 + 4) + 5 = 3 + (4 + 5)

c) 6 + 0 = 6

d) 6 * 0 = 0

Example 2: Which property can we use to find the missing number?

a) 8 + ? = 8

b) 1 * ? = 9

c) 10 * ? = 0

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Sequences

This lesson is the same as the Patterns Lesson we learned

in PSSA Coach!

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Step 1Are the numbers increasing or decreasing?

• If increasing

– Add,

– Multiply, or

– Square

• If decreasing

– Subtract or

– Divide

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Step 2

Compare consecutive numbers

• If increasing

1, 3, 5, 7, 9

3, 6, 12, 24, 48

• If decreasing

99, 88, 77, 66, 55

3125, 625, 125, 25

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Step 3

Use the rule!

Increasing numbers

1, 3, 5, 7, 9

• Increasing by the same number so

Add ?

• So the next number is ?

3, 6, 12, 24, 48

• Not increasing by the same

number so Multiply by ?

• So next number is ?

Decreasing numbers

99, 88, 77, 66, 55

• Decreasing by the same number

so Subtract ?

• So the next number is ?

3125, 625, 125, 25

• Not decreasing by the same

number so Divide by ?

• So next number is ?

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Practice Problems

Find the next three terms in these sequences:

a. 0, 2, 4, 6, 8, …

b. 2, 6, 18, 54, 162, …

c. 50, 46, 42, 38, …

d. 1, 4, 9, 16, …

Now do the Practice Set on page 12.

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Math 8/7 A

Lesson 3

Missing Numbers in

Addition, Subtraction,

Multiplication, and Division

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EquationsAn equation is a number sentence that says two quantities are

equal.

3 + 4 = 7 5 + a = 9

An equation has a left side and a right side.

In order for an equation to be true, the left side must equal the

right side.

When you see an equation, always ask yourself “Is it true?”

When an equation has a missing number, ask yourself “What

number will make this statement true?”

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Missing Numbers in AdditionVariables are used to represent missing numbers.

Missing Sum Missing Addend Missing Addend

2 + 3 = n 2 + a = 5 b + 3 = 5

What missing numbers will make these statements true?

Find the missing addends by using the inverse operation subtraction. Subtract the

known addend from the sum!

When there are more than two addends, subtract all the known addends from the sum.

3 + 4 + n + 7 + 8 = 40

40 – (3 + 4 + 7 + 8)

Find the missing number that will make each of these equations true:

(a) n + 53 = 75 (b) 26 + a = 61 (c) 3 + 4 + n + 7 + 8 = 40

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Missing Numbers in SubtractionWhat missing numbers will make these statements true?

Missing Minuend Missing Subtrahend Missing Difference

n – 3 = 2 5 – x = 2 5 – 3 = m

Find the missing minuends by using the inverse operation addition. Add the other two

numbers.

Find the missing subtrahend or difference by subtracting.

Find the missing number that will make each of these equations true:

(a) p – 24 = 17 (b) 32 – x = 14

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Missing Numbers in MultiplicationWhat missing numbers will make these statements true?

Missing Product Missing Factor Missing Factor

3 * 2 = p 3f = 6 r * 2 = 6

Find the missing product by multiplying the factors. Find the missing factors by using

the inverse operation division.

Find the missing number that will make each of these equations true:

(a) 12n = 168 (b) 7k = 105 (c) 2 * 3a = 30

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Missing Numbers in DivisionWhat missing numbers will make these statements true?

Missing Quotient Missing Divisor Missing Dividend

Find the missing quotient by dividing the dividend by the divisor.

Find the missing divisor by dividing the dividend by the quotient.

Find the missing dividend by using the inverse operation multiplication. Multiply

the quotient by the divisor.

Find the missing number that will make each of these equations true:

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Math 8/7 A

Lesson 4

Number Line

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The Number Line

The tick marks indicate the location of the integers.

The zero point on the number line is called the origin

The numbers to the right of the origin are called positive numbers

All positive numbers are greater than zero

The numbers to the left of the origin are called negative numbers

All negative numbers are less than zero

Zero is neither positive nor negative

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Comparing Numbers

Comparison Symbols

(=) Equal sign

(<) Less Than sign – points to the smaller numbers on the number line. -5 < 4

(>) Greater Than sign = points to the bigger numbers on the number line . 5 > -6

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Practice Problems

1. Arrange these numbers in order from least to greatest => 0, 1, -2

2. What is the correct comparison symbol? -5 _ 3

3. Show this addition problem on the number line => 3 + 2

4. Show this subtraction problem on the number line => 5 – 3

5. Show this subtraction problem on the number line => 3 – 5

6. Simplify: 376 – 840

Now do the Practice Set on page 24.

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Math 8/7 A

Lesson 5

Place Value Through Hundred Trillions

–

Reading and Writing Whole Numbers

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Hundred Trillions

The trillions place is the 13th place from the right.

Like all other place-value groups, the TRILLIONS also has a Ten-trillion and

Hundred-trillion place.

a. Which digit is in the trillions place in the number 32,567,890,000,000?

b. In 12,457,697,380,000, what is the place value of the digit 4?

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Expanded Notation

Expanded notation is writing each non-zero digit times its place

value.

5280 in expanded notation is (5 * 1000) + (2 * 100) + (8 * 10)

Write 25,000 in expanded notation

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Reading and Writing Whole Numbers

For whole numbers with more than three digits, use commas to make the number

easier to read. Use commas every three decimal places.

32,567,890,000,000

Hyphenate numbers between 20 and 100 that do not end in zero.

52 is written “fifty-two”

76 is written “seventy-six”

95 is written “ninety-five”

Practice problems:

1. Use words to write 1,380, 000,050,200

2. Use words to write 3406521

3. Use digits to write twenty trillion, five hundred ten million

4. Use only digits and commas to write 25 million

5. Write twenty four hundred

Now do the Practice Set on page 30