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Section 1 – Expressions
The following Mathematics Florida Standards will be covered in this section: MAFS.912.A-APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MAFS.912.A-SSE.1.1 Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. MAFS.912.A-SSE.1.2 Use the structure of an expression to identify ways to rewrite it. MAFS.912.N-RN.1.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. MAFS.912.N-RN.1.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. MAFS.912.N-RN.2.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Topics in this Section
! Topic 1: Using Expressions to Represent Real-World Situations Topic 2: Understanding Polynomial Expressions Topic 3: Algebraic Expressions Using the Distributive Property Topic 4: Algebraic Expressions Using the Commutative and Associative Properties Topic 5: Properties of Exponents Topic 6: Radical Expressions and Expressions with Rational Exponents Topic 7: Adding Expressions with Radicals and Rational Exponents Topic 8: More Operations with Radicals and Rational Exponents Topic 9: Operations with Rational and Irrational Numbers
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Section 1 – Topic 1
Using Expressions to Represent Real-World Situations
Jenny tweets 33 times a day. Antonio posts five tweets every day. Let " represent the given number of days_._
Use an algebraic expression to describe Jenny’s total posts after any given number of days.
Create an algebraic expression to describe Antonio’s total posts after any given number of days.
Write an algebraic expression to describe the combined total posts for Jenny and Antonio after any given number of days.
After five days, how many tweets have Antonio and Jenny posted altogether?
Let’s Practice!
- Mario and Luigi plan to buy a Wii U™ for . Wii U™ games cost each. They plan to purchase one console.
a. Use an algebraic expression to describe how much they will spend before sales tax based on purchasing the console and the number of games.
b. If they purchase one console and three games, how much do they spend before sales tax?
c. Mario and Luigi want to purchase some extra controllers for their friends. Each controller costs . Use an algebraic expression to describe how much they spend in total, before sales tax based on purchasing the console, the number of games, and the number of extra controllers.
- The Griffin family wants to buy an Xbox One + Kinect Sensor for 3. They also want to buy accessories and games. The wireless controllers cost each. The headsets cost each. The games cost each. Peter and Lois are trying to decide how many accessories and games to buy for their family. Let " represent the number of wireless controllers, # represent the number of headsets, and $ represent the number of games the Griffins will purchase. Which of the following algebraic expressions can be used to describe how much the Griffins will spend, before sales tax, based on the number of accessories and games they purchase?
A 3 H M" H # H $N
B 3 " H # H $
C 3 H " H # H $
D 3" H # H $ H
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Section 1 – Topic 2
Understanding Polynomial Expressions
A term is a constant, variable, or multiplicative combination of the two.
Consider 3" ,^ H # I $ H.
How many terms do you see?
List each term.
This is an example of a polynomial expression. A polynomial can be one term or the sum of several terms. There are many different types of polynomials.
A monarchy has one leader. How many terms do you think a monomial has?
A bicycle has two wheels. How many terms do you think a binomial has?
A triceratops has three horns. How many terms do you think a trinomial has?
!
Let’s recap:
Type of Polynomial Number of Terms Example
Monomial
Binomial
Trinomial
Polynomial
Some important facts:
The degree of a monomial is the sum of the ____________ of the variables.
The degree of a polynomial is the degree of the monomial term with the ____________ degree.
Sometimes, you will be asked to write polynomials in standard form.
Write the monomial terms in ________________ _________ order.
The leading term of a polynomial is the term with the ________________ _____________.
The leading coefficient is the coefficient of the _____________ _________.
Let’s Practice!
- Are the following expressions polynomials? If so, name the type of polynomial and state the degree. If not, justify your reasoning.
a. ",#- b.
,6?
c. - , "^.^ I^ "^ -^ H "^1 d.^
0 , H - I 1
e. H 35+^ H ,
!
BEAT THE TEST!
- Match the polynomial in the left column with its descriptive feature in the right column.
A. " -^ H " ,^ I " H I. Fifth degree polynomial
B. ,^ -^ II. Constant term of I
C. 3 ".^ I "-^ H "^3 III. Seventh degree polynomial
D. 0 ,^ H -^ I 1 IV. Leading coefficient of 3
E. " /^ I " -^ H " 1 V. Four terms
F. 3" -^ H " ,^ I VI. Eighth degree polynomial
G. ",^ I VII. Equivalent to "^3 H 3 ".^ I "-
Section 1 – Topic 3
Algebraic Expressions Using the Distributive Property
Recall the distributive property.
If &, ', and ( are real numbers, then & ' H ( K & H & Ǥ
One way to visualize the distributive property is to use models. Consider M H 3NM H N.
Now, use the distributive property to write an equivalent expression for M H 3NM H N.
Let’s Practice!
- Write an equivalent expression for 3M" H # I $N by
modeling and then by using the distributive property.
- Write an equivalent expression for M" I 3NM" I N by modeling and then by using the distributive property.
Try It!
- Use the distributive property or modeling to write an equivalent expression for H I 3.
Let’s look at some other operations and how they affect numbers.
Consider H H. What happens if you put parentheses around any two adjacent numbers? How does it change the sum?
Consider 3 . What happens if you put parentheses around any two adjacent numbers? How does it change the product?
This is the associative property.
The ____________ of the numbers does not change.
The grouping of the numbers can change and does not affect the ___________ or ______________.
If , and are real numbers, then H H K and/or K .
Does the associative property hold true for division or subtraction? If it does not, give a counterexample.
Let’s Practice!
Name the property (or properties) used to write the equivalent expression.
D O H I3 P H K H OMI3N H P
b. M N K M N
c. H H K H H
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!
! :LWK SURSHUWLHV ORRN FORVHO\ DW HDFK SLHFH RI WKH SUREOHP 7KH FKDQJHV FDQ EH YHU VXEWOH
!
Try It!
- Identify the property (or properties) used to find the equivalent expression.
a. M H N H K M H N H
b.! M# N K M! #N
c. M H N H K H M H N
d. J3 J K 3J J
- The following is a proof that shows M3"NM #N is equivalent to "#. Fill in each blank with either “commutative property” or “associative property” to indicate the property being used.
3" M #N = 3 " # _________________________________
= 3 " # _________________________________
= M3 NM" #N _________________________________
!
Let’s explore raising products to a power.
3 ,^ K
" -^ K
This is the power of a product property : MN;^ K
Let’s explore raising quotients to a power.
,*
, K
K
This is the power of a quotient property :
6 7
; K Ǥ
Let’s Practice!
1. Determine if the following equations are true or false.
Justify your answer.
a. 3 -^ 3.^ K M^
(^) N
M3 N
b. M ,N -^ K.^ *^
. C / E>
5+
Try It!
- Use the properties of exponents to match each of the
following with its equivalent expression.
BEAT THE TEST!
- Crosby and Adam were working with exponents.
Part A: Crosby claims that 3 -^ 3 ,^ K 3/^. Adam argues that 3 -^ 3 ,^ K 3^0. Which one of them is correct? Use the properties of exponents to justify your answer.
Part B: Crosby claims that
K 3. Adam argues that
- D -?^ K 3^. Which one of them is correct? Use the properties of exponents to justify your answer.
Try It!
- Use the rational exponent property to write an equivalent
expression for each of the following radical expressions.
a.! # b.! B^ # H I 3
- Use the rational exponent property to write each of the following expressions as integers. a. >?
b.
> @
c.
? @
BEAT THE TEST!
- Match each of the following to its equivalent expression.
I.!
> ? (^) I 3
II.! M3N
> ?
III.! M I 3N
> ?
IV.!
V.! @
VI.! @
!
Section 1 – Topic 7
Adding Expressions with Radicals and Rational
Exponents
Let’s explore operations with radical expressions and expressions with rational exponents.
H + , H + ,
H + , H + ,
3 I 3 3 +, I 3 +,
! To add radicals, the radicand of both radicals must be the same. To add expressions with rational exponents, the base and the exponent must be the same. In both cases, you simply add the coefficientV.!
Let’s Practice!
- Perform the following operations.
a. H (^3) b.
> ? (^) H 3
> ?
c. H H (^) d. >? (^) H >? (^) H >?
e. 3 H @ f. 3
> ? (^) H
> @
!
BEAT THE TEST!
1.! Which of the following expressions are equivalent to 3 Select all that apply.
> ? (^) H
> ? !!
> ? (^) H
> ? !! 3
> ? !! ! !! H
2.! Miguel completed the following proof to show that H 3 K 3
> ? (^) :
H 3
K
> ? (^) H 3
> ? K _________ K 3 3
> ? (^) H 3
> ?
K 3
,
Which equation can be placed in the blank to correctly complete Miguel’s work?
A! 3
> ? (^)
> ? (^) H 3
> ? (^) K 3
> ? (^) 3 H 3
> ?
B! 3
> ? (^) H 3
> ? (^) K
> ? (^) 3
> ? (^) H 3
> ?
C!
> ? (^) H 3
> ? (^) H 3
> ? (^) K 3 H 3
> ? (^) H 3
> ?
D! 3 -^
> ? (^) H 3
> ? (^) K
> ? (^) H 3
> ?
Let’s Practice!
1.! Use the properties of exponents to perform the following operations.
a.! "
> @
> ?
b.!
c.!
> ? (^)
? B (^)
? @ (^)
> ?
!! The properties of exponents also apply to expressions with rational exponents. !
Section 1 – Topic 8
More Operations with Radicals and Rational Exponents
Let’s explore multiplying and dividing expressions with radicals and rational exponents.
@^ +, +-
,
