Teaching to the Next Generation SSS

(2007)

Mathematics Pre-School Inservice (6 – 8) August 18, 2010

In today’s training, we will…

•  engage in activities and in-depth discussions that promote higher level thinking skills •  connect each standard in relation to equations with an example •  be aware of items that prior experience have clarified.

Comparison of Standards

How do we make sense of teaching deeply? " Provide a “focused” curriculum. " Collaboration of vertical and

horizontal teaching " Explore deeper content based

knowledge. Think of a swimming pool. Most people stay in the shallow end…we now need to venture to the unknown depths of knowledge.

" Develop connections from 6th grade math through pre-calculus

" Emphasis of higher level thinking- analysis, evaluation and synthesis

Self Worth-If by chance you were to be auctioned off today what would your price tag be?

n Equations as an example standard linked through the grade levels

n Fair game review through models and visuals

n Use cooperative learning strategies to introduce or remediate equations

n Represent equations using models, vocabulary, pictures, and real world situations

n Lead students to mastery level not memorization

Teaching For Depth

What is Rigor?

What Rigor is Not:

" a measure of the quantity of

content to be covered.

" a special program or curriculum for

select students.

" about severity or hardship.

" only about higher-order thinking.

Rigor

" Rigor is quality instruction that focuses on the depth of the learning not the breadth. It’s not more work; it’s meaningful, respectful work that requires the student to think deeply and critically to accomplish the assigned tasked.

Eric Bergholm, Hillsborough County Public Schools, Florida

MA. 6. A. 3. 1 Subject Grade

Level Body of

Knowledge Big Idea/

Supporting Idea

Benchmark

Coding Scheme for NGSSS

MA.6.A.3.1

Identify the threading standards for solving equations.

Grade Level/Course Benchmark

4 Describe mathematical relationships using

expressions, equations, and visual representations.

6 Write solve and graph one- and two- step linear equations and inequalities.

7 Formulate and use different strategies to solve

one and two step linear equations including equations with rational coefficients.

8 Use tables, graphs and models to represent,

analyze and solve problems related to systems of equations.

Algebra 1 Solve linear equations in one variable that include simplifying algebraic expressions

Geometry Use coordinate geometry to find slopes,

perpendicular lines, parallel lines and equations of lines.

Algebra 2

Solve logarithmic and exponential equations.

Pre-Calculus Solve trigonometric equations and real-world

problems involving applications of trigonometric equations using technology when appropriate.

The Common Thread

What are the NCTM Process Standards?

–  Problem Solving –  Reasoning and Proof –  Connections –  Communication –  Representation

12

NCTM Process Standards

" Problem Solving – Developing perseverance and critical thinking – Allow students think time to reach a solution

" Reasoning and Proof – Mathematical conjectures – Examples and counter examples

" Communication – Read, write, listen, think, and discuss – Increase the use of appropriate math

vocabulary

NCTM Process Standards

" Connections –  Integers, expressions, and equations – Other content areas, science – Real-world contexts

" Representation – Useful tools for building understanding – Concrete - Representational - Abstract – Tables, describe in words, draw a

picture, write and solve equations

Focus on Equations!

6th Grade Equation Strategies

Solving one-step algebraic equations

NGSSS: Equations(6th)

" Example: The height of a tree was 7 inches in the year 2000. Each year the same tree grew an additional 10 inches. Write an equation to show the height h of the tree in y years. Let y be the number of years after the year 2000. The most literal equation might be 7 + 10y = h.

MA.6.A.3.2

STRATEGIES

Dynamic Classroom Interactive Scale--Big Ideas website

Human Equations

Algebra Tiles

Magic Square

Dynamic Classroom Interactive Scale " This strategy gives students a visual

opportunity to see an algebraic equation solved using a scale. This demonstrates how equations must be balanced.

" Benefits: It allows the students to

enhance computer skills and provides a high interest activity with graphics.

Human Equations

" This is a kinesthetic activity that provides the students with an opportunity to move throughout the classroom while modeling how to solve one-step equations.

" Benefits: Allows students to

cooperate in a group activity while moving throughout the room .

How do I do this?

" Students are given cards with numbers, variables and symbols on them.

" An equation is placed on the board. X + 7 = 10 d – 8 = 9 a + 2 = 11

" Students will model the equation with their cards in the front of the class.

" Students in the class are called on to solve the problem by moving the cardholders as they go through the steps of solving the problem.

Algebra Tiles

" They are used to solve algebraic equations using colors and shapes.

" Benefits: Students see positive and

negative integers in colors. It allows them to see constants and variables in different shapes.

Algebra Tile Example

Try x – 4 = 8.

Magic Squares

" This is an enrichment option from page 289 of the 6th grade textbook.

" Benefits: This provides a

cooperative learning activity for students to explore and expand their comprehension of one-step equations and encourages higher-level critical thinking skills.

Magic Square…Your Turn! Use  equations  to  solve  the  magic  square.  

Hint:  all  lines  horizontal,  vertical  and  diagonal  have  the  same  sum.  

   

 

 

 

 

 

 

16   a   2   13  

b   10   11   8  

9   6   7   c  

4   15   d   1  

NGSSS: 7th grade MA.7.A.3.3

Example: It costs an initial fixed cost of $2 plus an additional $1.50 per mile to rent a taxi. Write an equation that represents the method for calculating the total cost of a taxi ride? What is the total cost for a 5-

mile trip?

0 + 1.50(0)

Systems of Equations

How many more do you need HTTP://BOOKSTORE.XLIBRIS.COM/PRODUCTS/SKU-0112143017/DEFAULT.ASPX

NGSSS: 8th

grade

Example: Jan decided to save some money. She already had $25. She received and

saved $5 on Friday each week for 8 weeks. Make a table and a graph of the money

she would have each week. If she continues with this same savings plan, how much money will she have after 2 years?

5

$25.00 +$5(8)

5

5

5 5

5

5

8

5 5 5

4

7 6

Weeks

8 7 6 5 4 3 2 1

$ (5) 10 15 20 25 30 35 40 Savings amount per week

+25

NGSSS: 8th grade

Example: Jan decided to save some money. She already had $25. She received and

saved $5 on Friday each week for 8 weeks. Make a table and a graph of the

money she would have each week. If she continues with this same savings plan, how much money will she have after 2

years?

$25.00 +$5(8)

5

5 5 5 5

5

5 6

8

5 5 5

4

7

Lesson: Systems of Equations

" Essential Question: How can you solve a system of linear equations? " Topic: Linear Systems " Next Generation Standard: " MA.8.A.1.3 & MA.8.A.1.4

Systems of Equations

MA.8.A.1.3 } Big Idea 1 - Analyze and represent linear functions, and

solve linear equations and systems of linear equations. } Benchmark: Use tables, graphs, and models to represent,

analyze, and solve real-world problems related to systems of linear equations.

MA.8.A.1.4 } Big Idea 1 - Analyze and represent linear functions, and

solve linear equations and systems of linear equations. } Benchmark: Identify the solution to a system of linear

equations using graphs

RIGHT-BRAIN VS. LEFT-BRAIN ALGEBRA FOR BABY HOW DOES RIGHT-BRAIN VS. LEFT-BRAIN THINKING IMPACT YOUR BABY'S LEARNING FROM A CHILD DEVELOPMENTAL STAND POINT ? Curriculum–In order to be more “whole-brained” in their orientation, schools need to give equal weight to the arts, creativity, and the skills of imagination and synthesis. "Baby Algebra" is Right-Brain vs. Left-Brain application. If you have made it to this Google Blog, You are ready for Change. Welcome to Algebra Change for Baby and you

Vocabulary 2-5 " System of Linear Equations – two or more

linear equations placed together " Solution of the System of Linear Equations –

any ordered pair in a system that makes all of the equations true.

" No solution – when two lines are parallel there

are no points of intersection. Since the two lines do not share a common point, there is no solution for parallel lines.

" Infinite solutions – a system of linear

equations has no limit to the pairs of solutions

Systems of Equations " Activities:

– Think-Pair- – Share – Game Closet: Tic-Tac-Toe " http://

www.bigideasmath.com/uploads/games/game_14.pdf

Systems of Equations " Lesson Objective: To solve systems of linear

equations using three different techniques

http://www.bigideasmath.com/protected/content/ipe/grade%208/02/g8_02_05.pdf

" Vocabulary: system of linear equations, solution

of a system of linear equations " Previous Learning: In Section 1.3n 8th grade

textbook, students learned how to solve equations with variables on both sides

TIC – TAC – TOE

Y = x + 4  

Y = 3x - 1  

Y = 1.5x - 2  

Y = -x + 13  

Y = 2/3x -3  

Y = -2x + 5  

Y = 2x + 9  

Y = 6 – x  

Y = -x – 4  

Y = 3/5x + 4  

Y = 2x + 5  

Y = 1/2x - 1  

Y + x = 27  

Y = x + 3  

Y = 2x + 3  

Y = x + 8  

X – y = 7  

0.5x + y = 5  

Think – Pair - Share Setup: Six note cards with the solving method

below: (2 cards per a method) 1. Solve Using Algebra. 2. Solve Using A Graph. 3. Solve Using A Table.

The teacher will give examples to the class, such

as: A middle school yearbook committee has 35 members. There are 7 more girls than boys. Use the models to write a system of linear equations. Then solve the system to find the number of boys x and the number of girls y.

Procedure " Students will be in teams of 4. A stack of note cards

will be in each team. Each team will choose a note card from the note card pile. Each student will allow 2-3 minutes to think and solve individually with the method that is shown on their card. When time is up, the students will share and check each others work and come up with a consensus before they share with the entire team. The teacher will then pick and review answers and methods from 2 groups before moving on to another example.

Where do they go from here? " Algebra MA.912.A.3.1: 3(2x+5) = 10x-3+2x

" Geometry MA.912.G.5.4: Are these equations parallel, perpendicular, or intersecting.

y = 3 (x - 3) + 9 y = 2.1 (x + 2) - 4

" Algebra II MA.912.A.6.5 : 3 log x = 4

" Pre-Calculus MA.912.T.3.4:

Solve: 2 sin(x) +1=0 on the interval [0, 2π)

Where do I find the Next Generation Sunshine State Standards(NGSSS)?

Textbook – Front of Text

Each Section www.FloridaStandards.org – Standards by Grade level

•  Step 1 – Subject: Mathematics •  Step 2 - Grade Level: 6,7, or 8

www.bigideasmath.com – Standards listed as NGSSS correlations

•  Step 1 – Teacher Resources Tab •  Step 2 – NGSSS Correlations

Academic Plans Quarterly Content Guide – Grade 6, Regular

Quarter 1 Quarter 2 1. Writing and Evaluating Expressions 2. Properties of Addition and Multiplication 3. Distributive Property 4. Use Formulas to Solve Problems 5. Multiplying Fractions and Mixed Numbers 6. Dividing Fractions and Mixed Numbers 7. Fraction-Decimal Equivalency

1. Multiplying Decimals 2. Dividing Decimals 3. Solve Real World Problems involving Decimals 4. Compare and Order Fractions, Decimals, and Percents 5. Finding the Percent of a Number 6. Estimate Percents for Reasonableness 7. Ratio and Rates 8. Mean, Median, Mode, and Range 9. Describe, Analyze, and/or Summarize a Data Set

Quarter 3 Quarter 4 1.  Concept of Pi 2.  Circumference and Area of Circles 3.  Perimeter and Area of Composite Figures 4.  Writing Equations in One Variable 5.  Solving One and Two Step Equations 6.  Finding Missing Dimensions of Plane Figures

and Prisms 7.  Writing and Graphing Inequalities 8.  Solving One- and Two-Step Inequalities

1.  Mapping Diagrams 2.  Functions, Tables, and Graphs 3.  Analyzing Graphs 4.  FCAT Testing 5.  Fractions on a Number Line 6.  Decimals on a Number Line 7.  The Coordinate Plane

Quarterly Content Guide – Grade 6, Advanced Quarter 1 Quarter 2

1. Writing and Evaluating Expressions 2. Properties of Addition and Multiplication 3. Distributive Property 4. Use Formulas to Solve Problems 5. Multiplying Fractions and Mixed Numbers 6. Dividing Fractions and Mixed Numbers 7. Fraction-Decimal Equivalency 8. Multiplying Decimals 9. Dividing Decimals

1. Writing Equations 2. Solving One and Two Step Equations 3. Fractions-Decimals-Percents Equivalency 4. Solve Real World Problems involving Fractions, Decimals and Percents 5. The Percent Equation 6. Interest, Discounts, Sales Tax, etc 7. Ratio and Rates 8. Mean, Median, Mode, and Range 9. Describe, Analyze, and Summarize Data Set

Quarter 3 Quarter 4 1.  Concept of Pi 2.  Circumference and Area of Circles 3.  Perimeter and Area of Composite Figures 4. Finding Missing Dimensions of Plane Figures

and Prisms 5. Writing and Graphing Inequalities 6. Solving One- and Two-Step Inequalities 7. Mapping Diagrams 8.  Functions, Tables, and Graphs 9.  Analyzing Graphs

1.  FCAT Review, Reference Sheet, Gridded Response

2.  FCAT Testing 3.  The Number Line 4.  Operations with Integers 5.  The Coordinate Plane 6.  Operations with Rational Numbers 7.  Solving One- and Two-Step Equations

Quarterly Content Guide – Grade 7 Quarter 1 Quarter 2

1. Operations with Integers 2. Operations with Rational Numbers 3. Solving One- and Two-Step Equations 4. Proportionality 5. Ratios and Rates 6. Slope

1. Writing and Solving Proportions 2. Converting Measures between Systems 3. Direct and Indirect Variation 4. Percent Equation 5. Percent Applications 6. Perimeter and Area of Similar Figures 7. Scale Drawings 8. Transformations

Quarter 3 Quarter 4 1. Surface Area of Prisms, Cylinders, Pyramids and Cones 2. Surface Area of Composite Solids 3. Volume of Prisms, Cylinders, Pyramids and Cones 4. Volume of Composite Solids 5. Stem-and-Leaf Plots 6. Histograms 7. Circle graphs 8. Samples and Populations

1. Theoretical and Experimental Probability 2. Independent and Dependent Events 3. FCAT 4. Exploring Equations and Functions 5. Solving Multi-Step Equations 6. Solving Equations with Variables on Both Sides 7. Extension of Slope 8. Linear Functions 9. My Big Ideas Projects (Optional)

Quarterly Content Guide – Grade 7 Advanced Quarter 1 Quarter 2

1. Review: Operations with Integers 2. The Coordinate Plane 3. Review: Rational Numbers and Operations with

Rational Numbers 4. Solving Multi-Step Equations 5. Solving Equations with Variables on Both Sides 6. Review: Ratios and Rates 7. Slope 8. Proportions 9. Linear Functions 10. Writing Equations Using Two Points 11. Converting Units of Measure 12. Direct and Inverse Variation 13. Domain and Range of a Function

1. The Percent Equation 2. Percent Applications 3. Simple Interest 4. Classifying Angles 5. Angles and Sides of Triangles 6. Angles of Polygons 7. Identifying Similar Figures 8. Perimeters and Areas of Similar Figures 9. Using Similar Triangles 10. Scale Drawings 11. Transformations 12. Parallel Lines and Transversals 13. Surface Area of Prisms and Cylinders

Quarter 3 Quarter 4

1. Surface Area of Pyramids and Cones 2. Surface Area of Composite Solids 3. Volumes of Prisms, Pyramids, Cylinders and Cones 4. Volumes of Composite Solids 5. Surface Area and Volumes of Similar Solids 6. Measures of Central Tendency 7. Stem-and-Leaf Plots 8. Histograms 9. Circle Graphs 10.Box-and-Whisker Plots 11.Scatter Plots and Lines of Best Fits

1. Introduction to Probability 2. Theoretical and Experimental Probability 3. Independent and Dependent Events 4. FCAT Testing 5. Finding Square Roots 6. The Pythagorean Theorem 7. Approximating Square Roots 8. Simplifying Square Roots 9. Writing, Graphing, and Solving Inequalities 10. Exponents and Powers Properties 11. Scientific Notation

Quarterly Content Guide – Grade 8 Quarter 1 Quarter 2

1. Solving Simple Equations 2. Solving Multi-Step Equations 3. Solving Equations with Variables on Both Sides 4. Rewriting Equations and Formula 5. Converting between US and Metric Systems 6. Graphing Linear Equations with Tables 7. Finding the Slope of a Line 8. Graphing Linear Equations in Slope-Intercept Form 9. Graphing Linear Equations in Standard Form 10. Solving Systems of Equations

1. Writing Equations in Slope-Intercept Form 2. Writing Equations using Point-Slope Form 3. Writing Systems of Linear Equations 4. Domain and Range of a Function 5. Discrete and Continuous Domains 6. Linear Function Patterns 7. Comparing Linear and Nonlinear Functions 8. Classifying Angles 9. Identifying Angles and Sides of Triangles 10. Angles of Polygons 11. Using Similar Triangles 12. Parallel Lines and Transversals

Quarter 3 Quarter 4 1. Approximating and Finding Square Roots 2. Using the Pythagorean Theorem 3. Simplifying Square Roots 4. Finding Measures of Central Tendency 5. Box-and-Whisker Plots 6. Scatter Plots 7. Choosing a Data Display 8. Writing and Graphing Simple Inequalities 9. Solving One-Step Inequalities 10.Solving Multi-step Inequalities 11.Using Exponents 12. Multiplying and Dividing With Exponents 13. Zero and Negative Exponents

1. Reading Scientific Notation 2. Writing Scientific Notation 3. FCAT 3. Financial Literacy: Simple and Compound Interest 4. Financial Literacy: Formula for Compound Interest 5. Financial Literacy: Installment Loans 6. Financial Literacy: Checking Accounts 7. Financial Literacy: Credit Cards 8. Financial Literacy: Payroll Deductions 9. Big Ideas Projects (optional)

Quarterly Content Guide – Grade 8, Advanced Quarter 1 Quarter 2

1. Converting Measures between Systems 2. Adding and Subtracting Fractions 3. Solving Multi-Step Equations 4. Solving Equations with Variables on Both Sides 5. Order of Operations 6. Graphing Linear Equations and Linear Systems 7. Slope of a Line 8. Graphing Linear Equations in Slope-Intercept Form 9. Graphing Linear Equations in Standard Form 10. Solving Systems of Equations

1. Discrete and Continuous Domains 2. Linear and Non-Linear Function 3. Classifying Angles and Sides of Triangles 4. Angles of Polygons Using Similar Triangle 5.  Parallel Lines and Transversals 6. Review: Comparing Decimals; Order of Operations 7.  Square Roots 8. The Pythagorean Theorem 9.  Measures of Central Tendency 10. Box-and-Whisker Plots 11. Scatter Plots 12. Lines of Best Fit

Quarter 3 Quarter 4 1. Graphing and Writing Inequalities 2. Solving One, Two and Multi-Step Inequalities 3. Review: Operations with Decimals 4. Exponents 5. Product of Powers Property 6. Quotient of Powers Property 7. Zero and Negative Exponents 8. and Writing Scientific Notation 9. Unit 4 Project

1.  Properties of Real Numbers 2.  Simplifying Expressions 3.  FCAT 4.  Adding and Subtracting Polynomials 5.  Multiplying Binomials 6.  Special Products of Polynomials 7.  Factoring Polynomials 8.  Factoring Special Products

Food For Thought " Liping Ma (1999) used this exact problem with

teachers in the U.S. and in China.

" Of 23 U.S. teachers: –  6 could not create a story problem –  16 created story problems with misconceptions

" -Confusing dividing by 1/2 with dividing by 2 " -Confusing dividing by 1/2 with multiplying by 1/2

–  1 provided a conceptually correct representation, but it contained pedagogical problems " -Fractional number of children

" 90% of Chinese teachers created a correct representation

Thank You!