Data Matters

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CITY YEAR CHICAGO
Math 101 Session Developer Mari Mermelstein
City Year Chicago
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Warm-Up
5 Minute Free-Write Write down anything and
everything you think about MATH
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Ultimate Goal

• Re-frame your mindset
• and attitude towards Math

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Putting Idealism to Work

• PITW 134 - A Positive, Can-Do Attitude Is the
  First Qualification for Being a Part Of City
  Year.
• This must be true for both corps and staff.
  Inspiring others and maintaining an environment
  in which idealism can flourish depends on all of
  us maintaining positive attitudes. This does not
  mean always being rah rah. But it does mean
  that we must all remain positive, constructive
  and inspired, even when being critical.

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Agenda

• Know Questions
• Common Misconceptions
• Foundational Skills
• Math Anxiety and its causes
• Common student struggles
• 1st month strategies

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Know Questions

• How to refute common misconceptions about
  mathematics
• How the Common Core State Standards can help you
  help your students
• How to identify common struggles that students
  have
• Computational Strategies

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Misconceptions

• Math is only about learning to compute
• Math is about following rules to guarantee
  correct answers
• Some people have the ability to do math and some
  dont
• Men are naturally better than women at math
• Learning math isnt important in the age of
  calculators and computers.

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Math is only about learning to compute

• It is a way of approaching new challenges
  through investigating, reasoning, visualizing and
  problem solving with the goal of communicating
  the relationships observed and problems solved to
  others.1
• Knowledge of mathematics and the ability to
  apply math skills to solve problems can be an
  empowering force for all studentsboth while in
  school and later in their lives.1
• 1Illinois State Board of Education
  http//www.isbe.state.il.us/ils/math/standards.htm
  

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Math is about following rules to guarantee
correct answers

• Some people do take comfort in the black and
  whiteness of Math, but NOT all.
• About the sense-making process.
• Learning to reason and understand why math
  works is just as important as finding the
  right answer.

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Some people have the ability to do math and some
dont

• Which is a person more likely to admit
• That they are Illiterate or Innumerate (math
  illiteracy)?
• If a student comes across a word they dont know,
  they dont say they cannot read but if a student
  comes across a problem they cant solve, they say
  they cannot do math.

Why is it socially acceptable to be math
illiterate, but not verbally illiterate?
By allowing this mindset, we are saying it is OK
for our students to fail in Math
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Men are naturally better than women at math

• Girls Sweep Google Science Fair
• A contrast to when women were largely excluded
  from the science world. Kenneth Chang
• It shows you that women are stepping up in
  science Shree Bose, Age 17 (Best in Show)
• This is just a reminder that women are fully
  capable of doing the same or better quality work
  than men can. Dr. Vint Cerf
• Chang, Kenneth. First-Place Sweep by American
  Girls at First Google Science Fair, July 18th,
  2011

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Learning math isnt important in the age of
calculators and computers

• Arithmetic vs Algebraic thinking
• The emphasis now is getting people to be
  sophisticated algebraic thinkers. You cannot
  become good at algebra without a mastery of
  arithmetic, because arithmetic is the gateway to
  algebra. But arithmetic itself is no longer the
  ultimate goal.2
• Need to be smarter than the machine
• Need to understand/be able to write the program
  that does the math for you
• Need to be able to understand the information the
  technology gives you
• 2NPR Interview with Keith Devlin, The Way You
  Learned Math Is So Old School March 5th, 2011

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Foundational Skills
(Lacking)

• Time to get up and move!

Activity Mental Math Strings
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Operations and Algebraic Thinking (including
Functions)
Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade
Understand addition as putting together and adding to. Represent and solve problems involving addition and subtraction Represent and solve problems involving addition and subtraction Represent and solve problems involving multiplication and division Use the four operations with whole numbers to solve problems. Write and interpret numerical expressions Apply and extend previous understandings of arithmetic to algebraic expressions Use properties of operations to generate equivalent expressions Work with radicals and integer exponents Perform arithmetic operations on polynomials
Understand subtraction as taking apart and taking from. Understand and apply properties of operations and the relationship between addition and subtraction Add and subtract within 20 Understand properties of multiplication and division. Gain familiarity with factors and multiples Analyze patterns and relationships Reason about and solve one-variable equations and inequalities Solve real-life and mathematical problems using numerical and algebraic expressions and equations Understand the connections between proportional relationships, lines, and linear equations Create equations that describe numbers or relationships. Be able to explain the problem solving process
  Add and subtract within 20 Work with equal groups of objects to gain foundations for multiplication Multiply and divide within 100 Generate and analyze patterns   Represent and analyze quantitative relationships between dependent and independent variables   Analyze and solve linear equations and pairs of simultaneous linear equations. Solve equations, inequalities, and systems of equations both algebraically and graphically
  Work with addition and subtraction equations.   Solve problems involving the four operations, and identify and explain patterns in arithmetic         Define, evaluate, and compare functions Understand the concept of a function, use function notation, and interpret real world applications of functions
                Use functions to model relationships between quantities. Build a function that models a relationship between two quantities
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Simplify 3(2x2 5x) - 7 - 3x 1 - x2
Reason
Steps
Calculation
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Operations and Algebraic Thinking (including
Functions)
3(2x2 5x) - 7 - 3x 1 - x2 5x2 12x 6
Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade
Understand addition as putting together and adding to. Represent and solve problems involving addition and subtraction Represent and solve problems involving addition and subtraction Represent and solve problems involving multiplication and division Use the four operations with whole numbers to solve problems. Write and interpret numerical expressions Apply and extend previous understandings of arithmetic to algebraic expressions Use properties of operations to generate equivalent expressions Work with radicals and integer exponents Perform arithmetic operations on polynomials
Understand subtraction as taking apart and taking from. Understand and apply properties of operations and the relationship between addition and subtraction Add and subtract within 20 Understand properties of multiplication and division. Gain familiarity with factors and multiples Analyze patterns and relationships Reason about and solve one-variable equations and inequalities Solve real-life and mathematical problems using numerical and algebraic expressions and equations Understand the connections between proportional relationships, lines, and linear equations Create equations that describe numbers or relationships. Be able to explain the problem solving process
  Add and subtract within 20 Work with equal groups of objects to gain foundations for multiplication Multiply and divide within 100 Generate and analyze patterns   Represent and analyze quantitative relationships between dependent and independent variables   Analyze and solve linear equations and pairs of simultaneous linear equations. Solve equations, inequalities, and systems of equations both algebraically and graphically
  Work with addition and subtraction equations.   Solve problems involving the four operations, and identify and explain patterns in arithmetic         Define, evaluate, and compare functions Understand the concept of a function, use function notation, and interpret real world applications of functions
                Use functions to model relationships between quantities. Build a function that models a relationship between two quantities
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Operations and Algebraic Thinking (including
Functions)
Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade
Understand addition as putting together and adding to. Represent and solve problems involving addition and subtraction Represent and solve problems involving addition and subtraction Represent and solve problems involving multiplication and division Use the four operations with whole numbers to solve problems. Write and interpret numerical expressions Apply and extend previous understandings of arithmetic to algebraic expressions Use properties of operations to generate equivalent expressions Work with radicals and integer exponents Perform arithmetic operations on polynomials
Understand subtraction as taking apart and taking from. Understand and apply properties of operations and the relationship between addition and subtraction Add and subtract within 20 Understand properties of multiplication and division. Gain familiarity with factors and multiples Analyze patterns and relationships Reason about and solve one-variable equations and inequalities Solve real-life and mathematical problems using numerical and algebraic expressions and equations Understand the connections between proportional relationships, lines, and linear equations Create equations that describe numbers or relationships. Be able to explain the problem solving process
  Add and subtract within 20 Work with equal groups of objects to gain foundations for multiplication Multiply and divide within 100 Generate and analyze patterns   Represent and analyze quantitative relationships between dependent and independent variables   Analyze and solve linear equations and pairs of simultaneous linear equations. Solve equations, inequalities, and systems of equations both algebraically and graphically
  Work with addition and subtraction equations.   Solve problems involving the four operations, and identify and explain patterns in arithmetic         Define, evaluate, and compare functions Understand the concept of a function, use function notation, and interpret real world applications of functions
                Use functions to model relationships between quantities. Build a function that models a relationship between two quantities
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Number and Operations in Base Ten (including
Fractions/Ratios/Proportions
Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade
Know number names and the count sequence Extend the counting sequence Understand place value Use place value understanding and properties of operations to perform multi-digit arithmetic Generalize place value understanding for multi-digit whole numbers Understand the place value system Compute fluently with multi-digit numbers and find common factors and multiples Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers Know that there are numbers that are not rational, and approximate them by rational numbers. Use properties of rational and irrational numbers
Count to tell the number of objects Understand place value Use place value understanding and properties of operations to add and subtract   Use place value understanding and properties of operations to perform multi-digit arithmetic Perform operations with multi-digit whole numbers and with decimals to hundredths Apply and extend previous understandings of numbers to the system of rational numbers     Reason quantitatively and use units to solve problems
Compare numbers Use place value understanding and properties of operations to add and subtract   Develop understanding of fractions as numbers Extend understanding of fraction equivalence and ordering. Use equivalent fractions as a strategy to add and subtract fractions Apply and extend previous understandings of multiplication and division to divide fractions by fractions Analyze proportional relationships and use them to solve real-world and mathematical problems   Extend the properties of exponents to rational exponents  
Work with numbers 11-19 to gain foundations for place value       Build fractions from unit fractions by applying and extending previous understanding of operations on while numbers Apply and extend previous understandings of multiplication and division to multiply and divide fractions Understand ratio concepts and use ratio reasoning to solve problems      
        Understand decimal notation for fractions, and compare decimal fractions.                      
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Geometry
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Measurement and Data/Statistics and Probability
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Math Anxiety

• Weak Computational Skills
• Poor conversation or academic language skills
• Lack of Confidence
• Unable/Unwilling to write out complete solutions

• Weak Conceptual Understanding
• Poor test-taking skills
• Low Work Completion Rate
• Poor English Language Skills

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Learning Differences

• People learn in different ways.
• Visual
• Auditory
• Kinesthetic
• Combo
• If the students learning style and the classroom
  teaching style are not compatible students may
  struggle

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Strategies for the 1st month

• Check List - watch for the above listed
  struggles
• Flash Cards
• Multiplication strategy - Elizabethan Boxes
• As a way to move toward the standard algorithm
• Division strategy - Clustering
• As a way to move toward the standard algorithm
• Fractions/Decimals/Percentages
• Order of Operations and Properties of Equalities

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Standard Multiplication Algorithm
156
? 72
312
? 1092?
11,232
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Multiplication StrategyElizabethan Boxes
(Lattice Method)if student is struggling with
the standard algorithm
Create the box grid
Answer 11,232
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Multiplication StrategyElizabethan Boxes
Are useful for multiplying polynomials of any
size (5x2 - 3x 2)(x2 3)
5x2 - 3x 2
5x4 - 3x3 2x2 x2
5x4 0x3 0x2 0x 0x
-3x3 15x2 - 9x 6 3
17x2 -9x 6
Answer 5x4 - 3x3 17x2 - 9x 6
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Standard Division Algorithm
Multiply, subtract, and repeat
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Division Strategy Clusteringif student is
struggling with the standard algorithm
Problem 128 ? 16 Example 1 Example 2 16 ?
5 80 16 ? 2 32 16 ? 2 32 16 ? 2
32 16 ? 1 16 16 ? 2 32 16 ? 2
32 5 2 1 8 2 2 2 2 8 (Check
80 32 16 128) (Check 32 32
32 32 128)
Guess 2 Check 128 - 32 96 96
remaining Guess 2 Check 96 - 32 64 64
remaining Guess 2 Check 64 - 32 32 64
remaining Guess 2 Check 32- 32 0 ?
Guess 5 Check 128 - 80 48 48
remaining Guess 2 Check 48 - 32 16 16
remaining Guess 1 Check 16 - 16 0 ?
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Fractions, Decimals, and Percentages
Adding Fractions Subtracting Fractions
Multiplying Fractions Dividing Fractions
Benchmark Values Benchmark Values
Conversions Conversions
Decimal to Percent Decimal to Fraction
Fraction to Decimal Fraction to Percent
Percent to Decimal Percent to Fraction
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Order of Operations
Order of Operations Order of Operations Order of Operations
     
Mnemonic Operation Symbol
Please Parentheses ( ) or
Excuse Exponents x3 or x5
My Multiplication or
Dear Division or ?
Aunt Addition
Sally Subtraction -
(fraction bar)
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Properties of Equality
Properties of Equality (These Properties are TRUE for ALL numbers!!) Properties of Equality (These Properties are TRUE for ALL numbers!!) Properties of Equality (These Properties are TRUE for ALL numbers!!)
     
Distributive a(bc) abac (bc)a baca a(b-c) ab-ac a(b-c) ba-ca
     
  Addition Multiplication
Identity a 0 0 a a
Inverse a (-a) 0 (-a) a 0
Zero  
Commutative a b b a ab ba
Associative a(bc) (ab)c a(bc) (ab)c