Florida GED PLUS College Preparation Program Algebraic Thinking The Beginning Steps

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Florida GED PLUS College Preparation Program
Algebraic ThinkingThe Beginning Steps

• Project Director
• Dr. Lucy Madsen Guglielmino
• Development and Research Consultants
• Susan Pittman
• Bonnie Vondracek

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Florida GED PLUS

• The Key to
• Success in Algebraic Thinking
• Critical Thinking and Problem-Solving Skills

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Florida GED PLUS - What Is Algebraic Thinking?

• Algebraic thinking is
• the ideas of algebra and the skill of being able
  to logically think.
• Algebraic ideas or building blocks include
• patterns
• variables
• expressions
• equations
• functions

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Florida GED PLUS Math Objectives

• Solve problems in different mathematical areas
• Understand the basics of algebraic thinking by
  completing operations/word problems
• Solve different testing style question formats
• Use mathematical skills in real-life contexts
• Understand different question types and
  strategies for solving problems

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Florida GED PLUS - Elementary Algebra the CPT

• Three Categories
• Operations with integers and rational numbers
  includes
• computation with integers and negative rationals
• ??the use of absolute values, and ordering
• Operations with algebraic expressions
• evaluation of simple formulas
• expressions, and adding
• ??subtracting monomials and polynomials,
• ?the evaluation of positive rational roots and
  exponents
• simplifying algebraic fractions, and factoring.
• Equation solving, inequalities, and word problems
  including
• solving verbal problems presented in algebraic
  context
• geometric reasoning
• the translation of written phrases into algebraic
  expressions
• graphing

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Florida GED PLUS The Process
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Florida GED Plus CPT Test Questions Types

• First Category of Questions on Elementary
• Algebra
• Operations with integers and rational numbers,
  includes computation with integers and negative
  rationals, the use of absolute values and
  ordering. 

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Florida GED Plus CPT Test Questions Types

• Sample Question
• If a number is divided by 4 and then 3 is
  subtracted, the result is 0.  What is the number?
  
• 12
• 4
• 3
• 2

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Florida GED Plus CPT Test Questions Types

• Second Category of Questions on
• Elementary Algebra
• Operations with algebraic expressions, which test
  skills in evaluating simple formulas and
  expressions, adding and subtracting monomials and
  polynomials, evaluating positive rational roots
  and exponents, simplifying algebraic fractions,
  and factoring. 

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Florida GED Plus CPT Test Questions Types

• Which of the following is a factor of x2 x
  12?a. (x 2)b. (x 3)c. (x 4)d. (x 6)

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Florida GED Plus CPT Test Questions Types

• Third Category of Questions on Elementary
• Algebra
• Tests skills in solving equations, inequalities,
  and word problems. Questions range from solving
  systems of linear equations, solving quadratic
  equations by factoring, verbal problems presented
  in algebraic context, geometric reasoning, the
  translation of written phrases into algebraic
  expressions and graphing. 

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Florida GED Plus CPT Test Questions Types

• Sample Question
• If A represents the number of apples purchased at
  .15 each and B represents the number of bananas
  purchased at .10 each, which of the following
  represents the total value of the purchases?
• A B
• 25 (A B)
• 10A 15B
• 15A 10B

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Florida GED PLUS - Implementing the Algebra
Curriculum

• Vocabulary
• Problem-Solving
• Critical thinking
• Processes
• Strategy application
• Question type identification
• Algebraic Concepts
• Teaching Strategies
• Graphic Organizers
• Manipulatives

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Florida GED PLUS - Algebra Definitions The
Basics

• Variable a letter or symbol that represents a
  number (unknown quantity). A variable can use any
  letter of the alphabet.
• n 5
• x 7
• w - 25
• 8 n 12

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Florida GED PLUS - Algebra Definitions The
Basics

• Algebraic expression a group of numbers,
  symbols, and variables that express an operation
  or a series of operations.
• m 8
• r 3

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Florida GED PLUS - Algebra Definitions The
Basics

• Equation A mathematical sentence stating that 2
  expressions are equal.
• 12 3 9
• 8 4 12

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Florida GED PLUS - Algebra Definitions The
Basics

• Evaluate an algebraic expression To find the
  value of an algebraic expression by substituting
  numbers for variables.
• m 8 m 2 2 8 10
• r 3 r 5 5 3 2

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Florida GED PLUS - Algebra Definitions The
Basics

• Simplify Combine like terms and complete all
  operations
• m 2
• m 8 m 2 m 8
• (2 x 2) 8 4 8 12

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Florida GED PLUS - Algebra Definitions The
Basics

• Inverse operation the opposite operation used
  to undo the first.
• 4 3 7 7 4 3
• 6 x 6 36 36 / 6 6

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Florida GED PLUS - Algebra Definitions The
Basics

• Any number or variable or product of numbers and
  variables is called a monomial. Each of the
  following is an example of a monomial5, -7, x,
  y, 6xyz, -9yz4, 2.3a3.
• A monomial or the sum of two or more monomials is
  called a polynomial. Each of the following is an
  example of a polynomial6xyz, 5a3-21, 4x2-9y2,
  2x3y4z, 5x26x7.
• A polynomial that has two (unlike) terms is
  called a binomial (e.g., 6a3-23 and 5x218y2).
• A polynomial that has three (unlike) terms is
  called a trinomial. (e.g., 3x5y7z and
  6x29x12).

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Theres a Need for Graphic Organizers
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Florida GED PLUS - Write Algebraic Expressions
for These Word Phrases
n 10

• Ten more than a number
• A number decrease by 5
• 6 less than a number
• A number increased by 8
• The sum of a number 9
• 4 more than a number

w - 5
x - 6
n 8
n 9
y 4
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Florida GED PLUS - Math Translation Guide
English Math Example Translation
What, a number x, n, etc. Three more than a number is 8. n 3 8
Equals, is, was, has, costs Danny is 16 years old. A CD costs 15 dollars. d 16 c 15
Is greater than Is less than At least, minimum At most, maximum gt lt ? ? Jenny has more money than Ben. Ashleys age is less than Nicks. There are at least 30 questions on the test. Sam can invite a maximum of 15 people to his party. j gt b a lt n t ? 30 s ? 15
More, more than, greater, than, added to, total, sum, increased by, together Kecia has 2 more video games than John. Kecia and John have a total of 11 video games. k j 2 k j 11
Less than, smaller than, decreased by, difference, fewer - Jason has 3 fewer CDs than Carson. The difference between Jennys and Bens savings is 75. j c 3 j b 75
Of, times, product of, twice, double, triple, half of, quarter of x Emma has twice as many books as Justin. Justin has half as many books as Emma. e 2 x j or e 2j j c x ½ or j e/2
Divided by, per, for, out of, ratio of __ to __ ? Sophia has 1 for every 2 Daniel has. The ratio of Daniels savings to Sophias savings is 2 to 1. s d ? 2 or s d/2 d/s 2/1
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Florida GED PLUS - Using a Transition Chart The
Beginning

• Jennifer has 10 fewer DVDs than Brad.
• j 10 b (common answer, but incorrect)
• Insert the words and see the difference in the
  equation.
• j (has) b (fewer) 10
• so
• j b 10

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Florida GED PLUS - Using a Transition Chart The
Beginning

• You Try It!
• Clay got 10 fewer votes than Kimberly. Reuben got
  three times as many votes as Clay. The three
  contestants received a total of 90 votes. Write
  an equation in one variable that can be used to
  solve for the number of votes Kimberly received.

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Florida GED PLUS - Using a Transition Chart The
Beginning

• Step 1 Pick which unknown will be represented by
  the variable. Since youre solving for Kimberly,
  let k be the number of votes Kimberly received.
• Step 2 Represent the other two unknowns in terms
  of k. Clay got 10 fewer votes so its k - 10 and
  Reuben got three times that so its 3(k - 10).
• Step 3 Set up the equation using all of the
  expressions to equal 90.
• k (k - 10) 3(k - 10) 90

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Florida GED PLUS - Using a Transition Chart The
Beginning

• A school is having a special event to honor
  successful alumni. The event will cost 500, plus
  an additional 85 for each alum who is honored.
  Write an equation that best represents the number
  of alumni that can be honored. (Select a maximum
  amount for the event as part of the equation.)

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Florida GED PLUS - Using a Transition Chart The
Beginning

• Step 1 The amount the school can spend is equal
  to or less than 1,000, so its ? 1,000.
• Step 2 The event has a fixed cost of 500 and a
  variable of 85 per alum so its 500 85a.
• Step 3 The equation then becomes 500 85a ?
  1,000.

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Florida GED PLUS - Using a Transition Chart The
Beginning

• A computer repair company charges 50 for a
  service call plus 25 for each hour of work.
  Write an equation that represents the
  relationship between the bill, b, for a service
  call, and the number of hours spent on the call,
  h.

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Florida GED PLUS - Using a Transition Chart The
Beginning

• Step 1 Some questions include a situation where
  there is more than one cost. One of them is fixed
  and one is variable. First identify the sum of
  the fixed and variable costs so b equals the
  total.
• Step 2 Next, identify the fixed cost of 50 and
  the variable cost of 25h (25 x the number of
  hours).
• Step 3 The equation then becomes 50 25h b.

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Florida GED PLUS Four Basic Skills for Thinking
About Patterns

• Finding patterns involves looking for regular
  features of a situation that repeats.
• Describing patterns involves communicating the
  regularity in words or in a mathematically
  concise way that other people can understand.
• Explaining patterns involves thinking about why
  the pattern continues forever, even if one has
  not exhaustively looked at each one.
• Predicting with patterns involves using your
  description to predict pieces of the situation
  that are not given.

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Florida GED PLUS - An Effective Problem Solver

• Reads the problem carefully, paying attention to
  detail
• Defines the type of answer that is required and
  eliminate extraneous information
• Identifies key words that will assist in choosing
  the correct operation or in the case of
  multi-step problems, the correct operations
• Uses a graphic organizer
• Sets up the problem correctly and remember the
  order of operations
• Uses mental math and estimation skills
• Checks the answer for reasonableness
• Uses a calculator with care and always
  double-check the answers

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Florida GED PLUS Obstacles to Word Problems

• Obstacles to Word Problems and Higher Level Math
• Knowledge and Skills in
• Using mental pictures to represent ap concepts
  must see the math problem
• The language of word problems relevant vs.
  irrelevant, sequence, terminology
• Breaking down word problems into simpler
  subproblems
• Using strategies, reasoning skills, and common
  sense problem-solving approaches
• Appropriate skills to successfully access higher
  math, such as algebra, geometry, statistics

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Florida GED PLUS - Using a Graphic Organizer to
Problem Solve

• Math Graphic Organizer
• Asks for the main idea
• What is happening in the problem?
• Asks the question
• What is the problem asking you to do?
• Lays out the facts
• What is pertinent, what is irrelevant?
• Checks to see if the answer relates
  to the question asked

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Florida GED PLUS - Using a Graphic Organizer to
Problem Solve
Main Idea (in your own words) Main Idea (in your own words)
Question Draw a Picture/Graph/Table
Pertinent Facts Irrelevant Information
Relationship Sentence (no numbers) Relationship Sentence (no numbers)
Equation (number sentence) Equation (number sentence)
Estimation (without computing) Estimation (without computing)
Computation Computation
Answer sentence Answer sentence
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Florida GED PLUS - Lets Try a Simple Problem

• Graphic Organizer
• Try It!
• Julia Child was roasting a turkey. It has been
  out of the oven for 20 minutes. The turkey was
  roasting for 4 hours and 15 minutes. The oven was
  preheated for 10 minutes. If it is 500 now, then
  what time did Julia put the turkey in the oven?

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Florida GED PLUS - Lets Try a Simple Problem

• Sample Relationship Sentence(s)
• From the current time, take the amount of time
  that the turkey was roasting in the oven and the
  amount of time it has been out of the oven. This
  will give you the time Julia put the turkey into
  the oven.

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Florida GED PLUS - A Beginning Algebraic Problem

• Graphic Organizer
• Try It!
• Samuel ate 100 cookies in five days. Each day he
  ate 6 more than the day before. How many cookies
  did he eat on the first day?

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Florida GED PLUS - A Beginning Algebraic Problem

• Graphic Organizer
• Try It!
• From the total number of cookies eaten, take a
  known amount of increase for a specified number
  of days and subtract that increase to get a total
  that can be divided by the number of days cookies
  were eaten to get the number of cookies eaten the
  first day.
• n (n 6) (n 12) (n 18) (n 24)
  100
• n 8 cookies eaten the first day

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Florida GED PLUS - A Beginning Algebraic Problem

• Graphic Organizer
• Try It!
• The 58 m SERF swimming pool can be filled in 4
  days using pipe A alone and in 3 days using both
  pipes A and B. How many days would be needed to
  fill the SERF pool using only pipe B?

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Florida GED PLUS - A Beginning Algebraic Problem
Filling mode No. of days to fill pool Fraction of pool filled in 1 day
Pipe A alone 4 ¼
Pile B alone X 1/x
Pipes A and B together 3 1/3

• From the last column above - ¼ 1/x 1/3
• Multiply by 12x, so that 3x 12 4x
• X 12

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Florida GED PLUS - FOIL
multiply First Outer Inner Last
F  (x 3)(x 2)
O  (x 3)(x 2)
I   (x 3)(x 2) 
L  (x 3)(x 2)
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Florida GED PLUS - Theres More Than FOIL!!!

• Help students to
• Visualize algebraic expressions and/or equations
• Understand effects of positive and negative

algebra tiles, algetiles, math tiles, virtual
tiles
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Florida GED PLUS - Algebra Tiles - Multiplying
Binomials

• The answer is x23x2

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Florida GED PLUS - Algebra Tiles - Multiplying
Binomials

• (x 2) (x 1)
• (x 5) (x 3)
• (2x 2) (2x 1)
• (3x 1) (1x 2)
• (2x 4) (3x 1)
• (3x 5) (2x 3)
• (6x 1)(1x 2)

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Florida GED PLUS - Algebra Tiles - Factoring
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Florida GED PLUS - Algebra Tiles - Factoring

• x2 4x 3
• x2 5x 4
• x2 5x 6
• x2 6x 8
• x2 7x 10
• x2 6x 9
• x2 11x 12
• x2 11x 18
• x2 11x 24
• x2 15x 36

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Florida GED PLUS - Algebra Tiles - Factoring
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Florida GED PLUS - Algebra Tiles - Factoring

• 2x2 5x 3
• 2x2 7x 3
• 2x2 7x 6
• 2x2 8x 6
• 4x2 13x 3
• 6x2 11x 4
• 9x2 15x 6
• 4x2 37x 9

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Florida GED PLUS - Algebra Tiles Squaring
Binomials
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Florida GED PLUS - Algebra Tiles Squaring
Binomials

• ( x 1 )2
• ( x 2 )2
• (2x 1)2
• (2x 3)2
• (3x 1)2
• (3x 2)2
• (4x 5)2

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Florida GED PLUS - Algebra Squares Factoring
Perfect Square Trinomials
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Florida GED PLUS - Algebra Squares Factoring
Perfect Square Trinomials

• x2 2x 1
• x2 4x 4
• 4x2 4x 1
• 4x2 12x 9
• 9x2 6x 1
• 9x2 12x 4
• 16x2 40x 25

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Florida GED PLUS - How About a Grid?

• Grids can also be used to teach students to
  multiply binomials.
• (x 2) (x 3)
• The answer is - x²5x6

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Florida GED PLUS Its Just the Beginning!
Its Algebra Jeopardy!
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• Do not worry about your difficulties in
  mathematics,
• I assure you that mine are greater."
• Albert Einstein

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2005 Florida GED PLUS College Preparation Program

• Florida Atlantic University2005 GED Project
  Project Director
• Dr. Lucy Madsen Guglielmino
• lguglie_at_fau.eduDevelopment and Research
  Consultants
• Susan K. Pittmanskptvs_at_aol.com
• Bonnie Vondracekbv73008_at_aol.com