start thinking of math as a language , not a pile of numbers
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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. - PowerPoint PPT PresentationTRANSCRIPT
Start thinking of math as a language, not a pile of numbersJust 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
A term that has no variables is often called a constant because it never changes.
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.
3 2x Has two terms.
3x 2and
5
Within each term, there can be two or more factors.
There are always at least two factors, though one of them may be the number 1, which isn't usually written.
The numbers and/or variables multiplied together.
3xHas two factors: 3 and x.
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.
Order is 13 2x Since the variable x in the first term has an exponent of 1 and there are no other terms with variables.
25 3 2 x x Order is 2
2 3 43 5 7 32 xy x y x y Order is 5
Now that we have our words, we can start putting them together and make expressions
Translate mathematical expressions into English
3 2x "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 A-SSE.A.1
1. 10 less than _______________ x
2. 5 more than _______________ d
3. the sum of 11 and _______________d
4. a number divided by 3 _______________t
10x
5d
11 d
3t 3t
11d
Practice 1.1 Variables and Expressions A-SSE.A.1
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. 3 less than the quotient of 20 and _______________ x
6. the quotient of 5 plus and 12 minus _______________d w
20 3x
512
dw
$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 SubmissionSubmitted 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 • Addition and Subtraction
In order from left to right
In order from left to right
SAMPLE PROBLEM #1
1122)13(416 3
1122)8(416 1122)2(416 3
1122)8(4 112232
23230
Parenthesis
Exponents
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)
More division
Subtraction
SAMPLE PROBLEM
265)32(3 2
265)5(3 2
265)25(3
2
6575 2
10 5Subtraction
Exponents
Remember the division symbol here is grouping everything on top, so work everything up there first….multiplication
Parenthesis
Division – because all the work is done above and below the line
Order of Operations-BASICSThink: PEMDAS
Please Excuse My Dear Aunt Sally
• Parenthesis• Exponents• Multiplication• Division • Addition• Subtraction
Practice 1.2 Order of Operations and Evaluating Expression A-CED.1
327 12 6. __________ 8 3
2
Simplify
1. 4 __________ 32. 5 __________ 35 3. __________
6
164. 4(5) __________ 2
35. 4 (5) 3(11) _________
4 4 16 5 5 5 125 5 5 5 1256 6 6 216
8 20 12 64(5) 33320 33
353315
5
33 27
Practice 1.2 Order of Operations and Evaluating ExpressionUsing the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 7. Area of a triangle: 6 and 14 . b in h in
1: 2
F A bh
1: (6)(14)2
S A
2: 42A A in
Practice 1.2 Order of Operations and Evaluating ExpressionUsing the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA
8. Volume of a pyramid: 18 and 8 .
B m h m 1: 3
F V Bh
1: (18)(8)3
S V
3: 48A V m
Practice 1.2 Order of Operations and Evaluating ExpressionUsing 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 1, 2 3a b and c
2 4: 2
b b acF xa
2( 2) ( 2) 4(1)( 3):
2(1)S x
2 4 12 2
x
2 4 32
x
2 4 12
x
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.
60hhours $
2
6
8
10
120
360
480
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
radicand radicand
radicand
Name the radicand of each of the following, then write in simplified form.
1. 64 ___________, 64 ________ 2. 3 25 ___________,3 25 ________
1 13. ___________, ____36 36
radicand
81 81____ 4. ___________, ________100 100
64 8 25 3 5 15
3616 81,100
910
Practice 1.3 Real Number and the Number Line
6. A ___________ is a well-defined collection of objects.
perfect square perfect square
Estimate the square root by finding the two closest perfect squares.
5. 51 < 51 < 51 _______49 64 7
7. Each objects is call an ________________ of a set.
set
8. A ____________ of a set consists of elements from the given set.
element
9. 2, 4,6,8 2,8 , is a subset of ? yes/no_________U and A A U
subset
10. 2,4,6,8 2,3 , is a subset of ? yes/no_________U and A A U
yes
no
Practice 1.3 Real Number and the Number Line
Circle all the statements that are true.
111. 9 rational 12. 5 irrational 13. integer 14. 0 whole3
15. rational 16. 25 irrational
2
9 17. whole 18. 0 natural3
10019. rational 20. 4 irrational 21. 2.56 rational 22. 2 irrational49
An is a mathematical sentence that compares the values of two expressions using aninequality symbol. The symbols are: ( >, <, , )
inequality
______, less than _______,less than or equal to ______, greater than _______,greater than or equal to
723. What is the order of 3.51, 2.1, 9, ,and 5 from least to greatest?2
2,
3 3.5
2
5, 9,7 ,2
3.51
Properties of Real Numbers(A listing)
• Associative Properties• Commutative Properties• Inverse Properties• Identity Properties• Distributive PropertyAll of these rules apply to Addition and Multiplication
Associative PropertiesAssociate = group
Rules:Associative Property of Addition
(a+b)+c = a+(b+c)
Associative Property of Multiplication
(ab)c = a(bc)
It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same!
Samples:Associative Property of Addition
(1+2)+3 = 1+(2+3)
Associative Property of Multiplication
(2x3)4 = 2(3x4)
Commutative PropertiesCommute = travel (move)
Rules:Commutative Property of Addition
a+b = b+a
Commutative Property of Multiplication
ab = ba
It doesn’t matter how you swap addition or multiplication around…the answer will be the same!
Samples:Commutative Property of Addition
1+2 = 2+1
Commutative Property of Multiplication
(2x3) = (3x2)
Stop and think!
• Does the Associative Property hold true for Subtraction and Division?
• Does the Commutative Property hold true for Subtraction and Division?Is 5-2 = 2-5? Is 6/3 the same as 3/6?
Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)?
Properties of real numbers are only for Addition and Multiplication
Inverse PropertiesThink: Opposite
Rules:Inverse Property of Addition
a+(-a) = 0
Inverse Property of Multiplication
a(1/a) = 1
Samples:Inverse Property of Addition
3+(-3)=0
Inverse Property of Multiplication
2(1/2)=1
What is the opposite (inverse) of addition?What is the opposite of multiplication?
Subtraction (add the negative)
Division (multiply by reciprocal)
Identity Properties
Rules:Identity Property of Addition
a+0 = a
Identity Property of Multiplication
a(1) = a
Samples:Identity Property of Addition
3+0=3
Identity Property of Multiplication
2(1)=2
What can you add to a number & get the same number back?What can you multiply a number by and get the number back?
0 (zero)
1 (one)
Distributive Property
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
If something is sitting just outside a set of parenthesis, you can distribute it through the parenthesis with multiplication and
remove the parenthesis.
A. Associative Property of Addition/Multiplication B. Commutative Property of Addition/MultiplicationC. Identity Property of Addition/MultiplicationD. Zero Property of MultiplicationE. MultiplicaWhat property is illustrated by each statement?_____1. 4 1 4 _____2. 3 ( 1 ) 3 ( ) _____3. 0_____4. 4( 1) ( 1)4 _____5. 5 (
x x p p m mx x x y
tion Property of -1
) (5 ) _____6. x y xyz yxz
Practice 1.4 Properties of Real Numbers
C E CB A B
: Give an exampleD
Practice 1.5 Adding and Subtracting Real Numbers
Find each sum.1. 8 5 2. 7 3 3. 6 4 4. 1 7 5. 2 9 6. 5 9
7. 10 6 8. 15 6 9. 8 10 10. 7 16 11. 2 9
12. 5 25
13. 10 1 14. 11 6 15. 8 5 16. 7 12 17. 12 10
3 10 2 6 11 4
4 21 18 9 7 30
11 5 13 5 2
Absolute Value.Simplify each expression.18. 8 5 19. 7 4 20. 6 4 21. 1 7 22. 2 9
8 513
1111
6 410
1 7
611
11Opposites: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
additive inverse
22
44
33
77
1010
Practice 1.6 Multiplying and Dividing Real Numbers
Find each product/quotient.1. 8 5 2. 7 3 3. 6 4 4. 1 7 5. 2 9 6. 5 9
7. 10 6 8. 15 6 9. 8 10 10. 7 16 11. 2 9
12. 5 25
12 1010 15 8 6 1213. 14. 15. 16. 17. 2 5 8 6 8
1 110 3 418. 19. 20. 21. 1 6 52 7
12 25
40 21 24 7 18 45
60 90 80 112 18 125
5 3 1 3
28
14
10( 2) 20
1 7 73 6 18
1 5 54 1 4
512 302
Practice 1.7 Distributive Property
What is the simplified form of each expression?11. 5( 7) 2. 12(3 ) 3. (0.4 1.1 )3 6
x x c
5 35x 36 2x 1.2 3.3c
27. 4(2 3 1) 8. 5 (2 5) 9. ( 3)
x x x x x x
4. (2 1)( ) 5. 4( 2 5) 6. ( 6)
y y x x 22y y 8 20x 6x
28 12 4x x 210 25x x 2 3x x
Practice 1.7 Distributive Property
210. Using the following expression: 3 4 2 . How many terms? _________ . List the coefficients: _________ . List the constants: _________
x xabc
3
3, 4 2
4 4 2 2 2
What is the simplified form of each expression?
11. 3 12. 7 5 13. 3
14. 3 4 15. 5 3 8
y y mn mn y x y x y
a b a b x y x y
16. 5 3 10 3 2
y y x x y
2y 412mn 2 22y x y
7b 2 3x y 4 7y x
Practice 1.8 An Introduction to Equations
Tell whether each equation is true, false or open. Explain.451. 14 22 2. 42 10 52 3. 7 8 15
x
Tell whether the given number is a solution of each equation.4. 3 8 13; 7 5. 4 7 15; 2 6. 12 14 2 ; 1
b x f
Open True False
?3( 7) 8 13
? 21 8 13
?4( 2) 7 15
?8 7 15
?15 15
Not
Yes
?12 14 2( 1)
?12 14 2
NO
Practice 1.8 An Introduction to Equations
Write an equation for each sentence.
7. The difference of a number and 7 is 8. _________________________________________
8. 6 times the sum of a number and 5 is 16. ________________________________________
7 8n
6( 5) 16n