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How to tour

When you first go to the site, the Navigation window will be visible.(It may be hiding behind and to the left of the Text window; just click on it to see it all. Or, on a PC, you might find a button for it along the bottom of your screen.) At the top of the Navigation window is a pull-down menu listing the various tours available. When you pick one, the Navigation window for that tour replaces the one you just chose from. If you're just beginning that tour, click on Problem statement. Otherwise, click on the appropriate link to resume work where you left off.

Generally in touring you work through the Problem and Modeling sections, then go on to the Explore section, with side trips as needed across to the Concepts section. Then you go to Summary. Finally, you move sideways and down among the three bottom paths until reaching the Critique section. You may go back to Concepts whenever you need to.

You don't have to keep going to the Navigation window to move along on a tour. As long as you're progressing within a section, Next arrows at the bottom of the Text window take you to the next pages of the tour. At the end of each section, you'll find instructions on how to go on to the other section.

You have a lot of freedom in touring. At the top of each Text window are links to the Gallery, Teaching, Help, and a Site map. You can follow links from within the text to more formal definitions of terms or descriptions of investigation techniques. Clicking on any of these links changes what's in the Text window. The background will change accordingly. Use Back on your browser to retrace your steps to the tour.

If you get too lost, just go back to the Navigation window and click on the name of the page you want to see.

Most computer screens are not large enough to have all windows fully visible at once. Try to arrange the windows so that a part of each of them shows; then clicking on that part will bring it to the front of the pile so you can see it. Alternatively, on a PC you can find buttons for the windows at the bottom of the screen.

Navigation window

The Navigation window for each tour gives you a picture of how the tour pages are organized. Clicking on any of the links puts the corresponding pages of the tour into the Text window.

The Navigation window begins to the left of your screen and extends into the bottom left corner. It hides behind the Text window and some times behind the GrafPad. You can make it show by clicking on any part of it.


The GrafPad is for experimenting with graphs. These instructions for using it also appear in the Help menu on the GrafPad itself. Its window resides to the top left of your screen.

The GrafPad has three modes: Construction mode, Trail mode, and Dominating set mode. These are chosen from the top left menu.

Most of the time you'll be in Construction mode.

To draw a new vertex, click the mouse in a blank spot in the drawing area. Try to click without wiggling the mouse too much. GrafPad will not allow you to put a vertex on an existing edge. Vertices are automatically labeled, using upper case letters (A to Z) first, then lower case. You are thus allowed at most 52 vertices. You can choose whether or not to display the vertex labels by clicking on Show Labels or Hide Labels in the Attributes menu.

To draw a new edge, first click the starting vertex of the edge, then click on the ending vertex. When you click on the starting vertex, it should turn pink. It will return to its proper color when you finish the edge.

If you wish, you may insert one or more turning points into an edge as you draw it, by clicking on blank spots around the graph after you click on the starting vertex and before you click on the ending vertex. GrafPad will not allow you to make an edge that passes through a vertex. It will also not permit bends to be too close to one another, nor too close to the edge of the pad.

To move a vertex, click on it, hold, and drag it to the new location. GrafPad will not allow you to put a vertex too near the edge of the pad. If you cause an edge to pass through a vertex that's not a vertex of the edge, the edge will be broken near the intermediate vertex as if to pass under it.

To add bends to an edge, drag a point on the edge to where you want the bend. You can make multiple bends in the same edge. You can also move a bend by dragging it.

To reduce the number of bends, drag one bend to another. This will consolidate the two bends, eliminating any bends in between. You can also drag a bend to the starting or ending vertex. For example, to eliminate all bends, drag the first bend to the ending vertex, or the last bend to the starting vertex.

To color a vertex or edge, drag from the color square of your choice to the part you want to color.

To delete a vertex, drag it to the trash can (at the left end of the color palette). To delete an edge, drag any point on it to the trash can.

Often you use GrafPad to modify a stock graph—either from the Gallery or from a page on a tour. If you wish to restore the graph to its appearance before you started modifying it, select Reset from the Edit menu. GrafPad will ask you to confirm your intention.

To erase the entire graph, select Erase All from the Edit menu. GrafPad will ask you to confirm your intention. Click on the Cancel button if you don't actually want to erase anything.

Every edge you draw has a direction, from its starting vertex to its ending vertex, determined by the way you drew the edge. You can choose whether or not to display arrowheads on the edges, showing their direction, by clicking on Show Arrows or Hide Arrows in the Attributes menu.

If you want to reverse the direction of an edge, you can double-click on the edge.

You can assign weights to the edges of a graph. When you create a graph, the possible weights are whole numbers from 0 through 30. When you first create an edge, the GrafPad automatically assigns it a weight of 15. You can choose whether or not to display weights on (or near) the edges by clicking on Show Weights or Hide Weights in the Attributes menu.

When you arrive at some pages while touring, graphs are put onto the GrafPad with weights in a different range. If you add an edge to such a graph, the edge will be given a weight in the middle of that range.

You can modify weights only when GrafPad is displaying them. To modify a weight, drag it (or the edge's arrowhead, if that is showing) along the straight segment of the edge where you find it.

All of the above takes place in Construction mode. Another mode is Trail mode, chosen from the Mode menu. All vertices and edges are black initially, and labels appear if they weren't already showing. You click on an initial vertex and then on succeeding vertices in a trail. Vertices and edges used on the trail turn blue. If weights are showing, the counter records the total of weights on the trail. Nothing will happen if you try to repeat an edge or click on a vertex that is not adjacent to the last chosen vertex. To begin again with a trail, choose Reset from the Edit menu.

A third mode is Dominating set mode, also chosen from the Mode menu. When you go into this mode, all vertices and edges become black and labels appear. (Original colors will be restored if you move back to Construction mode.) When you click on any vertex, that vertex and all adjacent vertices turn red. A list of vertices you have clicked on appears near the top of the pad.

When you first arrive at many pages, a graph will be drawn for you on the GrafPad. Occasionally you'll find pictures on the main text page that you can click on to have drawn on the GrafPad.

For further exploration, there's a Gallery containing many popular graphs. To see it, click on the button at the top of the Text window. Once there, you can click on any individual graph to have it drawn on the GrafPad.

Caution: Sometimes a graph doesn't appear on the GrafPad right away, or at all, but will show up if you choose Reset from the Edit menu of the GrafPad.


The Stepper is for experimenting with algorithms. Its window is initially behind that of the GrafPad. You might want to move the GrafPad to the top right of your screen when you want the Stepper to be visible as well.

When you arrive at certain pages in the tour, an algorithm is written out on the Stepper. To prepare the algorithm for running, click on the START button. One early instruction of the algorithm will turn blue. If you click on it, that instruction will be executed. To continue to run the algorithm one step at a time, keep clicking on the blue steps. Alternatively, you can click on the STEP button at the top.

To display some of the changes being made while running the algorithm, click on the green dot by Values in the bottom part of the Stepper.

For long algorithms, you may wish to click on the RUN button. The algorithm will run until done or you click on the PAUSE button or the END RUN button. If you want to run an algorithm through many steps, but stopping each time it comes to a particular step, and that step has a circle beside it, you can click on the circle to turn it red. Then click on RUN. The algorithm will run until the marked step is ready for execution. Clicking on RUN again will run through all steps until the marked one is again arrived at. Clicking on the circle again will remove the "stop sign" at the step.

When the algorithm finishes, a message toward the top of the GrafPad tells you the result, such as "Spanning tree in red."

When the algorithm is finished running, click on END RUN.

If the algorithm is being applied to a graph in the GrafPad, you can run many steps by working on the GrafPad yourself. For example, you can run a step such as "Choose any vertex and color it red" by merely clicking on the vertex. You are informed of what to do by a message toward the top of the GrafPad. Doing it yourself gives you a very good way of testing your understanding of the algorithm. If you make a mistake, an error box will appear.

An error box will also appear if the algorithm fails in some way, such as trying to choose a black edge when there isn't one. These messages help you test algorithms.

Caution: On some machines a single click on a step of the algorithm or on the GrafPad may not be sufficient, especially if you're moving back and forth between the two windows. The click merely gets the attention of the tool you want to use. A second click is needed to tell that tool what to do.

Also when running algorithms related to graphs, the GrafPad is under algorithm control from when you click on START until you on END RUN. During that time, the only changes you can make to the graph are those called for by the step being executed.


Note: The Help button on the Torrick at this time asks you to click on Help at the top of the text page. You must have done that, or you wouldnÕt be reading this.

Like the GrafPad, the Torrick can be used either for stand-along explorations or under the control of an algorithm on the Stepper.

Stand alone

To use in stand alone mode, you click on various buttons to control the Torrick's response to what you want to do.

Clicking on the new button opens another window, for specifying how you want to make lists. You have several options.

Mark list of combinations if you wish to list combinations, in which strings of items in different orders are considered the same string. Marking this choice on the specification window means that, when checking your work, the Torrick will report an error if there is more than one string with same characters (perhaps in different order). It will also notes if some combination of the characters is missing.

Mark list of permutations if you wish to list permutations, in which strings of items in different orders are considered to be different strings. Marking this choice will control what errors the Torrick reports when it's checking your work.

When you Type the characters and press enter, you type in digits or upper case letters. Do not use spaces or punctuation between the characters. You can type in repeated characters. For example, if you type AABC, then permutations of length 3 include AAB, AAC, ABA, CAA, etc. Combinations of length 3 include only AAB, AAC, and ABC.

After typing the characters to be used, be sure to press Enter or Return. The string of characters should show up on the slot labeled Current characters.

Minimum and maximum item sizes refer to the lengths of string the Torrick will allow. For all exploration in these tours, these numbers will be the same.

If you mark the box by Check here to allow repeated characters, the Torrick will allow any character to be repeated up to entire length of string. For permutations of length 2 from the string ABC, the Torrick will allow AA, AB, AC, BA, etc. For combinations of length 2 on ABC, it will allow only AA, AB, AC, BB, BC, and CC. If you have marked this item, balls will be replaced after being dropped.

The Distinguish between balls with the same letter/number item tells the Torrick to consider balls as different if they have different colors, even if they are labeled with the same character.

For making lists by hand, you'll usually use the one-tray method. Algorithms sometimes call for the other methods.

Click on OK to accept your specifications. If the specification window doesnÕt close, you have made an error, such as typing in lower case characters.

When the specification window closes, you can explore with the Torrick as described below ("Moving balls").

Control by an algorithm

When the algorithm starts, the Torrick's specifications are set up automatically, including monitor to check on demand. You should ignore Monitor and New buttons.

While the algorithm is waiting to execute a step, you can work with the Torrick itself as described below ("Moving balls"), depending on what method (one-tray, two-tray, or supply shop) has been chosen automatically.

Moving balls

In one-tray method:

Click on balls to have them drop to the ramp.

Click on the Clear button to clear the ramp of all balls if you made an error.

Click on the Dump button to dump balls from the ramp into a bin below.

In two-tray method:

Drag balls from one spot to any empty slot.

Click on one ball after another to interchange them.

Click on the Dump button to dump balls from the first tray into a bin below.

In supply-shop method:

You can click and drag balls between slots on the tray and the supply shop stock.

Click on one ball on the tray after another to interchange them.

Click on the Dump button to dump balls from the tray into a bin below.

For all methods:

Use the Undo button to remove balls from the rightmost bin.

Clicking on the Monitor button allows you to specify whether Torrick will check your list only when you click on Check, or will prevent you from dumping a string that doesnÕt fit the criteria youÕve set up.

The Check button asks the Torrick to check your list, when you've specified that it check on demand.

In checking, besides the specific check for combinations or permutations, the Torrick makes sure pile of balls doesnÕt have more repetition than that specified in the original list or by the check box.

Journal notes

Keeping notes about your touring experience is very valuable for your learning. But Web security restrictions prevent any software you get over the Web from recording extensive notes directly onto your own computer. So we've had to design a slightly contorted system for you. Your journal notes are kept on a temporary Web page, called Journal Notes, at the top center of the screen, almost always hiding behind the other windows.

But you don't need to find them often, because you don't enter the notes directly onto that page. The way you record your journal notes is through the response boxes into which you type your responses to the prompts. Besides responses to the prompts, you can type in anything you want. When you click on the Record button below the last response box on a page, your written thoughts will disappear from the response box and appear on the journal notes page.

The notes are still not saved on your own computer's hard disk, however. If you quit your browser, or the power goes out, they're lost, just like unsaved work in your word processor. To save your notes, click on the Journal notes page, select all of your work, copy it, and then paste it onto a page in your word processor. Then you can save it, edit it, and add graphs or anything else you want to append.


You can find tips about mathematical investigations and ideas for classroom teaching in the inquiry-based way consistent with these materials by clicking on the Teaching button at the top of each Text window.

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