Download UNIT-INTRODUCTION

MATHEMATICS
GRADE 7
UNIT - INTRODUCTION
1
VOCABULARY FOR FLASHCARDS
A.) Absolute Value: The absolute value of a number represents the
positive distance the number is away from the
origin (0). The symbol: x is used when the
absolute value of a number needs to be found.
Example 2:  2 = 2
Example 1: 2 = 2
B.) Associative Property: Addition and Multiplication
PLEASE NOTE: CHANGE IN GROUPING
Example 1: a + (b + c) = (a + b) + c
Example 2: 4 + (6 + 7) = (4 + 6) + 7
Example 3: a • (b • c) = (a • b) • c
Example 4: 2 x (3 x 4) = (2 x 3) x 4
C.) Commutative Property: Addition and Multiplication
PLEASE NOTE: CHANGE IN ORDER
Example 1: x + y = y + x
Example 2: 3 + 8 = 8 + 3
Example 3: x • y = y • x
Example 4:
3 1 1 3
  
4 3 3 4
2
D.) Distributive Property of Multiplication Over Addition(“sum of”):
a(b + c) = ab + ac OR ab + ac = a(b + c)
Distributive Property or Multiplication Over Subtraction(“difference of”):
a(b – c) = ab – ac OR ab – ac = a(b - c)
E.) Irrational Numbers: A number is irrational if it can not be written
as a fraction with an integer numerator and a
nonzero integer denominator.
*Three Major Categories We Study Are:
1) pi: 
2) Non-Perfect Square Roots:
3, 5, 6, 7,...
3) Decimals (Non-Terminating and Non-Repeating): 7.123145…,
.03003000300003000003…
F.) Order of Operations
PROCESS:
1. Use PEMDAS
(PARENTHESIS, EXPONENTS, MULTIPLICATION, DIVISION, ADDITION, SUBTRACTION)
2. PLEASE REMEMBER THAT DIVISION IS DONE FIRST IF IT COMES
BEFORE MULTIPLICATION
3. PLEASE REMEMBER THAT SUBTRACTION IS DONE FIRST IF IT
COMES BEFORE ADDITION
4. Underline the pair of numbers being worked on
5. Work down in a V-formation until expression is simplified
3
G.) Rational Numbers: A number is rational if it can be written as a
fraction with an integer numerator and a
nonzero integer denominator.
3
Examples: 0.4, 2. 9̅, 16 , , 5
4
4
Subsets of Real Numbers
A.) Counting Numbers or Natural Numbers: 1, 2, 3, 4, 5…
B.) Whole Numbers: 0, 1, 2, 3, 4, 5…
C.) Integers: (negative infinity) -  ,…..,-3, -2, -1, 0, 1, 2, 3, …..,  (infinity)
ADDITIVE INVERSE (OPPOSITE),
PROPERTIES AND IDENTITY ELEMENTS
A.) Additive Inverse (Opposite): A number and its opposite have a sum of
zero.
Example 1: g + (-g) = 0
Example 2: 5 + (-5) = 0
Example 3:
1  1 
   0
2  2 
5
B.) Addition Property of Zero: A number added to zero leaves the
number unchanged.
Example 1: g + 0 = g
Example 2: 5 + 0 = 5
Example 3:
1
1
0 
2
2
C.) Associative Property: Addition and Multiplication
PLEASE NOTE: CHANGE IN GROUPING
Example 1: a + (b + c) = (a + b) + c
Example 2: 4 + (6 + 7) = (4 + 6) + 7
Example 3: a • (b • c) = (a • b) • c
Example 4: 2 x (3 x 4) = (2 x 3) x 4
D.) Commutative Property: Addition and Multiplication
PLEASE NOTE: CHANGE IN ORDER
Example 1: x + y = y + x
Example 2: 3 + 8 = 8 + 3
Example 3: x • y = y • x
Example 4:
3 1 1 3
  
4 3 3 4
6
E.) Multiplication Property of One: For any whole number a,
a • 1 = a and 1 • a = a
Example 1: 1 x 5 = 5
Example 2: 6 x 1 = 6
F.) Multiplication Property of Zero: For any whole number a, a • 0 = 0 and
0 • a = 0
Example 1: 0 x 4 = 0
Example 2: 4 x 0 = 0
G.) Additive Identity Element of 0: 0 is the additive identity element
because anything added to 0 will be that number.
H.) Multiplicative Identity Element of 1: 1 is the multiplicative identity
because anything multiplied by 1 will be that number.
7
I.) Distributive Property of Multiplication Over Addition(“sum of”):
a(b + c) = ab + ac OR ab + ac = a(b + c)
Distributive Property or Multiplication Over Subtraction(“difference of”):
a(b – c) = ab – ac OR ab – ac = a(b - c)
HOW THE DISTRIBUTIVE PROPERTY CAN BE USED
FINDING PRODUCTS
Example 1: 4(3 + 2) = (4 x 3) + (4 x 2) = 12 + 8 = 20
Example 2: 2(8 – 3) = (2 x 8) – (2 x 3) = 16 – 6 = 10
Example 3: (9 x 20) + (9 x 7) = 9(20 + 7) = 9(27) = 243
Example 4: (5 x 7) – (7 x 2) = 7(5 – 2) = 7 x 3 = 21
CHANGING THE FORM OF AN ALGEGRAIC EXPRESSION
Example 1: 8(x + y) = 8x + 8y
Example 2: 3.4(a – b) = 3.4a – 3.4b
Example 3:
1
1
1
(9 x  6 x)  (  9 x   6 x)  4.5 x  3 x  1.5 x
2
2
2
8
ABSOLUTE VALUE
A.) Absolute Value: The absolute value of a number represents the
positive distance the number is away from the
origin (0). The symbol: x is used when the
absolute value of a number needs to be found.
Example 1: 2 = 2
Example 2:  2 = 2
Example 3: 0 = 0
Example 4:
Example 5:   2 =  2
Example 6:

2 = 2

62
6-2
4
Example 7:  7  3

7–3
-10
Example 8:
Example 9: 0   2
Example 10:

8  9
8+9
17
 2

2
0–2

2
4
9
2
2
IRRATIONAL AND RATIONAL NUMBERS
A. ) Rational Numbers: A number is rational if it can be written as a
fraction with an integer numerator and a
nonzero integer denominator.
3
Examples: 0.4, 2. 9̅, 16 , , 5
4
B. ) Irrational Numbers: A number is irrational if it can not be written
as a fraction with an integer numerator and a
nonzero integer denominator.
*Three Major Categories We Study Are:
1) pi: 
2) Non-Perfect Square Roots:
3, 5, 6, 7,...
3) Decimals (Non-Terminating and Non-Repeating): 7.123145…,
.03003000300003000003…
*AN EASY WAY TO REMEMBER IRRATIONAL NUMBERS*
An irrational number is a nonrepeating/nonterminating decimal (never ends!!!!!)
10
EXPONENTS
The example below has two parts: a base and an exponent(power).
The base is 5 and the exponent(power) is 3.
5
3
Example 1: 5 2 , means 5 x 5, read: five to the second power,
or five squared or 25
Example 2: 10 3 , means 10 x 10 x 10, read: ten to the third power,
or ten cubed or 1000
Example 3: 6 0 , read: six to the zero power or 1
ANY NUMBER, OTHER THAN ZERO, RAISED TO THE ZERO POWER IS 1
SQUARE ROOTS
Some counting numbers can be written as the product of two equal factors.
For example:
1 = 1 x 1
16 = 4 x 4
4 = 2 x 2
25 = 5 x 5
9 = 3 x 3
36 = 6 x 6
Counting numbers such as 1, 4, 9, 16, 25 and 36 are called square numbers
or perfect squares. These numbers can be written using the exponent of 2.
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
One of the two equal factors of a square number is called the square root of
the number. To show the square root of a number, a radical sign (√ ) is
used. Therefore the following is true:
√𝟏 = 𝟏
√𝟒 = 𝟐
√𝟗 = 𝟑
√𝟏𝟔 = 𝟒 √𝟐𝟓 = 𝟓
√𝟑𝟔 = 𝟔
11
RATIONAL NUMBER VOCABULARY
A.) Fraction: A number that has a numerator and denominator.
𝟐
Examples:
𝟕
,
𝟗 𝟏
, ,
𝟓 𝟏𝟏 𝟖 𝟑
B.) Proper Fraction: When the numerator is less than the denominator.
𝟏 𝟐
Examples:
, ,
𝟒
𝟐 𝟑 𝟏𝟏
C.) Improper Fraction: When the denominator is greater than the
numerator.
𝟗 𝟏𝟏 𝟏𝟎
Examples:
𝟖
,
𝟑
,
𝟒
D.) Mixed Number: Is a number that has a fraction attached to an integer.
𝟐
𝟒
𝟕
𝟏𝟏
Examples: 𝟏 , 𝟗
,𝟕
𝟑
𝟖
E.) Reciprocal: When the numerator and the denominator of a fraction is
flipped.
Example:
𝟐
𝟑
, reciprocal is
𝟑
𝟐
F.) Simplifying(Vertically and/or Diagonally): Is a method of simplifying two
or more fractions by simplifying a numerator and denominator. This may
be repeated more than once if able.
Example:
𝟏𝟎
Example:
𝟐
𝟏𝟐
𝟑
𝟑
𝟐
𝟏
𝟏
𝟏
𝟏
𝟓
𝟒
𝟏
𝟐
𝟏
𝟐
𝟏
𝟑
𝟑
𝟏
𝟓
𝟓
× = × = × =
×
𝟗
𝟏𝟎
= × =
12
ADDING, SUBTRACTING, BORROWING WITH FRACTIONS AND MIXED NUMBERS
METHOD (DENOMINATORS ARE THE SAME):
1. Perform the operation(s) asked
2. Simplify if asked
METHOD (DENOMINATORS ARE NOT THE SAME):
1. Rewrite the fractions with a common denominator
(be sure to adjust the numerator)
2. Perform the operation(s) asked
3. Simplify
EXAMPLES
5
8
1.
+
7
8
-
___
1
7
=
1
1 2
2
1
72
14
10
9
+2
3
14
= +2
3
3
 2
14
14

___
____
__________

3
1

9
3
8
8  3 24
=
=
9
9  3 27
3
3
3
+
= + =+
27
27
27
____
3. 1
____
12
4
1
1 1
8
8
2
4.
7
9
2.
___
5.
3
5
14
1
1 4
4
40
4
4
3
[BORROWING]
9
9 4
36
36
2
29
18
18
 2
 2
- 2  2
4
49
36
36
4
____ _____
____ ____
27
1
27
1
13
22  2
11
1
36  2
18
MULTIPLYING FRACTIONS AND/OR MIXED NUMBERS
METHOD:
1. Convert all mixed numbers to improper first
2. Simplify if able
3. Multiply the numerators together
4. Multiply the denominators together
EXAMPLES
1.
3 6

4 5
1
2
2. 2  3

5 10

2
3
3 3

2 5

5 5

1 3
9
10

3.
1
3
10 3 20
 
15 5 21
25
1
 8
3
3
3 8
4 9
4. 3   1

2 1 4
 
3 1 7
1
2
15 8 3
 
4 9 2

5 2 3
 
1 3 2
8
21


5 1 1
5 
  =
 5
1 1 1
1
5. 5 

9
20
5 9 1 9 9
1

 

 2
1 20 1 4
4
4
14
DIVIDING FRACTIONS AND/OR MIXED NUMBERS
METHOD:
1. Convert all mixed numbers to fractions
2. Change all division signs to multiplication signs and take the reciprocal of
all fractions following the changed operation sign(s)
3. Simplify if able
4. multiply fractions (numerator times numerator and denominator times
denominator)
EXAMPLES
1.
3 9

7 14
1
2
2. 2 

5 8

2 5
3 14

7 9

1 4  = 4
1 1
2
1 2
 =
3
1 3
2
5

27 1  27
2
 
 1
5
5
25
25
3. 5 5
5
8
4.
8 12 16


9 21 11
8 21 11


9  12 16
1 7 11 77
5
   
 1
3
72
12 2
72
5. 8  6
1
8
8  49

1
8
8
8
64
15

 
 1
1
49
49
49
15
Changing Fractions and Mixed Numbers To Decimals
Method:
1. Divide the numerator by the denominator
.8
5 4.0
4
1.
5
0.8
-40
0
2. 2
.75
4 3.00
3
4
2.75
-28
20
-20
0
Changing Decimals To Fractions and Mixed Numbers
Method:
1. Determine the place holder where the decimal terminates (ends)
2. Write the numeral of the decimal part over where the place holder where
the decimal terminated (ended)
3. Simplify if asked
Please Note: If the decimal has a whole number attached to it, write this in
front of the fraction. The decimal will convert to a mixed
number.
1. 0.7 =
7
10
2. 2.94 = 2
94  2
47
2
100  2
50
16
Changing Mixed Numbers To Improper Fractions
Method:
1. Multiply the denominator by the whole number and then add it to the
numerator (this becomes the new numerator)
2. Rewrite this new number over the original denominator
Please Note: If it is a negative mixed number, ignore the negative sign and
attach at the end.
1
4
1. 3 
(4  3)  1 13

4
4
9 (17 1)  9  26
2. 1 

17
17
17
Changing Improper Fractions To Mixed Numbers
Method:
1. Divide the numerator by the denominator (this determines the whole
number)
2. Write the remainder over the denominator
3. Write the whole number next to the fraction
Please Note: If the improper fraction is negative, ignore the negative sign
and attach at the end.
1.

17
3
5
3 17
-15
2
2.
2
5
3
9
8
1
89
-1
-8
1
17
1
8
Math Rap For Operations With Fractions by Mrs. Pirillo
Top Times Top
Bottom Times Bottom
Cancel If You Can
Reduce At The End
And When You Divide
You Flip The Other Side!!!!!
Adding And Subtracting Is A Different Game
The Bottoms Need To Be The Same
And If They’re Not, Don’t Bother Me
Go And Find The L.C.D!!!!!
18
ADDING AND SUBTRACTING DECIMALS
METHOD:
1. Write the given numbers one above the other with the decimal points in
line.
2. Add zeros to get the same number of decimal places and then add or
subtract as if the numbers were whole numbers.
3. Place a decimal point in the number for the sum or difference in position
under the decimal points in the given numbers.
Example 1: 6.47 + 340.8 + 73.523
6.470
340.800
+ 73.523
420.793
Example 2: 13.94 – 7.693
13.940
-7.693
6.247
19
ROUNDING
METHOD:
1. Find the place to which you wish to round and mark it with a line under
it.
2. Draw a vertical line segment after the place holder to rounded to.
3. If the digit to the right of the vertical line is 5 or greater, increase the
place holder being rounded to by 1. If the digit to the right of the
vertical line is less than 5, leave the marked digit unchanged.
4. If rounding a whole number, replace each digit to the right of the marked
place with 0.
5. If a mixed decimal, do not add 0 to the right of the marked place to be
rounded.
Complete the table below by rounding to the appropriate place
holder.
NUMBER
456
ROUNDING PLACE
TENS
ANSWER
460
456
HUNDREDS
0.87872
TEN-THOUSANDTHS
0.87872
HUNDREDTHS
0.88
12.111
TENTHS
12.1
12.111
ONES
7465
HUNDREDS
7500
7465
TENS
7470
500
0.8787
12
20
MULTIPLYING DECIMALS
METHOD:
1. Right justify the two numbers being multiplied
2. Write down to the right side of the numbers the amount of decimal
places needed to be moved in order to make it a whole number
3. Find the product of the two factors
4. Place the decimal point in the product so that the number of places to
the right of the decimal point in the product is the sum of the number of
places to the right of the decimal point in the factors
EXAMPLES
1. 14.92 x 7.2
14.92
x7.2
2984
+10444
107.424
2. 820.3 x 3.2
2 places
1 place
820.3
x3.2
16406
+24609
2624.96
3 places
21
1 place
1 place
2 places
DIVIDING DECIMALS
METHOD:
1. If needed, multiply the dividend and divisor by a power of ten that makes
the divisor a counting number.
2. Divide the new dividend by the new divisor.
3. Check by multiplying the quotient and divisor.
EXAMPLE
1.25
2.5232
3.14500
-250
654
-625
290
-250
400
-375
250
-250
0
22
COMPARING TWO MORE FRACTIONS
METHODS:
1. Cross multiplying technique
2. Change to fractions with same denominator
3. Change to decimals-stack them on top of each other and compare by
looking at columns
EXAMPLE: Which is bigger?
𝟓
𝟔
𝐨𝐫
𝟕
𝟖
CROSS MULTIPLYING
FRACTION
DECIMAL
𝟓
𝟕
𝐨𝐫
𝟔
𝟖
𝟓 𝟒𝟎
=
𝟔 𝟒𝟖
𝟓
= 𝟎. 𝟖𝟑𝟑 …
𝟔
40 < 42
𝟕 𝟒𝟐
=
𝟖 𝟒𝟖
𝟕
= 𝟎. 𝟖𝟕𝟓
𝟖
𝟓 𝟕
<
𝟔 𝟖
0.833…
0.875
𝟓
𝟕
<
𝟔
𝟖
𝟓 𝟕
<
𝟔 𝟖
23
ORDER OF OPERATIONS
PROCESS:
1. Use PEMDAS
(PARENTHESIS, EXPONENTS, MULTIPLICATION, DIVISION, ADDITION, SUBTRACTION)
2. PLEASE REMEMBER THAT DIVISION IS DONE FIRST IF IT COMES
BEFORE MULTIPLICATION
3. PLEASE REMEMBER THAT SUBTRACTION IS DONE FIRST IF IT
COMES BEFORE ADDITION
4. Underline the pair of numbers being worked on
5. Work down in a V-formation until expression is simplified
For Examples 1 – 6, use Order of Operations to simplify the expressions.
19 + 84 ÷ 4 x 8 – 11
(5.4 ÷ 0.6) x 5.5 – (3.6 + 6.7)
19 + 21 x 8 – 11
9 x 5.5 - 10.3
19 + 168 - 11
49.5 – 10.3
187 – 11
39.2
176
𝟖
2 x
𝟐
12 - ( + 𝟑 ÷ )
𝟓
𝟑
𝟖
𝟐
𝟑
𝟗
12 - ( + )
𝟓
𝟐
12 𝟓
𝟏
𝟑
÷ 2 x
÷ 2 x
𝟏 𝟏
×
𝟑 𝟓
𝟔𝟏
𝟏𝟎
𝟏
𝟏𝟓
𝟗
𝟏𝟎
24
𝟏
𝟓
𝟏
𝟓