Workshop I. Tools and Techniques

This workshop gives you experience at using WHS as a student, teacher, and homework author. You will also gain a working knowledge of Maple, the primary program we use to prepare homeworks.

Using WHS as a student

Most of the things a student can do in WHS are covered here, in the order they would do it for the first time. Some of the things would only be done once, such as registering in WHS. Some would be done infrequently, such as modifying your account or adding a class. But many of the things would be done on a regular basis during a course.

Single or infrequent use

Register as a student :

regular use

A student who is already registered for a class regularly logs into WHS and selects his class. Then he would

Check for announcements and the class-web page :

After you have requested registration in Communicating Mathematics, highlight the Announcements menu item to see the current announcements for the class. You can also see the class webpage from this item (click the link class webpage at the bottom of the table.

Retrieve the homework :

Now highlight the Assignments menu item, select the assignment 10 Sample Homework , and click the Display Homework button. This is a homework with several versions, so use the version number 2002 this time.

Work the homework :

Now we get to the part that all of this structure is for: working the homework and checking your answers. Working the problems can be done on scratch paper while sitting at the terminal; however, often students may want to print out the homework and work it on it at their desk.

Submit the homework :

After you have worked the problems, retrieve it again and submit your answers. Notice that there are various answer formats in this first sample homework: selection boxes, radio buttons, checkboxes, and text boxes. (There are other answer formats.) After you have chosen the answers, or typed them in, in the case of text boxes, scroll to the bottom of the homework and press the submit button.

Inspect the feedback sheet :

The page which is displayed after you submit your answers to a homework assignment is called the feedback sheet, because it tells you how you did problem by problem. In most cases, the correct answer is displayed in addition to the answer you submitted. With a little practice, you can read the feedback sheet like the answers in the back of the book.

Make comments or ask questions in the feedback sheet :

Note that for each problem you submit an answer to, there is a 'comment box' for that problem in the feedback sheet. Use this if you have a question about the problem you want to ask the teacher. Just for practice, make a comment here and submit it. It will be mailed to the teacher (me in this case).

Rework and resubmit homework :

To rework a homework, simply display it again, using the same version number, and select new answers, and then resubmit. Or to get a similar assignment, change the version number and display.

View homework scores .

If you are registered in a class, you can check you progress on the homework by highlighting Homework Scores and clicking the Show Scores button.

So that, in a nutshell, is how a typical student would use WHS. It provides a convenient method for the student to retrieve, work, and submit a homework assignment. It gives instant feedback on each problem submitted and allows for questions by the student. In addition, the student can keep up with class announcements and review their performance on past homeworks.

Exercise: Make certain that you have requested registration in Eberhart's Communicating Math. It can be found in the University of Kentucky classes.

Using WHS as a teacher

As a teacher, after you have logged into mathclass.org, you can highlight the Teacher Resource Menu item and access the buttons titled Manage Classes , and Access Records. Press the Manage Class button to bring up a page with three activities you can initiate:

Manage Class button

Create a class:

First type in a name for the class. For example, you could use the name X's Test Class , where X is your name.

Then select the school the course will be offered in (Morgan County High School)

Also select the homework list you will associate with the course. Use any homework list, and press the Creat Class button. Later, you can configure the class, and change the homework list to one you will create later.

Delete a class :

Any class you have created you can also delete. Just select it and press the delete button.

Configure a class:

This button will be used most often to update the announcements for the class, or to upload a webpage for the class.

Select the class you want to configure (X's Test Class) and press the button. You are led to a form where you can change several things: the name of the class, the school where the class is offered, a brief description of the class, the url of the class homepage, the current announcement, and the version of the class (use a version when you want to make sure the students all do exactly the same homework). The url can be any place where you can store a web page. The most convenient thing is to upload a zipfile containing the webpage, using the upload button at the bottom of the page. Then WHS installs the webpage on the server, and returns the url where it can be found. You type that into the homepage line of the form.

Access Records button

Registrations and Passwords

Click this button to open a page on which you can approve registration for students who have requested it.

You can register in your own class. After a student has registered in your class, his name appears on the class roll, along with a button (underneath his ssn) to access his individual homework progress, and a link to his email. At any time, you can remove a student from your class (so that he no longer appears on the class roll). Also, you can change his password, in case he forgot it or in the rare instance where you might need to block his access to WHS.

Student Comments

This button takes you to a page which lists the comments that students have submitted from the feedback sheet on a homework. Also, this page contains the student answers to handgraded questions (those with the answer format AX_) .

Update class grades

This button takes you to a page on which you can record exam scores for students on your roll. It is set up for 3 midterms, a final, and a grade for the course.

Assignment Submissions

This button leads to the page from which you can download comma separated files of student responses to the homework assignments. Most of this is summarized in the two buttons which follow.

Summaries by Student

This button brings up a page giving data on student performance (in your class)on the homework, by assigment.

Systemwide Summaries

This button brings up a page giving data on all student performance on your homework, even those in other classes.

Exercise: Make sure you have created a class of your own, either in Morgan County High or Morgan County Middle school.

Using WHS as an author

WHS coordinator/authors can create create homeworks, install them to WHS, and group them into homework lists, which then can be used by teachers as the homework list for their class. To get to the Homework Management page, first log into WHS, highlight the Teacher Resources Menu item, and click on the Authoring Tools button. You are led to a page with three tables: Homework Testing, Homework List Management , an d Homework Installation

The Homework Testing table

In this table, you can select a homework list (and press the Set Menus button) and preview the homeworks in that list, or you can select a homework author (and press the Set Menus button) and preview the homeworks installed by that author. You can also download a homework you like if the author has given permission.

Important: You select your homework lists and authors in this table when you are managing your homework list in the next table. This can be a point of confusion.

The Homework List Management table

In this table are gathered the activities associated with creating and maintaining a homework list.

Creating a homework list:

All you need to do to create a homework list is decide on a name. After you press the create list button, go back to the top table (Homework Testing) and set the homework list to the list you created. Then you can add homeworks to the list.

Adding homeworks to your homework list:

To add a homework, choose a display name you want to display it by in the list, then set the author of the homework in the Homework Testing table, then select the installation name in the Homework List Management Table, and press the Add to List button. (Note : It is a good idea to start the display name with a two digit number. You can use this device to order the homeworks in the order you prefer.)

Publishing a homework list :

You may press this button to toggle a publication flag for a homework list you have created, thereby allowing other teachers to use it as a homework list in one of their classes.

Deleting a homework list :

You may press this button to delete a homework list you have created at any time, provided it is not being used as a homework list in any class

Removing a homework from a list :

Press this button to remove a homework you have added to your list by mistake.

The Homework Installation table

This table is the one most used by authors. It has numerous buttons to initiate various authoring activities.

Creating a homework name:

This is merely giving a name to the homework you plan to install. The name must be the same as the zipfile that is used to install the homework, and also the same as the source html document in the zipfile.

Toggling the downloading flag :

After you have installed a homework, you can toggle the downloading flag. (Downloadable homeworks are not dimmed in the Toggle Downloading selection box.) This can be toggled temporarily by you when you want to make a change in the assignement. See below on modifying a downloaded assignement.

Deleting Homework :

Press this button to remove a homework you have installed from the server.

Uploading the homework zipfile:

Before you press this button, you must browse your computer to find the homework zipfile you want to install. When you have located the correct zipfile in the browse window, press the open button, then

Installing the homework :

press the Install button.

Showing the homework :

After you have installed the homework, you can inspect it with the Show homework button, or you can go to the top window and test it there.

Exercise: Make sure you have created a homework list (just the name, no assignments in it yet), and configured your class (created above) so that its homework list is the one you have created.

Installing your first WHS homework

The goal of this activity is to lead you through the process of creating and installing a homework in WHS using Maple. Instead of starting from scratch, you will download a homework zipfile from Communicating Mathematics and work with that. We will use the word zipit from the MCtools package to prepare the homework zipfile.

Here are the steps we will follow.

0 First log into WHS .

1 Download the homework zipfile:

Highlight the Teacher resources Menu item and click the Authoring Tools button.

In the top table, select the Homework List Eberhart's Workshop2003 and press the Set Menus button.

In the second row of the top table, select homework 10 First Homework and press the download button.

A window pops up. Choose to save the file, which is called comath10whs.zip. Save it in a directory (perhaps My Documents ) on your C drive.

2 Unzip the homework file:

If you are using Windows XP, you can click My Computer from the Start Button and find the zipfile you saved. Then rightclick the zipfile and choose the option Extract All . The zipfile will be unzipped into a directory comath10whs.

If you are using Windows 2000, you will have to unzip the file using winzip or some other unzipping program. You should still unzip it into a separate directory named something like comath10whs.

3 Load the source worksheet into Maple:

Locate the file comath10.mws in the directory and double click it. Maple will run and open the worksheet comath10.

4 Execute the MCtools package:

At the top of the worksheet, there is a closed section titled MCtools (June 8, 2002). Open that section, put your cursor anywhere in the red text you see and press return. This executes the MCtools package and enables you to use the vocabulary of words in it. Go to the top of the worksheet and close the MCtools section.

5 Edit the homework:

The worksheet comath10.mws is the source worksheet .for the homwork comath10whs. You are going to make a small modification to this homework and install it as your first homework. To keep it simple, change the title of the homework from Arithmetic Problems to Computing with Fractions. That will be our only change in the homework. There will be one additional change in the worksheet so that we can make zipfile..

Go to the top of the Maple worksheet and locate the line of red input text zipit("comath10","f://cfiles/cmath/comath1");

Change the second string to the correct path on your computer to the directory containing the files you have saved. So, for example, if you files are in the directory comath10whs located in the temp directory on your c drive, the correct string would be "c://temp/comath10whs".

6 Save and export the revised worksheet:

Now highlight the File menu at the top of Maple and click on Save to save the change in the title. Then highlight File again and click on Export As , then in the window that opens click on Html . In succession click on Save , Ok , and Yes to All . to export the changes to html files. Now you are ready to create the zipfile.

7 Use MCtools[zipit] to make a new zipfile:

Go to the zipit line at the top of the worksheet, put your cursor in the line and press return to execute the command. A window should open briefly, and a line telling where to find the zip file just created will be refreshed. At this point you are ready to to to WHS to complete the installation of the homework.

8 In WHS, create a name for your homework

Login to WHS, highlight the Teacher Resources menu item and click on the Authoring Tools Button.

Go to the bottom table in the page that opens, and in the top line of it, type in the name comath10whs and press the Create Homework button. This creates the homework name, and you can upload the homework zipfile now.

9 In WHS , Upload and Install the homework to your set of homeworks:

Go to the bottom table again and press the Browse button. Use the window that opens up to locate the zipfile comath10whs.zip on your computer. You can hightlight the name and click it to make it appear in the Open line. Then press the Open Button. The window disappears, and the path to the zipfile appears in the upload line of the bottom table. Press the Upload zipfile button. After a few seconds, you receive a message that the file was uploaded successfully.

To Install the homework, make sure its name appears in the next to last line of bottom table and then press the Install homework button. You will receive the message that the homework installed successfully.

10 In WHS, Preview the homework :

Go to the bottom table, select the homework in the last line of the bottom table and then press the Show homework button.

11 In WHS , Add the homework to your homework list:

If you haven't done so already, create a homework list (Go to the middle table in the Authoring Tools page and type in a name for your list in the middle line there and press the Create List button.)

In the top table , select your homework list and press the Set Menus button.

In the middle table , type a Display Name (such as 01 First Homework ) for the homework you just installed and make sure that homework appears in the selection box to the right of the name you typed. Then press the Add to List button.

The editing cycle

After you have installed a homework the first time, you may, and often will, find a need to make a change in the homework, to add a problem, correct the answer to a problem, etc. You are entering the editing cycle :
1. open the source worksheet for the homework. If you have it on your computer already, fine; otherwise, download the homework zipfile (you may need to toggle the download button on the homework installation page), and unzip it into its own directory.

2. execute the MCtools package at the top of the directory, then close that section.

3. make the changes source worksheet

4. save the source worksheet

5. export the worksheet to html

6. execute the zipit line at the top of the worksheet (if needed, change the inputs to zipit)

7. Go to WHS and (re)install the homework

8. Inspect the homework from WHS. If not satisfactory, go to 3, else quit.

Exercise: Make sure you have installed your first homework and have displayed it in your homework list.

Exercise: Request registration in your own class and also one other participants class. Then retrieve the first homework in each class and submit at least one answer. On the feedback sheet, make a comment and Mail it out to the teacher.

Introduction to Maple

In order to make much use of Maple when creating homework sets, you need a modest knowledge of the application to start, and have a way to expand your knowledge as the need dictates. In this activity, you will get a small dose of what Maple is, and what it can be used for.

You can go a long way with Maple if you just think of it as a fancy graphing calculator. You can make calculations, plot functions, and solve equations with Maple. First, however, you can save yourself time later by going through the basic editing features of a Maple worksheet.

Catechism on B asic Editing.

The answers below are not unique, but are the most convenient for me.

How do you open an input cell above the cursor? Answer : Press control J , that is, hold the control key down and press the j key

How do you open an input cell below the cursor? Answer: Press control K

How do you change an empty input cell into a text cell? Answer : Press F5

How do you typeset a mathematical expression in a text cell? Answer: Press control M, then type it in, then toggle the X and Maple leaf at the upper left.
How do you join two cells? Answer : Put the cursor in the top cell and press F4

How do you split a cell? Answer : Put the cursor at the front of the line just below the desired split and press F3

How do you copy and paste text? Answer : Use the cursor select the text, then press control C. Then place the cursor and press control V.

How do you put a new line in an input cell with out executing it? Answer : Do a shift return , that is, hold the shift key down and press enter.

How do you create a new section? Answer : Hold the alt key down and press I and S ( or U depending on desires indent) in succession

How do you copy and paste a section? Answer Close the section and select it. Control C and control V to copy and paste it.

Exercise: Practice each of these basic editing operations until you know them.

1. Entering arithmetic and algebraic expressions using calculator syntax (:

It is important to be able to look at a mathematical expressioin, and decide how to type that into Maple. The syntax is almost the same as typing it into a TI-83. Some examples: is entered as 4*(5+8), is x^2 + 3*x, sqrt(3)/(1+sqrt(2)) is and is sin(4/3*Pi)-1. When you type the expression into an input cell (starts with a prompt '>' and has red text), you need to end it with a semicolon ';'. Then when you press enter (with the cursor anywhere in that input cell), Maple will evaluate the expression and format the result in a following output cell. Examples:

> 4*(5+8);

> x^2 + 3*x;

> sin(4/3*Pi)-1;

To convert a number expressed symbolically to decimal form, we would use the maple word evalf.

> sin(4/3*Pi)-1 = evalf(sin(4/3*Pi)-1);

>

Hint : Use control J to open an input cell below the cursor and control K to open one above the cursor. It is usually a good idea to have an input cell directly under (or nearly so) a cell you execute. This is because the cursor will jump down to the next input cell, and if that is way down the worksheet you can lose your place in the worksheet.

Exercise: Type the number into an input cell. End the line with a semicolon. Put the cursor anywhere in the line and press return. Then convert it to a decimal number.

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2. Plotting functions:

The original popularity of the graphing calculator as an educational tool was based to a large extent on its capability to plot functions. Maple gives much better resolution graphs than any calculator. For example, suppose we want to plot the functions and to solve the equation . . The maple word plot is used. In the plot below, we have plotted both sides of the equation, specifying the range to be from x= -2 to x = 3, and the view window to be the rectangle with x range from -2 to 3 and y range from -2 to 2.5.

> plot({x^3-x-1,x^2-1},x=-2..3,view=[-2..3,-2..2.5]);

>

By looking at the graphs, we can see that the solutions to the equation are about , and .

Note: If we raised the left hand side of the equation by 1 to , we can see that two of the solutions disappear. (Actually, they become complex. See below)

Exercise: Solve the equation = by plotting both sides of the equation on the same axes.

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3. Solving equations algebraically

With a graphing calculator, you can solve equations graphically. For more accuracy, you can use the algera you know to try and get x by itself on one side of the equation. (This particular equation is easily solved by hand with pencil and paper.) These kinds of manipulations of the equations can be carried out in a Maple worksheet. A word to use is solve .

> sol :=solve(x^3-x-1=x^2-1,x);

So, we see that there are three solutions to the equation. We can convert the third one to a decimal approximation with evalf

> evalf(sol[3]);

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Note that if we were to lower the parabola down one unit to , we can see the resulting equation would only have one real solution. The other two solutions are complex numbers.

> solve(x^3-x-11=x^2-.2,x);

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Exercise. Solve the equation = by using the Maple word solve. (To get the solutions in decimal form, make at least one of the coefficients decimal.

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4. Naming and Using expression sequences, sets, and lists.

It is important to distinguish between three ways to input data into a Maple input line. Each way has its uses.

An expression sequence is a comma separated sequence of expressions. So for example

> (a+b)^2, 2*x^2-2, 4^10, 2*(x^2-1) ;

is an expression sequence. It has four things in it. We can give a name to the expression sequence so that we can refer to it later without having to type it in again. We'll call our sequence bill.

> bill :=(a+b)^2, 2*x^2-2, 4^10, 2*(x^2-1);

We can refer to terms in the sequence by using indices, so bill[2] refers to the second term in the sequence.

> bill[2];

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If you want to treat it as a single object, you can put square brackets around it. This is called a list .

> [(a+b)^2, 2*x^2-2, 4^10, 2*(x^2-1)];

If you enclose an expression sequence in curley brackets, the result is called a set. Duplicates are removed from a set and the order of the terms in a set may get changed by Maple.

> {(a+b)^2, 2*x^2-2, 4^10, 2*(x^2-1)};

You can name lists and sets, just like you can an expression sequence. In fact, you can give a name to almost any Maple output and use that to refer to it later in the worksheet.

Exercise: Give a name to list above and then multiply the first term in the list by the last term in the list.

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5. Using online Maple help.

There is extensive online help available in Maple, which you can use to expand and refresh your knowledge. For example, if you forget the syntax for a

Maple word, like plot, you can highlight the word plot and click on Help at the top of Maple. Then click on Help for plot. A page will come up. Down at the bottom are examples of how plot is used. Often that is enough.

To make a systematic investigation of Maple, you can use the help glossary. This gives you access to an orgainization tree of the various words and packages of words in Maple.

Exercise: How is the Maple word expand used?.

Exercise: What is a package and how do you use one?

6. Using the plots, plottools and MCtools packages

The plots, plottools, and MCtools packages all contain Maple words which are used to draw diagrams containing lines, circles, polygons, spheres, etc. Anytime we want to include a diagram in the statement of a WHS problem, we will use one or more words from one or more of these packages.

Example : Make a diagram of a square with an inscribed circle.

Solution. There are many ways to do this in Maple. Here is one. We draw (using the word polygon in plottools) the square with vertices [0,0], [1,0], [1,1], and [0,1]. Give it a name so that we can refer to it later.

> sq := plottools[polygon]([[0,0],[1,0],[1,1],[0,1]],color=grey);

Then draw the inscribed circle (using disk in plottools).

> circ := plottools[disk]([.5,.5],.5,color=yellow):

Now use plots[display] to put both of these objects on the same graph. Note the choice of options scaling = contained (that means draw to scale) and axes = none (that means don't show the x and y axes).

> plots[display](circ,sq,scaling=constrained,axes=none);

Exercise. Using the help page for plots[textplot], put a label in the center of the circle saying "yellow circle in grey square".

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The plots and plottools packages come with Maple. The MCtools package was written at UK specifically to help create WHS homework. It has several words to aid in drawing and labelling diagrams, plus some words to aid with formatting a problem for WHS. You will usually find a version at the top of any Maple worksheet which is the source of a WHS homework.

The procedure to use MCtools is to open the section which contains it and execute the cell containing MCtools, then close the section. MCtools help is different too. A list of the word in the package is available by executing mctools(); in an input cell.

> mctools();

MCtools, version 5/19/02.

Current choices are [ARRW, Axes, CARR, DL, DV, GP, GP2, GP3, MM, PA, PARAMS, PC, PP, PT, RANDANS, addlink, addsectionhint, colors, hashang, lineit, mcprint, mctools, roundto, tableit, tagit, zipit].

"mctools(X)" gives syntax and defaults for "X"

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Several of the graphing word in MCtools are written to simplify the use of some corresponding word in plots or plottools. For example, PT means Put Text and replaces plots[textplot]. So we could add the label in the above exercise with PT. Get the syntax.

> mctools(PT);

MCtools, version 5/19/02.

PT(Location,txt,fontsize=16,clr=black)

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> plots[display](circ,sq,PT([.5,.5],"yellow circle in gray square"),
scaling=constrained,axes=none);

Exercise: Subdivide the square into 4 subsquares and put inscribed circles into each of those squares. Use PC, Put Circle, from the MCtools package to draw the circles.

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Answers to exercises

Exercise: Type the number into an input cell. End the line with a semicolon. Put the cursor anywhere in the line and press return. Then convert it to a decimal number.

> sin(5/7*Pi)+sqrt(2);

> evalf(sin(5/7*Pi)+sqrt(2));

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Exercise. Solve the equation = by plotting both sides of the equation on the same axes.

> plot({2*x+1,x^3-1},x=-2..2);

By inspection, we see that the equation only has one real solution, approximately x = 1.78 (See this by putting your curson on the crossing point on the graph, clicking the left mouse button, and reading the approximate coordinates of the crossing point in the line just above the upper left corner of the worksheet.

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> solve(2*x+1=x^3-1,x);

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Exercise. Solve the equation = by using the Maple word solve. (To get the solutions in decimal form, make at least one of the coefficients decimal.

> solve(2*x+1 = x^3-1.,x);

So, we see one real solution at x=1.77 approximately, and two complex solutions.

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Exercise: Give a name to list above and then multiply the first term in the list by the last term in the list.

> sam := [(a+b)^2, 2*x^2-2, 4^10, 2*(x^2-1)];

> sam[1]*sam[4];

> expand(sam[1]*sam[4]);

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Exercise: How is the Maple word expand used?.

Highlight the word and click on Help. Then click Help on expand. A page comes up. Go to the bottom and skim the examples to see the usage. If you have any further questions, you can read the descriptive text at the top of the help sheet. We see that expand can be used to multiply out polynomials, among other things.

> p := x^2 + 3; q := 3*x^4 -x +1; p*q;

> expand(p*q);

Exercise: What is a package and how do you use one?

Highlight the word and click on Help, then on Help on package. You are given two choice. Choose the index of packages. Then click on the plots package . There are lots of words in this package. One you will use later is display. This is used to display several drawings in the same picture. Another word word in plots is textplot, which is used to put labels on drawings. See the next section.

> with(plots);

Warning, the name changecoords has been redefined

> ?plots[display]

Exercise. Using the help page for plots[textplot], put a label in the center of the circle saying "yellow circle in grey square".

Using the syntax as described in the examples, create the label, name it, and add the name to the display.

> labl := plots[textplot]([.5,.5,"yellow circle in grey square"]):

> sq := plottools[polygon]([[0,0],[1,0],[1,1],[0,1]],color=grey);

> circ := plottools[disk]([.5,.5],.5,color=yellow):

> plots[display](labl,circ,sq,scaling=constrained,axes=none);

Exercise: Subdivide the square into 4 subsquares and put inscribed circles into each of those squares. Use PC, Put Circle, from the MCtools package to draw the circles.

> sq1 := plottools[polygon]([[0,0],[1/2,0],[1/2,1/2],[0,1/2]],color=grey):
sq2 := plottools[polygon]([[0,1/2],[1/2,1/2],[1/2,1],[0,1]],color=red):
sq3 := plottools[polygon]([[1/2,0],[1,0],[1,1/2],[1/2,1/2]],color=blue):
sq4 := plottools[polygon]([[1/2,1/2],[1,1/2],[1,1],[1/2,1]],color=green):

> circ1 := PC([.25,.25],.25,color=yellow):
circ2 := PC([.25,.75],.25,color=magenta):
circ3 := PC([.75,.25],.25,color=turquoise):
circ4 := PC([.75,.75],.25,color=pink):

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> plots[display](circ1,circ2,circ3,circ4,sq1,sq2,sq3,sq4,scaling=constrained);

Now if we were to use the disk word from plottools, we could color the inside of the circle

> circ1 := plottools[disk]([.25,.25],.25,color=yellow):
circ2 := plottools[disk]([.25,.75],.25,color=magenta):
circ3 := plottools[disk]([.75,.25],.25,color=turquoise):
circ4 := plottools[disk]([.75,.75],.25,color=pink):

> plots[display](circ1,circ2,circ3,circ4,sq1,sq2,sq3,sq4,scaling=constrained);

Using Maple while making up and solving problems

We are all acustomed to writing, thinking, and working on paper, and will continue to do that. Being teachers, we can also add the blackboard work to the list of media we feel comfortable working and thinking with. Increasingly, we adding the monitor screen and keyboard to this list. The Maple worksheet provides a 'paper-like' place to work on mathematics problems, make calculations, draw diagrams, and write textual materials in a way that can be saved and retrieved later for further work.

Starting with a familiar problem.

To illustrate how you might use Maple to work on mathematics, lets take a familiar problem as the starting point.

Problem. Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How long does it take them to mow it together? (Assume they do not slow each other down, when they work together.)

Setting up the equation for the problem.

Let x be the length of time it takes when they mow together. Then in 1 hour, they mow of the yard. Now Bill mows of the yard in an hour and Jim mows of the yard. So the equation is = Taking reciprocals of both sides, we get = 1.2 hours log(3)

Exercise on inline formatting. When you type a mathematical expression into a text cell, you use calculator syntax. You can convert this to a typeset expression by painting the expression with your mouse cursor, then using the mouse to open the Format menu on the top line of Maple, then selecting the Convert to item (at the bottom of the menu), then selecting the Standard Math item in the menu that opens up. It sounds a little involved, and it is tedious, but it is important to be able to do it. Use the procedure to format the mathematical expressions in the next two lines.

1 Pythagoras' theorem says that in a right triangle with legs a and b, and hypotenuse c, c^2 = a^2 + b^2.

2 The sum of the reciprocals of x and y would be written 1/x + 1/y, the reciprocal of that is 1/(1/x +1/y).

Using the algebraic manipulation vocabulary in Maple to solve your equations.

We can also use Maple to solve the equation.

> eq := 1/x = 1/3+1/2;

> solve(eq ,x);

>

Using text cells to record your notes and comments on the solution.

It is very handy feature to have. To be able to record notes and comments in the same place you make computations works well. You can turn an empty input cell into a text cell by placing the cursor in it and pressing the F5 function key

Modifying the problem to a parameterized problem or to a related problem .

A parameter is a quantity which is regarded as known but unspecified in a problem.

We can parameterize this problem by replacing the numbers in the problem with parameters and then solving in terms of the parameters.

Parameterized Problem. Bill mows a yard in A hours. Jim can mow the same yard in B hours. How long does it take them to mow it together?

Solution. As before let x be the time needed when both are working together. In one hour, Bill mows of the yard and Jim mows of the yard. Together, they would mow of the yard. Solving for x, we get =

Checking the solution with Maple,

> eq := 1/x = 1/A + 1/B;

> solve(eq,x);

Exercise. Work this related problem. "Bill mows a yard in 5 hours. Together Bill and Jim can mow the yard in 2 hours. How long does it take Jim to mow it?" Then parameterize the problem and rework it.

We need to develop strategies for checking our work. For example, one good check is to work the problem using a different tool. So doing the problem with pencil and paper, and then using Maple to resolve the equation will sometimes expose errors. In geometric problems, drawing a scale picture can serve as a visual check to a computation. Here's an example.

Problem. Find the area of the circle inscribed in an isosceles right triangle with a hypotenuse of 10.

Solution outline. We need to find the radius of the inscribed circle. Call it R. The the area of the circle is . So lets find an equation in R to solve. We can make a rough sketch on paper of the triangle and its inscribed circle to help us with our thoughts. The circle touches the hypotenuse at the midpoint (why?). Using this fact and two congruent triangles (which ones?), we can write two expression for the length of a leg of the triangle: . Solve for R to get . So the area is . We can this to a decimal approximation,

> R:=evalf(sqrt(50)-5);

> Area := evalf(Pi*R^2);

Now, we can draw a scale diagram to check our work. First draw the triangle. We can situate it in the coordinate plane so that the vertex opposite the hypotenuse is at the origin [0,0], and the other two vertices are at [ ,0] and [0, ]. So, now we use the plottools and plots package words to draw and name the triangle and the circle, then display them together.

> tri := plottools[polygon]([[0,0],[sqrt(50),0],[0,sqrt(50)]],color=yellow):

> circ := plottools[disk]([R,R],R,color=turquoise):

> plots[display](circ,tri,scaling=constrained);

The visual evidence is sufficient to convince me of the correctness of my method.

Exercises.

1 Change the colors of the diagram.

2. Change the location of the triangle in the coordinate plane.

3. Interchange the order of tri and circ in the display line. What is the effect?

4. In the display line, remove the option scaling = constrained. What is the effect?

5. Get the help page on plot[options] and find out how to remove the axes from the diagram. (Hint: look at the second option.)

answers to exercises

Exercise on inline formatting. When you type a mathematical expression into a text cell, you use calculator syntax. You can convert this to a typeset expression by painting the expression with your mouse cursor, then using the mouse to open the Format menu on the top line of Maple, then selecting the Convert to item (at the bottom of the menu), then selecting the Standard Math item in the menu that opens up. It sounds a little involved, and it is tedious, but it is important to be able to do it. Use the procedure to format the mathematical expressions in the next two lines.

1 Pythangoras' theorem says that in a right triangle with legs a and b, and hypotenuse c, .

2 The sum of the reciprocals of x and y would be written , the reciprocal of that is .

Exercise. Work this similar problem. "Bill mows a yard in 5 hours. Together Bill and Jim can mow the yard in 2 hours. How long does it take Jim to mow it?" Then parameterize the problem and rework it.

Solution . Here in the parameterized probleme problem we could replace 5 with A and 2 with B. Then the equation would be . Solve for x to get = . So now, the solution to the specific instance of the problem with A = 5 and B = 2 is 10/3 =3.33 hours

We can make an arrow function here. (Get the help page for -> to see more examples)

> f := (A,B) -> A*B/(A-B);

Now to get the value of x for any values of A and B (with A > B > 0), just call the function with those values

> f(5.,2);

>

Exercises.

1 Change the colors of the diagram.

I like magenta and grey. (You may have other preferences)

> tri := plottools[polygon]([[0,0],[sqrt(50),0],[0,sqrt(50)]],color=magenta):

> circ := plottools[disk]([R,R],R,color=grey):

> plots[display](circ,tri,scaling=constrained);

You can make a function here too. This is a more complicated function, so we would use the word proc to define the function.

> pic := proc(clr1,clr2)
local tri,circ;
tri := plottools[polygon]([[0,0],[sqrt(50),0],[0,sqrt(50)]],color=clr1): circ := plottools[disk]([R,R],R,color=clr2):
plots[display](circ,tri,scaling=constrained);
end:

> pic(blue,red);

>

2. Change the location of the triangle in the coordinate plane.

This is a matter of changing the coordinates of the vertices of the triangle and the center of the disk. So, if we wanted to flip it about the x axis, the only change would be to move the point [0,sqrt(50)] to [0,-sqrt(50)], and the center of the circle to [R, -R].

> tri := plottools[polygon]([[0,0],[sqrt(50),0],[0,-sqrt(50)]],color=magenta):

> circ := plottools[disk]([R,-R],R,color=grey):

> plots[display](circ,tri,scaling=constrained);

>

3. Interchange the order of tri and circ in the display line. What is the effect?

The effect is that you don't see the color of the disk. What display does is lay down the parts of the picture from right to left. If you want to see the color of a part, lay it down after you lay down anything that covers it.

4. In the display line, remove the option scaling = constrained. What is the effect?

The effect is that the circle becomes an ellipse. Setting the scaling = constrained option forces the same scale to be used on both axes. If you don't set that, Maple chooses separate scales so as to make the picture as large as possible. Something tall and skinny would be stretched out in the horizontal direction.

5. Get the help page on plot[options] and find out how to remove the axes from the diagram. (Hint: look at the second option.)

Set the option axes = none in the display line.

Using MCtools[tagit] and MCtools[zipit] to format WHS homeworks

Now that we have had a brief introduction to Maple, and had some limited experience at problem solving in a Maple worksheet, let's go back and create a WHS homework with Maple.

The most important step in creating a homework problem is to state the problem in a clear and interesting manner. The next most important thing is to be solve the problem correctly (that usually means checking your work). Finally, you want to be able to put the problem in a WHS homework. That's what we are concentrating on here. We'll take the original lawn mowing problem and put it and variations of it into a WHS homework set. (most of the examples are variations, a few are not).

Problem. Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How long does it take them to mow it together? (Assume they do not slow each other down when they work together.)

Solution: Let x be the length of time it takes when they mow together. Then in 1 hour, they mow of the yard. Now Bill mows of the yard in an hour and Jim mows of the yard. So the equation is = Solving for x, we get = 1.2 hours

WHS tags

In order for WHS to display a homework set, the source document containing the homework problems must have certain tags inserted in the appropriate places. Generally speaking, tags should go on a line by themselves. The main tags are listed below.

H_ This tag signals the start of the homework.

H_[n] where n is a small positive integer like 3. This tag is like H_, except that when WHS compiles the homework, it only picks one from each group of n consecutive problems in the source document.

SKIP_ This tag tells WHS to ignore all lines below unless and untill it comes to another WHS tag.

T_ This tag signals that the following material is to be inserted just before the next question.

QM_[ precision and correct answer(s) go in here ] This tag signals the beginning of a question.

AS_[ alternative answers go in here ] This tag signals a multiple choice answer

AC-_[ alternative answers go in here ] This tag signals a numerical answer box.

Many other answer formats are available. A complete, 4 page discription of all the tags can be found at the bottom of the

Teacher Resources page under a button titled Author Documentation .

Exercise. In the Author Documentation page, find out the meaning of the answer format tag AL_.

Using tagit (and zipit) to format homework problems

It is quite possible to insert the WHS tags into the problem by typing them directly in, but we will avoid that by using the word tagit from the MCtools package. In order for tagit to format correctly, the MCtools[zipit] must be used to filter the html file that you export the homework source worksheet to.

You can refer back to the section on using zipit for how to do that.

Tagit has several options, but the ones which used most often are problem=, anstype=, rightanswers=, and, when you have multiple choice question , answers= .

Note the form of each option below.

problem = is a list. In this case the list has only one expression in it. That expression is itself a list whose only element is a string. Generally, the elements of problem= are lists, names, or plot structures. If an element of problem= is a list, then its elements are strings, or names, or plot structures.

anstype = is a name, which tells WHS which of its many answer formats you want to use for this problem.

answers = is a list of alternative answers to the question (for multiple choice questions)

rightanswers = is a list of 0's and 1's in case the answer format is a multiple choice format. It has the same length as answers=, a 1 indicates a correct answer, 0 an incorrect answer.

Here are some ways to format the lawnmowing problem or variations thereon with tagit.

Exercise. type mctools(tagit); in the input cell to get the complete list of options it has.

>

>

The help message for tagit is rather terse. In practice, it is useful to help jog your memory about the syntax of the word.

What follows are several examples illustrating options that are built into tagit.

anstype=radio

This is a multiple choice answertype. You supply a list of alternative answers in the option answers=, and indicate the correct answer in the option rightanswers=. In the WHS problem, a row of radio buttons appears with the alternatives (unformatted) beside them.

> tagit(
[problem=[["Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"]],
anstype=radio,
answers=[5,1/3+1/2,1/(1/3+1/2),2.5,1/5],
rightanswers=[0,0,1,0,0]]);

QM_[0;6/5]

AH_[0]

Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AL_[6/5;5;2.5;5/6;1/5]

SKIP_

>

Look at the output from tagit. It starts with a WHS tag QM_[0;5/6]. This signals the start of a problem. Inside the square bracket are two items separated by semicolons. The last item is the correct answer. On the next nonblank line is an AH_ tag which we haven't talked about. Then comes the statement of the problem. Then comes the answer tag AL_[5/6;2.5;1/5;6/5;5]. AL means radio button and inside the square brackets are the various alternatives we listed.

inserting a diagram

How do we insert a diagram into a WHS problem constructed with tagit? Well, construct and name it as shown in the Introduction to Maple, then just put the name into the problem= list. For example, for the yard problem, we could draw a rectangular yard, divided into 6 equal strips, then show Bill mowing 1/3 of it and Jim mowing 1/2 of it, with the title "After one hour".

> yard := plottools[polygon]([[0,0],[6,0],[6,1],[0,1]],color=green):
sticks := plot({[[1,0],[1,1]],[[2,0],[2,1]],[[3,0],[3,1]],[[4,0],[4,1]],[[5,0],[5,1]]},color=red):

> Bill := plottools[polygon]([[0,0],[2,0],[2,1],[0,1]],color=yellow):

> Jim := plottools[polygon]([[3,0],[6,0],[6,1],[3,1]],color=turquoise):

> Blabel := PT([1,1/2],"Bill mows 1/3"):
Jlabel:= PT([4.5,1/2],"Jim mows 1/2"):

> pic := plots[display](sticks,Blabel,Jlabel,Bill,Jim,yard,axes=none,
title="After one hour"):

> pic;

> tagit(
[problem=[["Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"],pic],
anstype=radio,
answers=[5,1/3+1/2,1/(1/3+1/2),2.5,1/5],
rightanswers=[0,0,1,0,0]]);

QM_[0;6/5]

AH_[0]

Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AL_[5/6;2.5;6/5;1/5;5]

SKIP_

>

randomize=no

In the above problem (and in all multiple choice formats), observe that the order the alternatives are listed in is changed from the original order That is built in. You can turn off changing the order of the alternatives by setting the option randomize =no.

Exercise: Select the tagit line in the problem above, then copy it with control C (or from the Edit Menu). Then put your cursor in the input cell just below and paste the problem into the line either with control V or from the Edit Menu. Then insert the option hidden=yes into the tagit line and re-execute the line.

anstype=radiospread (Alabels= )

This type is used when you want formatted mathematics. If your list of alternatives is too long for one line, you can increase the number of lines by changing the matsize option. Say you have 5 alternatives and you wanted two rows, set matsize=[2,3] or matsize=[2,4].
You can change the labels from A, B, etc to whatever with the option Alabels= Just make sure you have enough of them. Here we have changed the labels to small roman numerals.

> tagit(
[problem=[["Bill mows a yard in 5 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"]],
anstype=radiospread,
answers=[7,1/5+1/2,1/(1/5+1/2),7/2,1/7],
Alabels=["i)","ii)","iii)","iv)","v)"],
matsize=[3,2],
rightanswers=[0,0,1,0,0]]);

QM_[0;ii)]

AH_[0]

Bill mows a yard in 5 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AL_[i);ii);iii);iv);v)]

SKIP_

Exercise. Remove the matsize option from the tagit line above and re-execute the line. What is the difference in the formatting of the problem?

anstype=selection

This displays the alternative answers in a scrollbox menu. Comes in especially handy when you want students to choose the right word to complete a sentence

> tagit(
[problem=[["Bill mows a yard in 3 hours. So his mowing rate is 1/3 of the yard "]],
anstype=selection,
answers=["per hour.","per minute.","per day."],
rightanswers=[1,0,0]]);

QM_[0;per hour.]

AH_[0]

Bill mows a yard in 3 hours. So his mowing rate is 1/3 of the yard

AS_[per hour.;per minute.;per day.]

SKIP_

Exercise. Add the item "per year" at the end of the answers= list and modify rightanswers appropriately. Then rexecute the line.

hidden=yes option

You can tell WHS not to show the correct answer to a problem, but just to indicate whether the answer is correct. This is useful in quizzes, and other settings. The option can be used with all of the anstypes, not just radio.

Exercise: Select the tagit line in the problem above, then copy it with control C (or from the Edit Menu). Then put your cursor in the input cell just below and paste the problem into the line either with control V or from the Edit Menu. Then insert the option hidden=yes into the tagit line and re-execute the line. The only difference is that an N is appended after the underscore on the QM line. The effect is to hide the correct answer from the student. Of course, if it is a mulitple choice problem, then a determined student can still discover the correct answer by repeated submission.

>

>

>

anstype=checkbox

This option is used if you have more than 1 correct answer, say the fractional form of an answer and its decimal equivalent. In the problem below, we have the correct answer in the 3rd slot and a close approximation to it in the 5th slot, so righthanswers= has been set to [0,0,1,0,1]. Some indication that more than one answer is possible should be given in the statement of the problem or in the instructions at the top of the homework.

> tagit(
[problem=[["Sally mows a yard in 3 hours. Jim can mow the same yard in 5 hours. How many hours does it take them to mow it together? Assume they do not slow each other down when they work together.
Check each box that has the correct answer in some form or a close decimal approximation."]],
anstype=checkbox,
answers=[8,evalf(1/3+1/5,3),evalf(1/(1/3+1/5),3),(1/3+1/5),1/(1/3+1/5)],
matsize=[2,3],
rightanswers=[0,0,1,0,1]]);

QM_[0;;;;15/8;1.88]

AH_[0]

Sally mows a yard in 3 hours. Jim can mow the same yard in 5 hours. How many hours does it take them to mow it together? Assume they do not slow each other down when they work together.

Check each box that has the correct answer in some form or a close decimal approximation.

AB_[8;8/15;.533;15/8;1.88]

SKIP_

pretext=

The pretext option is used to put text or diagrams before the question number in the WHS homework.

Exercise. One might like to put the above sentence "Check each box .... approximation." in front of the question number. Do this by inserting the option pretext=[["Check each box ... approximaton."]] into the tagit line, and re-executing the tagit line. (Also remove the the sentence from its position in the problem= option.)

anstype=checkboxspread

This answer type is like radiospread in that the alternatives are labelled and put into a spreadsheet for automatic formatting. But more than 1 correct alternative is possible as in the case of checkbox.

Notice the use of the backquote symbol around digits in the answers= line. This turns 2 and 3 into symbols (rather than numbers) and is done to keep Maple from carrying out the simplification of 1/3+1/2 to 5/6. (Check the input line just below to see what I mean.)

> 1/3+1/2= 1/`2`+1/`3`;

> tagit(
[problem=[["Sandra mows a yard in 3 hours. Bill can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"]],
anstype=checkboxspread,
answers=[5,1/`3`+1/`2`,1/(1/`3`+1/`2`),5/6,6/5],
matsize=[2,3],
rightanswers=[0,0,1,0,1]]);

QM_[0;;;;D;E]

AH_[0]

Sandra mows a yard in 3 hours. Bill can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AB_[A;B;C;D;E]

SKIP_

Exercise: Change matsize to 5 by 1 (ie matsize=[5,1]) and re-execute. What is the effect?

displaying a mathematical expression or diagram

If you have a mathematical expression that you want to be displayed nicely (not in calculator syntax), don't put it in quotes. Instead break it out of the problem statement and list it by itself. So if the problem was [["What is the value of 1/(1/a + 1/b) for a = 2 and b = 3?"]], you would break out the expression as follows [["What is the value of "],1/(1/a + 1/b),["when a = 2 and b = 3?"]]

> tagit(
[problem=[["What is the value of "],1/(1/a + 1/b),["when a = 2 and b = 3?"]],
anstype=radio,
answers=[1/5,2/5,1/6,5/6,1],
rightanswers=[0,0,0,1,0]]);

QM_[0;5/6]

AH_[0]

What is the value of

when a = 2 and b = 3?

AL_[2/5;1;1/5;5/6;1/6]

SKIP_

Exercise. Suppose the problem were [["Write 1/(1/2 + 1/3) as a simple fraction."]]. How would you break out the expression in this case? Hint: Look at the example illustrating checkboxspread.

Now, suppose you wanted to display the expression nicely 'inline', that is on the same line with the text. This can be done with the MCtools word lineit. Lineit can be used inside the problem option (or the pretext or aftertext options). The input for lineit is a list of items, each of which is a list, a plot structure, or a mathematical expression..

> tagit(
[problem=[lineit([["What is the value of "],1/(1/a + 1/b),["when a = 2 and b = 3?"]])],
anstype=radio,
answers=[1/5,2/5,1/6,5/6,1],
rightanswers=[0,0,0,1,0]]);

QM_[0;5/6]

AH_[0]

lt_table gt_tr_td_

What is the value of

dt_td_

dt_td_

when a = 2 and b = 3?

dt_

rt_ lt_/table gt_

AL_[1/5;1;1/6;2/5;5/6]

SKIP_

The formatted problem has some HTML pretags (they are converted to real HTML tags by zipit) which make a the line a table with one line and three cells. The middle cell in this case has the mathematical expression. You can't really see the effect until you look at the posted WHS homework.

You can use lineit to place a diagram to the side of the problem. Here's an example.

> tagit(
[problem=[lineit([["Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"],pic])],
anstype=radio,
answers=[5,1/3+1/2,1/(1/3+1/2),2.5,1/5],
rightanswers=[0,0,1,0,0]]);

QM_[0;6/5]

AH_[0]

lt_table gt_tr_td_

Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

dt_td_

dt_

rt_ lt_/table gt_

AL_[6/5;5;1/5;2.5;5/6]

SKIP_

anstype=textbox (precision=`.01` and txtboxsize=3)

This type is the first non-multiple choice answer format. When this is used as the answer type, then WHS puts a textbox in the homework where the student is expected to type a numerical answer.

The option rightanswers is set to the correct answer in this answertype, instead of a list of 0's and 1's.

There are two additional options which are important when anstype=textbox

precision= This option determines the error you will permit on the student input. The default is precision=0. The student has to type the exact answer in in this case. Use this only when the answer is an integer. If precision=1/100 or precision=`.01` then if the student answer is within 1/100 of the right answer WHS counts it as correct.

txtboxsize= This option determines the length of the textbox the student is given to type in the answer. The default is 3 characters. If txtboxsize=10, then a textbox is created that will allow the input of a 10 digit number.

>

> tagit(
[problem=[["Sandra mows a yard in 3 hours. Bill can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.) "]],
anstype=textbox,
rightanswers=6/5,
precision=`.01`,
txtboxsize=5]);

QM_[.01;6/5]

AH_[0]

Sandra mows a yard in 3 hours. Bill can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AC_[5]

SKIP_

anstype=wordbox

This type is also a non-multiple choice answer format. When this is used as the answer type, then WHS puts a textbox in the homework where the student is expected to type a string answer.

The option rightanswers is set to the correct answer, which should be enclosed in double quotes. Alternative versions of the correct answer are separated by #'s within the double quotes.

When anstype is wordbox, you will usually need to set txtboxsize.

txtboxsize= This option determines the length of the textbox the student is given to type in the answer. The default is 3 characters. If txtboxsize=10, then a textbox is created that will allow the input of a 10 character word.

>

> tagit(
[problem=[["Sandra mows a yard in 3 hours. Bill can mow the same yard in 2 hours. So together, assuming they do not slow each other down when they work together, they can mow the yard in 1.2 "]],
anstype=wordbox,
rightanswers="hours#hrs#hours.#hrs.",
txtboxsize=8]);

QM_[0;hours#hrs#hours.#hrs.]

AH_[0]

Sandra mows a yard in 3 hours. Bill can mow the same yard in 2 hours. So together, assuming they do not slow each other down when they work together, they can mow the yard in 1.2

AW_[8]

SKIP_

anstype=mfunctionbox (and funstuff= )

This is an answer format that requires the student type in an expression, in calculator syntax, with one or more variables. For example, on the lawnmowing problem, if we let x be the time that it takes Bill to mow the yard and y be the time it takes Jim to mow the yard, then the expression 1/(1/x + 1/y) represents the time it takes to mow the yard when both Bill and Jim are mowing. The WHS formatting of a question designed to elicit this expression (or an equivalent expression) from the student might look like this:

> tagit([
problem=[["Bill can mow a yard in x hours. Jim can mow the same yard in y hours. How many hours does it take Bill and Jim to mow the yard together, assuming they do not interfere with each other?"]],
anstype=mfunctionbox,
funstuff="2;x;y;5;1;3;3;5",
txtboxsize=15,
rightanswers=1/(1/x+1/y),
precision=`.01`]);

QM_[.01;1/(1/x+1/y)]

AH_[0]

Bill can mow a yard in x hours. Jim can mow the same yard in y hours. How many hours does it take Bill and Jim to mow the yard together, assuming they do not interfere with each other?

AE_[15;2;x;y;5;1;3;3;5]

SKIP_

The option funstuff = is used to tell WHS how to check to see if the student answer is correct. The right hand side of funstuff = is a string of semicolon separated values. The first value (2 in this case) tells how many variables the expression has. This is followed by the names of the variables (order is not important here). Then comes a positive integer (5 in this case) followed by a sequence of intervals (in this case two intervals 1;3 and 1;5). WHS compares the values of the student answer and the correct answer at 5*5 randomly selected points (x,y) where x is between 1 and 3 and y is between 3 and 5.

If they all differ by no more than the designated precision (in this case `.01`), the student answer is deemed correct. Note: Surprizingly, this actually works most of the time.

anstype=spreadbox or anstype=tablebox (ansmat= incidmat=)

These answer formats are similar. They provide a rectangular array of boxes, where some of the boxes are textboxes to be filled in by the student. (and the rest are filled in already. One use of the format is to have the student fill in a table of function values. Note: With this answer format, use the option ansmat to record the complete rectangular array you want, and use the option incidmat to indicate which entries you want the student to type in (1 means the student should fill it in.)

>

> tagit([
problem=[["Complete the following table of function values:"]],
anstype=spreadbox,
txtboxsize=3,
ansmat=matrix([[x,f(x)=x^2+x],[1,2],[2,6],[3,12],[4,20]]),
incidmat=matrix([[0,0],[0,1],[0,0],[0,1],[0,1]])
]);

QM_[0;2;12;20]

AH_[0]

Complete the following table of function values:

TABLEOPTIONS_border=0_

TABLEOPTIONS_border=0_

SKIP_

anstype=crossword (uses ansmat and incidmat)

This answer format provides a rectangular array of boxes, where some of the boxes are wordboxes and possibly some are textboxes. It can be used to create crossword puzzle like questions. The format uses ansmat and incidmat to tell WHS what goes in the crossword array. Each cell of the array is a list. If an entry of incid is 0, the the corresponding cell in ansmat has 1 or 2 items, the first a string and the second a cell color. If the entry of incid is 1, then the corresponding cell in ansmat has 4 or 5 items: the first is a The formatted problem has HTML pretags which must be filtered by zipit before you install the homework.

> tagit([
anstype=crossword,
problem=[["Complete the crossword puzzle."]],
ansmat=matrix([
[[ "Roses are" ,"pink"],["W",8,"red#Red#RED",red]],
[[ "Violets are" ,"turquoise"],[ "W",8,"blue#Blue#BLUE",blue]],
[[ "Sugar is" ,"yellow"],[ "sweet" ,"gray" ]],
[[ "And so are" ,"violet" ],[ "you" ,"tan"]]]),incidmat=matrix([[0,1],[0,1],[0,0],[0,0]]),
tableoptions="border= 1"]);

QM_[0;red#Red#RED;blue#Blue#BLUE]

AH_[0]

Complete the crossword puzzle.

lt_table border= 1 gt_

tr_

lt_td bgcolor=pink gt_

Roses are

dt_

lt_td bgcolor=red gt_

AW_[8]

dt_

rt_

tr_

lt_td bgcolor=turquoise gt_

Violets are

dt_

lt_td bgcolor=blue gt_

AW_[8]

dt_

rt_

tr_

lt_td bgcolor=yellow gt_

Sugar is

dt_

lt_td bgcolor=gray gt_

sweet

dt_

rt_

tr_

lt_td bgcolor=violet gt_

And so are

dt_

lt_td bgcolor=tan gt_

you

dt_

rt_

lt_/table gt_

SKIP_

We can use crossword to do tables of function values.

> tagit([
anstype=crossword,
problem=[["Complete the table of function values."]],
ansmat=matrix([
[[ x ],[f(x)=x^2 + x]],
[[ [1] ,"yellow"],[ "C",8,2,blue]],
[[ [2] ,"yellow"],[ "C",8,6,blue ]],
[[ [3] ,"yellow"],[ "C",8,12,blue ]]]),
incidmat=matrix([[0,0],[0,1],[0,1],[0,1]]),
tableoptions="border= 1"]);

QM_[0;2;6;12]

AH_[0]

Complete the table of function values.

lt_table border= 1 gt_

tr_

lt_td bgcolor=white gt_

dt_

lt_td bgcolor=white gt_

dt_

rt_

tr_

lt_td bgcolor=yellow gt_

1

dt_

lt_td bgcolor=blue gt_

AC_[8]

dt_

rt_

tr_

lt_td bgcolor=yellow gt_

2

dt_

lt_td bgcolor=blue gt_

AC_[8]

dt_

rt_

tr_

lt_td bgcolor=yellow gt_

3

dt_

lt_td bgcolor=blue gt_

AC_[8]

dt_

rt_

lt_/table gt_

SKIP_

>

>

anstype = ungraded

Use this answer format to allow students to submit a sentence or two in answer to a question. WHS does not count the problem as correct or incorrect, but the teacher can access the answer and 'grade' it by hand. Probably not the answer format to use with a class of 300 students!

> tagit([
problem=[["State the theorem of Pythagoras."]],
anstype=ungraded,
txtboxsize=3]);

QN_[0;Your answer will be mailed to the teacher.]

AH_[0]

State the theorem of Pythagoras.

AX_[3]

SKIP_

multipart problems

tagit will construct problems with more than one part. WHS counts the problem as correctly answered only when each part is correctly answered in one submission. So, students must be told this. If the answers are not hidden

Say you have 3 parts in the same problem. The syntax is tagit([part 1],[part 2],[part 3]). Here's a two part example:

> tagit(
[problem=[["Choose the selection which makes the following sentence true: Two lines in a plane are "]],
anstype=selection,
answers=["parallel","perpendicular"],
rightanswers=[1,0]],
[problem=[[" if they "]],
anstype=selection,
answers=["intersect","do not intersect"],
rightanswers=[0,1]]);

QM_[0;parallel;do not intersect]

AH_[0]

Choose the selection which makes the following sentence true: Two lines in a plane are

AS_[parallel;perpendicular]

AH_[0]

if they

AS_[intersect;do not intersect]

SKIP_

Exercise. The options pretext= and aftertext= are used to insert text in front of the question and after the answer, respectively.
i) Modify the tagit line above so that the sentence "Choose the selection ... sentence true:" occurs before the question number. You can do this by removing that sentence from the problem= option, and inserting the option pretext=[["Choose ... true."]] into the first list in the tagit line.
ii) Then modify the tagit line so that answers=["do","do not"]. This means you will need to insert the option aftertext=[[" intersect."]] into the second list in the tagit line.

Homeworks with versions -- Parameterized problems

What and why?

A homework with versions is a homework which has a certain fixed number (5 is a good number) of versions of each problem in it. You signal to WHS that a homework has versions by including the number of instances in the header tag: for example, a homework with 5 versions of each problem would have the header tag H_[5]. When the homework is retrieved, if you do not specify a version number for the homework WHS pick a version number and constructs the assignment by selecting at random one version of each problem and then randomizing the order in which those problems are listed in the assignment. In either case, you can retrieve the same homework again by retrieving the assignment with that version number. This allows for individual homework assignements for students, and also allows for additional practice for them.

How do you construct 'versions' of a problem? You would parameterize the problem, and solve the parameterized problem. Then you would define a problem generator , a Maple procedure whose inputs are the parameters of the problem and whose output on any given call is a formatted WHS problem with whatever values you gave the parameters in the call. Then the versions of the problem would be obtained by a specified number of calls to the procedure, changing one or more of the values of the parameters each time.

Construction of a problem generator

We will illustrate a procedure for turning an ordinary problem formatted with tagit into a problem generator by using our familiar lawn mowing problem: Bill mows a yard in 3 hours, Jim mows it in 2. How long for both to mow it? We have already parameterized it and solved the more general parameterized problem.

Parameterized Problem. Bill mows a yard in A hours. Jim can mow the same yard in B hours. How long does it take them to mow it together?

Solution.

As before let x be the time needed when both are working together. In one hour, Bill mows of the yard and Jim mows

of the yard. Together, they would mow of the yard. Solving for x, we get =

What we have below is the solution to this problem with A=3 and B=2, which is copied down from above. We will show how to turn this into a procedure for formatting different versions of the parameterized problem.

> tagit(
[problem=[["Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"]],
anstype=radiospread,
answers=[5,1/3+1/2,1/(1/3+1/2),2.5,1/5],
rightanswers=[0,0,1,0,0]]);

>

The method for turning a specific problem into a procedure is illustrated here. The procedure is named in the proc line , which goes at the top (no punctuation at the end of the proc line). The parameters of the problem (A and B in this case) are enclosed in paraentheses after the word proc. Note how the parameters occur in the body of the procedure (the tagit line, in this case). We have replaced 3 by the parameter A and broken it out of problem statement, that is 3 has been replaced by ",A,". This needs to be done, because an A inside of a string is just printed as A. Note that we did not replace 3 by "],A,[", although we could have. I just like it better the first way.
Note how the answers line has changed: When we make up alternative answers, we try to think of ways the problem could be worked wrong. So for example, one might just add the times it take each boy (A+B), or add the rates at which they worked (1/A + 1/B). We just then put those into the answers line, and mark the right answer.

The at the bottom goes the end proc statement (ending with a semicolon)

> lawn := proc(A,B)
tagit(
[problem=[["Bill mows a yard in ",A," hours. Jim can mow the same yard in ",B," hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)"]],
anstype=radiospread,
answers=[A+B,1/A+1/B,1/(1/A+1/B),(A+B)/2,1/(A+B)],
rightanswers=[0,0,1,0,0]])
end:

Now to get specific instances of the problem formatted for WHS, make calls to the procedure as shown below.

> lawn(3,2);

QM_[0;C]

AH_[0]

Bill mows a yard in 3 hours. Jim can mow the same yard in 2 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AL_[A;B;C;D;E]

SKIP_

>

When we are doing multiple choice problems, we can put names in for the parameters as in.

> lawn(x,3);

QM_[0;B]

AH_[0]

Bill mows a yard in x hours. Jim can mow the same yard in 3 hours. How many hours does it take them to mow it together? (Assume they do not slow each other down when they work together.)

AL_[A;B;C;D;E]

SKIP_

>

Exercise: There is nothing that says you can't make things other than numbers into a parameter for the problem. ( Such a parameter would only affect the statement of the problem and would not affect the rightanswers. So you could call it a formatting parameter .) For example, we could parameterize the name of the first mower. Copy the current lawn procedure into the input cell below and change the defintion to break out the name of the first mower as a third parameter lawn(A,B,name). Check your procedure by making the call lawn(3,2,"Jack");

>

>

>

Creating a WHS homework from scratch

How do you create a WHS homework from scratch? Here is a procedure I use.

1 Open a Maple worksheet

2 Copy the latest MCtools you have (At least June 8, 2002) into the new worksheet. Do this by opening a worksheet with an MCtools section, then using the cursor, right click the boundary of the section icon to select the section without opening it. (The border will disappear), then copy with a control C, then put your cursor in the new Maple worksheet and paste the section in.

3. Click on the Format Menu at the top of Maple to open it, then click Merge Existing at the lower left of the Style Management Window that opens up. A window titled Merge Style Set opens up. Browse in there to find a previous homework (comath10.mws will do). Highlight it and press the Open button, then press Done. This is a convenient way to modify the styles so that zipit will work as it is supposed to.

4. Make a title for the homework. This involves putting a heading tag H_ on one line, then the title (plus whatever introductory text you want), then a SKIP_ tag on its own line so that you can compute and jot notes without it appearing in the homework.

5. I like to put problems in their own sections so that I can eliminate them from the homework easily (just close off the section). So I create a problem template like the one below. These can be as elaborate as you want. Another way is to copy a problem template from a previous homework, or just load an entire homework worksheet, and proceed to eliminate the problem sections you don't want to modify.

problem generator template

>

> prob := proc(a)
tagit(
[problem=[[a,"+1=?"]],
anstype=textbox,
rightanswers=a+1]
)
end:

>

>

Then you can make a copy of the problem template section and paste it in however many problems you plan for the homework.

Now you are ready to crank out the homeworks.

MCtools (June 8, 2002)