Unit 1 - Algebra Foundations - Weebly

Unit Essential Questions

How can you represent quantities, patterns, and relationships?

How are properties of real numbers related to algebra?

Williams Math Lessons

Algebra 1 Algebra Foundations -8-

VARIABLES AND EXPRESSIONS MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

RATING LEARNING SCALE

4 I am able to

• write algebraic expressions and apply them to real world situations or more challenging problems that I have never previously attempted

3 I am able to • write algebraic expressions

2 I am able to

• write algebraic expressions with help

1 I am able to • understand that algebra uses symbols to represent quantities that are unknown or may vary

WARM UP Which of these situations have a value that varies?

a) The population of this school

b) The number of classrooms in this school

c) The time it takes you to get to your next class

d) Your high school GPA

KEY CONCEPTS AND VOCABULARY

_____________________________________ – anything that can be measured or counted

________________________ – a symbol, usually a letter, that represents the value of a variable quantity

__________________________________ – a mathematical phrase that includes one or more variables

__________________________________ – a mathematical phrase involving numbers and operation symbols, but no variables

EXAMPLES

EXAMPLE 1: REWRITING A WORD EXPRESSION (ADDITION OR SUBTRACTION)

Write an algebraic expression for each word phrase.

a) 3 more than f b) 10 less than c c) 5 decreased by p

Addition Subtraction Multiplication Division

sum plus added to more than increased by

difference minus subtract less than decreased by less fewer than

product times multiply multiplied by of double/ triple

quotient divide shared equally divided by divided into

TARGET


EXAMPLE 2: REWRITING A WORD EXPRESSION (MULTIPLICATION OR DIVISION)

Write an algebraic expression for each word phrase. a) the quotient of 9 and k b) the product of 15 and y

c) r divided by 5 d) twice a number s

EXAMPLE 3: REWRITING A WORD EXPRESSION (WITH VARIOUS OPERATIONS)

Write an algebraic expression for each word phrase.

a) The sum of 4 and twice y b) 7 less than the product of y and z

c) 5 minus the quotient of x and y d) 4 more than twice the number z EXAMPLE 4: REWRITING AN ALGEBRAIC EXPRESSION

Write the word phrase for each algebraic expression.

a) d + 5 b) p – 3 c) 2x

d) x/7 e) 100 + 6y f) 2c - 85

EXAMPLE 5: WRITING AN ALGEBRAIC EXPRESSION FOR REAL WORLD SITUATIONS

A car salesman gets paid a weekly salary of $300. They are also paid $100 for each car that they sell during the week. Write a rule in words and as an algebraic expression to model the relationship in the table.

RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)

Circle one: 4 3 2 1

Cars Sold Total Earned

0 $300 + (0 x $100)

1 $300 + (1 x $100)

2 $300 + (2 x $100)

n ☐


ORDER OF OPERATIONS MACC.912.A.SSE.A.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity


4 I am able to

• evaluate algebraic expressions and apply them to real world situations or more challenging problems that I have never previously attempted

3 I am able to

• simplify expressions involving exponents • use the order of operations to evaluate expressions

2 I am able to

• simplify expressions involving exponents with help • use the order of operations to evaluate expressions with help

1 I am able to • understand that you can use powers to shorten how you represent repeated multiplication

WARM UP Find the greatest common factor of each pair of numbers.

1) 2 and 6 2) 9 and 15 3) 3 and 13 4) 12 and 18


_________________________________ – has two parts, base and exponent

_________________________________ – a number that shows repeated multiplication

_________________________________ – a number that is multiplied repeatedly

_________________________________ – to replace an expression with its simplest form

_________________________________ – to substitute a given number for each variable, and then simplify

ORDER OF OPERATIONS

P PARENTHESIS – perform any operations inside grouping symbols, such as parenthesis ( ), brackets [ ], and a fraction bar.

E EXPONENTS – simplify powers

M MULTIPLY AND DIVIDE – from LEFT TO RIGHT (not multiplication before division)

D

A ADDITION AND SUBTRACTION – from LEFT TO RIGHT (not addition before subtraction)

S

EXAMPLES

EXAMPLE 1: EVALUATING AN EXPRESSION

What is the simplified form of each expression?

a) 24 b) 85 c) d) (0.3)3

14

⎛⎝⎜

⎞⎠⎟

2

TARGET


EXAMPLE 2: ORDER OF OPERATIONS

What is the simplified form of each expression? a) b)

c) d)

EXAMPLE 3: EVALUATING AN ALGEBRAIC EXPRESSION

What is the value of the expression for x = 1 and y = 3?

a) b)

c) d) EXAMPLE 4: WRITING AND EVALUATING AN EXPRESSION FOR REAL WORLD SITUATIONS

You receive a weekly allowance. Every week you deposit ¼ of your allowance into a savings account. Evaluate the amount of spending money you have if your weekly wage is:

a) $60 b) $ 100 c) $200


Circle one: 4 3 2 1

(5− 3)4 ÷ 4 1+ 2× 6

14 − 20 ÷ 5

32 − 52

x + y 2

3xy + x 2

y + y 2 − 8 ÷ x (x + y )2 ÷ (y + 1)


REAL NUMBERS AND THEIR SUBSETS MACC.912.N-RN.B.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.


4 I am able to

• provide counterexamples involving the sum and product of real number subsets

3 I am able to

• classify real numbers into real number subsets • understand the sum and product of real number subsets

2 I am able to

• classify real numbers into real number subsets with help • understand the sum and product of real number subsets with help

1 I am able to • understand that real numbers can be divided into subsets

WARM UP Use order of operations to simplify.

1) 3 ÷ 4 + 6 ÷4 2) 5[(2 + 5) ÷3] 3) 40 + 24 ÷8 – 22 – 1


EXAMPLES

EXAMPLE 1: IDENTIFYING SUBSETS OF REAL NUMBERS

Your math class is selling pies to raise money to go to a math competition. Which subset of real numbers best describes the number of pies p that your class sells?

SUBSETS OF REAL NUMBERS

NAME DESCRIPTION EXAMPLES

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

TARGET


EXAMPLE 2: CLASSIFYING NUMBERS INTO SUBSETS OF REAL NUMBERS

For each number, place a check in the column that the number belongs to. Remember the numbers may belong to more than one set.

# Number Real Whole Natural Integer Rational Irrational

a) –9

b) 4

c) 81

d)

25

e)

102

f) 0

g)

− 42

h) 3π + 1

EXAMPLE 3: OPERATIONS OF REAL NUMBERS

Show each statement is false by providing a counterexample. a) The difference of two natural numbers is a natural number.

b) The product of two irrational numbers is irrational.

c) The product of a rational number and an irrational number is rational.

d) The sum of a rational number and an irrational number is rational.


Circle one: 4 3 2 1


WARM UP Simplify each expression.

1) 4 + 5 × 2 2) 6 + 12 ÷ 6 3) (3 + 4) 8 4) 3 + 10 × 2


Two algebraic expressions are _______________________________________________ if they have the same value for

all values of the variables.

COMMUTATIVE PROPERTY

The order in which you add or multiply does not matter. For any numbers a and b,

a + b = b + a and ab = ba

ASSOCIATIVE PROPERTY

The way three or more numbers are grouped when adding or multiplying does not matter. For any numbers a, b, and c,

(a + b) + c = a + (b + c) and (ab)c = a(bc)

ADDITIVE IDENTITY

For any number a, the sum of a and 0 is a

a + 0 = 0 + a = a or 7 + 0 = 0 + 7 = 7

MULTIPLICATIVE IDENTITY

For any number a, the product of a and 1 is a.

ADDITIVE INVERSE

A number and its opposite are additive inverses of each other,

MULTIPLICATIVE INVERSE (RECIPROCALS)

For every number, , there is exactly

one number such that the product is one.

MULTIPLICATIVE PROPERTY OF ZERO

Anything times zero is zero.

a ⋅1 = 1 ⋅a = a or 14 ⋅1 = 1 ⋅14 = 14

a + (−a) = 0 or 5+ (−5) = 0

ab

, where a,b ≠ 0

ba

ab⋅ba= 1 or

23⋅ 32= 1

a ⋅0 = 0 ⋅a = 0 or 2 ⋅0 = 0 ⋅2 = 0

PROPERTIES OF REAL NUMBERS MACC.912.A.SSE.A.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity


4 I am able to

• use properties of real numbers to provide counterexamples or solve more challenging problems that I have never previously attempted

3 I am able to

• identify and use properties of real numbers

2 I am able to • identify and use properties of real numbers with help

1 I am able to

• understand that relationships that are always true for real numbers are called properties

TARGET


= (3)+ 5 ⋅ 15− 3 ⋅1

= 3+ 1− 3 ⋅1= 3+ 1− 3

= 1

EXAMPLES

EXAMPLE 1: IDENTIFYING PROPERTIES

What property is illustrated?

a) b)

c) d)

EXAMPLE 2: USING PROPERTIES TO FIND UNKNOWN QUANTITIES

Name the property then find the value of the unknown. a) n x 12 = 0 b) 7 + (3 + z) = (7 + 3) + 4

c) 0 + n = 8 d) 6h = 6

EXAMPLE 3: IDENTIFYING THE PROPERTY USED IN EACH STEP

Name the property used in each step.

Step 1) ___________________________

Step 2) ___________________________

Step 3) ___________________________

Step 4) ___________________________

EXAMPLE 4: PROVIDING A COUNTEREXAMPLE

Is the statement true or false? If it is false, give a counterexample. a) For all real numbers, a + b = ab

b) For all real numbers, a(1) = a


Circle one: 4 3 2 1

z ⋅47 = 47z 9 ⋅ 1

9= 1

(f + 5)+ 13 = f + (5+ 13) 3xyz + 0 = 3xyz

(3+ 0)+ 5 ⋅ 1

5− 3 ⋅1


WARM UP

Name the property that each statement illustrates. 1) 8 + 0 = 8 2) 2(–4 ) = –4(2) 3) x + (y + 3) = (x + y) + 3


THE DISTRIBUTIVE PROPERTY For any numbers a, b and c

a (b + c) = ab + ac and (b + c)a = ab + ac

and

a(b – c) = ab – ac and (b – c)a = ab – ac

________________ – is a number, a variable, or the product of a number and one or more variables

_________________________ – is a term that has no variable

_________________________ – is a numerical factor of a term

____________________________________ – have the same variable factors (same variables raised to the same power)

An expressions is in __________________________________ when it contains no like terms or parenthesis

EXAMPLES

EXAMPLE 1: USING THE DISTRIBUTIVE PROPERTY OVER ADDITION AND SUBTRACTION

Use the distributive property to write in simplified form.

a) 12(y + 3) b) –2(xy + 8y – 3) c) –h(3h – 7)

THE DISTRIBUTIVE PROPERTY MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients


4 I am able to

• use the Distributive Property to rewrite fraction expressions or solve more challenging problems that I have never previously attempted

3 I am able to

• use the Distributive Property to simplify expressions

2 I am able to • use the Distributive Property to simplify expressions with help

1 I am able to

• understand that I can use the Distributive Property to simplify expressions

TARGET


EXAMPLE 2: WRITING EXPRESSIONS IN SIMPLEST FORM

Simplify the following.

a) 2(a – 7) + 3a + a b) 9y – 5 + 8 + 2y – 11y

c) d)

EXAMPLE 3: REWRITING FRACTION EXPRESSIONS

Write each fraction as a sum or difference.

a) b)

EXAMPLE 4; WRITING AND SIMPLIFYING EXPRESSIONS

Use the expression twice the sum of 4x and y increased by six times the difference of 2x and 3y. a) Write an algebraic expression for the verbal expression.

b) Simplify the expression.


Circle one: 4 3 2 1

3h2 − 5h + 4 − 8h2 − 12 2x 2 − 3

4x + x

4

11x + 24

6 − 3x9


WARM UP

Identify and correct the error. 1) 2) 10(a − 3) = 10(a)+ 10(−3) = 10a + 7


A mathematical statement that contains algebraic expressions and symbols is an ______________________________.

An _________________________ is a mathematical sentence that uses an equal sign (=).

A __________________________ of an equation containing a variable is a value that makes the equation true.

_________________________- an equation that is true for every value of the variable.

EXAMPLES

EXAMPLE 1: IDENTIFYING SOLUTIONS OF AN EQUATION

Determine if the given value is a solution to the equation. a) Is x = 7 a solution of the equation 2x + 10 = 23?

b) Is x =10 a solution of the equation ?

5(x + 2) = 5(x ) ⋅5(2) = 50x

12

x − 2 = 3

AN INTRODUCTION TO EQUATIONS MACC.912.A-CED.A.1: Create equations and inequalities in one variable and use them to solve problems.


4 I am able to

• write and solve equations with one variable and apply them to real world situations or more challenging problems that I have never previously attempted

3 I am able to

• write and solve equations with one variable

2 I am able to • write and solve equations with one variable with help

1 I am able to

• understand that an equation is a mathematical sentence

TARGET


EXAMPLE 2: APPLYING THE ORDER OF OPERATIONS

Solve. a) b) c) d) EXAMPLE 3: USING MENTAL MATH TO FIND SOLUTIONS

What is the solution of each equation?

a) b) c)

EXAMPLE 4: WRITING EQUATIONS

Write an equation for each sentence. a) The sum of 3x and – 5 is 13. b) The product of x and 4 is 64.

EXAMPLE 5: WRITING EQUATIONS FOR REAL WORLD SITUATIONS

A grocery store cashier makes $1.50 more per hour than a bagger. Write an equation that relates the amount x that a bagger earns each hour if a cashier makes $10.25 per hour.


Circle one: 4 3 2 1

8 + (22 − 1)÷ 3 = x b = 27 ÷ (42 − 7)

3(x + 1)− 5 = 3x − 2 6 ⋅3 ⋅c + 9 ⋅5 = (22− 4) ⋅c − (2 ⋅5)

d + 4 = 9 3 ⋅m = 6

t5= 35


WARM UP Ticket prices for admission to a museum are $8 for adults, $5 for children, and $6 for seniors.

a) What algebraic expression models the total number of dollars collected in ticket sales?

b) If 20 adult tickets, 16 children’s tickets, and 10 senior tickets are sold one morning, how much money is collected in all?


A __________________________________________________ is formed by the intersection of two number lines.

__________________ - the horizontal axis

__________________ - the vertical axis

The ____________________ is the point where the x and y axes intersect.

_____________________________ - names the location of the point in the plane, usually written (x, y)

___________________________ - a set of ordered pairs

__________________________ - the set of all inputs (x-coordinates)

__________________________ - the set of all outputs (y-coordinates)

RELATIONS MACC.912.A-REI.D.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the

coordinate plane, often forming a curve (which could be a line)


4 I am able to

• represent a relation and interpret graphs of relations that apply real-world situations

3 I am able to

• represent a relation • interpret graphs of relations

2 I am able to

• represent a relation with help • interpret graphs of relations with help

1 I am able to • understand how to read and plot points on a coordinate plane

TARGET

y - axis

x - axis origin

Ordered Pair (4,7)


EXAMPLES

EXAMPLE 1: REPRESENTING A RELATION

Express the relation as a table, a graph, and a mapping.

EXAMPLE 2: DETERMINING DOMAIN AND RANGE

Determine the domain and range for each relation. a) {(2, 3), (-1,5), (-5, 5), (0, -7)} b)

c) d)

Ordered Pairs

Mapping Diagram Table Graph

(0, 0)

(-1, 3)

(2, 5)

(-4, -2)

(0, -7)

Ordered Pairs

Mapping Diagram Table Graph

(5, 0)

(-2, 5)

(1, 3)

(-6, 1)

(-4, -1)

x y

1 0

2 3

3 -4

4 12


EXAMPLE 3: ANALYZING A GRAPH

What are the variables? Describe what happens in the graph. a) The graph shows the volume of air in a balloon as Alyssa blows it up, until it pops.

b) Rocco rides his bike to the park. The graph represents the distance he travels.

c) The graph represents the height of a basketball after Hadley dropped it from the top of a ladder.


Circle one: 4 3 2 1

Time

Vo

lum

e

Time

Dis

tanc

e

Time

Hei

ght


WARM UP

Simplify.

1) 2)


A _____________________________ is a relationship that pairs each input value with exactly one output value.

In a relationship between variables, the _________________________ variable changes

in response to the ________________________________ variable.

_________________________________ - is a test to see if the graph

represents a function. If a vertical line intersects the graph more

than once, it fails the test and is not a function.

Equations that are functions can be written in a form called __________________________________. It is used to

find the element in the range that will correspond the element in the domain.

Equation Function Notation

y = 4x − 10 f (x ) = 4x − 10

Read: y equals four x minus 10 Read: f of x equals four x minus 10

32(2h)

14

12− 12( ) + 3 15÷ 5− 2( )

FUNCTIONS MACC.912.F-IF.A.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the

output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). MACC.912.F-IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function

notation in terms of a context.


4 I am able to

• use function values to solve more challenging problems that I have never previously attempted

3 I am able to

• determine whether a relation is a function • find function values

2 I am able to

• determine whether a relation is a function with help • find function values with help

1 I am able to • understand that there are special relations called functions

TARGET

A properly working vending machine is an example of a function. You put in a code (input B15) and it gives you exactly one item (output Mountain Dew).


EXAMPLES

EXAMPLE 1: IDENTIFYING A FUNCTION

Determine whether each relation is a function. a) {(0, 1), (1, 0), (2, 1), (3, 1), (4, 2)} b) {(4, 9), (4, 3), (4, 0), (4, 4), (4, 1)}

EXAMPLE 2: USING THE VERTICAL LINE TEST

Use the vertical line test. Which graphs represent a function? a) b) c)

EXAMPLE 3: EVALUATING FUNCTION VALUES

Evaluate each function for the given value. a) f (x) = −2x + 11 for f(5), f(-3), and [3 – f(0)]

b) f (x ) = x 2 + 3x − 1 for f(2), f(-1), and [f(0) + f(1)] EXAMPLE 4: EVALUATING FUNCTION VALUES FOR REAL WORLD SITUATIONS

Write a function rule to model the cost per month of a cell phone data plan. Then evaluate the function for given number of data.

Monthly service fee: $24.99 Rate per GB of data uses: $5

GB of data used: 13


Circle one: 4 3 2 1


WARM UP The cost of one scoop of ice cream is $3.50 and the cost of two scoops of ice cream is $5.75. Write and evaluate an expression to find the cost of 3 one-scoop ice creams and 4 two-scoop ice creams.


________________________________ – the point in which the graph intersects the x-axis

________________________________ – the point in which the graph intersects the y-axis

A function whose graph is a straight line is a ___________________________________________

A function whose graph is not a straight line is a ___________________________________________

A function has ________________________________ on some vertical line if each half of the graph on either side of

the line matches exactly

EXAMPLES

EXAMPLE 1: DETERMINING THE DOMAIN OF A FUNCTION GIVEN ITS GRAPH

Identify the domain of the function. a) b)

INTERPRETING GRAPHS OF FUNCTIONS MACC.912.F-IF.B.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in

terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship MACC.912.F-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.


4 I am able to

• interpret intercepts and symmetry of graphs of functions and apply them to real world situations or more challenging problems that I have never previously attempted

3 I am able to

• interpret intercepts and symmetry of graphs of functions • identify the domain of a function

2 I am able to

• interpret intercepts and symmetry of graphs of functions with help • identify the domain of a function with help

1 I am able to

• understand the definition of an intercept and symmetry

TARGET


EXAMPLE 2: DETERMINING IF A GRAPH IS LINEAR VS. NON-LINEAR

Identify the function as linear or non-linear. Explain.

EXAMPLE 3: IDENTIFYING INTERCEPTS AND DETERMINING SYMMETRY

Estimate the intercepts and determine if the graph has symmetry. a) b)

c) d)


Circle one: 4 3 2 1

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Unit 1 - Algebra Foundations - Weebly