Solving Linear Equations in Algebra I

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Solving Linear Equations in Algebra I

• Jim Rahn
• James.rahn_at_verizon.net
• www.Jamesrahn.com

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What must all Algebra I students know about
solving and working with Linear Equations?
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Goals from the New Jersey Algebra I Core Content
Standards

• Students should
• Understand the big ideas of equivalence and
  linearity
• Modeling real situations with variables
• Use appropriate tools such as algebra tiles and
  graphing calculators, and spreadsheets regularly
• Understand that geometric objects can be
  represented algebraically (lines can be described
  using coordinates), and algebraic expressions can
  be interpreted geometrically (systems of
  equations and inequalities can be solved
  graphically)

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• EQUIVALENCE
• Numbers, expressions, functions, or equations
  have many different but equivalent forms.
  These forms differ in their efficacy and
  efficiency in interpreting or solving a problem,
  depending on the context.
• Algebra extends the properties of numbers to
  rules involving symbols to transform an
  expression, function, or equation into an
  equivalent form and substitute equivalent forms
  for each other.
• Solving problems algebraically typically involves
  transforming one equation to another equivalent
  equation until the solution becomes clear.

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• Linearity
• The relationship between two quantities can often
  be represented graphically by a linear function.
• Linear functions can be used to show a
  relationship between two variables with a
  constant rate of change
• Linear functions can be used to show the
  relationship between two quantities that vary
  proportionately.
• Linear functions can also be used to model,
  describe, analyze, and compare sets of data.
• Understanding linear functions should be
  prominent in the Algebra I content.

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Benchmarks Related to Linear Relationships

• Recognize, describe and represent linear
  relationships using words, tables, numerical
  patterns, graphs and equations.
• Describe, analyze and use key characteristics of
  linear functions and their graphs.
• Graph the absolute value of a linear function and
  determine and analyze its key characteristics.
• Recognize, express and solve problems that can be
  modeled using linear functions. Interpret
  solutions in terms of the context of the problem.

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• Solve single-variable linear equations and
  inequalities with rational coefficients.
• Solve equations involving the absolute value of a
  linear expression.
• Graph and analyze the graph of the solution set
  of a two-variable linear inequality.
• Solve systems of linear equations in two
  variables using algebraic and graphic procedures.
  
• Recognize, express and solve problems that can be
  modeled using single-variable linear equations
  one- or two-variable inequalities or
  two-variable systems of linear equations.

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Building Understanding for Writing and Evaluating
Expressions
Algebra should become a language through which we
can describe various situations
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• What is three plus five times two?

Try entering this problem on the homescreen of
the graphing calculator
How many ways can you enter it on the homescreen?
Is there an order for the operations when the
problem is written horizontally? 432
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Order of Operations

1. Evaluate expressions within parentheses or other
   grouping symbols.
2. Evaluate all powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

How would you have to write 4 32 so the
answer is 14?
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Learning to build mathematical expressions
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• Lets try performing a string of operations to
  see what we get.
• On paper
• Start with 6.
• Multiply 2 times a starting number, then add 6,
  divide this result by 2, and then subtract your
  answer from 10.
• Start with 20.
• Multiply 2 times a starting number, then add 6,
  divide this result by 2, and then subtract your
  answer from 10.
• Start with -4
• Multiply 2 times a starting number, then add 6,
  divide this result by 2, and then subtract your
  answer from 10.

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These problems appear pretty simple because we
are giving all the directions in short steps and
you are performing them in the order in which
they are described.
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Lets see if we can learn to write expressions
through a similar activity. Start with a chart
and complete each line based on the directions
given.
Description Expression





How is what we did in the four steps equivalent
to this one relationship?
Is there any relationship between the starting
number and the resulting answer?
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Description Expression
Start with a number
Multiply the number by 2
Add 6
Divide by 2
Subtract the result from 10
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After we have written the expression we can test
it

• On the graphing calculator homescreen type
• 6 gt x10-(2x6)/2

gt means STO
General Format Your Number gt x10-(2x6)/2
Confirm 20 and -4
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Set up the expression for this problem

• Using the Description/Expression Template
• Pick any number
• Divide the number by 4
• Add 7
• Multiply the result by 2
• Subtract 8
• Find the value of your expression when x2, -5, 8

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Number Tricks

• Each person pick any number from 1 to 25.
• Add 9 to it.
• Multiply the result by 3.
• Subtract 6 from the current answer.
• Divide this answer by 3.
• Now subtract your original number.
• Compare your results.
• Will the answer be the same regardless on the
  number you begin with?
• Why is this?
• Write out the algebraic expression for this
  number trick.

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Description Expression
Start with a number
Add 9
Multiply the result by 3
Subtract 6 from the result
Divide by 3
Subtract the original number
This is a pretty complex expression. Can we put
these in an equation and solve for x?
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If you were told the expression on the left
describes several operations that were performed
to a given number and that the result equals to
7, describe all the operations that were
performed on x and what order they were performed
to arrive at the answer 7?
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You create a trick

• Create your own trick that has at least 5 stages.
  
• Test it on your calculator with at least four
  different numbers to make sure all the answers
  are the same.
• When you think your trick works, test it on your
  other group members.

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Analyzing a Number Trick

• Write in words the number trick that is described
  above.
• Test the number trick to be sure you get the same
  result no matter what number you choose.
• Can you explain why this number trick work?

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• Given the expression on the left, you might want
  to think of subtraction as adding the opposite
  and re-write the expression
• Write, in words, the number trick that is
  described above.
• Test the number trick to be sure you get the same
  result no matter what number you choose.
• Which operations that undo previous operations
  make this number trick work?

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• Daxun, Lacy, Claudia, and Al are working on a
  number trick. Here are the number sequences their
  number trick generates

a. Describe the stages of this number trick in
the first column. b. Complete Claudias
sequence. c. Write a sequence of expressions for
Al in the last column.
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In Chapter 2

• Lessons 2.1 and 2.2 review proportions and
  introduce the idea of undoing to solve a
  proportion.
• Lesson 2.3 deriving linear expressions from
  measurement
• Lesson 2.4 introduces direct variation equations
  as an alternative to solving proportions (a
  special linear function), create a scatter plot
  of a real data set, model with a line through the
  points, and write an equation in the form ykx to
  describe that line.
• Lesson 2.5 introduces the related topic of
  inverse variation (not a linear function)
• Lesson 2.7 rules for order of operations by
  analyzing how the steps in linear expressions
  that describe number tricks undo each other to
  end with the same number
• Lesson 2.8 write linear equations to represent
  sequences of steps and solve those equations by
  undoing.

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What does it mean to solve an equation?
Is it any more than just undoing the procedure of
building an equation?
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• Choose a secret number.
• Now choose four more non-zero numbers and in any
  random order
• add one of them,
• multiply by another,
• subtract another, and
• divide by the final number
• Record in words what you did and your final
  result on the communicator with a blank Building
  and Evaluating an Expression or Equation
  template. (Do not record your secret number.)
• Switch communicators and have another students
  find your secret number.

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2,5,4,8
x
?
Add 2, Multiply by 5, Subtract 4, And divide by 8
Ans2
Add 2
X2
Multiply by 5
Ansx5
Subtract 4
Ans-4
Reveal the results
Divide by 8
Ans 8
What was my starting number?
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To some number, add 3, multiply by 2, add 18, and
finally divide by 6. a. Convert the description
into an expression, and write an equation that
states that this expression is equal to 15. b.
Find the starting number if the final result is
15. c. Test your solution to part b using your
equation from part a.
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Solve Equations is Just Undoing Operations

• Use the Build and Undo Expression or Equations
  Chart to complete the following number trick.
  Complete the first three columns only.
• Pick a number
• Divide the number by 4
• Add 7
• Multiply the result by 2
• Subtract 8

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Solve Equations is Just Undoing Operations

Description / Sequence Expression
Pick a number ? X
Divide by 4 /4 x4
Add 7 7 -7
Multiply by 2 x2 /2
Subtract 8 -8 8 28
44
11
18
36
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Instead of building an equation, lets start with
an equation and solve it

• Here is an equation. What is it saying? First
  build the equation, then well solve it.

Pick a number
x
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Try solving these equations

• Place the Building and Undoing an Expression
  Template in your Communicator.
• Record an equation in the cell at the top.
• Complete the description column using the order
  of operations.
• Complete the undo column.
• Finally, work up from the bottom of the table to
  solve the equation.
• Write a few sentences explaining why this method
  works to solve an equation

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Simplifying the Technique of Solving an Equation
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x
143
-3
3
140
35
x4
4
42
-7
7
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• An equation is a statement that says the value of
  one expression is equal to the value of another
  expression.
• Solving equations is the process you used to
  determine the value of the unknown that makes the
  equation true. This is called the solution.

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Chapter 3

• In Chapter 3, students use equations to model
  linear growth and graphs of straight lines and
  learn the balancing method for solving equations.
  This chapter builds toward the concept of
  function, which is formalized in Chapter 8.

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• Lesson 3.1 development of linear growth with
  recursive sequences.
• Lesson 3.2 linear plots.
• Lesson 3.3 walking instructions to study motion
• Lesson 3.4 intercept form of a line with
  starting value and rate of change
• Lesson 3.5 rates of change
• Lesson 3.6 balancing technique for solving
  equations
• Lesson 3.7 Model real-world data with linear
  equations

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Solving Equations by Balancing Equations
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The figure illustrates a balanced scale. This is
because 4 yellow square tiles balances with 4
square yellow square tiles. Build this scale in
front of you.
Lets discover some things we can do to balanced
scale that will keep it that keep the scale in
figure 1 balanced.
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___What would happen if you added 2 yellow
squares tiles to both sides of the figure
? ___What would happen if you added 1 red square
tile to both sides of the figure ? ___What would
happen if you added 1 red square to the left side
and one yellow square tile to the right side of
the figure ? ___What would happen if you added
double the number of tiles on both sides of the
figure ? ___What would happen if you removed one
yellow square from the left side and added one
red square to the right side of the figure
? ___What would happen if you cut the number of
tiles in half on each side of the figure
? ___What would happen if you doubled the left
side and divided the right side by 2 in the
figure ? ___What would happen if you added one
red square to the left side only in the
figure? ___What would happen if you added one
yellow square to the right side only in the
figure? ___What would happen if you added red
square to the left and removed one yellow square
from the right in the figure?
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The figure illustrates a balanced scale. Build
this on your scale. How many red or yellow
squares would the green rectangle be equal
to? Using one of the ideas from above, we can
show that the green rectangle is equal to 2
yellow squares. Show at least two ways this can
be accomplished.
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Make a sketch of the balance scale that matches
with this equation. Solve the equation by
using the algebra tiles.
-4 2(x 2)
x -3 -4
x 4 -2x -2
2x -3 5
3x -3 2(x ?1)
-2x -3 -4 -1x
-3 x 2x 1
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Making the Transition to solving an Equation
Algebraically with symbols
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If the equation was 1 2x 3 7 you would have
built the balance scale in the figure. One step
you might do first is combine the like terms.
This would result in the next figure. This figure
says that 2x4 8. Now you might think about
remove 4 yellow squares from both sides.
This would leave you with the next figure. This
figure says that 2x 4. Then you would have
divided both sides into two equal groups so the
green rectangle equals 2 yellow squares or x 2.
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This time our steps will be more algebraic, but
based upon what we did with the balance scale.
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Chapter 4

• Chapter 4 emphasizes slope in the context of
  finding lines of fit.

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• Lesson 4.1 formula for determining slope
• Lesson 4.2 use the intercept form to fit lines
  to data
• Lessons 4.3 and 4.4 point-slope form through
  application
• Lesson 4.5Use the point-slope form to fit lines
  to data
• Lessons 4.6 and 4.7 method for determining lines
  of fit
• Lesson 4.8 activity day for reviewing lines of
  fit.

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• In Chapter 5, students look at systems of linear
  equations and consider linear inequalities. Then
  they put these two ideas together to think about
  systems of linear inequalities.

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• Lessons 5.1 to 5.4 five ways to solve a system
  of equations tables, graphs, the substitution
  method, the elimination method, and row
  operations on matrices.
• Lesson 5.5 Inequalities in one variable are
  introduced
• Lesson 5.6 graph inequalities in two variables
• Lesson 5.7 graph and solve systems

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• Students, through using Discovering Algebra are
  going to discover and learn much useful algebra
  along the way.
• Learning algebra is more than learning facts and
  theories and memorizing procedures and then
  trying to apply them through applications
  sections.
• Through the text students be involved in
  mathematics and in learning how to do
  mathematics.
• Success in algebra is a gateway to many varied
  career opportunities

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• Through the investigations, students will make
  sense of important algebraic concepts, learn
  essential algebraic skills, and discover how to
  use algebra.

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Michael T. Battista of Kent State University
writes

• that algebra teaching should focus on the basic
  skills of today, not those of 40 years ago.
  Problem solving, reasoning, justifying ideas,
  making sense of complex situations, and learning
  new ideas independentlynot paper-and-pencil
  computationare now critical skills for all
  Americans. In the Information Age and the Web
  era, obtaining the facts is not the problem
  analyzing and making sense of them is.
• The Mathematical Miseducation of Americas
  Youth, The Phi Delta Kappan. February, 1999

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In Discovering Algebra

• technology, along with applications, is used to
  foster a deeper understanding of algebraic ideas.
  
• The explorations emphasize symbol sense,
  algebraic manipulations, and conceptual
  understandings.
• The investigative process encourages the use of
  multiple representationsnumerical, graphical,
  symbolic, and verbalto deepen understanding for
  all students and to serve a variety of learning
  styles.
• Explorations from multiple perspectives help
  students simplify and understand what formerly
  were difficult algebraic abstractions.
• Investigations actively engage students as they
  make personal and meaningful connections to the
  mathematics they discover.

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• Traditional algebra teaches skills and ideas
  before examples and applications.
• The investigative approach works the other way.
• Interesting questions and simple hands-on
  investigations precede the introduction of
  formulas and symbolic representations.
• By providing meaningful contexts for students,
  the investigations motivate relevant algebraic
  concepts and processes.
• The investigations are accessible. They use
  inexpensive and readily available materials,
  require little prerequisite technical knowledge,
  and follow simple procedures. Students can
  conduct them with a minimum of direction and
  intervention from you.

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• Teaching with Discovering Algebra decreases the
  time students spend on rote memorization, teacher
  exposition, and extended periods of
  paper-and-pencil drill.
• It changes the rules for what is expected of
  students and what they should expect of their
  teacher.
• Teaching from Discovering Algebra requires
  nontraditional thinking and behavior and a
  nontraditional classroom. Success depends on your
  sensitivity, patience, enthusiasm, and
  determination.