NAME:

ALGEBRA I

UNIT 1:

MULTIPLE REPRESENTATIONS

(Resource Guide)

UNIT 1 – MULTIPLE REPRESENTATIONS
ASSIGNMENT/ACTIVITY CHECKLIST
ASSESSMENT
Unit 1 – Pre-Test
SECTION 1 – Variables, Expressions, and Equations
Topics: Variables, Expressions, and Equations
Review the notes in the resource guide.
Complete the journal questions and KWL charts.
Review PowerPoint presentations.
Review video tutorials.
Review relevant websites and links.
Post initial response and responses to classmates for DQ #1.
Post initial response and responses to classmates for DQ #2.
Complete homework assignments (see wiki for details):
Section 1-1 / Section 1-2
Section 4-1 / Section 1-3
Word Problem Worksheet
Complete Code Breaking Worksheet #1.
Complete Code Breaking Worksheet #2.
Complete the Unit 1 – Section 1 – Quiz.
SECTION 2 – Words to Symbols (and vice versa)
Topics: Translating Between Words/Problems to Expressions/Equations
Review the notes in the resource guide.
Complete the journal questions and KWL charts.
Review PowerPoint presentations.
Review video tutorials.
Review relevant websites and links.
Post initial response and responses to classmates for DQ #1.
Post initial response and responses to classmates for DQ #2.
Complete homework assignments (see wiki for details):
Section 1-4 / Section 1-5
Section 1-6 / Section 1-7
Section 2-7 / Word Problem Worksheet
Complete Baseball Stats Worksheet/Spreadsheet.
Complete the Unit 1 – Section 2 – Quiz.
SECTION 3 – Functions
Topics: Functions
Review the notes in the resource guide.
Complete the journal questions and KWL charts.
Review PowerPoint presentations.
Review video tutorials.
Review relevant websites and links.
Post initial response and responses to classmates for DQ #1.
Post initial response and responses to classmates for DQ #2.
Complete homework assignments (see wiki for details):
Section 8-6 / Section 8-6 (application)
Section 8-7 / Word Problem Worksheet
Complete Drug Elimination Rate Worksheet.
Complete the Unit 1 – Section 3 – Quiz.
SECTION 4 – Review
Topics: Review for Unit 1
Review the notes for all sections in the resource guide. Be sure to review any of the PowerPoint presentations, video tutorials, websites, old homework assignments, and so forth that would be helpful.
Post initial response and responses to classmates for the Wrap-up DQ.
Complete homework assignments (see wiki for details):
Review Worksheet / Problem-Solving Practice
Complete “Everyday Math: When will I ever use this stuff?” activity. (This can be done at any time throughout the course of the unit.)
GROUP ACTIVITY
Unit 1 – Group Activity (Multiple Representations of Data)
ASSESSMENT
Unit 1 – Assessment – Written Test
Unit 1 – Assessment – Notes Quiz
Unit 1 – Assessment – Individual Assessment
Unit 1 – Assessment – Problem-Solving Activity
Unit 1 – Assessment – ePortfolio
Unit 1 – Post-Test

UNIT 1 – SECTION 1

TEXTBOOK: / Algebra: Structure and Method – Book 1 (McDougal Littell)
SECTIONS: / 1-1; 1-2; 4-1; and 1-3

K-W-L
K / KNOW: What do you know about the terms listed above?
W / WOULD: What would you like to knowafter this lesson?

GOALS AND EXPECTATIONS

In this section, you will be expected to:

  • Define what a variable is and describe several purposes for variables.
  • Define what an expression and write examples of expressions.
  • Define power, identify the key components of a power, and simplify a power.
  • Describe the steps in the “Order of Operations.”
  • Simplify numerical expressions and evaluate algebraic expressions using the “Order of Operations.”
  • Define what makes “something” an equation.
  • Transform expressions into simple equations.
  • Solve problems utilizing algebraic expressions and equations.

NOTES

NUMERICAL EXPRESSION – an expression that equals a number

VALUE OF A NUMERICAL EXPRESSION – a numerical answer to an expression

  • Examples include:

a)4 + 2 = 6

b)7  3 = 21

  • An EQUAL SIGN (=) is used to show that expressions and numbers have the same value.
  • An INEQUALITY () maybe used to show that two things are not equal.

SIMPLIFYING – writing an expression with variables in an equivalent form that uses the fewest terms possible

ORDER OF OPERATIONS – the steps that are followed when simplifying a multi-step problem

  • The steps in the order of operation are:

a)Grouping Symbols (parentheses, brackets, squiggly brackets, etc.)

b)Exponents

c)Multiply and Divide (from left to right)

d)Add and Subtract (from left to right)

  • GROUPING SYMBOL – a symbol used to enclose an expression and indicate that that expression should be simplified first
  • Examples of grouping symbols include:

1)Parentheses – ( )

2)Square brackets – [ ]

3)Squiggly brackets – { }

4)Fraction bar –

5)There are many other symbols that can also behave like grouping symbols. These will be discussed more later on…

  • Every grouping symbol has the characteristic that, in addition to grouping things together, it indicates some other operation to perform. For instance, parentheses group things together and then tell you to multiply. A fraction bar groups expression and then tells you to divide.
  • Grouping symbols are simplified in the first step of order of operations.
  • If a problem has grouping symbols within other grouping symbols always start with the innermost ones first.
  • When working inside grouping symbols, follow the remaining steps in the order of operations to simplify.
  • Examples of simplifying include:

a)(48  3) + 4 = (16) + 4 = 16 + 4 = 20

b)

VARIABLE – a symbol used to represent one or more numbers

  • Letters are typically used as variables.
  • The most common letters include x, y, and n. However, just about any symbol can be used as a variable.
  • In the context of a word problem, it is usually best to use a letter that is associated with what the variable represents (i.e. h = hours, a = apples).

VALUE OF A VARIABLE – the number that a variable stands for

ALGEBRAIC EXPRESSION – an expression that contains a variable

  • Examples of variable expressions include:

a)4.5h

b)2x

c)4x + 7y + 2

SUBSTITUTION – replacing an expression with another expression or number that is equivalent (a.k.a. has the same value)

EVALUATING A VARIABLE EXPRESSION – replacing each variable in an expression with a number and simplifying to get the answer

  • Examples of evaluating include:

a)4xy, if x = 2 and y = 3  4(2)(3) = 24

b)a + 5, if a = 4  4 + 5 = 9

  • If a number is next to a variable or if variables are next to each other in the expression, then you need to multiply.

POWER – an expression that represents repeated multiplication of the same number (factor)

An example of a power is:

EXPONENT – in a power, it’s the number that tells you how many times to multiply a number by itself; the “small” number

BASE – in a power, it’s the number that it being multiplied; the “big” number

  • Examples of exponents and expressions written in EXPONENTIAL FORM include:

a)45

b)24

c)32

  • To simplify a numerical expression with exponents, you must multiply:

a)45 = 44444 = 1024

b)24 = 2222 = 16

c)32 = 33 = 9

  • To write a variable expression in EXPONENTIAL FORM, count up the number of each variable:

a)xxxxxxxxxxxx = x12

b)2xx3yyyy = 6x2y4

c)–6abab3abbb = –18a3b5

  • Be careful when simplifying expressions that contain exponents, parentheses, and negative signs (or combinations of all three).
  • Simplify each of the following problems:

a)(–3)2 = –3  –3 = 9 …the parentheses indicate that the negative sign should stay attached to the 3 and –3 should be taken to the second power…

b)–32 = –(3  3) = –9 …according to order of operations, the power is done first and the negative sign is applied second…

c)–(1 + 3)3 = –(4)3 = –(4  4  4) = –64 …according to order of operations, parentheses first, the power second, and the negative sign is applied last…

d)1 + 52 = 1 + 25 = 26 …according to order of operations, the exponent is done before the addition…

  • Evaluate each of the following problems:
  • A good rule of thumb is to replace each variable with a set of parentheses first. Then plug in the numbers.
  • This will help you keep track of the negative signs and where they fit into the order of operations.

a)ax2 if a = 2 and x = –3 ( )( )2(2)(–3)2 = 2(9) = 18

b)5x3 + y if x = –1 and y = 4 5( )3 + ( )  5(–1)3 + (4) = 5 (–1) + 4 = –5 + 4 = –1

c) if x = 1 and y = 3 

d) if x = –4 and y = –1 

EQUATION – a statement formed by placing an equal sign between two numerical or variable expressions

  • Examples of equations include:

1)x + 5 = 9

2)y = 3x – 5

3)5x + 6 = x – 7

  • SIDES OF AN EQUATION – the expressions or numbers separated by the equal sign
  • Every equation has two sides.
  • OPEN SENTENCE – an equation containing variables
  • DOMAIN – numbers that the variable(s) may represent
  • SOLUTION – the answer to an equation
  • Synonyms for solution include: ANSWER, ROOT, and SOLUTION SET.

JOURNALING

Compare and contrast the meanings of “simplifying an expression” and “evaluating an expression.”

Explain (in your own words) the purpose of “Order of Operations” in math.

Explain what the following statement means: “Just like multiplying is a fast way to add the same number over and over, powers are a fast way to multiply the same number over and over.”

What are the Three Golden Rules for Solving Equations? Explain what each of the rules means in your own words.

What is the difference between an equation and an expression? Include an example of each. Can you solve for a variable in an expression? Explain. Can you solve for a variable in an equation? Explain. Write an example of an expression or an equation for someone else to solve.

K-W-L
L / LEARN: What did you learnin this lesson?

UNIT 1 – SECTION 2

TEXTBOOK: / Algebra: Structure and Method – Book 1 (McDougal Littell)
SECTIONS: / 1-4; 1-5; 1-6; 1-7; and 2-7

K-W-L
K / KNOW: What do you know about the terms listed above?
W / WOULD: What would you like to knowafter this lesson?

GOALS AND EXPECTATIONS

In this section, you will be expected to:

  • Translate verbal phrases, sentences, and word problems into expressions and equations.
  • Identify words that indicate various basic operations.
  • Utilize and apply formulas to solve problems.
  • Utilize the “Problem Solving Plan” to solve word problems.
  • Complete problems that involve consecutive integers.

NOTES

WORDS THAT MEAN ADDITION

1)Sum4) Increase7)

2)Add5) More (than)8)

3)Plus6) Greater (than)9)

  • SUM – the answer to an addition problem
  • In an expression, the word “number” represents the variable.
  • When adding, it IS okay to switch things around (i.e. x + 8 = 8 + x).
  • Examples 

PHRASE / TRANSLATION
The sum of a number and 4 / N + 4 or 4 + N
A number increased by 7 / N + 7 or 7 + N
8 more than a number / N + 8 or 8 + N

WORDS THAT MEAN SUBTRACTION

1) Difference4) Decrease7) Differ

2) Subtract5) Less than**8)Less

3) Minus6)Subtracted from**9)Take Away

  • DIFFERENCE – the answer to a subtraction problem
  • In an expression, the word “number” represents the variable.
  • When subtracting, it IS NOT okay to switch things around (i.e. x – 8  8 – x). It is really important to pay close attention to the order of terms in a subtraction expression.
  • When using “less than” or “subtracted from”, you must switch the order of the pieces from how they are written in the sentence.
  • All other words for subtraction stay in the same order.
  • Examples 

PHRASE / TRANSLATION
The difference of a number and 4 / N – 4
A number decreased by 7 / N – 7
8 less than a number / N – 8

WORDS THAT MEAN MULTIPLICATION

1) Product4) Of (when used alone)7) Half**

2) Multiply5) Twice**8)

3) Times6) Triple**9)

  • PRODUCT – the answer to a multiplication problem
  • In an expression, the word “number” represents the variable.
  • When multiplying, it IS okay to switch things around (i.e. x  8 = 8  x).
  • Twice, triple, quadruple, half, etc. are terms that indicate multiplying by a specific number.
  • Examples 

PHRASE / TRANSLATION
The product of a number and 4 / 4N or N4
A number times 7 / 7N or N7
Twice a number / 2N or N2

WORDS THAT MEAN DIVISION

1) Quotient4) Half**7)

2) Divide5) Third**8)

3) Divided by6) Fourth**9)

  • QUOTIENT – the answer to a division problem
  • In an expression, the word “number” represents the variable.
  • When dividing, it IS NOT okay to switch things around (i.e. x  8  8  x). It is really important to pay close attention to the order of terms in a division expression.
  • Half, third, fourth, etc. are terms that indicate dividing by a specific number.
  • When using division words, the order of the pieces in your expression is the same as the order or the words in the sentence.
  • Examples 

PHRASE / TRANSLATION
The quotient of a number and 4 / N  4 or
A number divided by 7 / N  7 or
8 divided by a number / 8  N or

FORMULA – an equation that states a rule about a relationship between two or more variables

  • Common examples of formulas include:

1)Area of Rectangle  A = lw

2)Perimeter of a Rectangle  P = 2l + 2w

3)Rate, time, and Distance  D = Rt

4)Cost  C = np

  • When working with formulas, you must evaluate to find the answer.
  • Formulas are equations only in variable form.
  • Notice that the different letters represent different measurements or values.
  • Pay attention to the “units” that are associated with the variables and the answer.

EQUATION – a statement formed by placing an equal sign between two numerical or variable expressions

IS – in math, is means equals

  • In a sentence, the word “is” represents the equals sign and splits the sentence into the two sides of the equation.
  • Other words for equals include: is, was, the result is, the answer is, etc.

TRANSLATING SENTENCES INTO EQUATIONS

  • In an expression, the word “number” represents the variable.
  • Examples for translating sentences into equations include:

SENTENCE / TRANSLATION
The product of a number and 4 is 12. / 4x = 12
The sum of twice number and 7 is 15. / 2x + 7 = 15
5 less than a number is 20. / x – 5 = 20
  • Look out for equations that have more than one step and require grouping symbols.
  • The word “quantity” usually indicates that parentheses are needed.

SENTENCE / TRANSLATION
Twice the sum of a number and 3 is 12. / 2(x + 3) = 12
The quantity of a number minus 6 is divided by 4. The result is 3. /
A number is increased by 4. This sum is multiplied by 5. The result is 30. / 5(x + 4) = 30
  • HINTS for writing equations:

1)Find the word “is” and locate the equal sign first.

2)Look for the word “number.” That is your variable.

3)Start with word “number” and write down the operations that are literally closest to that word in the sentence first. Then work your way outward.

WORD PROBLEM – a description of a situation in which certain numbers (either known or unknown) are related to each other

PROBLEM SOLVING PLAN – a series of steps that provide a way to organize the process of solving any type of word problem

  • The following plan can be used to SOLVE any problem:

1)ORGANIZE – Read the problem carefully and organize the info. This step includes:

a)Decide what you KNOW.

b)Decide what you DO NOT KNOW.

c)Drawing a SKETCH (if necessary).

2)VARIABLE – Define an appropriate variable that can be used to solve the problem.

3)EQUATION – Write an equation that relates your variable to the info in the problem.

4)SOLVE – Solve the equation to get your solution.

5)CHECK – Check your answer to make sure that the math is correct, that your answer makes sense, and that you are answering the question.

CONSECUTIVE INTEGERS – a group of integers (nice, round numbers) that are in order

  • Examples of consecutive integers include:

a){2, 3, 4, 5}

b){–3, –2, –1, 0}

c){–1, 0, 1}

  • In a word problem, the key is to write the equation using one variable.
  • To write a list of consecutive integers using one variable, start with your variable and add:

N, N + 1, N + 2, N + 3, N + 4,…

CONSECUTIVE EVEN INTEGERS – a group of even integers (nice, round numbers) that are in order

  • Examples of consecutive even integers include:

a){2, 4, 6, 8}

b){–6, –4, –2, 0}

c){–2, 0, 2}

  • In a word problem, the key is to write the equation using one variable.
  • To write a list of consecutive even integers using one variable, start with your variable and add:

N, N + 2, N + 4, N + 6, N + 8,…

CONSECUTIVE ODD INTEGERS – a group of odd integers (nice, round numbers) that are in order

  • Examples of consecutive odd integers include:

a){1, 3, 5, 7}

b){–7, –5, –3, –1}

c){–3, –1, 1}

  • In a word problem, the key is to write the equation using one variable.
  • To write a list of consecutive odd integers using one variable, start with your variable and add:

N, N + 2, N + 4, N + 6, N + 8,…

JOURNALING

Explain why expressions that involve addition and multiplication can be reversed (i.e. x + 6 is the same as 6 + x) whereas expressions that involve subtraction and division cannot.

What do you think are some of the pros and cons of the “Problem Solving Plan”? Explain.

Can you think of any parallels between the “Problem Solving Plan” in mathematics and procedures used in other content areas (i.e. science, writing, etc.)? Explain.

K-W-L
L / LEARN: What did you learnin this lesson?

UNIT 1 – SECTION 3

TEXTBOOK: / Algebra: Structure and Method – Book 1 (McDougal Littell)
SECTIONS: / 8-6 and 8-7

K-W-L
K / KNOW: What do you know about the terms listed above?
W / WOULD: What would you like to knowafter this lesson?

GOALS AND EXPECTATIONS

In this section, you will be expected to:

  • Define what a function is.
  • Compare and contrast functions and equations.
  • Determine what relationships are examples of functions.
  • Apply the proper terminology when discussing functions.
  • Represent functions in a variety of different forms.
  • Interpret meanings from graphs, tables, and charts.

NOTES

RELATION – a correspondence between two groups

FUNCTION – a relationship or correspondence between two different groups or variables such that every member of the first group matches up with only a single member of the second group

  • A function can be described in a variety of different ways:

a)Mappings

b)Tables

c)Graphs and Charts

d)Equations

e)Etc.

  • DOMAIN – the group of numbers that are input into a function
  • The domain can be also thought of as the:

a)x-values

b)the independent variable

c)the input

  • On a graph or chart, the domain corresponds with the data that goes along the x-axis.
  • RANGE – the group of numbers that are output from a function
  • The range can be also thought of as the:

a)y-values

b)the dependent variable

c)the output

d)the answers

  • On a graph or chart, the range corresponds with the data that goes along the y-axis.

TYPES OF FUNCTIONS

MAPPING – a way to illustrate a function using arrows to indicate which values are paired together

  • The following example illustrates how a function can be visualized as a mapping:
  • The key thing to note is that each value in the domain is matched with only one value in the range.

BAR GRAPH – a type of graph that is helpful for visually depicting a relationship between two variables

  • A bar graph is particularly useful when the domain is non-numerical.
  • An example of a bar graph is: