Region 11: Math

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Region 11 Math Science Teacher
CenterSolving Equations
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Where weve been
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Patterns Review - Justify It!
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Patterns Review - Justify It!
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Patterns Review

• With a partner, talk about how a third grader
  would talk about this problem
• Draw the next figure in the pattern
• How many dots will be in the next figure?
• Describe how make the pattern

Adult Challenge How many dots would be in the
nth figure?
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Patterns Review

• Share in Grade Level Groups
• What strategies did you see kids use?
• What did the students find most challenging?
• What growth did you see?
• What surprised you?

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Patterns Review
Watch video and record Good questions
Not-so-good questions
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Patterns Review
Discuss video at your tables What questions
helped most to get your students to explain the
explicit rule for different patterns? What did
you learn about scaffolding questions? How do
you know when you are done asking questions?
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Solving Equations

• Goals
• Identify a developmental sequence for solving
  equations
• Structure math talk about expressions
  equations
• Emphasize equivalence and how we communicate
  this with students
• Discuss properties of equations

Work with a partner to solve problems on page 1
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Solve.
Find all values that make the statement true.
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Solve.
Find all values that make the statement true.
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Solve.
Find all values that make the statement true.
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Solve.
Find all values that make the statement true.
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Solve.
Find all values that make the statement true.
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Solve.
Find all values that make the statement true.
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What does solve mean?
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Break
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Think/Pair/Share

• Define expression
• Define equation

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Big Idea!
We solve equations because we can make them
true or false. We dont solve expressions
because we cant make them either true or false.
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Expression vs. Equation

• Share and discuss in your group as you work on
  page 2
• How would you write the directions?
• How would you want/expect students to show
  their work?What would you write on the board?

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Expression vs. Equation
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Expression vs. Equation
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Expression vs. Equation
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Expression vs. Equation
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Expression vs. Equation
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Expression vs. Equation
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Expression vs. Equation

• Directions make a big difference!
• Directions depend on context and where you are
  in the curriculum
• Expressions are not equations
• No one right way to show work
• Most middle school textbook authors have
  thought carefully about what strategies to use to
  solve equations

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CGI Algebra Video Clips
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Benchmarks in Student ThinkingAbout The Equal
Sign
Note These benchmarks are a guide, not a firm
sequence
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CGI Algebra Video Clips
Solving the Equation
(4th grader, 2 minutes, 42 seconds)
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CGI Algebra Video Clips
Solving the Equation
(4th grader, 1 minutes, 51 seconds)
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CGI Algebra Video Clips
Solving the Equation
(4th grader, 39 seconds)
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CGI Algebra Video Clips
Solving the Equation
(4th grader, 51 seconds)
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Benchmarks in Student ThinkingAbout The Equal
Sign
Note These benchmarks are a guide, not a firm
sequence
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Lunch
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Methods for solving linear equations of the form
ax b cx d
Traditional Approach vs. Functions Approach
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Traditional approach for solving linear equations
of the form ax b cx d

• Use of number facts
• (solve mentally)
• Example
• 3 x 7
• Not-so-good for
• 3x 7 5x 14

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Traditional approach for solving linear equations
of the form ax b cx d
2) Generate and evaluate (guess and check or
trial and error substitution) Example 2x 3
4x 7 Not-so-good for 3x 7 10 4x
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Traditional approach for solving linear equations
of the form ax b cx d
3) a. Undoing (or working
backwards) Example 20 3x 4 24 3 x 8
x Not-so-good for 3x 7 5x 14
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Traditional approach for solving linear equations
of the form ax b cx d

• 3) b. Undoing
• 17 3p 1

x 3
- 1
p
3p
3p 1
17
18
6
1
3
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Traditional approach for solving linear equations
of the form ax b cx d
4) Cover-up Example k k 13 k 20 k
k 13 k 13 7 k 7 Not-so-good for 3x
7 25 5x
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Traditional approach for solving linear equations
of the form ax b cx d
5) Transposing (change side-change
sign) Example 3x 8 5x
15 x 3 Not-so-good for
7
2x
2x
7
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Traditional approach for solving linear equations
of the form ax b cx d
6) Equivalent equations (performing the
same operation on both sides) Example
17 3x 7 17 7 3x 7 7 24 3x 24/3
3x/3 8 x
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Traditional approach for solving linear equations
of the form ax b cx d
Group Task 1 Break into six table groups
Write one or two good problems for each method on
the table When the bell rings, move to the
next table
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Traditional approach for solving linear equations
of the form ax b cx d
Group Task 2 Move to the table where you
started Work the problems using that particular
method Star any problems that cant be solved
easily with the method Determine the minimum
benchmark level needed to solve these problems
Be ready to report out
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Functions approach for solving linear equations
of the form ax b cx d
1) Table using graphing calculator (similar to
guess and check) Example 3x 4 x 6 x 5
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Functions approach for solving linear equations
of the form ax b cx d
2) Graphing Example 3x 4 x 6 x 5
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• Why standards?

By viewing algebra as a strand in the
curriculum from prekindergarten on, teachers can
help students build a solid foundation of
understanding and experience as a preparation for
more-sophisticated work in algebra in the middle
grades and high school. - NCTM, 2000, p.37
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True or False

• Your textbook determines the algebra concepts
  and skills that you should cover at a particular
  grade level.

?TRUE FALSE? ? dont
know

• False In a standards-based system, the focus is
  shifted from what is TAUGHT to what is LEARNED.
• The standards tell us what students should know
  and be able to do.

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True or False

• Algebra content has been shifted down and now
  starts in the middle grades.

?TRUE FALSE? ? dont
know
False Algebra and algebraic thinking are
integrated across K-11 in the state standards.
Every teacher has to do his/her part to give
students the opportunity to learn the grade-level
content.
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Sort the Standards Involving Solving Equations

• In your groups,
• Look through individual standards
• Classify for grades 1 - 8
• Ask for an answer key when finished
• Take time to reflect on standards
• with your group members

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BIG IDEA
TRADITIONAL ALGEBRA I
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BIG IDEA
TRADITIONAL ALGEBRA I
8th GRADE STANDARDS
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Break
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Equivalence - Expressions
 On page 7 State directions for each
problem Simplify each expression or equation
one operation at a time, leaving a trail down of
equivalent expressions or equivalent equations.
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Equivalence - Expressions
 Directions  
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Equivalence - Expressions
 Directions  
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Equivalence - Expressions
 Directions   Check
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Equivalence - Expressions
Expressions are equivalent
if every line has the same value for the same
value of x
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Equivalence - Expressions
Equations are equivalent
if they have the same solution set (but each
line has a different value for a given value of
x).
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Equivalence - Expressions
 Can I add 5? 10?  
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STRETCH
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Balance metaphor for equations
Same weight on both sides
page 8 of the handout
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Balance metaphor for equations
Build a balance to solve 4 x 4 12
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Balance metaphor for equations
Build a balance to solve x y 7
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Balance metaphor for equations
Build a balance to solve 2 x 4 2 x
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Problems with balance metaphor
Negatives
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Problems with balance metaphors
Subtraction
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Metaphors for equations Balance (same weight
on both sides)
1. Open Number sentences (CGI)
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Metaphors for equations Balance (same weight
on both sides)

• Algebra Tiles (McDougal Littell) or
• Bags of Gold (CMP)

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Metaphors for equations Balance (same weight
on both sides)
3. Equation Mat (CPM Algebra Connections)
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STRETCH
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Baseline Assessment
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Baseline Assessment
1. Find the value of m that makes the number
sentence below true. 12 4 m
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Baseline Assessment
2. Find the value of b that makes the number
sentence below true. 15 3b 42
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Baseline Assessment
3. Find the value of x that makes the number
sentence below true. 12x - 10 6x 32
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Baseline Assessment
4. Find the value of n that makes the number
sentence below true. Show your steps to
demonstrate how you solved the problem. 4
n - 2 5 11 3 5
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Baseline Assessment
5. Balance A is balanced. The amount on the left
side of Balance A is tripled for Balance B. Draw
in what should appear on the right side of
Balance B to be balanced. Explain why
your answer works.
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Baseline Assessment
6. Determine which of the equations below are
equivalent to 3b 4 b 6 Circle yes, no or
do not know for each part.
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Properties of Equations
Addition Property of Equality Words Adding the
same number to each side of an equation produces
an equivalent equation Algebra from McDougal
Littell Math Course 2
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Properties of Equations
What other properties maintain equivalence? Words
Algebra
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Big Ideas

• We want to
• Learn the different methods that children
  use to solve linear equations
• Learn how to select examples to push kids
  from less sophisticated methods (i.e. guess
  and check) to more algebraic methods
• Learn different metaphors for helping
  students solve equations

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Evaluation

• On an index card, please record
• P one positive from todays work
• M one minus or concern from
• todays work
• I something that you found interesting or
• intriguing from todays work.
• Thanks for your feedback.