Tuesday, October 2, 2012

Tuesday, September 11, 2012

Percent Dictionary

To solve percent problems, use the following dictionary.

Replace 'what' with x.

Replace 'is' with the equal sign.

Replace 'percent' with divide by 100, /100.

Replace 'of' with the multiplication symbol, *.

What is twenty percent of seventy?
x = 20/100*70

Thirty is what percent of ninety?
30 = x/100*90

Twenty is sixty percent of what number?
20 = 60/100*x

Converting Units with Maple

In the Maple worksheet from the toolbar select [Tools], [Assistants], [Units Calculator].
Follow the instructions there.

What Is A Dimensionally Homogeneous Equation?

What Is A Dimensionally Homogeneous Equation?

In the post, Solving Equations with Maple, I noted the Manning equation for flow in an open channel was only valid when used with US Customary units.  The units on the left side of the equation were cubic feet per second.  The units on the right hand side were feet raised to the 8/3 power.  Units are hidden in the constant, 1.486.

When the units on the two sides are not the same, the equations is said to be not dimensionally homogeneous.  Such an equation can only be used with the units for which it was derived.

When the units on the two sides of an equation are the same, the equation is said to be dimensionally homogeneous.  Such an equation can be used with any system of units. 

Solving Equations with Maple

Solving Equations with Maple

[Keep scrolling.  To get JPG images on blog required going from Maple to Word to Paint to JPG to MAC to blog.  Lots of unintended space.  Is there a better way?]

Open the Maple worksheet.  From the toolbar, select Text to type a comment.

Select Math to type executable commands which appear in italics.

Place the cursor on the command and press [Enter] to execute the command.

Algebraic Operations
To add, use the plus sign, +.
To subtract, use the dash, -.
3 - 2
Use the asterisk, *, to multiply.  It appears as a raised dot in Maple input.  The symbol can also be selected from the Expression template.
Use the forward slash, /, to divide.  Use the right arrow key to exit the denominator.


The evalf command converts a rational number to decimal (floating point) format.
If a decimal point is placed in any number in an expression, Maple gives a decimal answer.
If all entries are integers, Maple gives the exact answer.
The % sign refers to the last answer calculated.

To raise a value to a power, use the caret, ^.  Use the right arrow key to exit the exponent.

There is no implicit multiplication in Maple.  The multiplication symbol, *, must be used for all multiplication.  In Math input it appears as a raised dot.
3 ∙ (2 ∙ x ∙ y – 7)

6 x y - 21
The solve command will solve equations and inequalities.
solve(3∙x – 7 = 5, x)

solve({3∙x – 7 = 5},{ x})

x = solve(3∙x – 7 = 5, x)

x = 4
Write in slope-intercept form the equation of the line that passes through (-2,3) and has slope of 3/4.

y = solve(y – 3 = ¾*(x+2),y)

Braces, {}, are used for sets.  Parentheses are used as grouping symbols.
Brackets, [], are used for lists and matrices.

Now let’s use math.
One day in 1978 at New Braunfels, Texas, almost a foot of rain fell in one hour.  Let’s design a drainage ditch to remove that water from a 10-acre parking lot in one hour.  Assume the slope of the ditch is three inches per hundred feet.  Let the ditch be trapezoidal with sides sloping 45 degrees from horizontal and let the depth of flow be equal to the width of the bottom.  The bottom and sides are unfinished concrete.
Caveat:  The example is for teaching purposes only and not to be used as an actual design.
We’ll use Manning’s formula for flow in an open channel.

in which Q is the flow in cubic feet per second, A is the cross sectional area of flow, R is the hydraulic radius which is the area divided by the wetted perimeter, S is the slope in feet per foot, and n is the roughness coefficient.
We have ten acres one foot deep in water to remove in 3600 seconds.  To solve, place the cursor on the equation and press [Enter].

Let b = width of the bottom and d = the depth of flow.
Then the cross-sectional area of flow A is



Manning’s roughness coefficient for unfinished concrete is 0.014.

We have a set of seven equations in seven unknowns.  Add one more for average velocity of flow.  In Maple, sets are enclosed in braces.  Because we are interested in only real, positive answers, we use the fsolve command instead of the solve command.  Place the equations in braces, separated by commas and the unknowns in braces, also separated by commas.

The bottom of the ditch is almost three feet wide.  The depth of flow is almost three feet.  The velocity of flow is just over seven feet per second.  The width of the top of the ditch is almost nine feet.
All we needed was geometry, the Pythagorean Theorem, and a Google search for flow in open channels.
Manning’s equation is valid for only US Customary units.  It won’t work with metric units.

Generating a Randomized Test

Generating a Randomized Test
Teachers wishing to not have to stare down the back row can easily generate many versions of a test.
In the following, there are 1250 possible versions of problem 1 and 1,889,568 possible versions of problem two.
The first two blocks of code create two procedures.  The third block generates the test.
That generates a test without answers followed by the same test with answers.
The program is best run on the Classic Worksheet because it allows one to block and print the blocked selection.
Generate more versions by increasing the range of k.
> ComplexFraction[c2p19]:=proc(answer) 
local r1,r103,r103a,r103b,r103c,r103d,r103e,plist103,s103,s103a,s103b,s103c,s103d,s103e,s103f,s103g,s103h,clist103,ans103:
printf("Simplify the complex fraction.\n");#with(combinat):
if answer = 1 then
end proc:

local r1103a,r1103b,r1103c,r1103d,r1103e:
if answer = 1 then
fi:end proc:

> printf("\nA Randomized Test with and without Answers\n");
for k from 1 to 2 do
for j from 1 to 2 do
printf("Problem 1, Version %d\n",k);
printf("Problem 2, Version %d\n",k);
end do:end do:

A Randomized Test with and without Answers
Problem 1, Version 1
Simplify the complex fraction.

Problem 2, Version 1

Problem 1, Version 1
Simplify the complex fraction.

Problem 2, Version 1

Problem 1, Version 2
Simplify the complex fraction.

Problem 2, Version 2

Problem 1, Version 2
Simplify the complex fraction.

Problem 2, Version 2

Sunday, September 9, 2012

History of Math Tools

History of math tools...

Through the ages, mankind has used various tools to perform calculations.  Some use fingers. 

Others used an abacus (Not this one.  It’s missing a bead.).  

In 1634, William Oughtred invented the slide rule.  This one has more scales, some of which were added by Amadee Mannheim, a French cavalry officer.   The slide rule was the best tool available for 300 years.

In 1968, Hewlett Packard produced the first scientific calculator.  Through the years more features have been added by a number of manufacturers.   I use the TI-89.

The latest and greatest tool is the computer algebra system.  The development of computer algebra systems began in the 1960s.  
Two popular commercial systems are Maple and Mathematica.

The essential math software tool

These programs will solve equations and draw pretty colored graphs.  They are programmable, complete with do loops and if-else statements.    I was introduced to Maple first and will use it in this blog.  Go to maplesoft.com and, if you are a student, buy the student edition.
Most current math teachers were not taught the pencil and paper procedure for getting the square root of a number by grouping digits in groups of two in both directions from the decimal point.  They are perfectly happy to use, and let students use, a calculator to obtain the square root of 231.67. 
Those same teachers begin to hyperventilate when I suggest we no longer need to teach long division, a very similar procedure.
Next Blog: Solving equations.