city year chicago math 101 session developer: mari mermelstein city year chicago

CITY YEAR CHICAGOCITY YEAR CHICAGOCITY YEAR CHICAGO

Math 101Session Developer: Mari Mermelstein

City Year Chicago

Warm-Up

5 Minute Free-Write

Write down anything and everything you think

about

MATHWrite about your feelings

Write about your experiences

Write about all the things you wanted to say to your teachers but never did.

Ultimate Goal

• Re-frame your mindset

and attitude towards Math

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.

Agenda

• “Know” Questions

• Common Misconceptions

• Foundational Skills

• Math Anxiety and it’s causes

– Common student struggles

• 1st month strategies

“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

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.

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

1Illinois State Board of Education: http://www.isbe.state.il.us/ils/math/standards.htm

http://www.isbe.state.il.us/ils/math/standards.htm








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.

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

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

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

2NPR Interview with Keith Devlin, “The Way You Learned Math Is So Old School” March 5 th, 2011

Foundational Skills

Time to get up and move!

Activity: Mental Math Strings

(Lacking)

Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th GradeUnderstand 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


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)

Simplify 3(2x2 + 5x) - 7 - 3x + 1 - x2

Steps Calculation Reason

1. Rewrite the expression

3(2x2 + 5x) - 7 - 3x + 1 - x2

2. DistributeOrder of Operations: Multiplication and division should be solved before addition and subtraction. Always work from right to left.

6x2 + 15x – 7 – 3x + 1 - x2

3. Gather like terms together 6x2 – x2 + 15x – 3x – 7

+ 1Ensures proper simplification

4. SimplifyOrder of Operations: Addition and subtraction from left to right, do whichever operation appears first.

5x2 + 12x – 6



































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 GradeKnow 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

Geometry

Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade Identify an d describe shapes

Reason with shapes and their attributes



Draw and identify lines and angles, and classify shapes by properties of their lines and angles

Graph points on the coordinate plane to solve real-world and mathematical problems

Solve real-world and mathematical problems involving area, surface area, and volume

Draw, construct and describe geometrical figures and describe the relationships between them

Understand congruence and similarity using physical models, transparencies, or geometry software

Explain volume formulas and use them to solve problems

Analyze, compare, create, and compose shapes

Classify two-dimensional figures into categories based on their properties

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume

Understand and apply the Pythagorean Theorem

Visualize relationships between two-dimensional and three-dimensional objects

Solve real-world and mathematical problems involving volume of cylinders, cones and spheres

Apply geometric concepts in modeling situations

Measurement and Data/Statistics and Probability

Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade Describe and compare measurable attributes

Measure lengths indirectly and by iterating length units

Measure and estimate lengths in standard units

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit

Convert like measurement units within a given measurement system

Develop understanding of statistical variability

Use random sampling to draw inferences about a population

Investigate patterns of association in bivariate data

Calculate expected values and use them to solve problems

Classify objects and count the number of objects in categories

Tell and write time

Relate addition and subtraction to length

Represent and interpret data



Summarize and describe distributions

Draw informal comparative inferences about two populations

Use probability to evaluate outcomes of decisions


Work with time and money

Geometric measurement: understand concepts of area and relate area to multiplication and to addition

Geometric measurement: understand concepts of angle and measure angles

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition

Investigate chance processes and develop, use, and evaluate probability models


Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures

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

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

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

Standard Multiplication Algorithm

156 72

312 1092

11,232

Multiplication Strategy:Elizabethan Boxes (Lattice

Method)*if student is struggling with the standard algorithm*

Multiply each singledigit pair

Add along the diagonals

Create thebox grid

Answer: 11,232

Multiplication Strategy:Elizabethan 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

Standard Division Algorithm

Multiply, subtract, and repeat

Division Strategy: Clustering*if student is struggling with the standard algorithm*

Problem: 128 16Example 1 Example 216 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: 2Check: 128 - 32 = 96 96 remaining



Guess: 2Check: 32- 32 = 0



Guess: 1Check: 16 - 16 = 0

Fractions, Decimals, and Percentages

Adding FractionsSubtracting Fractions

Multiplying Fractions

Dividing Fractions

Benchmark Values

Conversions:

Decimal to PercentDecimal to

Fraction

Fraction to Decimal

Fraction to Percent

Percent to Decimal Percent to Fraction

Order of OperationsOrder of Operations

Mnemonic Operation Symbol

Please Parentheses ( ) or [ ]

Excuse Exponents x^3 or x5

My Multiplication × or ·

Dear Division ÷ or ―

Aunt Addition +

Sally Subtraction ─

(fraction bar)

Properties of EqualityProperties 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

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+(b+c) = (a+b)+c a(bc) = (ab)c€

a ⋅1

a=

1

a⋅a =1

€

a ⋅1 = 1⋅a = a

€

a ⋅0 = 0 ⋅a = 0

city year chicago math 101 session developer: mari mermelstein city year chicago

Documents

math illiterate

whiteness of math

math works

math skills

innumerate math illiteracy

correct answerssome

age of calculators

science world