Your Students…Where Are They Headed?

What Do They Need?(with your help)

Jeff MorganChair, Department of Mathematics

Director, Center for Academic Support and Assessment

University of Houston

Shameless Advertising

• High School Mathematics Contesthttp://mathcontest.uh.edu

• Houston Area Calculus Teachers Association AP Calculus Workshopshttp://www.HoustonACT.org

• EatMath Algebra I Teacher Workshopshttp://www.EatMath.org

(continued)

• Online Practice AP Testing in Calculus and Statistics (info at http://www.HoustonACT.org)

• Online Practice Algebra II EOC Testing(Fall 2011) – email bekki@math.uh.edu

• Online Help Materials (see http://online.math.uh.edu)

• teachHOUSTON

Technology Tool Tips

• PDF Annotator• Mimio Notebook• Geogebra• Google SketchUp• WinPlot• Bamboo Tablet

Freshmen Level Mathematics at UH(a snapshot)

Freshmen Math Enrollment

Allison

Calculus I, II, III Enrollment

Calc I

Calc II

Calc III

Enrollment Below Calculus

0

500

1000

1500

2000

2500

3000

F96 F97 F98 F99 F00 F01 F02 F03 F04 F05 F06 F07 F08 F09 F10

1300 Fundamentals of Mathematics

1310 College Algebra

1311 Elementary Math. Modeling

1312 Intro. To Math. Reasoning

1313 Finite Math

1314 Elements of Calculus

1330 Precalculus

Faculty Concerns

• Prerequisite Knowledge

• Student Performance

Previous Faculty Attitude

They are adults. Let them find their own way.

Current Attitude

They are young. Encourage them to work hard and be responsible.

Essentially no course

coordination, and very few

grades.

Total course coordination,

and many grades.

The Bottom Line

Students will only work as hard as you make them work.

If you want more from your students, then demand more.

Actions In Freshmen Mathematics(since 2003)

• Improved Placement Testing - Fall 2007• No Adjunct and Essentially No Graduate Student

Instruction• Mandatory Attendance• Daily Class Grades• Daily Homework• Weekly Online Quizzes• Online Course Materials• Increased Tutoring Availability• Common Exams and Common Grading• No Curves!!• Improved Instruction and Course Coordination• Faculty Peer Pressure

Performance Data - College Algebra(College Algebra = Algebra II)

1310 Fall A B C D F+W

% 32 20 20 12 16

Cum. % 32 52 72 84 100

According Data - Responsible students succeed in College Algebra.

Performance Data - PreCalculus

1330 Fall A B C D F+W

% 24 18 15 12 31

Cum. % 24 42 57 69 100

This course is traditionally hard at every university.

Performance Data - Calculus I

1431 Fall A B C D F+W

% 21 23 21 13 22

Cum. % 21 44 65 78 100

Which Missing Skills Are Crippling Students in College?

(…the same ones that cripple them in your classroom…)

• Arithmetic• Simple Algebraic Manipulation• Graphing and Understanding Basic

Shapes• Simple Geometry• Critical Thinking• Work Ethic• Responsibility

A Nontrivial Number of Students Blindly Accept the

Following

( )

2 2

21 1

a b a b

a b a b

x x

+ = +

+ = +

+ = +

In General…

• Students do not know how to experiment.• Students have seen so many

mathematical topics that they do not know any of them very well.

• Students struggle to organize their work and explain what they have done.

• Students have learned processes that they do not understand.

…and it’s not your fault…

Important Geometric Concepts

• Pythagorean Theorem• Areas of triangles, rectangles and circles.• Circumference of rectangles and circles.• The sum of the angles in a triangle.• Similar triangles.• Isosceles and equilateral triangles.• Triangle trigonometry (if they are

proceeding to calculus).Honestly, that’s it!

In General, We Are Happy If Students

• Can do simple arithmetic and basic algebra.• Know everything about lines and parabolas.• Can solve linear and quadratic equations, and linear

inequalities.• Can factor simple quadratics, complete the square and

use the quadratic formula.• Understand asymptotes (for calculus).• Know the shapes of basic functions.• Understand functions.• Know the area formulas for circles, rectangles and

triangles, and know the perimeter of a circle and a rectangle.

• Know basic trigonometry and the values of the trig functions at the special angles (for calculus).

• Can organize their work.• Can act responsibly and are willing to work hard.

Of Course…

We would be thrilled if students could read and write mathematics!

Discussion Point: Why is this so difficult?

Mathematical Explorations

Is Something Wrong Here?

Original Figure Re-organized Figure

Geometry ChallengeSomething to Sleep On

Is it possible to cut a circular disk into 2 or more congruent pieces so that at least one of the pieces does not “touch” the center of the disk?

Exploration 1

Consider the figure below. What can you conclude?

Exploration 2

( )The line 8 is graphed and the point 1,2

is chosen on this line. A new point is formed by adding

to the coordinate of and to the coordinate of .

Discuss the relation

3

between h

3

t

2

2

e l

x y P

Q

x P y P

+ = =

ine segment and the

line 8.

Discuss any possible generalizatio

2

n

3

.

PQ

x y+ =

Exploration 3

A rectangle with sides parallel to the and axes

has its lower left hand vertex at the origin and its

upper right hand vertex in the first quadrant along

the line 10 2 . Give the dimensions of

x y

y x= − the

rectangle so that it has the largest possible area.

Exploration 3 – Figure

Exploration 4

( ) ( )

( )( ) ( )

( ) ( )

1

2 1 3 2

1Graph both 1 and . Find their point of

3intersection, and then explore the sequence of values 0 ,

, , ... etc.

Let 1 1. Graph both 1 and . Find

their po

f x x g x x

a f

a f a a f a

m f x m x g x x

= + =

=

= =

− < < = + =

( ) ( ) ( )1 2 1 3 2

int of intersection, and then explore the sequence of values

0 , , , ... etc.a f a f a a f a= = =

Exploration 5( )

( )

I. A line with slope 3 passes through the point 2,3 . Give the equation

of the line in slope-intercept form.

II. A line with slope 3 passes through the point 2 ,4 1 . Give the

equation of the l

a a a

−

− −

( )( )

ine in slope-intercept form.

III. A line with slope 3 passes through the point 2 ,4 1 . Is there a

value of for which 1, 2 is on the line?

IV. For each real number , a line is created wia

a a a

a

a L

− −

− −

( )th slope 3 that passes

through the point 2 ,4 1 . Are there any points that fail to be on any of

these lines?

a

a a

−−

4 1 5 3 3 5 2 4

3 2 2 5 1 5 2 5

2 4 2 1 3 4 2 3

3 5 4 3 2 3 3 3

1 1 1 3 5 5 5 5

1 2 1 5 5 5 3 3

Pick a value in the first row. Then move forward that number from left to right and top to bottom. Keep going until you cannot complete a process.

In this case, you will always land on the 4th entry in the last row.Question: Can you create a grid where the last value can be different

depending upon where you start?

Exploration 5 - Figure

Exploration 6

The ancient Egyptians used the following interesting figure to approximate the area of a circle. What value do you think they used to approximate π?

Exploration 7

The previous example used simple shapes (triangles and

rectangles) to approximate the area of a more complicated

shape. Explore using this idea to approximate the area of

the region between the axis x 2and the graph of 1 . y x= −

Exploration 7 - Figure

Exploration 8

Sketch the triangle with vertices (1,1), (3,-1) and (2,6). Find the coordinates of the midpoints of each of the sides of the triangle and the coordinates of the centroid of the triangle.

Generalize the process.

Exploration 9

Sketch the triangle with vertices (1,1), (3,1) and (2,5). Sketch the line segments connecting the vertices to the midpoints of the opposite sides. Discuss the figure.

Generalize the process.

Exploration 9 – Figure(generalization)

Can you use coordinates of vertices to find areas of triangles?

Exploration 10Discuss the figure below. Feuerbach 9 Point Circle

Exploration 11

Suppose that either a circle, a square, or both a circle and square

are to be created having a total circumference of 10 m.

Discuss the total area that is enclosed.

For which choices will this area be as small as possible?

For which choices will this area be as large as possible?

Exploration 12

( )

21Graph the parabola . Then draw the vertical line segment

2from the point ,20 to the point where it intersects the parabola

for several values of between 4 and 4. Now imagine that

each of

y x

a

a

=

− these vertical line segments is a path of a laser beam that

is shown towards the parabola, and then reflects off of the parabola

towards the axis. Discuss the points of intersection of the reflectedy

laser with the axis, and the total length of the beam's path from

its origin to the axis.

y

y

Exploration 12 - Figure

Exploration 13We can use the TI calculator to simulate 100 experiments of flipping a coin 100 times, where we keep track of the maximum streak length in each experiment.

The program on the right uses J to keep track of the experiment. In each experiment, a list of length 100 containing random 0’s and 1’s is stored in L1. Then L2 is created to keep track of the streak lengths. Finally, the maximum value of the streaks for the experiment is recorded in L3(J). We display J at each step to keep track of the progress.

For(J,1,100)randInt(0,1,100) → L1

1 → L2(1)For(K,2,100)If L1(K) = L1(K-1)ThenL2(K-1)+1 → L2(K)Else1 → L2(K)EndEndmax(L2) → L3(J)Disp JEnd

Exploration 13 – Comments

• The calculator is SLOW!!! The process will take a few minutes. When you are done, use either the Table feature or Stat Plots to view the output in L3.

• This process can be performed in Excel VERY FAST. I have posted an associated Excel file, and I will post a video demonstration.

Exploration 14

• What are the different numbers of regions that can be determined by 2 distinct circles in the xy-plane? Draw representative sketches.

• What are the different numbers of regions that can be determined by 3 distinct circles in the xy-plane? Draw representative sketches.

• For each natural number n, let m(n) be the smallest number of regions that can be determined by n distinct circles in the xy-plane. Give the value of m(n) for each n. If you are able to give a formula in terms of n for m(n), then prove that the formula is correct.

• For each natural number n, let M(n) be the largest number of regions that can be determined by n distinct circles in the xy-plane. Give the value of M(n) for each n. If you are able to give a formula in terms of n for M(n), then prove that the formula is correct.