1. CITY YEAR CHICAGO Math 101 Session Developer: Mari Mermelstein City Year Chicago

2. Warm-Up 5 Minute Free-Write Write down anything and everything you think about MATH Write about your feelings Write about your experiences Write about all the things you wanted to say to your teachers but never did.

3. Ultimate Goal Re-frame your mindset and attitude towards Math

4. 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.

5. Agenda “Know” Questions Common Misconceptions Foundational Skills Math Anxiety and it’s causes –Common student struggles 1st month strategies

6. “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

7. 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 don’t Men are naturally better than women at math Learning math isn’t important in the age of calculators and computers.

8. 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 students—both while in school and later in their lives.” 1 1 Illinois State Board of Education: http://www.isbe.state.il.us/ils/math/standards.htm http://www.isbe.state.il.us/ils/math/standards.htm

9. 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.

10. Some people have the ability to do math and some don’t Which is a person more likely to admit: That they are Illiterate or Innumerate (math illiteracy)? If a student comes across a word they don’t know, they don’t say they cannot read; but if a student comes across a problem they can’t 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

11. 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 18 th, 2011

12. Learning math isn’t 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 2 NPR Interview with Keith Devlin, “The Way You Learned Math Is So Old School” March 5 th, 2011

13. Foundational Skills Time to get up and move! Activity: Mental Math Strings (Lacking)

14. Kindergarten1 st Grade2 nd Grade3 rd Grade4 th Grade5 th Grade6 th Grade7 th Grade8 th Grade9 th Grade Understand addition as putting together and adding to. 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 Operations and Algebraic Thinking (including Functions)

15. Simplify 3(2x 2 + 5x) - 7 - 3x + 1 - x 2 Steps Calculation Reason 1. Rewrite the expression 3(2x 2 + 5x) - 7 - 3x + 1 - x 2 2. Distribute Order of Operations: Multiplication and division should be solved before addition and subtraction. Always work from right to left. 6x 2 + 15x – 7 – 3x + 1 - x 2 3. Gather like terms together 6x 2 – x 2 + 15x – 3x – 7 + 1 Ensures proper simplification 4. Simplify Order of Operations: Addition and subtraction from left to right, do whichever operation appears first. 5x 2 + 12x – 6

16. Kindergarten1 st Grade2 nd Grade3 rd Grade4 th Grade5 th Grade6 th Grade7 th Grade8 th Grade9 th Grade Understand addition as putting together and adding to. 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 Operations and Algebraic Thinking (including Functions) 3(2x 2 + 5x) - 7 - 3x + 1 - x 2 = 5x 2 + 12x – 6

17. Kindergarten1 st Grade2 nd Grade3 rd Grade4 th Grade5 th Grade6 th Grade7 th Grade8 th Grade9 th Grade Understand addition as putting together and adding to. 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 Operations and Algebraic Thinking (including Functions)

18. Kindergarten1 st Grade2 nd Grade3 rd Grade4 th Grade5 th Grade6 th Grade7 th Grade8 th Grade9 th 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. Number and Operations in Base Ten (including Fractions/Ratios/Proportions

19. Geometry

20. Measurement and Data/ Statistics and Probability

21. 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

22. Learning Differences People learn in different ways. –Visual –Auditory –Kinesthetic –Combo If the student’s learning style and the classroom teaching style are not compatible students may struggle

23. 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

24. Standard Multiplication Algorithm 156  72 312  1092  11,232

25. Multiplication Strategy: Elizabethan Boxes (Lattice Method) *if student is struggling with the standard algorithm* Multiply each single digit pair Add along the diagonals Create the box grid Answer: 11,232

26. Multiplication Strategy: Elizabethan Boxes Are useful for multiplying polynomials of any size: (5x 2 - 3x + 2)(x 2 + 3) 5x 2 - 3x2 5x 4 - 3x 3 2x 2 x2x2 5x 4 0x 3 0x 2 0x -3x 3 15x 2 - 9x63 17x 2 -9x6 Answer: 5 x 4 - 3x 3 + 17x 2 - 9x + 6

27. Standard Division Algorithm Multiply, subtract, and repeat

28. Division Strategy: Clustering *if student is struggling with the standard algorithm* Problem: 128  16 Example 1Example 2 16  5 = 8016  2 = 32 16  2= 32 16  2 = 32 16  1= 16 16  2 = 32 16  2 = 32 5 + 2 + 1 = 82 + 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 

29. Fractions, Decimals, and Percentages Adding FractionsSubtracting Fractions Multiplying FractionsDividing Fractions Benchmark Values Conversions: Decimal to PercentDecimal to Fraction Fraction to DecimalFraction to Percent Percent to DecimalPercent to Fraction

30. Order of Operations MnemonicOperationSymbol P lease Parentheses ( ) or [ ] E xcuse Exponents x^3 or x 5 MyMy Multiplication × or · D ear Division ÷ or ― A unt Addition + S ally Subtraction ─ (fraction bar)

31. Properties of Equality (These Properties are TRUE for ALL numbers!!) Distributive  a(b+c) = ab+ac  (b+c)a = ba+ca  a(b-c) = ab-ac  a(b-c) = ba-ca AdditionMultiplication Identity a + 0 = 0 + a = a Inverse a + (-a) = 0 (-a) + a = 0 Zero Commutative a + b = b + aab = ba Associative a+(b+c) = (a+b)+ca(bc) = (ab)c