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Algebra for Precalculus

and Calculus Students

Mindful Manipulation

Philip Cheifetz, Ellen Schmierer

Nassau CC

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1960-1970s New Math Viet Nam War Deferments Open Enrollment: less talented student pool

1980s Graphing Calculators High salaries for computer science and

engineering grads attract mathematics majors Less talented mathematics teachers

“Recent” Events

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“Recent” Events1990s Symbolic manipulators TIMMS Report (1995)

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Really Recent Events2000s No Student Left Behind = Lowering of Standards Rescaling SATs 2007 NYS 9th Grade Algebra Exam

44% = passing 2008 NYS 9th Grade Algebra Exam

34% = passing 50% of all elementary & middle school teachers take

ALL the mathematics they will ever take at a TYC.

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Who are some of the students we teach? (A true story)

Student:  "I need to find my math textbook." Librarian: "Sure, which course are you taking?" Student: "Premeditated math." Librarian: " Premeditated math?" Student: "On no, that's not it…. I know, it's

primordial math!" Librarian: "Primordial math?  Could you possibly

mean remedial math?" Student: "That's it!!! ( smiling brightly). Remedial

Math 001. That's it!"

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Recognition After a Unit on Number Systems:

There are 10 kinds of mathematicians. Those who can think in binary and those who can't...

NO recognition

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Mistakes My Students Truly Made Last

Month in Calc II

2 2x y 3 3t t dt t dtt

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3 2x x3 1 12 2 23 3dtt t t dt

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Mistakes My Students Truly Made Last Month in Calc II

2 212 1

12 1

t tdt dt

t t

6 4 2cos cos cosx xdx xdx

3 3

0 0 0

t tt e dt t dt e dt

3 3

0 0

t tt e dt te dt

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Mistakes My Students Truly Made Last Month in Calc II

2 2sin 5 5 sinxdx x dx

3arctan arctan

3 3

w w

6 136 1

3

x

x

29 1

10( 9) 1

zdz dz

zz

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Issues My Students Routinely Have Not knowing when to distribute (when to, UGH,

“FOIL”.) GET RID OF FOIL!!

Pattern recognition

Reading tables of integrals

Copying what I write rather than what the problem solving strategy is

Refusing to think

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Mistakes My Students Routinely Make Everything is commutative Everything is distributive Things behave the way we want them to as in:

(fg)’=f ’g’ since it “works” with sums Everything is linear: f(a+b) = f(a)+f(b) Canceling Fractions I’m ANGRY. Who Is doing the teaching, and

what is being taught??

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Mistakes with Cancelling My Students Routinely Make (A Joke)

1sin ?

1sin

6

xn

xnsix

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The chef instructs his apprentice: "You take two thirds of water, one third of

cream, one third of broth..."

The apprentice: "But that makes four thirds already!“

"Well - just take a larger pot!"

Mistakes with Fractions My Students Routinely Make (A Joke)

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Mistakes People Make (NOT A Joke)

Item costs $200

There is a 40% discount

Then…….

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Teacher: "Who can tell me what 7 times 6 is?"

Student: "It's 42!"

Teacher: "Very good! - And who can tell me what 6 times 7 is?"

Same student: "It's 24!"

Mistakes with Commutativity Young Students Make (Not a Joke)

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UNFORTUNATELY,

Students truly believe, and indeed expect, that if they come to class, they will receive a passing grade and, more likely than not, the grade will be a C or better.

Students believe that coming to class is their sum responsibility as students.

The vast majority of students do no homework, ever.

Students have come to expect long review sessions before an exam, and that the exam will contain only the problems covered in the review, with altered constants.

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Mediocrity is now acceptable and even encouraged.

Partial credit for mostly incorrect work encourages mediocrity.

Having a cursory knowledge of a topic is considered sufficient.

Multi-step reasoning is an unfair expectation.

UNFORTUNATELY,

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UNFORTUNATELY,

Students do not have a number sense.

Therefore, they do not have an algebra sense,

Nor do they have the sense to ask “Does the answer seem reasonable?”

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“No student left behind” has produced expectations aimed at the lowest common denominator. Skills have become so weak that examinations have had to be rescaled.

Making students and parents feel good is more important than setting realistic goals and limits.

The concept of “on time” is very loosely applied.

Better: Each child to his/her maximum potential

The Fall of Rome

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If a student does poorly on an exam, it is deemed the fault of the exam.

If a student fails to live up to expectations, it is deemed the fault of the teacher.

A college degree today is significantly different from such a degree of 25 years ago.

Many experienced, well-trained teachers are being replaced by faculty who are weaker.

I’m Angry!

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A non-trivial number of high school mathematics teachers and/or supervisors:

do not have an undergraduate degree that has

the word “mathematics” in it, or, do not have an graduate degree that has the

word “mathematics” in it, or, have neither an undergraduate nor graduate

degree that has the word “mathematics” in it.

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What to do?

We are urged to have higher retention rates! I, for one, chose to be part of the cure. No student of mine will ever be asked by a

professor “Who did you have in the previous course?”

Visit www.matcmp.sunynassau.edu/

~cheifp/Itseemstome1.htm

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Which of the following three questions would you deem most important for your algebra students?

What algebraic law would you use to solve the equation 3(x + 5) = 27?

For what value of x is equal to zero?

Use an algebraic expression to prove or disprove that the sum of any three consecutive integers is a multiple of three.

7( 3)

2 1

x

x

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In the following expressions, a and x are positive. What is the effect of increasing a on the value of each expression?

1. 1ax

2. x a

3. x a

4. 1x

a

15. x

a

16. ax

a

7. (2 )a x a

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Without solving, decide whether the following equations have a positive solution, a negative solution, a zero solution, or no solution.

1. 4

32

w

2. 2 5

13 5

x

x

3. 3 4 6 4t t

4. 1

3 4z

5. 1 1

2 1 2 1

x x

x x

6. 2 5

43 5

x

x

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A car trip costs $1.50 per fifteen miles for gas and 30¢ per mile for other expenses, plus $20 for car rental. The total cost for the trip is given

by the equation Cost 1.5 0.3 2015

dd

1. Explain what each term represents in terms of the trip.

2. What units for cost and distance are being used?

3. Is the equation linear?

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The number of people who attend a concert is 160 – p when the price of a ticket is $p.

What is the practical interpretation of the 160? Why is it reasonable that the p term has a negative sign? The number of people who attend a movie at ticket price $p

is 175 – p. If tickets are the same price, does the concert or the movie draw a larger audience?

The number of people who attend a dance recital at ticket price $p is 160 – 2p. If tickets are the same price, does the concert or the dance recital draw a larger audience?

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A company uses two different sized trucks to deliver sand. The first truck can transport x cubic yards and the second y cubic yards. The first truck makes S trips to the job site and the second makes T trips. What do the following expressions represent in practical terms?

1. S T 2. x y 3. xS yT

4. ( )

( )

xS yT

S T

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You plan to drive 300 miles at 55 miles per hour, stopping for a two-hour rest. You want to know t, the number of hours the journey will take. Which of the following equations would you use?

1. 55 190t 2. 55 2 300t 3. 55( 2) 300t 4. 55( 2) 300t

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Precalculus: Fostering a deeper understanding

What is it? For whom? What topics do we need in such a course?

Calculus bound?Liberal arts?

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Understanding the information a graph imparts

I.

III.

II.

a. Two real solutions

b. One real solution

c. No real solutions

Match each graph with a description of its zeros

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Function Notation If f (x) = 5 then f (2) = 10

If f (x) = 5 then 2f (x) = 10

If f (x) = 6 then f (2x) = 12

f (a + b) = f (a) + f (b)

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Reading and Interpreting Graphs

If f (x) = h(x) – g(x) then the y-intercept of f is at

a. 0 b. 1 c. 2 d. 3 e. none of these

g(x)

h(x)

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Reading and Interpreting Graphs

If f (x) = h(x) – g(x) then the x-intercept of f is at

a. 0 b. 1 c. 2 d. 3 e. none of these

g(x)

h(x)

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Deeper Understanding of Symbols

In the equation 3x + 2y = 9, if x increases by 2 then y

a. Increases by 2

b. Decreases by 6

c. Increases by 3

d. Increases by 6

e. Decreases by 3

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Understanding Exponential Growth

Match the equation to the description

a. Astronomical growth

b. A 6% increasing growth rate

c. A 6% decay rate

d. Quickly approaches zero

A. 1000 1.06

B. 2(0.94)

C. 0.5(106)

. (0.06)

t

t

t

t

P

P

P

D P

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How to Solve it?Classify each equation as

0.03

0.06

0.055

a. 2.5 3.7

b. 2.5 7.5

c. 2 5d.80(1.08) 40e.80(1.08) 40f. ln(2 1)g. ln(0.05 1) 7.5h. log 4

t

t

t

t

e

e

e t

x xx

x

a. Solvable using the exponential function.b. Solvable using the log functionc. Not solvable using the log function or the

exponential function but a solution existsd. No real number is a solution

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Reading Tables

Which table shows an increasing function? a. t 0 24 48 72

h (t ) 75 75 75 75

b. t 4 3 2 1r (t ) 5 10 15 20

c. t 0 2 4 8s (t ) -25 -20 -15 -10

d. t 0 -24 -48 -72m (t ) 5 10 15 20

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Reading Tables

Which of the following tables shows data from an invertible function?

a. t 0 24 48 72h (t ) 75 75 75 75

b. t 4 3 2 1r (t ) 5 10 15 20

c. t 0 2 4 8s (t ) 25 20 -15 -10

d. t 0 -24 -48 -72m (t ) 5 10 5 20

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The Relationship Between Equations and Their Graphs

Which of the following are possible equations for the graph below?

d

a b c

2

2

2

a.

b. ( )

c. ( )( )

d.

e. None of these

y x b d

y x a d

y x a x c

y ax bx c

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In Closing: Where are we, as a mathematics

community, going? What are we going to do about it? Be an ambassador for common sense Talk to anyone you can about the sorry

state of education in the US and in particular, about improving mathematical reasoning

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Thank you for coming

Phil: philip.cheifetz@ncc.edu

Ellen: ellen.schmierer@ncc.edu