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

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 3

t11d

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

203

x

5

12

d

w

$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

2

65)32(3 2

2

65)5(3 2

2

65)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 __________ 3

5 3. __________

6

164. 4(5) __________

2

35. 4 (5) 3(11) _________

4 4 16 5 5 5 125 5 5 5 125

6 6 6 216

8 20 12 64(5) 33320 33

3533

15

5

33 27

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: 6 and 14 . b in h in

1:

2F A bh

1: (6)(14)

2S A

2: 42A A in

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: 18 and 8 .

B m h m 1

: 3

F V Bh

1: (18)(8)

3S V

3: 48A V m

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 1, 2 3a b and c

2 4:

2

b b acF x

a

2( 2) ( 2) 4(1)( 3):

2(1)S x

2 4 12

2x

2 4 3

2x

2 4 1

2x

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

361

6 81,1009

10

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 whole

3

15. rational 16. 25 irrational

2

9 17. whole 18. 0 natural

3

10019. rational 20. 4 irrational 21. 2.56 rational 22. 2 irrational

49

An is a mathematical sentence that compares the values of two expressions using an

inequality 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 Property

All 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/Multiplication

C. Identity Property of Addition/Multiplication

D. Zero Property of Multiplication

E. Multiplica

What 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 m

x 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

10

10

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

2

81

4

10( 2) 20

1 7 7

3 6 18

1 5 5

4 1 4

512 30

2

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

6x 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 x

a

b

c

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 15Not

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