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“The definition of insanity is doing the same thing over and over again and expecting different results.” - Albert Einstein PowerPoint PPT Presentation


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“The definition of insanity is doing the same thing over and over again and expecting different results.” - Albert Einstein Presenter: Tony Hollowell [email protected] The 80/20 Principle. 20% of the inputs cause 80% of the outputs

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“The definition of insanity is doing the same thing over and over again and expecting different results.” - Albert Einstein

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“The definition of insanity is doing the

same thing over and over again and

expecting different results.”

- Albert Einstein

Presenter: Tony Hollowell

[email protected]


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The 80/20 Principle

  • 20% of the inputs cause 80% of the outputs

  • “The trivial many vs. the critical few”

    • 20% of motorists cause 80% of accidents

    • 20% of your carpet gets 80% of the wear

    • 20% of the population has 80% of the wealth

    • 20% of a businesses’ customers determine 80% of the total profits


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What are the “critical few” for the Core 40 Exam?

  • Write down all the things your students do in a given year in the classroom

    • Tasks, projects, activities, etc.


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The “Critical Few” for the Core 40 Exam

  • Example Problems

    • Students do not need to do more problems. They need to do more specific types of problems.

    • Quality and consistency are important, not quantity

  • Graphing Calculator Activities/Enrichment

  • Assessments (summative)


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Graphing Calculators in the State Standards


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What I’m Proposing (and what I did in my own classroom)

  • Students should have daily/weekly access to graphing calculators in an Algebra I classroom


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Why introduce graphing calculators in Algebra I?

  • Preparation for future math courses and assessments (AP Calc, SAT, ACT)

  • Benefits

    • Reduction of computational errors

    • Real data analysis

    • Hands-on interaction with the content

    • Multiple representations of function

    • They are allowed on the exam!!


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The Fundamental Law of Graphing Calculators…

  • Graphing Calculators cannot think for the student. They can only supplement what they already know.

  • Corollary to the Fundamental Law of Graphing Calculators: Your students need to know Algebra for a graphing calculator to be useful


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Introduction to Graphing Calculators

  • Graphing Calculator Scavenger Hunt


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Fundamental Law of Technology

  • My students, as a whole, will always know more about technology than I do individually


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Graphing Calculator Activities

  • Three major content strands today

    • Linear Equations

    • Systems of Linear Equations

    • Quadratics

    • Work on them individually or with a partner


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Break!!

  • Questionnaire


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Discussion of the Activities

  • Build critical thinking

  • Involve real data

  • Can be used to cover ALL major content strands in Algebra I

  • Easy to make!

    • NCTM Illuminations:

    • TI Activities Exchange:


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Core 40 Questions with a Graphing Calculator

  • What you may do to solve these problems may not be the same as someone else

  • The calculator is only a tool (usually just helps check an answer)


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Graphing a Line

  • Using equations that are in slope-intercept form

    • Ex: Graph the function y = -2x – 3

  • Corollary: learn to get equations in slope-intercept form!


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Finding Intercepts

  • Find the x and y intercepts of the equation 2y + x = 4


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Slope through two points

  • Write the complete equation of a line that goes through the two points (-3,4) and

    (-5,10)


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Systems of Equations

  • Which value of b will make the graphs

    y = -2x +1 and y = x + b intersect at the point (-2, 5)?

  • -2

  • 7

  • 5

  • 2


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Exponents

  • Simplify the expression


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Quadratics

  • Which of the following are solutions of the equation (t + 6)(2t – 9)= 0 ?

    I. t = 5 II. t = -6III. t = 4½

  • I

  • II

  • III

  • II and III


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Quadratics

  • How many solutions are there to the equation x2 + 8 = -3x?


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Authentic Testing Environment

  • Creating tests that mimic the expectations on the Core 40 exam

    • Allow formula sheet

    • Allow graphing calculators

    • Timed (50 minutes)

    • Use of computers depends on what your students will do when they take the ECA

  • How often is this feasible at your school?


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Building Assessments that remove Graphing Calculator dependency

  • Students should learn how to think without their calculator

    • SAT and ACT allow for a graphing calculator throughout the entire exam

    • the AP calc exam does not.

    • Enforces the idea that technology is only a tool


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Building Assessments that remove Graphing Calculator dependency

  • Graphing Calculators are allowed on almost every assessment (mine and standardized tests)

  • Test items are designed KNOWING that graphing calculators are available

  • Requires some creativity (usually the inclusion of variables, but not always)


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Building Assessments that remove Graphing Calculator dependency

  • Standard A1.4.2

    • Find the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line.

  • Construct a question for this standard that will assess student comprehension irregardless of the use of a graphing calculator.


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A note on critical thinking

  • If we do not teach critical thinking, our country is in trouble.

  • Any skill that is downloadable, wikipediable or googleable is not a skill that will provide economic viability

  • Teaching memorization is easy, teaching critical thinking is hard.


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A note on critical thinking

  • Content should ALWAYS be subservient to critical thinking/analysis

  • This is nothing “new”

  • “The ‘missing link’ in all systems of education today may be found in the failure of educational institutions to teach their students how organize and use knowledge after they acquire it.”

    -Napoleon Hill, Think and Grow Rich (published in 1920)


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Assessment Construction

  • Review the assessments in the handouts

  • Do you think allowing a graphing calculator for these problems will cause calculator dependency?


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Implementation

  • A year-long process

  • Which type of graphing calculators to use?

  • How to get funding?

    • First, you need to be convinced of their use and effectiveness


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80/20 analysis revisited

  • If the “critical few” is only 20% of your time, what do you do with the rest of your time?

  • Get Creative! Play Games! Analyze! Do Projects!

  • Side Note: If what you are doing is working, you don’t need to change much


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Stop the insanity. Do something different.

  • “The definition of insanity is doing the same thing over and over again and expecting different results.”

    - Albert Einstein


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Transforming your classroom

  • Algebra vs. the Cockroaches

  • Finger painting and solving equations

  • Colored pencil math

  • 36 cards and puzzles

  • If you think of something but you are not sure if it will work, just imagine handing them a worksheet.

  • REFUSE to do things that are ineffective


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My “Critical Few”

  • Example Problems

    • Students do not need to do more problems. They need to do more specific types of problems.

    • Quality and consistency are important, not quantity

  • Graphing Calculator Activities/Enrichment

  • Assessments (summative)


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  • "It is not the critic who counts: not the man who points out howthe strong man stumbles or where the doer of deeds could have done better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood, who strives valiantly, who errs and comes up short again and again, because there is no effort without error or shortcoming, but who knows the great enthusiasms, the great devotions, who spends himself for a worthy cause; who, at the best, knows, in the end, the triumph of high achievement, and who, at the worst, if he fails, at least he fails while daring greatly, so that his place shall never be with those cold and timid souls who knew neither victory nor defeat."

    –Teddy Roosevelt


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A note on leadership

  • Leadership is not a position

  • There is a difference between “being in charge” and “leading”


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Remember…

  • Your students are not a number

  • The Core 40 Exam cannot measure every important skill our students need to be successful in the world


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