Just the facts: Order of Operations and Properties of real
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Transcript Just the facts: Order of Operations and Properties of real
Start thinking of math as a language, not a pile of numbers
Just like any other language, math can help us communicate thoughts and ideas with
each other
An expression is a thought or idea communicated by language
In the same way, a mathematical expression can be considered a mathematical
thought or idea communicated by the language of mathematics.
Mathematics is a language, and the best way to learn a new language is to immerse
yourself in it.
A SSE 1
Just like English has nouns, verbs, and adjectives,
mathematics has terms, factors, and coefficients.
Well, sort of.
TERMS are the pieces of the expression that are separated by plus or minus signs,
except when those signs are within grouping symbols like parentheses,
brackets, curly braces, or absolute value bars.
Every mathematical expression has at least one term.
3x 2
Has two terms.
3x and 2
5
A term that has no variables is often called a constant because it never
changes.
Within each term, there can be two or more factors.
The numbers and/or variables multiplied together.
3x
Has two factors: 3 and x.
There are always at least two factors, though one of them may be the number 1,
which isn't usually written.
Finally, a coefficient is a factor (usually numeric) that is multiplying a variable.
Using the example, the 3 in the first term is the coefficient of the variable x.
The order or degree of a mathematical expression is the largest sum of the exponents
of the variables when the expression is written as a sum of terms.
3x 2
Order is 1
Since the variable x in the first term has an exponent of 1 and there are no other terms
with variables.
Order is 2
5 x 3x 2
2
3xy 5x y 7 x 32 y
2
3
4
Order is 5
Now that we have our words, we can start putting them together and make
expressions
3x 2
Translate mathematical expressions into English
"the sum of 3 times a number and 2,"
"2 more than three times a number"
It's much easier to write the mathematical expression than to write it in English
Practice 1.1 Variables and Expressions
1. 10 less than x _______________
x 10
2. 5 more than d _______________
d 5
3. the sum of 11 and d _______________
11 d
d 11
t
t 3 3
4. a number t divided by 3 _______________
A-SSE.A.1
Practice 1.1 Variables and Expressions
A-SSE.A.1
5.
20
3
x
3 less than the quotient of 20 and x _______________
6.
5 d
12 w
the quotient of 5 plus d and 12 minus w _______________
7. Write a rule in words and as an algebraic expression to model the
relationship in each table. The local video store charges a monthly
membership fee of $5 and $2.25 per video.
$5 plus $2.25 times the number of videos;
5 2.25v
Just the facts:
Order of Operations and
Properties of real numbers
A GEMS/ALEX Submission
Submitted by: Elizabeth Thompson, PhD
Summer, 2008
Important things to remember
• Parenthesis – anything grouped… including information
above or below a fraction bar.
• Exponents – anything in the same family as a ‘power’…
this includes radicals (square roots).
• Multiplication- this includes distributive property
(discussed in detail later).
Some items are grouped!!!
• Multiplication and Division are GROUPED from left to
right (like reading a book- do whichever comes first.
• Addition and Subtraction are also grouped from left to
right, do whichever comes first in the problem.
So really it looks like this…..
•
•
•
•
Parenthesis
Exponents
Multiplication and Division In order from left to right
Addition and Subtraction In order from left to right
SAMPLE PROBLEM #1
16 4(3 1) 22 11
3
16 4(2) 22 11
3
Parenthesis
Exponents
16 4(8) 22 11
4(8) 22 11
32 22 11
This one is tricky!
Remember: Multiplication/Division are grouped from left to right…what comes 1st?
Division did…now do the multiplication (indicated by parenthesis)
32 2
More division
Subtraction
30
SAMPLE PROBLEM
3(5) 65
3(2 3) 65
2
2
2
2
Exponents
Parenthesis
75 65 10
3(25) 65
2
2
2
Remember the division
symbol here is grouping
everything on top, so
work everything up there
first….multiplication
Subtraction
Division –
because all the
work is done
above and
below the line
5
Order of Operations-BASICS
Think: PEMDAS
Please Excuse My Dear Aunt Sally
•
•
•
•
•
•
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
Practice 1.2 Order of Operations and Evaluating Expression
Simplify
1.
42 __________
4 4 16
5 5 5 125
2. 53 __________
16
8 20 12
4. 4(5) __________
2
3
15 33 27
3
27 12
6.
__________
5
83
A-CED.1
5 5 5 125
5
3. __________
6 6 6 216
3
6
64(5) 33
5. 43 (5) 3(11) _________
320 33
353
Practice 1.2 Order of Operations and Evaluating Expression
Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the
given values of the variables. FSA
7. Area of a triangle: b 6in and h 14in.
F:
1
A bh
2
S:
1
A (6)(14)
2
A:
A 42 in2
Practice 1.2 Order of Operations and Evaluating Expression
Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the
given values of the variables. FSA
8. Volume of a pyramid: B 18m and h 8m.
F:
1
V Bh
3
S:
1
V (18)(8)
3
A:
V 48 m3
Practice 1.2 Order of Operations and Evaluating Expression
Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the
given values of the variables. FSA
9. Find the value of x using the quadratic formula with a 1, b 2 and c 3
F:
S:
b b2 4ac
x
2a
(2) (2) 2 4(1)(3)
x
2(1)
2 4 12
x
2
24
x
3
2
24
x
1
2
10. The cost to rent a hall for school functions is $60 per hour. Write an expression for the cost
of renting the hall for h hours. Make a table to find how much it will cost to rent the hall for
2, 6, 8, and 10 hours.
60h
hours
$
2
120
6
360
8
480
10
600
Lesson Extension
• Can you fill in the missing operations?
1. 2 - (3+5) + 4 = -2
2. 4 + 7 * 3 ÷ 3 = 11
3. 5 * 3 + 5 ÷ 2 = 10
Practice 1.3 Real Number and the Number Line
Name the radicand of each of the following, then write in simplified form.
1.
64
8
64 ___________,
64 ________
radicand
3.
1
1
___________,
36
radicand
36
36
1
____
6____
25
3 5 15
2. 3 25 ___________,3
25 ________
radicand
4.
81
81
81, 100
___________,
radicand
100
100
9
________
10
Practice 1.3 Real Number and the Number Line
Estimate the square root by finding the two closest perfect squares.
5.
51
<
49
perfect square
51 <
64
perfect square
7
51 _______
set
6. A ___________
is a well-defined collection of objects.
element
7. Each objects is call an ________________
of a set.
8. A ____________
subset of a set consists of elements from the given set.
9. U 2,4,6,8 and A 2,8, is A a subset of U? yes/no_________
yes
no
10. U 2,4,6,8 and A 2,3, is A a subset of U? yes/no_________
Practice 1.3 Real Number and the Number Line
Circle all the statements that are true.
11.
9 rational
15. rational
19.
100
rational
49
12.
5 irrational
13.
1
integer
3
16.
25 irrational
17.
9
whole
3
20.
4
2
irrational
21. 2.56 rational
14. 0 whole
18. 0 natural
22. 2 irrational
An inequality is a mathematical sentence that compares the values of two expressions using an
inequality symbol. The symbols are: ( >, <, , )
______, less than
_______,less than or equal to
______, greater than _______,greater than or equal to
3.5
7
23. What is the order of 3.51, 2.1, 9, , and 5 from least to greatest?
2
2
3
2,
7
5, 9, 2 , 3.51
Properties of Real Numbers
(A listing)
•
•
•
•
•
Associative Properties
Commutative Properties
Inverse Properties
Identity Properties
Distributive Property
All of these rules apply to Addition and Multiplication
Associative Properties
Associate = group
It doesn’t matter how you group (associate) addition
or multiplication…the answer will be the same!
Rules:
Samples:
Associative Property of Addition
Associative Property of Addition
(a+b)+c = a+(b+c)
(1+2)+3 = 1+(2+3)
Associative Property of Multiplication
Associative Property of Multiplication
(ab)c = a(bc)
(2x3)4 = 2(3x4)
Commutative Properties
Commute = travel (move)
It doesn’t matter how you swap addition or
multiplication around…the answer will be the same!
Rules:
Samples:
Commutative Property of Addition
Commutative Property of Addition
a+b = b+a
1+2 = 2+1
Commutative Property of Multiplication
Commutative Property of Multiplication
ab = ba
(2x3) = (3x2)
Stop and think!
• Does the Associative Property hold true
for Subtraction and Division?
Is (5-2)-3 = 5-(2-3)?
Is (6/3)-2 the same as 6/(3-2)?
• Does the Commutative Property hold true
for Subtraction and Division?
Is 5-2 = 2-5?
Is 6/3 the same as 3/6?
Properties of real numbers are only for Addition and Multiplication
Inverse Properties
Think: Opposite
What is the opposite (inverse) of addition?
What is the opposite of multiplication?
Rules:
Inverse Property of Addition
a+(-a) = 0
Subtraction (add the negative)
Division (multiply by reciprocal)
Samples:
Inverse Property of Addition
3+(-3)=0
Inverse Property of Multiplication
Inverse Property of Multiplication
a(1/a) = 1
2(1/2)=1
Identity Properties
What can you add to a number & get the same number back? 0 (zero)
What can you multiply a number by and get the number back? 1 (one)
Rules:
Identity Property of Addition
a+0 = a
Samples:
Identity Property of Addition
3+0=3
Identity Property of Multiplication
Identity Property of Multiplication
a(1) = a
2(1)=2
Distributive Property
If something is sitting just outside a set of parenthesis, you can
distribute it through the parenthesis with multiplication and
remove the parenthesis.
Rule:
a(b+c) = ab+bc
Samples:
4(3+2)=4(3)+4(2)=12+8=20
• 2(x+3) = 2x + 6
• -(3+x) = -3 - x
Practice 1.4 Properties of Real Numbers
A.
B.
C.
D.
E.
Associative Property of Addition/Multiplication
Commutative Property of Addition/Multiplication
Identity Property of Addition/Multiplication
Zero Property of Multiplication
Multiplica tion Property of -1
What property is illustrated by each statement?
C
_____1.
4 x 1 4 x
_____2.
3 (1 p) 3 ( p)
E
_____4.
4( x 1) ( x 1)4
_____5.
5 ( x y ) (5 x) y
B
A
D : Give an example
C
_____3.
m0 m
_____6.
xyz yxz
B
Practice 1.5 Adding and Subtracting Real Numbers
Find each sum.
1. 8 5
3
7. 10 6
4
13. 10 1
11
2. 7 3
3. 6 4
10
4. 1 7
6
2
8. 15 6
9. 8 10
21
18
14. 11 6
5
5. 2 9
15.
10. 7 16
11
11. 2 9
7
9
8 5
13
16.
7 12
5
6. 5 9
4
12. 5 25
30
17. 12 10
2
Absolute Value.
Simplify each expression.
18. 8 5
85
13
19. 7 4
11
11
20. 6 4
64
10
21. 1 7
22. 2 9
1 7
11
6
11
Opposites:
additive inverse
A number and its opposites are called _________________________________.
State the opposite of result of each statement.
23. 3 5
24. 5 9
25. 6 (9)
26. 5 2
27. 2 8
2
2
4
4
3
3
7
7
10
10
Practice 1.6 Multiplying and Dividing Real Numbers
Find each product/quotient.
1. 8 5
2. 7 3
40
7. 10 6
8.
15 6
10
2
5
18.
10
1
2
10(2)
20
9.
90
14.
15
5
4.
24
21
60
13.
3. 6 4
8 10
10.
3
1
19. 3
6
7
1 7
7
3 6
18
5. 2 9
7
18
45
7 16
11. 2 9
12. 5 25
8
8
16.
6 12
6
17.
3
1
1
4
5
20.
15
5
41
4
21.
6. 5 9
18
112
80
15.
1 7
12 10
8
125
2
8
1
4
12
2
5
5
12 30
2
Practice 1.7 Distributive Property
What is the simplified form of each expression?
1. 5( x 7)
5 x 35
4. (2 y 1)( y)
2y 2 y
7. 4(2 x 2 3x 1)
8x2 12 x 4
1
2. 12(3 x)
6
36 2x
5. 4(2 x 5)
8 x 20
8. 5x(2 x 5)
10 x 2 25 x
3. (0.4 1.1c)3
1.2 3.3c
6. ( x 6)
x 6
9. x( x 3)
x 2 3x
Practice 1.7 Distributive Property
10.
Using the following expression: 3x 2 4 x 2
3
a. How many terms? _________
3, 4
b. List the coefficients: _________
2
c. List the constants: _________
What is the simplified form of each expression?
11. 3 y y
2y
14. a 3b a 4b
7b
12. 7 mn 4 5mn 4
12mn4
15. x 5 y 3 x 8 y
2x 3 y
13. 3 y 2 x y 2 x y 2
2y 2 x y 2
16. 5 y 3 y 10 x 3 x 2 y
4 y 7x
Practice 1.8 An Introduction to Equations
Tell whether each equation is true, false or open. Explain.
1.
45
14 22
x
Open
2. 42 10 52
True
False
Tell whether the given number is a solution of each equation.
4. 3b 8 13; 7
5. 4 x 7 15; 2
?
3(7) 8 13
?
21 8 13
Not
3. 7 8 15
?
4(2) 7 15
?
8 7 15
?
15 15
Yes
6. 12 14 2 f ; 1
?
12 14 2(1)
?
12 14 2
NO
Practice 1.8 An Introduction to Equations
Write an equation for each sentence.
n7 8
7. The difference of a number and 7 is 8. _________________________________________
6(n 5) 16
8. 6 times the sum of a number and 5 is 16. ________________________________________