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Chapter 1
Decimals and Integers
MATH 7
Using Estimation Strategies
LESSON 1-1
Definition
Numbers that are easy
to compute mentally
Facts/Characteristics
Also called “friendly
numbers” as mental
calculations are easy
Vocabulary
Word
3.67 x 42.5
4 x 40 = 160
3.67 x 42.5 ≈ 160
Examples
Compatible
Numbers
10.93 + 3.25
11 + 3
10.93 + 3.25 ≈ 14
Non-Examples
Definition
Add the “front-end
digits,” estimate the
sum of the remaining
Facts/Characteristics
Gives a higher estimate
as involves the cents, less
likely to be short of money
Vocabulary
Word
$3.98
6.49 then look
9.08 at the cents
+3.47 adjust the estimate
21
Front-End
Estimation
$3.98 + $6.4
4 + 7 = 11
$3.98 + $6.49 ≈ $11.00
21 + about $2 = $23
Examples
Non-Examples
Adding and Subtracting Decimals
LESSON 1-2
COMMUTATIVE PROPERTY OF ADDITION

Changing the order of the addends does not
change the sum.
5.78 + 9.3 = 9.3 + 5.78
a+b=b+a
ASSOCIATIVE PROPERTY OF ADDITION

Changing the grouping of the addends does not
change the sum
(3.2 + 8) + 4 = 3.2 +( 8 + 4)
(a + b) + c = a + (b + c)
IDENTITY PROPERTY OF ADDITION

The sum of 0 and any number is that number
4.5 + 0 = 0 + 4.5 = 4.5
a + 0 = 0 + a =a
FINDING DECIMAL SUMS

Line up the decimal points!!!
Add: 3.842
2.450 write zeros so that all
+1.300 decimals have the same
7.592
number of digits to the
right of the decimal point
FINDING DECIMAL DIFFERENCES

Line up the decimal points!!
7 10
Subtract: 68.0
- 51.8
16.2
rename as 7 and 10 tenths
Estimate answers first then check for
reasonableness!
Multiplying and Dividing Decimals
LESSON 1-3
COMMUTATIVE PROPERTY OF MULTIPLICATION

Changing the order of the factors does not
change the product.
5.78 x 9.3 = 9.3 x 5.78
a∙b=b∙a
ASSOCIATIVE PROPERTY OF MULTIPLICATION

Changing the grouping of the factors does not
change the product
(3.2 x 8) x 4 = 3.2 x ( 8 x 4)
(a ∙ b) ∙ c = a ∙ (b ∙ c)
IDENTITY PROPERTY OF MULTIPLICATION

The product of 1 and any number is that
number
4.5 x 1 = 1 x 4.5 = 4.5
a∙1=1∙a
ZERO PROPERTY OF MULTIPLICATION

The product of 0 and any number is 0
6x0=0x6=0
a∙0=0∙a=0
DECIMAL X DECIMAL
0.25
x .015
125
+ 025
.00375
two decimal places
three decimal places
move the decimal point five places to
………………………………………………..the left
Add the number of decimal places in the factors to
find the number of decimal places in the product.
WHOLE NUMBER X DECIMAL
0.25
x 5
1.25
two decimal places
0 decimal places
Add the number of decimal places in the factors
to find the number of decimal places in the
product.
DECIMAL ÷ DECIMAL

Multiply both the dividend and the divisor by
the same number, (multiple of 10), so the
divisor becomes a whole number.
ANNEXING ZEROS TO DIVIDE
Measuring in Metrics
LESSON 1-4
USING METRIC UNITS
Type
Unit
Reference Example
Length
millimeter (mm)
centimeter (cm)
meter (m)
kilometer (km)
about the thickness of a dime
about the width of your little finger
about the distance from the doorknob to the floor
about the length of 11 football fields
Capacity
milliliter (mL)
liter (L)
a small spoon holds about 5 mL
a little more than 1 quart
Mass
milligram (mg)
gram (g)
kilogram (kg)
about the mass of a mosquito
about the mass of a paper clip
about the mass of a bunch of bananas
MASS VS. WEIGHT

Mass is a measurement of how much matter is
in an object; weight is a measurement of how
hard gravity is pulling on that object. Your mass
is the same wherever you are--on Earth, on the
moon, floating in space--because the amount
of stuff you're made of doesn't change. But
your weight depends on how much gravity is
acting on you at the moment; you'd weigh less
on the moon than on Earth, and in interstellar
space you'd weigh almost nothing at all.
CONVERTING METRIC UNITS

Kids Have Dropped Over Dead Converting
Metrics
HOW DOES THIS WORK?
K H
D O
5 liters
D
C
M
5 L = ____________ mL
Start at Over and move to the right as mL are
smaller than L. (Think of O as original unit)
We moved 3 places. 5 L = 5000 mL
HOW DOES THIS WORK?
K
H D O D
5 liters
C
M
5 L = _____?____ kL
Start at Over and move to the left as kL are larger
than L. (Think of O as original unit)
We moved 3 places. 5 L = .005 kL
CAUTION:
Remember all whole numbers have a decimal point
Comparing and Ordering Integers
LESSON 1-6
Definition
Facts/Characteristics
Two numbers that are the
same distance from 0 on a
number line, but in opposite
directions
Another name for opposites is
additive inverse as a number
+ its additive inverse = 0
Vocabulary
Word
Opposites
7 + (-7) = 0
-0.3 + 0.3 = 0
Ordinal numbers do
not have opposites. There is no
negative 3rd.
Examples
Non-Examples
Definition
The set of positive whole
numbers, their opposites,
and zero.
Facts/Characteristics
Fractions are not integers
Decimals are not integers
Vocabulary
Word

Integers

Examples
Non-Examples
Definition
Facts/Characteristics
The distance the
number is from 0
Always use bars
to show you mean
absolute value |a|
Vocabulary
Word
Absolute
Value
|3| = 3
|-3| = 3
Both are 3 places from 0
Examples
(3) [3}
these notations do not
mean absolute value
Non-Examples
ORDERING INTEGERS
Positive numbers are always greater than
negative numbers
 Negative numbers are always less than positive
numbers
 When using a number line, numbers increase
as you move to the right
 When using a number line, numbers decrease
as you move to the left

INTEGERS ON A NUMBER LINE
Adding and Subtracting Integers
LESSON 1-7
ADDING INTEGERS WITH SAME SIGN

The sum of two positive numbers is positive.
3+5=8
The sum of two negative numbers is negative.
-3 + (-5) = -8
ADDING INTEGERS WITH DIFFERENT SIGNS

Find the absolute value of each. Subtract the
lesser absolute value from the greater. The sum
has the sign of the integer with the greater
absolute value
-3 + 5 = 2
3 + (-5) = -2
(that is, subtract the lower number from the
higher and keep the higher sign)
SUBTRACTING INTEGERS

To subtract integers, add its opposite
Examples:
3- 5 = 3 + (-5) = -2
-7 – (-3) = -7 + 3 = -4
4 – 5 = 4 + (-5) = -1
-6 – 1 = -6 + (-1) = -7
Multiplying and Dividing Integers
LESSON 1-8
MULTIPLYING INTEGERS

The product of two integers with the same sign
is positive
3(4) = 12
-4(-6) = 24
The product of two integers with different
signs is negative
3(-4) = -12
-4(6) = -24
DIVIDING INTEGERS

The quotient of two integers with the same sign
is positive
14 ÷ 7 = 2
-14 ÷ (-7) = 2
The quotient of two integers with different
signs is negative
-14 ÷ 7 = -2
14 ÷ (-7) = -2
HOW ABOUT THIS
-
-35
-7 = 5
25
-5 = -5
-
Since a fraction is a division problem, the rules
apply
Order of Operations and the Distributive Property
LESSON 1-9
PLEASE EXCUSE MY DEAR AUNT
SALLY
Do all operations within parentheses first
 Work the exponents
 Multiply and divide in order from left to right
 Add and subtract in order from left to right

USING THE DISTRIBUTIVE PROPERTY

The distributive property lets you multiply a
sum by multiplying each addend separately and
then add the products.

4 ∙ 25 = 100
or
4(20 + 5)
4(20) + 4(5)
80 + 20 = 100
CONTINUED…….
other examples
6(7 – 5) =12
or
6(7 – 5)
6(7) – 6 (5)
42 – 30 = 12
AND WITH DECIMALS
5(6.8) = 34
or
5(6.8)
5(7 - .2)
5(7) – 5(.2)
35 – 1 = 34
Why do this? The Distributive Property makes
numbers easier to work with and mental
calculations become easier.
Mean, Median, Mode and Range
LESSON 1-10
Definition
The mean of a set of data
is the average: sum of the
set divided by the number
of items
Facts/Characteristics
You can use the mean to
find your grade average
Vocabulary
Word
Grades:
90, 95, 88
Mean:
273 ÷ 3 = 91
Examples
Mean
Not
some one who is
selfish or unkind
Non-Examples
Definition
The median is the middle
value when the data
are in numerical order
Facts/Characteristics
The median always separates
the data into two groups of
equal size
Vocabulary
Word
Median
23, 36, 45, 46, 89
The median is 45
Examples
46, 23, 89, 45, 36
The median is not 89. Arrange
data in numerical order FIRST
Non-Examples
Definition
Facts/Characteristics
The mode of the data set is
the item that occurs with
the greatest frequency
There can be more
than one mode in a
data set
Vocabulary
Word
Mode
1,2,2,3,4,4,5,6,7,7
the mode is 2, 4, 7
Examples
There is no mode
when all data items occur
the same number of times
Non-Examples
Definition
Facts/Characteristics
The range of a data set is
the difference between the
greatest and least values
Order the data from least
to greatest before finding
the range
Vocabulary
Word
Range
14, -12, 7, 0, -5, -8, 17,
order least to greatest
-12, -8, -5, 0, 7, 14, 17
Range is 17- (-12) =
17 + 12 = 29
Examples
Non-Examples