1. Place Value, Names for Numbers, and Reading Tables
Module 1
Place Value, Names for Numbers, and Reading Tables

2. Place Value
The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 can be used to write numbers.
The whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …
The position of each digit in a number determines its place value.

3. Whole Numbers in Words and Standard Form
The whole number 1,083,664,500 is written in standard form.
Each group of three digits is called a period.
Writing a Whole Number in Words
To write a whole number in words, write the number in each period followed by the name of the period. (The ones period is usually not written.) This same procedure can be used to read a whole number.

4. Whole Numbers in Words and Standard Form
Writing a Whole Number in Standard Form
To write a whole number in standard form, write the number in each period,
followed by a comma.

5. Writing a Whole Number in Expanded Form
The expanded form of a number shows each digit of the number with its place value.

6. Rounding and Estimating

7. Rounding Whole Numbers
Rounding a whole number means approximating it.
26 is closer to 30 than 20, so 26 is rounded to the nearest ten is 30.
52 is closer to 50 than 60, so 52 is rounded to the nearest ten is 50.

8. Rounding Whole Numbers
25 is halfway between 20 and 30. It is not closer to either number. In such a case, we round to the larger ten, that is, to 30.

9. Rounding Whole Numbers
Rounding Whole Numbers to a Given Place Value
Step 1: Locate the digit to the right of the given place value.
Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and replace each digit to its right by 0.
Step 3: If this digit is less than 5, replace it and each digit to its right by 0.

10. Estimating Sums and Differences
Estimate the sum by rounding each number to the nearest hundred.
Example
Objective B

11. Dividing Whole Numbers

12. Dividing Whole Numbers
The process of separating a quantity into equal parts is called division.
Objective A

13. Properties of 1 Division Properties of 1
The quotient of any number (except 0) and that same number is 1. For example,
The quotient of any number and 1 is that same number. For example,

14. Properties of 0
The quotient of 0 and any number (except 0) is 0. For example,
The quotient of any number and 0 is not a number. We say that
are undefined.

15. Solving Problems by Dividing
Key Words or Phrases
Example
Symbols
Divide
Divide 8 by 4
8 ÷ 4 or
Quotient
The quotient of 64 and 8
64 ÷ 4 or
Divided by
12 divided by 4
12 ÷ 4 or
Divided or Shared
Equally Among
$75 divided equally among three people
75 ÷ 3 or
per
100 miles per 2 hours
Objective C

16. Exponents and Order of Operations

17. Using Exponential Notation
In the product 3  3  3  3  3, notice that 3 is a factor several times.
Objective A
The exponent, 5, indicates how many times the base, 3 is a factor.

18. Using Exponential Notation
Expression
In Words 32
“three to the second power” or “three squared.” 33
“three to the third power” or “three cubed” 34
“three to the fourth power”
Objective A Continued

19. Examples
Evaluate.
1. 92
2. 45
3. 5  32
Objective B

20. Using the Order of Operations
1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars or square roots, starting with the innermost set.
2. Evaluate any expressions with exponents.
3. Multiply or divide in order from left to right.
4. Add or subtract in order from left to right.
Objective D

21. Introduction to Variables, Algebraic Expressions, and Equations

22. Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra
Algebraic Expressions
A combination of operations on letters (variables) and numbers is called an algebraic expression.
Algebraic Expressions
5 + x  y  y – 4 + x
4x means 4  x
and
xy means x  y

23. Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra
Algebraic Expressions
Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable.

24. Chapter 1 / Whole Numbers and Introduction to Algebra
Equation
Statements like = 7 are called equations. An equation is of the form expression = expression An equation can be labeled as
Equal sign
x = 9
left side
right side

25. Chapter 1 / Whole Numbers and Introduction to Algebra
Solutions
When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable. A solution of an equation is a value for the variable that makes an equation a true statement.

26. Chapter 1 / Whole Numbers and Introduction to Algebra
Keywords and Phrases
Keywords and phrases suggesting addition, subtraction, multiplication, division or equals.
Addition
Subtraction
Multiplication
Division
Equal Sign
sum
difference
product
quotient
equals
plus
minus
times
into
gives
added to
less than of
per
is/was/ will be
more than
less
twice
divide
yields
total
decreased by
multiply
divided by
amounts to
increased by
subtracted from
double
is equal to

27. Helpful Hint Phrase Translation a number decreased by 5 x – 5
Remember that order is important when subtracting. Study the order of numbers and variables below.
Phrase
Translation
a number
decreased by 5
x – 5
subtracted from 5
5 – x

28. Introduction to Integers

29. Positive and Negative Numbers
Numbers greater than 0 are called positive numbers. Numbers less than 0 are called negative numbers.
negative numbers
zero
positive numbers 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6

30. Negative and Positive Numbers
–3 indicates “negative three.”
3 and + 3 both indicate “positive three.”
The number 0 is neither positive nor negative.
zero
negative numbers
positive numbers 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6

31. Chapter 1 / Whole Numbers and Introduction to Algebra
Comparing Integers
We compare integers just as we compare whole numbers. For any two numbers graphed on a number line, the number to the right is the greater number and the number to the left is the smaller number.
<
means
“is less than”
>
means
“is greater than”

32. Graphs of Integers 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6
The graph of –5 is to the left of –3, so –5 is less than –3, written as – 5 < –3 .
We can also write –3 > –5.
Since –3 is to the right of –5, –3 is greater than –5. 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 –1 –2 –3 –4 –5 –6

33. Absolute Value
The absolute value of a number is the number’s distance from 0 on the number line. The symbol for absolute value is | |.
is 2 because 2 is 2 units from 0. 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6
is 2 because –2 is 2 units from 0.

34. Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
Since the absolute value of a number is that number’s distance from 0, the absolute value of a number is always 0 or positive. It is never negative.
zero
a positive number

35. Opposite Numbers
Two numbers that are the same distance from 0 on the number line but are on the opposite sides of 0 are called opposites.
5 units
5 and –5 are opposites. 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6

36. Chapter 1 / Whole Numbers and Introduction to Algebra
Opposite Numbers
5 is the opposite of –5 and –5 is the opposite of 5.
The opposite of 4 is –4 is written as
–(4) = –4
The opposite of – 4 is 4 is written as
–(– 4) = 4
If a is a number, then –(– a) = a.

37. Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
Remember that 0 is neither positive nor negative. Therefore, the opposite of 0 is 0.

38. Adding Integers

39. Adding Integers
Adding integers can be visualized by using a number line. A positive number can be represented on the number line by an arrow of appropriate length pointing to the right, and a negative number by an arrow of appropriate length pointing to the left.
Objective A 39

40. Add using a number line. −2 + (−3)
Objective A 40

41. Add using a number line. −8 + 3
Objective A 41

42. Adding Signed Numbers Adding Two Numbers with the Same Sign
Step 1: Add their absolute values.
Step 2: Use their common sign as the sign of the sum.
Objective A 42

43. Adding Signed Numbers Adding Two Numbers with Different Sign
Step 1: Find the larger absolute value minus the smaller absolute value.
Step 2: Use the sign of the number with the larger absolute value as the sign of the sum.
Objective A 43

44. Opposites If a is a number, then −a is its opposite. Also, Objective A44

45. Subtracting Integers

46. Subtracting Integers Subtracting Two Numbers
If a and b are numbers, then a – b = a + (–b).
Objective A 46

47. Evaluating Expressions
Evaluate 3x ‒ y for x = 2 and y = – 6.
Replace x with 2 and y with –6 in 3x + y.
3x + y = 3 · 2 ‒ (–6)
Objective B
= 6 ‒ (–6)
= 6 + 6
= 12

48. Multiplying and Dividing Integers

49. Consider the following pattern of products.
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Integers
Consider the following pattern of products.
First factor
decreases by 1
each time.
3  5 = 15
Product
decreases by 5
each time.
2  5 = 10
1  5 = 5
0  5 = 0
This pattern continues as follows.
– 1  5 = - 5
– 2  5 = - 10
– 3  5 = - 15
This suggests that the product of a negative number and a positive number is a negative number.

50. Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Integers
Observe the following pattern.
2  (– 5) = - 10
Product increases by 5 each time.
1  (– 5) = - 5
0  (– 5) = 0
This pattern continues as follows.
– 1  (– 5) = 5
– 2  (– 5) = 10
– 3  (– 5) = 15
This suggests that the product of two negative numbers is a positive number.

51. Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Integers
The product of two numbers having the same sign is a positive number.
2  4 = 8
– 2  (–4) = 8
The product of two numbers having different signs is a negative number.
2  (–4) = – 8
–2  4 = – 8

52. Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Integers
Product of Like Signs
( + )( + ) = +
(–)(–) = +
Product of Different Signs
(–)( + ) = –
( + )(–) = –

53. Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
If we let ( – ) represent a negative number and ( + ) represent a positive number, then
( – ) ( – ) = ( + )
The product of an even number of negative numbers is a positive result.
( – ) ( – ) ( – ) = ( – )
The product of an odd number of negative numbers is a negative result.
( – ) ( – ) ( – ) ( – ) = ( + )
( – ) ( – ) ( – ) ( – ) ( – ) = ( – )

54. Division of integers is related to multiplication of integers.
Chapter 1 / Whole Numbers and Introduction to Algebra
Dividing Integers
Division of integers is related to multiplication of integers. 3 2 6 =
because =
– 3 2
– 6
because
– 3
(– 2) 6 =
because
– 2 = 3
– 6
because
(– 2)
– 6
– 2

55. Chapter 1 / Whole Numbers and Introduction to Algebra
Dividing Integers
Chapter 1 / Whole Numbers and Introduction to Algebra
The quotient of two numbers having the same sign is a positive number.
12 ÷ 4 = 3
– 12 ÷ (– 4 ) = 3
The quotient of two numbers having different signs is a negative number.
– 12 ÷ 4 = – 3
12 ÷ (– 4) = – 3

56. Chapter 1 / Whole Numbers and Introduction to Algebra
Dividing Numbers
Chapter 1 / Whole Numbers and Introduction to Algebra
Quotient of Like Signs
Quotient of Different Signs

57. Example Divide. a. b. = 0 because 0(–7) = 0
is undefined because there is no number that gives a product of 8 when multiplied by 0.

58. Evaluating Expressions
Evaluate 7xy for x = –5 and y = 8.
Replace x with –6 and y with 8 in 7xy.
7xy = 7 (–6)(8)
= – 42 · 8
Objective B
= –336

59. Order of Operations

60. Chapter 1 / Whole Numbers and Introduction to Algebra
Order of Operations
Chapter 1 / Whole Numbers and Introduction to Algebra
1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set.
2. Evaluate any expressions with exponents.
3. Multiply or divide in order from left to right.
4. Add or subtract in order from left to right.

61. Chapter 1 / Whole Numbers and Introduction to Algebra
Helpful Hint
Chapter 1 / Whole Numbers and Introduction to Algebra
When simplifying expressions with exponents, parentheses make an important difference.
(–5)2 and –52 do not mean the same thing.
(–5)2 means (–5)(–5) = 25.
–52 means the opposite of 5 ∙ 5, or –25.
Only with parentheses is the –5 squared.