Math 8 Unit 1

1
Math 8 Unit 1

• Operations with Integers
• Exponents
• Scientific Notation
• Square Roots
• Estimating Square Roots
• Order of Operations with Square Roots
• Evaluating Expressions
• Writing Expressions and Equations
• Problem Solving with Word Problems

2
Integers

• S1 C2 PO 1. Select the grade level
  appropriate operation to solve word problems.
• S1 C2 PO 2. Solve word problems using grade
  level appropriate operations and numbers.
• S1 C2 PO 6. Apply grade level appropriate
  properties to assist in computation

3
Integers

• Integers any positive or negative whole number
• ,-4,-3,-2,1,0,1,2,3,4,

4
When Am I Ever Going To Use This?
You can use integers to keep score in sports.
You can use integers to keep track of
Temperature Change.
You use integers to measure sea level.
5
When Am I Ever Going To Use This?
You can use integers to keep track of the Stock
Market.
You can use integers to record Weight Gain or
Loss.
You can use integers to record money withdrawals
or deposits to your banking accounts.
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Basic Vocabulary

• Integers less than 0 are negative integers.
  (Negative not MINUS!)
• Integers greater than 0 are positive integers.
• Zero is neither positive or negative.
• Positive integers are usually written without the
  sign, so 5 and 5 are the same.
• All Positive numbers are larger than any negative
  number.

7
Basic Vocabulary

• Two numbers are opposites of one another if they
  are represented by points that are the same
  distance from 0, but on opposite sides of 0.
• All opposites add to zero.
• -3 3 0
• (21) (-21) 0

8
Number Lines

• On the number linethe numbers have a greater
  value as you go to the right.
• The larger the negative number, the SMALLER it
  is.
• - 1 is the largest negative integer.
• A number line is drawn by choosing a starting
  position, usually 0, and marking off equal
  distance from that point
• Although only a portion of the number line is
  shown, the arrowheads indicate that the line and
  the set of numbers continue

5
9
Integers
Example Put in order from greatest to
least. -3, 0, 5, -25, 36, -1, 7
36, 7, 5, 0, -1, -3, -25
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Adding Integers Using Chips

• Red Chips are negative, yellow chips are
  positive.
• Consider the following two problems.
• -3 -4 3 4

-7
7

How many yellow chips do I have?
How many red chips do I have?

• What are some observations about the problems
  modeled above?

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Rules for addition ()

• Adding integers with the same sign
• you ADD the number parts and keep the sign.

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Examples
Ex. 2 25 (-33)

• Ex. 1 4 (16)

-58
-20
Ex. 4 8 11
Ex. 3 5 (-7)
19
-12
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Using Chips
Note If you dont have counters, you can model
the problems using for positive integers and
for negative integers.

• What happens if there is one positive and one
  negative number?
• Remember Zero Pair? A zero pair with chips is
• 3 -4 -3 4

-1
1
Cross out Zero Pairs.
Cross out Zero Pairs.
Which color do I have more of? How many more?
Which color do I have more of? How many more?

• What are some observations about the problems
  modeled above?

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Rules for addition ()

• Adding integers with different signs
• you SUBTRACT the number parts and keep the sign
  of the larger number.

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Examples
Ex. 6 25 (-50)

• Ex. 5 16 (46)

-25
30
Ex. 8 7 (-9)
Ex. 7 3 5
-2
2
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Subtracting Integers

• Another strategy for adding and subtracting
  integers is using a number line. Last year you
  knew this activity as the Hiker.
• Consider the following problem

4 - 7
17
4 - 7
-3
The ending position is your answer.
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-1 - 2
-3
The ending position is your answer.
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-1 (-7)
6
The ending position is your answer.
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1 (-4)
5
The ending position is your answer.
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Think about it

• Study the four example we looked at. Do you see
  a pattern?

• 4 7 -3
• -1 2 -3

• -1 (-7) 6
• 1 ( -4) 5

Rule for Subtraction

• Add the Opposite of the 2nd integer
• This is called the additive inverse.
• So, change all subtraction problems to addition
  problems!
• Some of you might know it as Chop Chop

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Examples
Change to adding the opposite.
Ex. 2 5 1

• Ex. 1 2 6

-8
-6
Ex. 3 6 (-11)
Ex. 4 8 (-8)
5
16
Ex. 5 2 13
Ex. 6 0 ( -3)
-15
3
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Rules for Multiplication and Division
Two Cases

• Same Signs Answer Positive
• ?
• - ? -

• Different Signs Answer Negative
• ? - -

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Triangle Shortcut

• You can use the triangle to help you remember.
  The triangle only works for multiplying and
  dividing. Put a positive in the top angle, and
  two negatives in the bottom.
• Cover the signs of the two value in your problem.
  The sign that is left is the sign on your
  answer.

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Examples

• Ex. 1 -7 ? -3

21
Ex. 2 -9 ? 2
-18
Ex. 3 4 ? -3
-12
1
Ex. 4 -2 -2
-5
Ex. 5 15 -3
4
Ex. 6 20 5
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Summary

• We have looked at several strategies for you to
  use when adding integers.
• Counters
• Number Line
• Rules (also called algorithm)
• You may use whatever strategy you are comfortable
  with, just be consistent.

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EXPONENTS AND SQUARE ROOTS
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Defn When 2 or more numbers are multiplied,
these numbers are called factors of the product.
 EX. 4 x 2 8
4 and 2 are factors of 8
8 is the product
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Defn When the same factor is repeated, you may
use an exponent to simplify notation.
EX. 16
2 ? 2 ? 2 ? 2
Exponent -how many bases you multiply
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Base (Big Number) -the number you multiply
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Ex 1 Write 5 ? 5 ? 5 ? 5 ? 5 ? 5 using
exponents.
56
EX. 2 Write 3 ? 3 ? 5 ? 5 ? 6 ? 6 ? 6 using
exponents.
32 ? 52 ? 63
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EX. 3 Find 83.
8 ? 8 ? 8
64 ? 8
512
EX. 4 Find 23 ? 34.
2 ? 2 ? 2 ? 3 ? 3 ? 3 ? 3
8 ? 81
648
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Special Case
any number to the zero power is 1

• 401
• 701
• 11001
• 12701

Why????
What is 8 8?
If I factor 8, I get
Divide out all common factors.
You are left with 1.
33
Scientific Notation

• S1 C2 PO 10. Convert standard notation to
  scientific notation and vice versa.

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Powers of 10

• 101 10
• 102 100
• 103 1000
• 104 10, 000
• 105 100, 000

10 -1 0.1 10-2 0.01 10-3 0.001
10-4 0.0001 10-5 0.00001

• When you multiply by 10 to a positive number, the
  decimal moves to the right. The value of the
  number INCREASES.
• When you multiply by 10 to a negative number, the
  decimal moves to the left. The value of the
  number DECREASES.

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Vocabulary

• Scientific Notation a way of expressing numbers
  as the product of a number between 1 an 10 and a
  power of 10
• Purpose a way for scientists to write very
  large or very small numbers in order to make them
  easier to compute
• If a number is not written in scientific
  notation, we say it is written in Decimal
  Notation or Standard Form

Decimal Notation 5, 900, 000, 000
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Examples Is the number in scientific notation?
If no, explain why.

• a. 11.8 ? 107

b. 6.9 ? 105
c. 0.7 ? 1018
d. 1.2 ? 53
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To change a number from decimal notation to
scientific notation

• Place the decimal in a location that creates a
  number that is greater than zero but less than
  ten. (You want the digit to be a 1-9)
• Count the number of decimal places you moved AND
  notice which direction you moved. If the
  original number is greater than 1, the exponent
  is positive. If the original number is less than
  1, the exponent is negative.
• Multiply your new number by 10 to a power (the
  number of decimal places you counted above).

38
Examples Write the following numbers in
scientific notation.

• EX. A 7,900,000

Step 2 I moved the decimal 6 places, and the
original number was greater than one so my
exponent will be positive.
? 106
Answer 7.9
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EX. B 0.0045
I had to move the decimal 3 places to get to my
desired location.
? 10-3
4.5
40
EX. C 2390
2.39
? 103
EX. D 0.0000563209
? 10-5
5.63209
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To change a number from scientific notation to
decimal notation

• Step 1 Look at the exponent on the power of
  ten. This tells you how many places to move.
• If the exponent is positive you will want to move
  the right (makes the number larger) and if it is
  negative you want to move the decimal point to
  the left (makes number smaller).
• Step 2 Place your pencil on the decimal point
  in the decimal value and move the number of
  places determined above.
• Step 3 Fill in any empty spaces with zeros.

42
Examples Express Each number in standard form.
EX. E 2.76 ? 107
2.76
27, 600, 000

• EX. F 1.8 ? 10-4

1.8
0.00018
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Examples Express Each number in standard form.

• EX. G 9.3 ? 102

930
9.3
EX. H 2.34 ? 10-3
0.00234
2.34
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1.6 Notes Square Roots
Key Term
Perfect square the square of a number Perfect
squares relate to the number of squares.
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To find the area of a squareA s ? s or A s2
EX. 1
EX. 2
If 52 25, then 5 is a square root of 25.
If 32 9, then 3 is a square root of 9.
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square root A square root is one of two equal
factors of a number.
Def If x2 y, then x is a square root of y.
Symbol This symbol is called a square root
12
Is there another number that is a square root of
144?
47
Further information

• Every number has two square roots.
• The radical symbol most of the time represents
  the principle (positive) square root of a number.
  
• The negative square root of a number is shown
  with the following symbol.
• We use the following symbol to represent the two
  square roots of a number

48
Examples
-5
9
Example 6 If the area of a square is 256 square
inches, what is the length of one of the sides?
16
49
What about fractions, decimals and large numbers?

• For fractions, take the square root of the
  numerator, then the square root of the
  denominator.

9
50
What about decimals and large numbers?
Why?

• There is a short cut for decimals and large
  numbers.
• If a large number has an even number zeros, take
  the square root of the non-zero numbers and then
  cut the zeros in half.
• If a number has an even number of decimal places,
  and is a perfect square, take out the decimal
  point and square root the number. The answer
  will have half the number of decimal places as
  the original number.

51
Examples
The answer will have 3 zeros, because half of 6
is 3.
Answer 4000
52
Examples
The answer will have 2 decimal places.
Answer 0.12
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Table 1 Relationship between squaring a number
and taking the square root
1
1
1
11
121
11
12
2
4
2
144
12
3
9
3
13
169
13
4
16
4
14
196
14
5
25
5
15
225
15
6
36
6
16
256
16
7
49
7
17
289
17
8
64
8
18
324
18
9
81
9
19
361
19
10
100
10
20
400
20
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What is the relationship between squaring a
number and taking the square root

• Inverse Operations!
• In other words, one undoes the other.

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Group ActivityEstimating Square Roots

• Directions
• Put all Square Roots in Numerical Order.
• Once in Numerical Order, Estimate the value of
  the Square Root to the tenths place on the number
  line.
• Show how you found your estimation.

56
Estimating Square Roots

• Most numbers are NOT perfect squares.
• We have approximate their square roots.

EX. 1 Estimate to the nearest whole number.
90 is between the perfect squares
81
and 100.
Which one is it closer to?
Therefore, the best whole number estimate is
9.
Question Is a whole number estimate the most
accurate?
No, the more decimal places in your answer, the
more accurate your answer.
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EX. 2 Estimate the following to the nearest
tenth.
55 is between the perfect squares
49
and 64.

• So, my estimate will be more than 7, but less
  than 8.

• But, which one is it closer to?

• When you are estimating to the nearest tenth,
  you want to pick a digit between 1 and 9 to
  represent where your value would be on the number
  line.

• 55 is 6 units away from 49 and 9 units away from
  64.

Estimate 7.
4
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EX. 2 Estimate the following to the nearest
tenth.
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Order Of Operations
S1 C2 PO 6. Apply grade level appropriate
properties to assist in computation S1 C2 PO
11. Simplify numerical expressions using the
order of operations with grade appropriate
operations on number sets
60
Order Of Operations
-The order of operations is a set of directions
for evaluating an expression so that the
expression has only one value.
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Order Of Operations
-Steps to follow when you do a problem
Please Excuse My Dear Aunt Sally
dd
ivide
ubtract
xponents
ultptiply
arentheses
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Four Steps!!!
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Order Of Operations

• First Perform all calculations inside grouping
  symbols.
• These include parenthesis (), brackets , and
  braces , and fraction bars.
• Next Perform multiplication, and division in
  order from left to right. Simplify
  exponents/square roots before any other
  multiplications.
• Then Perform addition and subtraction in order
  from left to right.

64
Who is correct?
Method 1
Method 2
65
Who is correct?
The rules for order of operations exist so that
everyone can perform the same consistent
operations and achieve the same results. Method
2 is the correct method.
66
Example 1
Follow the left to right rule First solve any
multiplication or division parts left to right.
Then solve any addition or subtraction parts left
to right.
Divide
A good habit to develop while learning order of
operations is to underline the parts of the
expression that you want to solve first. Then
rewrite the expression in order from left to
right and solve the underlined part(s).
Multiply
Add
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Example 2
Example 3
Square roots are considered exponents in the
order of operations. However, if there is an
expression that needs to be simplified, you must
do it first before you can take the square root.
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Example 4
Exponents (powers)
Work above the fraction bar
Multiply
48
Grouping symbols
Subtract
Work below the fraction bar
Add
Simplify Divide
69
Example 5
16 3 ? 4 5
16 12 5
4 5
9
70
Example 6
71
Evaluating Expressions
S3 C3 PO 1. Evaluate algebraic expressions by
substituting rational values for variables.
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Key Terms

• Variable number we dont know (use a letter)
• Expression mathematical phrase (does not have an
  equals sign)
• Evaluate Find the answer
• Equation mathematical sentence (HAS an equals
  sign)
• Substitute replace

73
Evaluate the following if a 12 and b -3.
Example 1
Replace all as with 12.
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Evaluate the following if a 12 and b -3.
Example 2
Replace all bs with -3.
75
Evaluate the following if a 12 and b -3.
Example 3
Replace all as and bs.
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Writing Expressions and Equations

• S3 C3 PO 2. Use variables in contextual
  situations.
• S3 C3 PO 3. Translate a written sentence or
  phrase into an algebraic equation or expression
  and vice versa
• S3 C3 PO 4. Translate a sentence written in
  context into an algebraic equation involving two
  operations.
• S3 C3 PO 6. Identify an equation or
  inequality that represents a contextual situation.

77
Verbal Clues!

• With your group brainstorm as many ideas that you
  can that mean the following symbols.

78
Verbal Phrases (Clues! ?)
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Verbal Phrases (Clues! ?)
80
Write Algebraic Expressionsfor These Word Phrases
x 6

• The sum of x and six
• Twice a number, minus 11
• The quotient of b and 45
• Three times a number
• Ten minus, the quantity of the product of y and 3
  
• 3 less than x
• 3 less x

2y - 11
b45
3n
10 - (y?3)
x - 3
3 - x
81
Write Algebraic Expressionsfor These Word Phrases
n 10

• Ten more than a number
• A number decreased by 5
• 6 less than a number
• A four times a number increased by 8
• The sum of a 3 times a number and 9
• The quantity of x minus 2, divided by 10

w - 5
x - 6
4n 8
3n 9
(x 2)10
82
Write an Algebraic Equation for Equation for
These Situations

• Scotts brother Alan is 2 years younger than
  Scott
• The sum of two numbers is 12
• The difference between two numbers is 5

A S - 2
b c 12
m n 5
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EXPRESSIONS
A NUMBER X INCREASED BY 9
X 9
84
EXPRESSIONS
A NUMBER Y DIVIDED BY 8
Y / 8 OR
85
EXPRESSIONS
ELEVEN LESS B
11 - B
86
EXPRESSIONS
THE MONEY LEFT OVER FROM 10 AFTER SPENDING J
10 - J
87
EXPRESSIONS
5 LESS THAN X
x - 5
88
EXPRESSIONS
2/3 OF A NUMBER R MINUS TWENTY-THREE
2/3 R - 23
89
EXPRESSIONS
A NUMBER 21 DIVIDED BY P, PLUS 16
21p 16
90
EXPRESSIONS
THE PRODUCT OF SIX AND Y INCREASED BY FIVE
6Y 5
91
EXPRESSIONS
THE DIFFERENCE WHEN 32 IS SUBTRACTED FROM Z
Z - 32