foundations for...
TRANSCRIPT
Foundations for Algebra
Introduction to Algebra I
Variables and Expressions
Objective: To write algebraic expressions.
Objectives
1. I can write an algebraic expression for addition, subtraction, multiplication, and division from an algebraic phrase.
2. I can write algebraic expressions using more than one operation.
3. I can write algebraic phrases when I am given an algebraic expression.
4. I can write a rule to describe a pattern using both algebraic phrases and algebraic expressions.
Vocabularyβ’ A mathematical quantity is anything that can be measured or counted.
Some quantities remain constant. Others change, or vary, and are called variable quantities.
β’ Algebra uses symbols to represent quantities that are unknown or that vary. You can represent mathematical phrases and realβworld relationships using symbols and operations.
β’ A variable is a symbol, usually a letter, that represents the value(s) of a variable quantity.
β’ An algebraic expression is a mathematical phrase that includes one or more variables.
β’ A numerical expression is a mathematical phrase involving numbers and operation symbols, but no variables.
Key Words
β’ When you see the words more than, what do you think of? Are there any other words?
β’ When you see the words less than, what do you think of? Are there any other words?
β’ What do you think of when you see product? Are there any other words?
β’ What comes to mind if you see quotient? Are there any other words?
Math Vocabulary
+ -
Γ Γ·
Writing Expressions With Addition and Subtraction
What is an algebraic expression for the word phrase?
1. 32 more than a number n
2. 58 less than a number n
3. 18 more than a number n
4. 8 minus a number y
Practice
Write an algebraic expression for each phrase.
1. 4 more than a number
2. 10 less than a number
3. The difference of a number x and 1
2
4. The sum of a number m and 7.1
Writing Expressions With Multiplication and Division
What is an algebraic expression for the word phrase?
1. 8 times a number n
2. Quotient of a number n and 5
3. The product of 7 and a number p
4. Quotient of a 8 and a number c
Practice
Write an algebraic expression for each phrase.
1. Quotient of 5 and a number
2. Product of 8 and a number
3. The product of 9 and a number t
4. The quotient of 207 and a number m
Writing Expressions With Two Operations
What is an algebraic expression for the word phrase?
1. 9 more than a number times 10
2. Difference of 1 and the product of 5 and a number
3. The sum of 3 and the quotient of a number and 9
4. 9 less than the quotient of 8 and a number
Practice
Write an algebraic expression for each phrase.
1. A number times 4 plus 2
2. 2 less than the quotient of 7 and a number
3. 6.7 more than the product of 5 and n
4. 9.85 less than the product of 37 and y
Using Words for an Expression
What word phrase can you use to represent the algebraic expression?
1. 9 + r
2. y β 6
3. 3m
4. 2 Γ· w
5. 10x + 9
6. 5x β 1
Practice
Write a word phrase for each algebraic expression.
1. 9 β 2π
2. 8 +π
3
3.π¦
5
4.π§
8β 9
Vocabulary
β’ You have learned how to write an algebraic expression from a phrase.
β’ You have also learned how to write an algebraic phrase from an expression.
β’ You can use words or an algebraic expression to write a mathematical rule that describes a realβlife pattern.
Writing a Rule to Describe a Pattern
The table shows how the height above the floor of a house of cards depends on the number of levels.
What is a rule for the height? Give the rule in words and as an algebraic expression.
Numberof Levels
Height (in.)
2 (3.5 x 2) + 24
3 (3.5 x 3) + 24
4 (3.4 x 4) + 24
n ?
Writing a Rule to Describe a Pattern
A group of students built another house of cards that has 10 levels. Each card was 4 inches tall, and the height from the floor to the top of the house of cards was 70 inches. How tall would the house of cards be if they built an 11th level?
Writing a Rule to Describe a Pattern
Another group of students built a third house of cards with n levels. Each card was 5 inches tall, and the height from the floor to the top of the house of cards was 34 + 5n inches. How tall would the house of cards be if the group added 1 more level of cards?
Practice
While on vacation, you rent a bicycle. You pay $9 for each hour you use it. It costs $5 to rent a helmet while you use the bicycle. Write a rule in words and as an algebraic expression to model the relationship in each table.
Number of Hours
Rental Costs
1 ($9 x 1) + $5
2 ($9 x 2) + $5
3 ($9 x 3) + $5
n ?
Order of Operations and Evaluating Expressions
Objective: To simplify expressions involving exponents.
To use the order of operations to evaluate expressions.
Objectives
1. I can simplify powers of numbers by multiplying them out based on the power.
2. I can simplify a numerical expression by using the order of operations.
3. I can evaluate algebraic expressions using the given value for a variable in the equation.
4. I can evaluate and simplify when using real β world examples.
Vocabulary
β’ You can use powers to shorten how you represent repeated multiplication, such as 2 x 2 x 2 x 2 x 2 x 2.
β’ A power has two parts, a base and an exponent.
β’ The exponent tells you how many times to use the base as a factor.
β’ The base is the bottom (or repeated number).
β’ If you simplify a numerical expression you replace it with its single numerical value.
Simplifying Powers
What is the simplified form of the expression?1. 107
2. (0.2)5
3. 34
4. (2
3)3
5. (0.5)3
Practice
Simplify each expression.1. 35
2. (1
2)4
3. (0.4)6
4. 108
5. (β3)3
6. (β5)4
Vocabulary
β’ When simplifying an expression, you need to perform operation in the correct order.
β’ Order of Operations:1. Perform any operation(s) inside grouping symbols, such as
parentheses ( ) and brackets [ ]. A fraction bar also acts as a grouping symbol.
2. Simplify powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.
Saying for Order of Operations
β’ Please p stands for parenthesis
β’ Excuse e stands for exponents
β’ My m stands for multiply
β’ Dear d stands for divide
β’ Aunt a stands for add
β’ Sally s stands for subtract
β’ Remember to multiply and divide from left to right. The same is true for addition and subtraction.
Simplifying a Numerical Expression
What is the simplified form of each expression?1. (6 β 2)3Γ· 2
2.24β1
5
3. 5 Γ 7 β 42 Γ· 2
4. 12 β 25 Γ· 5
5.4+34
7β2
Practice
Simplify each expression.1. 5 Γ 23 Γ· 2 + 8
2. 20 β 2 Γ 32
3. 52 + 82 β 3(4 β 2)3
4.2Γ7+4
9Γ·3
5. 42 β 5 Γ 4 + 6
6. 3 + 62 Γ· 12 β 7
Vocabulary
β’ When two or more variables, or a number and variables, are written together, treat them as if they were within parentheses.
β’ You evaluate an algebraic expression by replacing each variable with a given number.
β’ Then simplify the expression using the order of operations.
Evaluating Algebraic Expressions
β’ What is the value of the expression for x = 5 and y = 2?
1. π₯2 + π₯ β 12 Γ· π¦2
2. (π₯π¦)2Γ· (π₯π¦)
β’ What is the value of the expression when a = 3 and b = 4?
1. 3π β π2
2. 2π2 β 7π
Practice
Evaluate each expression for s = 4 and t = 8.
1. (π + π‘)3
2. π 4 + π‘2 + π Γ· 2
3.(3π )3π‘+π‘
π
4. 2π π‘2 β π 2
Evaluating a Real-World Expression
What is an expression for the spending money you have left
after depositing 2
5of your wages in savings? Evaluate the
expression for weekly wages of $40, $50, $75, and $100.
Wages (w)
πβπ
ππ
Total Spending Money
40
50
75
100
Evaluating a Real-World Expression
The shipping cost for an order at an online store is 1
10the cost of
the items you order. What is an expression for the total cost of a given order? What are the total costs for orders of $43, $79, $95, and $103?
Online Order
π +π
πππ
Total Cost for Order
43
79
95
103
Practice
Write an expression for the amount of change you will get when you pay for a purchase p with a $20 bill. Make a table to find the amounts of change you will get for purchases of $11.59, $17.50, $19.00, and $20.00.
Real Numbers and the Number Line
Objective: To classify, graph, and compare real numbers. To find and estimate square roots.
Objectives
1. I can simplify square roots and square root expressions without a calculator.
2. I can estimate square roots.
3. I an classify real numbers.
4. I can compare and order real numbers.
Vocabulary
β’ Square Root:β A number a is a square root of a number b if π2 = π.
β Example: 72 = 49, so 7 is the square root of 49.
β’ You can use the definition of the square root to find the exact square roots of some nonnegative numbers. You can approximate the square roots of other nonnegative numbers.
β’ The radical symbol β indicates a nonnegative square root.
β’ The expression under the radical symbol is called the radicand.
Vocabulary
β’ πππππππ π π¦ππππ β π β ππππππππ
β’ Together, the radical symbol and radicand form a radical.
β’ The square of an integer is called a perfect square.
β’ When a radicand is not a perfect square, you can estimate the square root of the radicand.
Simplifying Square Root Expressions
What is the simplified form of each expression?1. 81
2.9
16
3. 64
4. 25
5.1
36
6.81
121
Practice
What is the simplified form of each expression?
1. 121
2.81
144
3. 169
4. 225
5.1
196
6.16
100
Estimating a Square Root
Lobster eyes are made of tiny square regions. Under a microscope, the surface of the eye looks like graph paper. A scientist measures the area of one of the squares to be 386 square microns. What is the appropriate side length of the square to the nearest micron?
What is the value of 34 to the nearest integer?
What is the value of 242 to the nearest integer?
What is the value of 61 to the nearest integer?
Practice
Estimate the square root. Round to the nearest integer.
1. 35
2. 17
Find the approximate sine length of each square figure to the nearest whole number.
1. A game board with an area of 160 inΒ²
2. A mural with an area of 18 mΒ²
Vocabularyβ’ A set is a well-defined collection of objects.
β’ Each object is called an element of the set.
β’ A subset of a set consists of elements from the given set.
β’ A set of real numbers is formed by rational and irrational numbers.
Vocabularyβ’ A rational number is any number that you can write in the form
π
π,
where a and b are integers and b cannot equal 0. Rational numbers can also be repeating decimals.
β’ An integer can be positive or negative.
β’ A whole number starts at 0 and is counted up.
β’ A natural number starts at 1 and is counted up using whole numbers.
β’ An irrational number cannot be represented as the quotient of two integers. Usually is a nonrepeating decimal or other number that in not in
π
πform.
Examples of Real Numbers
Real NumbersRATIONAL IRRATIONAL
INTEGER
WHOLE
NATURAL
Classifying Real Numbers
Determine the set of real numbers each belongs to.
1. 25
2. 21
3. 91
4. 121
5. 40
6.3
5
7.2
3
8. -7
Practice
Name the subset(s) of the real numbers to which each number belongs.
1. π
2. β3
3. 196
4. 12
5.4
5
6.10
12
Vocabulary
β’ An inequality is a mathematical sentence that compares the values of two expressions using an inequality symbol.
β’ The inequality symbols are:
β < Less than
β > Greater than
β β€ Less than or equal to
β β₯ Greater than or equal to
Comparing Real Numbers
Determine the correct inequality symbol.1. 7 -9
2. 2 6
3. 5 22
4. 75 7.5
5. 400 20
6.9
160.75
Practice
Compare the numbers in each using an inequality symbol.
1. β3.1, β16
5
2.4
3, 2
3. β7
11, β0.63
4. 184, 15.56987β¦
Graphing and Ordering Real Numbers
What is the order from least to greatest?
1. 4, 0.4,β2
3, 2, β1.5
2. 3.5,β2.1, 9, β7
2, 5
3. β1
6, β0.3, 1, β
2
13,7
8
4. β3, 31, 11, 5.5,β60
11
Practice
Order the numbers in each from least to greatest.
1.1
2, β2, 5, β
7
4, 2.4
2.10
3, 3, 8, 2.9, 7
3. β6, 20, 4.3, β59
3
4. β1
6, β0.3, 1, β
2
13,7
8
Exit Ticket
1. Solve: 169
2. Round to the nearest integer: 3203. Name the subsets the following belong to:
π
144 β2.38 β6
19
100
4. Compare: β22
25, 2
5. Order from greatest to least: β6, 20, 4.3,β59
9
Properties of Real Numbers
Objective: To identify and use properties of real numbers.
Objectives
β’ I can identify the different properties of real numbers.
β’ I can use the properties to solve problems and to write equivalent expressions.
β’ I can use deductive reasoning and find counterexamples.
Vocabulary
β’ Two algebraic expressions are equivalent expressions if they have the same value for all values of the variable(s).
a. 3 + 5 β 2 ___ 6
b. 0 + 180 ___ 180
c. 12 Γ· 1 ___ 13
d. 2 Γ 5 ____ 5 Γ 2
e. 45 β 1 ____ 45
Communitive Property
β’ Changing the order of the addends does not change the sum.
β’ Changing the order of the factors does not change the product.
Algebra Example
Addition a + b = b + a 18 + 54 = 54 + 18
Multiplication a Γ b = b Γ a 12 Γ 1 = 1 Γ 12
Associative Property
β’ Changing the grouping of the addends does not change the sum.
β’ Changing the grouping of the factors does not change the product.
Algebra Example
Addition (a + b) + c = a + (b + c) (23 + 9) + 4 = 23 + (9 + 4)
Multiplication (a Γ b) Γ c = a Γ (b Γ c) (7 Γ 9) Γ 10 = 7 Γ (9 Γ 10)
Identity Properties
β’ The sum of any number and zero is the original number.
β’ The product of any number and one is the original number.
Algebra Example
Addition a + 0 = a 5.75 + 0 = 5.75
Multiplication a Γ 1 = a 67 Γ 1 = 67
Multiplication Properties
β’ Zero Property of Multiplication is the product of a number and zero which always equals zero.
β’ Multiplication Property of Negative One states that the product of negative one and a number is that number negative.
Algebra Example
Zero Property of Multiplication
a Γ 0 = 0 18 Γ 0 = 0
MultiplicationProperty of β1
β1 Γ a = βa β1 Γ 9 = β9
Identifying Properties
What property is illustrated by each statement?
a. 8 + 9 = 9 + 8
b. 88 Γ -1 = -88
c. (12 + 3) + 5 = (3 + 5) + 12
d. 7 Γ 1 = 1 Γ 7
e. 4 Γ 0 = 0
f. (2 Γ 4) Γ 6 = (6 Γ 2) Γ 4
g. 6 + 0 = 6
h. 13 Γ 1 = 13
Practice
What property is illustrated by each statement?
a. β3 Γ 1 = β3
b. β14 Γ β1 = 14
c. (β2 Γ β3) Γ β8 = (β3 Γ β8) Γ β2
d. β7 Γ 8 = 8 Γ β7
e. β20 Γ 0 = 0
f. 14 + (β8) = (β8) + 14
g. β77 + 0 = β77
h. (7 + β3) + β5 = (β3 + β5) + 7
Using Properties for Mental Calculations
A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs $1.25. What is the total cost for a ticket, a drink, and popcorn? Use mental math.
A can holds 3 tennis balls. A box holds 4 cans. A case holds 6 boxes. How many tennis balls are in 10 cases? Use mental math.
Practice
Simplify each expression.
a. 21 + 6 + 9
b. 10 x 2 x 19 x 5
c. 55.3 + 0.2 + 23.8 + 0.7
d. 0.25 x 12 x 4
Writing Equivalent Expressions
Simplify each expression.
a. 5 + (3n β 7)
b. (4 + 7b) + 8
c.6xy
y
d. 2.1(4.5x)
e. 6 + (4h + 3)
f.8m
12mn
Practice
Simplify each expression. Justify each step.1. 2 + 3π₯ + 9
2. 4 Γ π₯ Γ 6.3
3.13π
ππ
4.33π₯π¦
3π₯
5. 8 + (9π‘ + 4)
Vocabulary
β’ Deductive reasoning is the process of reasoning logically from given facts to a conclusion.
β’ To show that a statement is not true, find an example for which it is not true. An example showing that a statement is false is a counterexample.
Using Deductive Reasoning and Counterexamples
Is the statement true or false? If it is false, give a counterexample.
β’ For all real numbers a and b, π Γ π = π + π.
β’ For all real numbers a, b, and c, π + π + π = π + π + π .
β’ For all real numbers j and k, π Γ π = π + 0 Γ π.
β’ For all real numbers m and n, π π + 1 = ππ + 1.
Practice
Use deductive reasoning to tell whether each statement is true or false. If it is false, give a counterexample. If true, use properties of real numbers to show the expressions are equivalent.
β’ For all real numbers r, s, and t, π Γ π Γ π‘ = π‘ Γ π Γ π
β’ For all real numbers p and q, π Γ· π = π Γ· π
β’ For all real numbers x, π₯ + 0 = 0
β’ For all real numbers a and b, βπ Γ π = π Γ (βπ)
Exit Ticket
1. Name the property:
a. 75 + 6 = 6 + 75
b. (2 Γ 3) Γ 4 = 3 Γ (4 Γ 2)
c. h + 0 = h
d. 9 Γ -1 = -9
e. 27Γ0 = 0
f. 3 Γ 2 = 2 Γ 3
g. 5 Γ 1 = 5
h. (3 + 4) + 9 = 4 + (3 + 9)
2. Simplify Using Mental Math: 10 Γ 2 Γ 5 + 7
3. Simplify the Expression: (2 + 3x) + 9
4. Tell if the expressions are equivalent: 9y Γ 0 and 1
Adding and Subtracting Real Numbers
Objective: To find sums and differences of real numbers.
Objectives
β’ I can use number lines to model addition and subtraction.
β’ I can add real numbers.
β’ I can subtract real numbers.
β’ I can solve real world problems using addition or subtraction.
Using Number Line Models
What is each sum? Use a number line.
a. 3 + 5
b. 3 + (β5)
c. β3 + 5
d. β3 + (β5)
e. β8 + 4
Practice
Use a number line to find each sum.
a. β6 + 9
b. 4 + (β3)
c. 2 + 5
d. β3 + (β4)
e. 5 + (β7)
Vocabulary
β’ The absolute value of a number is its distance from zero on a number line. Absolute value is ALWAYS nonnegative.
β’ Adding numbers with the same sign:
β To add two numbers with the same sign, add their absolute values. The sum has the same sign as the addends.
β Examples: 3 + 4 = 7 β3 + (β4) = β7
β’ Adding numbers with different signs:
β To add two numbers with different signs, subtract their absolute values. The sum has the same sign as the addend with the greater absolute value.
β Examples: β3 + 4 = 1 3 + (β4) = β1
Adding Real Numbers
Find the sum of each.
a. β12 + 7
b. β18 + (β2)
c. β4.8 + 9.5
d.3
4+ β
5
6
e. β16 + (β8)
f. β11 + 9
g. 9 + (β11)
h. β6 + (β2)
Practice
Find each sum.
a. β2 + 7
b. 3.2 + 1.4
c. β14 + β10
d.1
2+ β
7
2
e. 11 + 9
Vocabulary
β’ To numbers that are the same distance from zero on a number line but lie in opposite directions are opposites.
β’ A number and its opposite are called additive inverses.
β’ To find the sum of a number and its opposite, you can use the Inverse Property of Addition.
Vocabulary
β’ Inverse Property of Addition:β For every real number a, there is an additive inverse βa such that
a + (βa) = βa + a = 0.
β Example: 13 + (β13) = 0
β’ You can also use opposite to subtract real numbers.
β’ Subtracting Real Numbers:β To subtract real numbers us the following expression a β b = a + (βb).
β Example: 15 β 5 = 15 + (β5) = 10
Subtracting Real Numbers
Find the difference.a. β8 β (β13)
b. 3.5 β 12.4
c. 9 β 9
d. 4.8 β (β8.7)
e. β7 β (β5)
f.1
8β
3
4
g. β2.5 β 17.8
h. 36 β (β12)
Practice
Find each difference.
a. 5 β 15
b. β3.5 β 14.8
c. β7 β β3
d. 36 β β14
e. 3.5 β 1.9
f.5
8β
1
2
Adding and Subtracting Real Numbers
β’ A reef explorer dives 25 feet to photograph brain coral and then rises 16 feet to travel over a ridge before diving 47 feet to survey the base of the reef. Then the diver rises 29 feet to see an underwater cavern. What is the location of the cavern in relation to sea level?
β’ A robot submarine dives 803 feet to the ocean floor. It rises 215 feet as the water gets shallower. Then the submarine dives 2619 feet into a deep crevice. Next, it rises 734 feet to photograph a crack in the wall of the crevice. What is the location of the crack in relation to sea level?
Practice
A stockβs starting price per share is $51.47 at the beginning of the week. During the week, the Price changes by gaining $1.22, then losing $3.47, then losing $2.11, then losing $0.98, and finally gaining $2.41. What is the ending stock price?
Exit Ticket
Find the sum or difference.
1. 2 + 5
2. 4 + (β3)
3. β2 + 7
4. β9 + β2
5. 5 β 16
6. 8.5 β 7.6
7. β2.5 β 17.8
8. β7 β (β5)
Objective: to find products and quotients of real numbers.
Multiplying and Dividing Real Numbers
Objectives
β’ I can multiply real numbers.
β’ I can simplify square root expressions.
β’ I can divide real numbers.
β’ I can divide fractions.
Vocabulary
β’ The rules for multiplying real numbers are related to the properties of real numbers and the definitions of operations.
β’ Multiplying real numbers:β The product of two real numbers with different signs is negative.
β Examples: 2(β3) = β6 β2 x 3 = β6
β The product of two real numbers with the same sign is positive.
β Examples: 2 x 3 = 6 β2(β3) = 6
Multiplying Real Numbers
What is each product?
1. 12(β8)
2. 24(0.5)
3. β3
4Γ
1
2
4. (β3)2
5. 6(β15)
6. 12(0.2)
7. β7
10
3
5
8. (β4)2
Practice
Find each product. Simplify, if necessary.1. β7 Γ 11
2. 10 β2.5
3. β1
9β
3
4
4. 8 12
5. 6 β1
4
6. (β1.2)2
Simplifying Square Root Expressions
What is the simplified form?
1. β 25
2. Β±4
49
3. 64
4. Β± 16
5. β 121
6. Β±1
36
Practice
Simplify each expression.
1. 400
2. Β± 0.25
3.36
49
4. β 16
5. β1
9
Vocabulary
Dividing Real Numbers
β The quotient of two real numbers with different signs is
negative.
β 16 Γ· 4 = β4
β The quotient of two real numbers with the same sign is
positive.
32 Γ· 8 = 4
β The quotient of zero and any real number is o.
0 Γ· 4 = 0
β The quotient of any real number and o is undefined.
4 Γ· 0 = π’ππππππππ
Dividing Real Numbers
β’ A sky diverβs elevation changes by -3600 feet in 4 minutes
after the parachute opens. What is the average change in the
sky diverβs elevation each minute?
β’ You make five withdrawals of equal amounts from your bank
account. The total amount you withdraw is $360. What is the
change in your account balance each time you make a
withdrawal?
Practice
Find each quotient. Simplify, if necessary.1. 48 Γ· 3
2.β121
11
3. β39 Γ· β13
4. β8.1 Γ· 9
5.β63
β21
6.46
2
Vocabulary
β’ The Inverse Property of Multiplication describes the
relationship between a number and its multiplicative inverse.
β’ Inverse Property of Multiplication:
β For every nonzero real number a, there is a multiplicative
inverse1
πsuch that a(
1
π) = 1.
β Example: The multiplicative inverse of β4 ππ β1
4because
β 4 β1
4= 1
β’ The reciprocal of a nonzero number in the form π
πis
π
π.
β’ The product of a number and its reciprocal is 1.
β’ The reciprocal of a number is its multiplicative inverse.
Dividing Fractions
β’ What is the value of π₯
π¦when x =
3
4and y = β
2
3?
β’ What is the value?
β3
4Γ· β
5
2
β9
10Γ· β
4
5
β β12
13Γ·
12
13
β’ What is the value of π₯
π¦when x =
2
7and y = β
20
21?
β’ What is the value of π₯
π¦when x = β
5
6and y =
3
5?
Practice
β’ Find each quotient. Simplify, if necessary.
1.2
5Γ·
8
5
2. 20 Γ· β1
4
3. β5 Γ· β5
3
β’ Find the value of the expression π₯
π¦for the given values of x and
y. Write your answer in the simplest form.
1. π₯ = β2
3πππ π¦ = β
1
4
2. π₯ =3
8πππ π¦ =
3
4
Exit Ticket
β’ What is the reciprocal of 1
5?
β’ What is the product of 2
3and its reciprocal?
β’ Solve: 4 x -5
β’ Solve: -3 x -8
β’ Solve: 7 x 9
β’ Solve: 1
4Γ·
1
2
β’ Solve: 2
3Γ·
4
3
The Distributive Property
Objective: To use the distributive property to simplify expressions.
Objectives
β’ I can use the distributive property to simplify expressions.
β’ I can use the distributive property to rewrite fraction expressions.
β’ I can use the multiplication property of β1 to simplify expressions.
β’ I can use the distributive property to solve mental math problems.
β’ I can combine like terms using the distributive property.
β’ The distributive property is another property of real numbers that helps you to simplify expressions.
β’ When you are using the distributive property, you are combining like terms.
β’ You can use the distributive property to simplify the product of a number and a sum or difference.
Vocabulary
Let a, b, and c be real numbers.
a(b + c) = ab + ac
4(20 + b) = 4(20) + 4(b)
(b + c)a = ba + ca
(20 + b)4 = 20(4) + b(4)
a(b β c) = ab β ac
7(30 β x) = 7(30) β 7(x)
(b β c)a = ba β ca
(30 β x)7 = 30(7) β x(7)
Distributive Property
What is the simplified form of each expression?
1. 3(x + 8)
2. 7(5 β r)
3. 5(b β 4)
4. 2(c + w)
5. β3(8 + y)
6. β4(βe β 2p)
Simplifying Expressions
Practice
Use the distributive property to simplify each expression.
1. 6 π + 10
2.1
44π + 12
3. 2 7 β π
4.1
3(9 β 3π)
5. β5(4π β 2π)
Vocabulary
β’ Recall that a fraction bar may act as a grouping symbol.
β’ A fraction bar indicates division.
β’ Any fraction π
πcan also be written as π
1
π.
β’ You can use this fact and the Distributive Property to rewrite some fractions as sums or differences.
What is the sum or difference that is equivalent?
1.7x+2
5
2.4x β16
3
3.11+3x
6
4.15+6x
12
5.4 β2x
8
6.β 2+4x
5
Rewriting Fraction Expressions
Practice
Write each fraction as a sum or difference.
1.25β8π‘
5
2.10π+12
2
3.42π€+14
7
4.18π₯β51
17
Vocabulary
β’ The Multiplication Property of β1 states that β1 x r = βr.
β’ To simplify an expression such as β(x + 6), you can rewrite the expression as β1(x + 6).
What is the simplified form?
1. β(8 + 2e)
2. β(4r β 3b)
3. β(v + re)
4. β(2y β 3x)
5. β(a + 5)
6. β(βx + 31)
7. β(4x β 12)
8. β(6m β 9n)
Using Multiplication Property of -1
Practice
Simplify each expression.
1. β 20 + π
2. β 4π₯ β 7
3. β β5 β 4π¦
4. β(β2π + π)
5. β π + 5 + π
6. β(βπ + 3π¦ β π₯)
β’ Deli Sandwiches cost $4.95. What is the total cost of 8 sandwiches? Use mental math.
β’ Julia commutes to work on the train 4 times each week. A round-trip ticket costs $7.25. What is her weekly cost for tickets? Use mental math.
Using Distributive Property for Mental Math
Practice
β’ You buy 50 of your favorite songs from a Web site that charges $0.99 for each song. What is the cost of 50 songs? Use mental math.
β’ One hundred and five students see a play. Each ticket costs $45. What is the total amount the students spend for tickets? Use mental math.
β’ Suppose the distance you travel to school is 5 miles. What is the total distance for 197 trips from home to school? Use mental math.
β’ A term is a number, a variable, or the product of a number and one or more variables.
β’ A constant is a term that has no variable.
β’ A coefficient is a numerical factor of a term.
β’ Like terms have the same variable factors.
Vocabulary
8x2 - 4xy + 7y β 14
β’ What are the terms?
β’ What are the constants?
β’ What are the coefficients?
β’ What are the like terms?
Combining Like Terms
Simplify each.
1. 8x2 + 2x2
2. 7y3z β 6yz3 + y3z
3. 3y β y
4. -5w2 + 12w2
5. -7mn4 β 5mn4
6. -7h + 3h2 β 4h β 3
Combining Like Terms
Practice
Simplify each expression by combining like terms.
1. βπ + 4π
2. 2π₯2 β 9π₯2
3. β4π₯3 + 6π₯3
4. β7β + 3β2 β 4β β 3
5. 10ππ + 2ππ2 β 9ππ
6. 2π + 1 β 4π β π
What is the simplified form?
β’ (j + 2)7
β’ -8(x β 3)
β’ 3a β 5a
β’ 12(2j β 6)
β’ -(-m + n + 1)
β’ Use mental math: 3 Γ 7.25
β’ You buy 50 of your favorite songs from a Web site
that charges $.99 for each song. What is the total
cost of 50 songs? Use mental math.
Exit Ticket
An Introduction to Equations
Objective: To solve equations using tables and mental math.
Objectives
β’ I can classify different type of equations.
β’ I can identify solutions of an equation.
β’ I can write an equation.
β’ I can use mental math to find solutions of equations.
β’ I can use a table to find a solution of an equation.
β’ I can estimate a solution of an equation.
Vocabulary
β’ An equation is a mathematical sentence that uses an
equal sign (=).
β’ An equation is true if the expressions on either side of
the equal sign are equal. 2 + 5 = 9 β 2
β’ An equation is false if the expressions on either side of
the equal sign are not equal. 12 β 4 = 5 + 2
β’ An equation is an open sentence if it contains one or
more variable and may be true or false depending on
the values of its variables. 4x + 5 = 13
Classifying Equations
Is the equation true, false, or open?
1. 24 + 14 = 9 + 29
2. 4 Γ 6 = 22
3. 5v + 13 = -7
4. 3y + 6 = 5y β 8
5. 16 β 7 = 4 + 5
6. 32 Γ· 4 = 2 Γ 3
Practice
Tell whether each equation is true, false, or open. Explain.
1. 85 + β10 = 95
2. 225 Γ· π‘ β 4 = 6.4
3. β8 β2 β 7 = 14 β 5
4. 14 + 7 + β1 = 21
5. 5π₯ + 7 = 17
6. 29 β 34 = β5
Vocabulary
β’ A solution of an equation containing a variable is
the value of the variable that makes the equation
true.
β’ In real-world problems, the word is can indicate
equality.
Identifying Solutions of an Equation
β’ Is x = 6 a solution of the equation 32 = 2x + 12?
β’ Is m = 1
2a solution of the equation 6m β 8 = -5?
β’ Is x = 3 a solution of the equation 8x + 5 = 29?
β’ Is a = -8 a solution of the equation 9a β (β72) = 0?
Practice
Tell whether the given number is a solution of each equation.
1. 2 = 10 β 4π¦; 2
2. β6π + 5 = 1;1
2
3.3
2π‘ + 2 = 4;
2
3
4. 6 = 2π β 8; 7
5. 7 + 16π¦ = 11;1
4
6. 9π β β72 = 0;β8
Writing an Equation
β’ An art student wants to make a model of the
Mayan Great Ball Court in ChichΓ©n Itza, Mexico.
The length of the court is 2.4 times its width. The
length of the studentβs model is 54 inches. What
should the width of the model be?
β’ The length of the ball court at La Venta is 14 times
the height of its walls. Write an equation that can
be used to find the height of a model that has a
length of 49 centimeters.
Practice
β’ The sum of 4x and β3 is 8.
β’ The product of 9 and the sum of 6 and x is 1.
β’ The manager of a restaurant earns $2.25 more each hour than the host of the restaurant. Write an equation that relates the amount h that the host earns each hour when the manager earns $11.50 each hour.
Using Mental Math to Find Solutions
What is the solution of each equation? Use mental
math.
1. x + 8 = 12
2.π
8= 9
3. 12 β y = 3
4. 4m = 16
Practice
Use mental math to find the solution of each equation.
1. 2 β π₯ = β5
2.π₯
7= 35
3. 20π = 100
4. 6π‘ = 36
5. 18 + π = 24
6. 4 = 7 β π¦
Using a Table to Find a Solution
What is the solution of 5n + 8 = 48? Use a table.
N 5n + 8 Value of 5n + 8
5 5(5) + 8 33
6
7
8
9
Using a Table to Find a Solution
What is the solution of 25 β 3p = 55? Use a table.
P 25 β 3p Value of 25 β 3p
Practice
Use a table to find the solution of each equation.
1. 12 = 6 β 3π
2. 5π₯ + 3 = 23
3.1
2π₯ β 5 = β1
4. 0 = 4 + 2π¦
Estimating a Solution
What is an estimate of the solution of -9x β 5 = 28?
Use a table.
x -9x β 5 Value of -9x β 5
-1 -9(-1) β 5 4
-2
-3
-4
Estimating a Solution
What is the solution of 3x + 3 = -22? Use a table.
x 3x + 3 Value of 3x + 3
Practice
Use a table to find two consecutive integers between which the solution lies.
1. 6π₯ + 5 = 81
2. 3.3 = 1.5 β 0.4π¦
3. β115π + 80 = β489
Exit Ticket
1. Determine if open, true or false:
85 + (β10) = 95
4a β 3b = 21
5(2) + 7 = 17
2. Determine if a solution: x = 7, 6 = 2x β 8
3. Write an equation: The sum of 4x and -3 is 8
4. Use mental math: 20a = 100
5. Use a table to solve: 8a β 10 = 38
Graphing in the Coordinate Plane
Vocabulary
β’ Two number lines that intersect at right angles form a coordinate plane.
β’ The horizontal axis is the x-axis and the vertical axis is the y-axis.
β’ The axes intersect at the origin and divide the coordinate plane into four sections called quadrants.
β’ An ordered pair of numbers names the location of a point in the plane.
Vocabulary
β’ These numbers are the coordinates of the point.
β’ πβπ π₯ β πππππππππ‘π β 2, 4 β πβπ π¦ β πππππππππ‘π.
β’ To reach the point (x, y), you use the x-coordinate to tell how far to move right (positive) or left (negative) from the origin. You then use the y-coordinate to tell how far to move up (positive) or down (negative).
Graph
Patterns, Equations, and Graphs
Objective: To use tables, equations, and graphs to describe relationships.
Objectives
β’ I can identify solutions of a two-variable equation.
β’ I can use a table, an equation, and a graph to solve an equation.
β’ I can extend a pattern using a table, equation, and a graph.
Vocabulary
β’ Sometimes the value of one quantity can be found if you know the value of another. You can represent the relationship between the quantities in different ways, including tables, equations, and graphs.
β’ You can use an equation with two variables to represent the relationship between two varying quantities.
β’ A solution of an equation with two variables x and y is any ordered pair (x, y) that makes the equation true.
Identifying Solutions of a Two-Variable Equation
β’ Is (3, 10) a solution of the equation y = 4x?
β’ Is (5, 20) a solution of the equation y = 4x?
β’ Is (-5, -20) a solution of the equation y = 4x?
β’ Is (1.5, 6) a solution of the equation y = 4x?
Practice
Tell whether the given equation has the ordered pair as a solution.
1. π¦ = π₯ + 6; 0,6
2. βπ₯ = π¦; β3.1,3.1
3. π¦ = π₯ +2
3; 1,
1
3
4. π¦ = 1 β π₯; 2,1
5. π¦ = π₯ β3
4; (2,1
1
4)
Using a Table, an Equation and a Graph
Both Carrie and her sister Kim were born on October
25, but Kim was born 2 years before Carrie. How can
you represent the relationship between Carrieβs age
and Kimβs age in different ways?
Carrie 1 2 3 4 5
Kim 3 4
Using a Table, an Equation and a Graph
Will runs 6 laps before Megan joins him at the track.
They then run together at the same pace. How can
you represent the relationship between the number
of laps Will run and the number of laps Megan runs
in different ways. Use a table, an equation, and a
graph.
Will
Megan
Practice
Use a table, an equation, and a graph to represent each relationship.
1. Ty is 3 years younger than Bea.
2. The number of checkers is 24 times the number of checkerboards.
3. The number of triangles is 1
3the number of sides.
4. Gavin makes $8.50 for each lawn he mows.
Vocabulary
β’ Inductive reasoning is the process of reaching a conclusion based on an observed pattern. You can use inductive reasoning to predict values.
Extending a Pattern
The table shows the relationship between the number of blue tiles and the total number of tiles in each figure. Write an equation and draw a graph.
Number of Blue Tiles, x Total Number of Tiles, y
1 9
2 18
3 27
4 36
5 45
Extending a Pattern
The table shows amounts earned for pet sitting. How
much is earned for a 9 day job?
Days, x Dollars, y
1 17
2 34
3 51
4 68
Practice
Use the table to draw a graph and answer the question.
β The table shows the height in inches of stacks of tires. Extend the pattern. What is the height of a stack of 7 tires?
β The table shows amounts earned for pet sitting. How much is earned for a 9βday job?
Number of Tires, x
Height of Stack, y
1 8
2 16
3 24
4 32
Days, x Dollars, y
1 17
2 34
3 51
4 68
Exit Ticketβ’ Is the ordered pair a solution?
β y = x + 6 (0, 6)
β y = -4x (-2, 8)
β’ Use a table, an equation, and a graph:
β Ty is 3 years younger than Bo.
β Gavin makes $8.50 for each lawn he mows.
β’ Use the table to draw a graph and answer the
question: The table shows the height in inches of
stacks of tires. Extend the pattern. What is the
height of the stack of 7 tires?
Number of Tires, x 1 2 3 4
Height of Stack, y 8 16 24 32