Authors choose which FluidMath capabilities are enabled in a document.
Welcome to FluidMath!
FluidMath provides a fluid user interface to mathematics. It brings computations and interactive visualizations to your math writing surface.
FluidMath's features currently include:
The tutorial. This tutorial will get you up and running quickly through a series of basic lessons. It should take about 5-10 minutes.
Live practice area. Note to the right of the tutorial is a "live", interactive FluidMath region in which you can try out concepts presented in this tutorial.
Which devices? FluidMath is designed for use on iPads, Interactive Whiteboards, and Tablet PC's. It may also work on other devices.
FluidMath provides computation and computer algebra system (CAS) support.
In general, to numerically approximate an expression, write a "double arrow" to the right of the expression.
To simplify an expression, write a "single arrow" to the right of the expression.
See the image to the right for examples.
Try it! Try writing a few expressions and numerically approximating and simplifying them now.
Practice: What is the numeric approximation of ? What does simplify to?
FluidMath can solve for one or more variables.
To solve for a variable, first write some math that uses the variable and then write a new expression consisting only of the variable you want to solve for. Next write a single-arrow or double-arrow to the right of the variable and the solution will be shown.
In the example to the right, 'x' is being solved for on the top and 'a' and 'b' are being solved for on the bottom.
Try it! Try writing a few math expressions and solving for one or more variables now.
The "Clear" button is useful but it erases everything on the screen. Sometimes you just want to edit a few strokes.
FluidMath provides a "scribble-erase gesture" that can erase just a few strokes. A gesture is a stroke that is interpreted as a command rather than a stroke that is left on the page.
To erase a stroke, scribble over it. In this example, the scribble and the "3" symbol will disappear leaving just the "2" symbol.
Let's try it! First click "Clear" to erase the page. Then write the number "23". Next, draw a "scribble-like" stroke back and forth several times over the "3", as shown in the image to the right. The "3" should disappear.
Be sure to touch the ink of the stroke you want to erase. Also, be sure to go back-and-forth several times (3-4 times) or the stroke may be interpreted as math rather than an erase gesture.
Tip: If the scribble stroke and the target strokes do not disappear and the scribble mark is added to the page, then just try again. Draw another scribble stroke in the same place but try adding more "back-and-forths" and/or sharper turns as illustrated in the figure to the left.
Practice: Now practice writing some multi-symbol math expressions and then scribbling out one or more symbols. For example, try writing "y = x + 2" and then change the math expression to "y = x - 4".
This lesson will teach you how to write a single symbol.
The right half of this page is a "live" FluidMath page. In it, write the number "2".
In general, when you finish writing a math stroke several things happen. First, strokes close to each other will be grouped into a single math expression. Each math expression will have a yellow box drawn around it.
The expected result from drawing a "2". The yellow box contains the expression's strokes. The blue typeset indicates the system's recognition of the strokes. And the green box is a drag handle for moving the expression.
Also, when you stop writing, a green progress bar is shown below the math expression. When it completes, the recognition result will appear in blue typeset below your writing. As you use FluidMath more, you will just glance at the blue typeset as you work to verify your strokes were recognized correctly, or to know there is a problem that needs correcting.
Notice too the green square at the top-left of each yellow-box. You can move a math expression by clicking and dragging the green square.
Returning to our goal of writing and recognizing a "2", if the recognition is wrong press the "Clear" link (at the top-right of the page) and try again.
Practice: Once you get a "2" recognized, please try writing a few more individual symbols — maybe a "3", "8", and "y". At first write one symbol at a time and push the "Clear" link after each is recognized.
Next try writing multiple symbols on the same page. Notice if you write symbols near each other that they are grouped into a single math expression (visually indicated by a yellow box that contains neighboring symbols) and if you write symbols far apart there will be multiple yellow boxes, each containing the strokes of separate expressions.
For your reference, below are demonstrations of the allographs supported in FluidMath for each of the basic symbols. Strokes have been exaggerated to show when one, two, or three strokes were used.
FluidMath will recognize additional math symbols as well, but the most common ones are shown here.
You now know the techniques needed to write and edit math expressions:
Practice: Try writing some increasingly complex math expressions. First try writing some numbers or simple assignments like "a=12". Then move on to simple and complicated expressions like fractions, exponents, square roots, etc.
1) A "plus" symbol was recognized as an "f". 2) To correct the problem, scribble erase the misrecognized strokes and rewrite them. 3) The plus symbol is recognized correctly.
Editing expressions. At times, you will want to modify an expression. This is a powerful feature of FluidMath because all calculations and graphs on the page will update in response to the change. You can perform the modification by simply scribbling out the symbols you want to change and then writing new symbols.
Correcting a misrecognition. Sometimes, FluidMath will not recognize a part of an expression correctly. As an example, see the top image to the right. Although the handwritten strokes look somewhat like a "plus", it was recognized instead as a "f".
If this happens, you do not need to erase the whole expression. Instead, first find the incorrect symbol in the blue typeset and then locate the strokes associated with that symbol. Scribble erase those strokes as shown in the middle image, and then rewrite the symbol as shown in the bottom image.
Graphing is easy with FluidMath!
To graph an expression, first write the expression, and then draw the graphing gesture. Plot a point by writing a coordinate pair and then graphing it.
The sequence of images below demonstrate the process. (To zoom in on the illustration, click it and then click 'back' to return to this page.)
1) Start graph gesture inside yellow box near equal sign. 2) Graph gesture should have looping-curve to it, as shown. 3) End graph gesture on existing graph or empty part of page to create a new graph.
The graphing gesture is a stroke that starts inside the yellow bounding box and ends on an empty part of the page. See for example the orange stroke in the second image from the top to the right.
The graphing gesture is a stroke that starts inside an expression's yellow bounding box, has a looping shape like a "U", and ends on an empty part of the page.
FluidMath can graph inequality equations such as y > x or y ≤ 2x + 3.
Try it! Try graphing the expression "y = x + 1".
NOTE: If you draw a graphing gesture and do not get a graph, then scribble erase the stroke left on the page and try drawing the graphing gesture again. If you continue to not get a graph, try starting your graphing gesture on top of the equal sign.
Interacting with Graphs. Graphs can be interactively manipulated.
Deleting a graph. To delete a graph, click the "X" button in the top-right of the graph window. You can also scribble-delete a graph.
Table of Values. Click the "Table" button to view a table of values for the expressions in a graph. Click on the top-left cell to change the min/max/step values for the table.
Graph menu. Click a graph's ▼ button for a list of additional graph commands including reset view, edit/import data points, toggle features (intersections, intercepts, local min/max), and axis style (e.g., radian tick marks).
Import/Edit data points. You can add or edit a graph's data points by clicking the graph's menu ▼ and then "Edit Data". Next, type or paste coordinate pairs into the pop-up window that appears. For example, paste data from an Excel spreadsheet you access on your iPad. Data must be in 2-column format and represent x, y coordinate pairs. To edit data later, select the "Edit Data" menu item and you can edit the data points you entered previously.
Advanced - empty graph gesture. Draw a large pair of axes ("+"-like shape) to create an empty graph at a particular location and size.
Sliders provide a way to quickly vary the value of a parameter. One useful application of sliders is varying one or more parameters of a graphed math expression and observing how the plot responds.
To create a slider in FluidMath, first write a simple assignment of the form "variable = value", such as "b = 3". Next, to create a slider for "b" draw the "slider gesture". The slider gesture is a stroke that starts on the equals sign, extends directly down, and ends outside the expression's yellow box. The expression "b=3" will be replaced with a slider.
The slider gesture is a stroke that starts on an equal sign, extends directly downwards, and ends outside an expression's yellow bounding box. The slider gesture only applies to expressions of the form "variable = value".
1) Start slider gesture inside yellow box on the equal symbol. 2) Draw slider gesture straight down from equal sign. 3) End slider gesture outside yellow box below math expression.
A slider widget replaces the math handwritten "b=3" math expression.
Try it! Press the "Clear" button and write "b=3" then draw the slider gesture to create a slider for "b".
To vary the value, click on the tear-drop-shaped widget handle labeled with the variable name and drag it left or right.
To change the minimum and maximum of the slider's range, click and drag the axis background horizontally to offset the range. Click and drag the background vertically to scale the range.
Slider values can be continuous or constrained to discrete or values at tick marks. Click the slider's ▼ button and then the "Values" option to review the current mode (a checkmark will appear next to it) or to set the current mode.
To move the slider, click on the titlebar and drag it to the desired position.
To delete the slider, click on the "X" button in the top-right of the slider's window.
Bring your sketches to life by applying math expressions to sketches!
Example math sketch. Consider the following example problem:
Car a is moving with constant velocity and car b is moving with constant accelleration. At what time t will car b catch up to car a?
To setup the problem, two cars were drawn, labeled, and the math equations describing the motion of the two cars was written on the page. A slider for t was created and then you can either run the animation or manually drag the t slider widget to explore the state of the animation for various values of t. Graphical and numeric representations of the equations can further help investigate this problem.
How to create your own animation:
iPad users. You can send your FluidMath page to others as bitmap images. Capture your FluidMath screen by pressing-and-holding the iPad home button and then pressing the suspend button. An image of the screen will be added to your Photos. You can then email or share the image as you would any photo.
Mac OS users. You can capture the entire screen by pressing Command-Shift-3. Or, you can capture a specific part of the screen by pressing Command-Shift-4 and dragging a rectangle to surroung the area you want to capture.
Windows users. Snipping Tool is available on Windows 7. Pressing Alt-PrtScn will copy the active window to your clipboard. Ctrl-PrtScn will capture the entire screen to your clipboard.
Additional options for sharing FluidMath content are in the plans.
Turn on this feature (in the "..." menu at the top) to set the background color of equivalent expressions or equations to be the same color.
Simplifying an expression. On the left side of the image, the student first tries to simplify, but makes a mistake in the first step and the student gets feedback because the expressions are different colors. On the right side of the image the student corrects the mistake and continues. The student gets feedback that no mistake has been made because all three expressions have the same background color.
Solving an equation. On the left side of the image the student tries to solve the equation, but makes a mistake in the first step and the student gets feedback because the expressions are different colors. On the right side of the image the student corrects the mistake and continues. The student gets feedback that no mistake has been made because all three equations have the same background color.
With this option set, all variable settings are used when highlighting equivalences. With the option not set, only variable settings in parentheses are used.
In the first image, highlight equivalence and automatic context are both turned on, and the user has written two derivations on the same page. Because each derivation has a final line of the form \(x=\ldots\), there are multiple apparent definitions of \(x\) on the page. The system tries to guess which value the user wants, but in this case the user doesn't want anything substituted, and the highlighting appears wrong. In the second image, automatic context has been turned off, and the highlighting is now as expected. In the final image, the user is still able to get a variable substituted by writing the definition in parentheses. (Values defined by sliders are also always substituted.)
This section describes several topics you may want to learn about after you have mastered the previous topics.
"Shrinking Ink". You may find you run out of space after adding content to a page. You can shrink the size of math expressions by converting them from hand-drawn math expressions to more compact typeset ones.
How to shrink ink. An expression must be recognized (i.e., it must have blue typeset showing its recognition below it) before you can shrink it. To shrink an expression click on the blue typeset button at the lower-left of a handwritten math expression. Once converted, you can click-and-drag to move the expression on the page or tap it to edit the math strokes in a pop-up overlay. Scribble over the typeset math expression to erase it.
Shortcut. Click on a letter symbol to create a slider for that symbol.
(Above) Click on blue typeset to convert handwritten math to a more compact form (see below).
Drag-and-drop the typeset math expression widget to move it or add it to a graph. Scribble over typeset math widget to erase it.
Empty graph gesture. Draw a large "+" on an empty part of the page to create an empty graph.
Draw a large "+" on an empty part of the page (see above) to create an empty graph (see below).
Congratulations! You have finished the FluidMath tutorial. You are ready to start using FluidMath. We hope that you enjoy it!
Fluidity Software also provides a stand-alone version of FluidMath.