Teaching to the Teaching to the Next Generation SSSNext Generation SSS

(2007)(2007)

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

In today’s training, we will…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 grade levels have learned

Comparison of StandardsComparison of Standards

Grade LevelGrade Level OldOld GLE’s GLE’s New New BenchmarksBenchmarks

KK 6767 1111

11stst 7878 1414

22ndnd 8484 2121

33rdrd 8888 1717

44thth 8989 2121

55thth 7777 2323

66thth 7878 1919

77thth 8989 2222

88thth 9393 1919

How do we make sense of teaching deeply?How do we make sense of teaching deeply?

Provide a “focused” curriculum.Provide a “focused” curriculum.Collaboration of vertical and Collaboration of vertical and horizontal teachinghorizontal teachingExplore deeper content-based Explore deeper content-based knowledge. Think of a swimming knowledge. Think of a swimming pool. Most people stay in the pool. Most people stay in the shallow end…we now need to shallow end…we now need to venture to the unknown depths venture to the unknown depths of knowledge.of knowledge.Develop connections from 6Develop connections from 6thth grade math through pre-calculusgrade math through pre-calculusEmphasis on higher level Emphasis on higher level thinking- analysis, evaluation thinking- analysis, evaluation and synthesisand synthesis

Equations as an example standard linked Equations as an example standard linked through the grade levelsthrough the grade levels

Fair game review through models and Fair game review through models and visualsvisuals

Use cooperative learning strategies to Use cooperative learning strategies to introduce or remediate equationsintroduce or remediate equations

Represent equations using models, Represent equations using models, vocabulary, pictures, and real world vocabulary, pictures, and real world situationssituations

Lead students to mastery level not Lead students to mastery level not memorizationmemorization

Teaching For Depth

What is Rigor?

What Rigor is Not:What Rigor is Not:a measure of the quantity of a measure of the quantity of

content to be covered.content to be covered.

a special program or curriculum a special program or curriculum

for select students.for select students.

about severity or hardship.about severity or hardship.

only about higher-order thinking.only about higher-order thinking.

RigorRigor

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

Eric Bergholm, Hillsborough County Public Schools, Florida

MA.MA. 6.6. A.A. 3.3. 11SubjectSubject GradeGrade

LevelLevelBody of Body of

KnowledgeKnowledgeBig Idea/ Big Idea/

Supporting Supporting IdeaIdea

BenchmarkBenchmark

Coding Coding Scheme for NGSSSScheme for NGSSS

MA.6.A.3.1

Identify the threading Identify the threading standards for solving standards for solving

equations.equations.

Grade Level/CourseGrade Level/Course BenchmarkBenchmark

44Describe mathematical relationships using Describe mathematical relationships using

expressions, equations, and visual expressions, equations, and visual representations.representations.

66Write solve and graph one- and two-step linear Write solve and graph one- and two-step linear

equations and inequalities.equations and inequalities.

77Formulate and use different strategies to solve Formulate and use different strategies to solve

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

88Use tables, graphs, and models to represent, Use tables, graphs, and models to represent,

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

Algebra 1Algebra 1Solve and graph simple and compound inequalities in Solve and graph simple and compound inequalities in

one variable and be able to justify each step in a one variable and be able to justify each step in a solution. solution.

GeometryGeometryUse coordinate geometry to find slopes, Use coordinate geometry to find slopes,

perpendicular lines, parallel lines and equations perpendicular lines, parallel lines and equations of lines.of lines.

Algebra 2Algebra 2 Solve logarithmic and exponential equations.Solve logarithmic and exponential equations.

Pre-CalculusPre-CalculusSolve trigonometric equations and real-world Solve trigonometric equations and real-world

problems involving applications of trigonometric problems involving applications of trigonometric equations using technology when appropriate.equations using technology when appropriate.

The Common Thread

What are the What are the NCTM Process Standards?NCTM Process Standards?

– Problem SolvingProblem Solving

– Reasoning and ProofReasoning and Proof

– ConnectionsConnections

– CommunicationCommunication

– RepresentationRepresentation

12

NCTM Process StandardsNCTM Process Standards

Problem SolvingProblem Solving– Developing perseverance and critical Developing perseverance and critical

thinkingthinking– Allow students think time to reach a solutionAllow students think time to reach a solution

Reasoning and ProofReasoning and Proof– Mathematical conjecturesMathematical conjectures– Examples and counter examplesExamples and counter examples

CommunicationCommunication– Read, write, listen, think, and discussRead, write, listen, think, and discuss– Increase the use of appropriate math Increase the use of appropriate math

vocabularyvocabulary

NCTM Process StandardsNCTM Process Standards

ConnectionsConnections– Integers, expressions, and equations Integers, expressions, and equations – Other content areas, scienceOther content areas, science– Real-world contextsReal-world contexts

RepresentationRepresentation– Useful tools for building understandingUseful tools for building understanding– Concrete - Representational - AbstractConcrete - Representational - Abstract– Tables, describe in words, draw a Tables, describe in words, draw a

picture, write and solve equationspicture, write and solve equations

Focus on Equations!Focus on Equations!

66thth Grade Equation Grade Equation StrategiesStrategies

Solving one-step Solving one-step

algebraic equationsalgebraic equations

NGSSS: Equations(6NGSSS: Equations(6thth))

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

MA.6.A.3.2

STRATEGIESSTRATEGIES

Dynamic Classroom Interactive Dynamic Classroom Interactive Scale--Big Ideas websiteScale--Big Ideas website

Human EquationsHuman Equations

Algebra TilesAlgebra Tiles

Magic SquareMagic Square

Dynamic ClassroomDynamic Classroom Interactive ScaleInteractive Scale

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

Benefits: Benefits: It allows the students to It allows the students to enhance computer skills and provides a enhance computer skills and provides a high interest activity with graphics. high interest activity with graphics.

Human EquationsHuman Equations

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

Benefits:Benefits: Allows students to Allows students to cooperate in a group activity while cooperate in a group activity while moving throughout the room .moving throughout the room .

How do I do this?How do I do this?Students are given cards with numbers, Students are given cards with numbers, variables and symbols on them.variables and symbols on them.

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

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

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

Algebra TilesAlgebra Tiles

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

Benefits:Benefits: Students see positive and Students see positive and negative integers in colors. It allows negative integers in colors. It allows them to see constants and variables them to see constants and variables in different shapes.in different shapes.

Algebra Tile ExampleAlgebra Tile Example

Try x – 4 = 8.

Magic SquaresMagic Squares

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

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

Magic Square…Your Turn!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: NGSSS: 77thth grade gradeMA.7.A.3.3MA.7.A.3.3

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

mile trip?mile trip?

Systems of EquationsSystems of Equations

Big Ideas Math TextbookBig Ideas Math Textbook

Aligned to the Aligned to the Next Generation SSSNext Generation SSSwww.floridastandards.orgwww.floridastandards.org

NGSSS: NGSSS: 88thth grade grade

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

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

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

years?years?

Lesson: Systems of EquationsLesson: Systems of Equations

Essential QuestionEssential Question: How can you : How can you solve a system of linear equations?solve a system of linear equations?

Chapter: 2Chapter: 2

Section: 5Section: 5

Topic: Linear SystemsTopic: Linear Systems

Next Generation Standard : MA.8.A.1.3 & Next Generation Standard : MA.8.A.1.3 & MA.8.A.1.4MA.8.A.1.4

Systems of EquationsSystems of EquationsMA.8.A.1.3MA.8.A.1.3Big Idea 1 Big Idea 1 - Analyze and represent linear functions, - Analyze and represent linear functions,

and solve linear equations and systems of linear and solve linear equations and systems of linear equations.equations.

Benchmark: Benchmark: Use tables, graphs, and models to Use tables, graphs, and models to represent, analyze, and solve real-world problems represent, analyze, and solve real-world problems related to systems of linear equations.  related to systems of linear equations.  

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

and solve linear equations and systems of linear and solve linear equations and systems of linear equations.equations.

Benchmark:Benchmark: Identify the solution to a system of Identify the solution to a system of linear equations using graphslinear equations using graphs

Vocabulary 2-5Vocabulary 2-5

System of Linear Equations System of Linear Equations – two or more linear – two or more linear equations placed togetherequations placed together

Solution of the System of Linear Equations Solution of the System of Linear Equations – – any ordered pair in a system that makes all of any ordered pair in a system that makes all of the equations true.the equations true.

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

Infinite solutions Infinite solutions – a system of linear equations – a system of linear equations has no limit to the pairs of solutionshas no limit to the pairs of solutions

Systems of EquationsSystems of Equations

Activities:Activities:

– Taking Math Deeper TE pg. T-81Taking Math Deeper TE pg. T-81Incorporates Kagan strategy, Think-Pair-Share Incorporates Kagan strategy, Think-Pair-Share

– Game Closet: Tic-Tac-ToeGame Closet: Tic-Tac-Toehttp://www.bigideasmath.com/uploads/gameshttp://www.bigideasmath.com/uploads/games/game_14.pdf/game_14.pdf

Lesson Objective: Lesson Objective: To solve systems of linear To solve systems of linear equations using three different techniquesequations using three different techniques

http://www.bigideasmath.com/protected/content/ipe/grade%20http://www.bigideasmath.com/protected/content/ipe/grade%208/02/g8_02_05.pdf8/02/g8_02_05.pdf

Vocabulary: Vocabulary: system of linear equations, solution system of linear equations, solution of a system of linear equationsof a system of linear equations

Previous Learning: Previous Learning: In Section 1.3n 8In Section 1.3n 8thth grade grade textbook, students learned how to solve textbook, students learned how to solve equations with variables on both sidesequations with variables on both sides

Y = x + 4Y = x + 4Y = 3x - 1Y = 3x - 1

Y = 1.5x - 2Y = 1.5x - 2Y = -x + 13Y = -x + 13

Y = 2/3x -3Y = 2/3x -3Y = -2x + 5Y = -2x + 5

Y = 2x + 9Y = 2x + 9Y = 6 – xY = 6 – x

Y = -x – 4Y = -x – 4Y = 3/5x + 4Y = 3/5x + 4

Y = 2x + 5Y = 2x + 5Y = 1/2x - 1Y = 1/2x - 1

Y + x = 27Y + x = 27Y = x + 3Y = x + 3

Y = 2x + 3Y = 2x + 3Y = x + 8Y = x + 8

X – y = 7X – y = 70.5x + y = 50.5x + y = 5

Think – Pair - ShareThink – Pair - Share

Setup: Six note cards with the solving methods Setup: Six note cards with the solving methods

below: (2 cards per a method)below: (2 cards per a method)

1. Solve Using Algebra.1. Solve Using Algebra.

2. Solve Using A Graph.2. Solve Using A Graph.

3. Solve Using A Table.3. Solve Using A Table.

The teacher will give examples to the class, suchThe teacher will give examples to the class, such

as: as: A middle school yearbook committee has 35A middle school yearbook committee has 35

members. There are 7 more girls than boys. Usemembers. There are 7 more girls than boys. Use

the models to write a system of linear equations.the models to write a system of linear equations.

Then solve the system to find the number of boyThen solve the system to find the number of boy

x and the number of girls y.x and the number of girls y.

ProcedureProcedureStudents will be in teams of 4. A stack of note cards Students will be in teams of 4. A stack of note cards will be in each team. Each team will choose a note will be in each team. Each team will choose a note card from the note card pile. Each student will allow card from the note card pile. Each student will allow 2-3 minutes to think and solve individually with the 2-3 minutes to think and solve individually with the method that is shown on their card. When time is up, method that is shown on their card. When time is up, the students will share and check each others work the students will share and check each others work and come up with a consensus before they share with and come up with a consensus before they share with the entire team. The teacher will then pick and review the entire team. The teacher will then pick and review answers and methods from 2 groups before moving on answers and methods from 2 groups before moving on to another example.to another example.

Where do they go from here?Where do they go from here?

AlgebraAlgebra MA.912.A.3.1 :MA.912.A.3.1 : 3(2x+5) = 10x-3(2x+5) = 10x-3+2x3+2x

GeometryGeometry MA.912.G.5.4 : Are these MA.912.G.5.4 : Are these equations parallel, perpendicular, or intersectingequations parallel, perpendicular, or intersecting.

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

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

Pre-CalculusPre-Calculus MA.912.T.3.4 : MA.912.T.3.4 : Solve 2 sin(x) +1=0 on Solve 2 sin(x) +1=0 on the interval [0, 2the interval [0, 2))

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

What does the School District of Lee County mandate I teach and when?

You will need to check your Academic Plan by grade leveland subject (intensive, regular, advanced).

Go to… - Learn Page- District Resources - Academic Plans

- Click on Navigation Bar on left- Scroll down to your grade/subject and click

Once the plan is loaded, you have three choices:• 1st Tab – Quarterly Guides• 2nd Tab – Complete Plan• 3rd Tab – Vocabulary Key Terms (also

located on Bigideasmath.com, Teacher Resources, Glossary flash cards)

What Materials will I receive? - Teacher Edition - Student Textbook - Record and Practice Journal and Answer Key - Advance Topics Book for Advanced Courses (store extras for next year – not consumable!) - Assessment Book (6th and 7th)

- Online Resources- Math Resources by Chapter TE book

Available on CD Rom - TE CD - Lesson Planning Tools - Lesson Tutorials - Dynamic Assessment - Dynamic Teaching - Editable Ancillary can be edited from CD

What do I do if a student is absent or needs extra help?

- Have students go to www.bigideasmath.com

- Student Resources

Lesson Tutorial (podcast of lesson)

Graphic Organizer

Progress Check

Vocabulary Check

Chapter Test

What can Parents do to help students at home?

Go to www.BigIdeasmath.com and click on Parent tab

Help with Homework (English, Spanish, Haitian-Creole)

Parent Letters

Basic Skills Handbook

Skills Review Handbook

Planning Time!

- Get with your grade level / subject counterpart

- Using resources you have been given,talk about and plan a section from Chapter 1

Curriculum & Staff Development CenterCurriculum & Staff Development CenterRequires a Course Evaluation!Requires a Course Evaluation!

Don’t forget to complete your Don’t forget to complete your online evaluation!online evaluation!

Thank You!

Food For ThoughtFood For ThoughtLiping Ma (1999) used this exact problem with Liping Ma (1999) used this exact problem with teachers in the U.S. and in China. teachers in the U.S. and in China.

Of 23 U.S. teachers:Of 23 U.S. teachers:– 6 could not create a story problem6 could not create a story problem– 16 created story problems with misconceptions16 created story problems with misconceptions

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

– 1 provided a conceptually correct representation, 1 provided a conceptually correct representation, but it contained pedagogical problemsbut it contained pedagogical problems

-Fractional number of children-Fractional number of children

90% of Chinese teachers created a correct 90% of Chinese teachers created a correct representationrepresentation