The lens tool provides templates for a Convex lens, a concave lens and a mirror.
Global description
The symmetrical tool could be described as a traditional house where the top part is curved. The first floor is one big window where to top side is curved. The second floor has a window where the top and bottom sides are curved in opposite directions.
In the top and bottom sides of the tool small indents are provided to align the tool on the median light beam.
Pushpin markers are provided at the top surface.
The SinTang tool combines graphs for three functions; the sin(X), the cos(X) and the tan(X). More precise, a half period of the sine graph and a quarter of the tangent graph. The sine graph, is the ‘hill’ contour which is the first half period of the sine graph. The cosine graph is the for 90 degrees shifted to the left sine graph.
When placing the tool looking at the hill, the top side of the tool is a left to right downhill slope. After rotating 90 degrees clockwise it represents a quarter of the tangent graph. The vertical left and right hand sides have an indented centimetre indication. Pushpin markers are provided at the top surface. For sine and tangent values of 1, the distance along the Y-axis is 4 centimetres. These dimensions provide sufficient ‘tactile space’.
Detailed description of the sine tool
The range from 0 to 180 degrees X values is distributed over a length of eight centimetres (aprox. 3 inches). At 90 degrees is the top of the graph (maximum amplitude) at an height of four centimetres. Small indents are provided in the curve at 30, 45, 60, 90, 120, 135, 150 degrees positions. Pen blockers are provided at 0 and 180 degrees positions.
Detailed description of the tangent tool
The tangent part of the tool has to be rotated for 90 degrees clockwise and counter clockwise. The contour of the tangent has small indents indicating the 30, 45 and 60 degrees positions on the X axis. The tan (45) is 1, the height of the graph is 4 centimetres. Pen blockers are provided at the 0 degrees and towards the 90 degrees position. A length of four centimetres along the X axis covers a range of 90 degrees.
Photo: The first half of the sine graph has been drawn, the tool is rotated 180 degrees so the second half can be created.
Draw an X axis with the ruler. Mark the X=0 and X=180 degrees positions. The distance between the two marks is eight centimetres.
Align the sides with the two centimetre indications of the tool with the X axis.
Draw the contour line for the sine from the X value of 0.
Start at the pen blocker to avoid glitches. Draw along the contour until the pen blocker at the X value of 180 degrees which is on the X axis.
Hold the pen on this position and rotate the tool around this pen position so the tool is upside down.
Continue drawing the second half of the tool until reaching the pen blocker again on the X axis.
For convenience place a pushpin at the 180 X position first and rotate the tool around the pushpin.
You can also place pushpin(s) in the pushpin marker positions to hold the tool in place.
Cosine graph
Photo: The first quarter of the cosine graph is drawn. The tool is rotated 180 degrees clockwise, so the ‘hill’ is upside down for the negative part of the graph.
Basically the shape of the cosine has the similar shape as the sine. However the graph is shifted to the left for 90 degrees.
Draw an X axis with the ruler. Mark the 0, 90 and 180 degrees positions. A section of 90 degrees has a length of four centimetres.
Align the tool with the X axis where the right pen blocker is positioned at the 90 degrees position.
Draw along the contour starting at the top (X=0) downwards to the X axis.
Rotate the tool 180 degrees clockwise. Align the tool with the X axis and draw the negative part of the graph from 90 to 270 degrees.
Rotate the tool once again. Align the tool with the X axis and draw the last quarter of the graph up from the X axis on to the top of the graph.
Tangent graph
Photo: The X and Y axis are drawn as well as the first quarter of the tangent graph. The tool is positioned so the second quarter can be created. The pen blocker is hooked with a pushpin.
Draw an X axis with the ruler. Mark the 0, 90, 180, 270 and 360 degrees positions. The distance between each position is four centimetres, which covers 90 degrees.
Draw two asymptote lines: One at 90 degrees and one at 270 degrees.
The distance for the first asymptote line is four centimetres from the 0 degrees position on the X axis, the next one is at a 12 centimetres distance.
For the first section of the graph align the short (four centimetres) side with the X axis. Start at the pen blocker (0 degrees) and draw along the tangent contour upwards until you approach the first asymptote.
For the second section of the graph find the 180 degrees position (preferably provided with a pushpin).
Hook the pen blocker with the pushpin while the tangent curve is pointing down. Align the tool with X axis and draw the lower part of the graph.
For the third section rotate the tool around the pushpin 180 degrees counter clockwise. The tool has the same position as for the first section.
The fourth section is similar to the second section. Find the 360 degrees position, hook the pen blocker to the pushpin and align the tool with the X axis and draw the curve going down untill you approach the second asymptote.
The hyperbole represents the function 1 divided by X (1/x). The tool is L-shaped. Both parts of the L have a length of 10 centimetres. The inner corner of the tool is curved. By nature of the formula, the tails of the tool become very thin. To maintain the sturdiness of the tool, additional material has been added. For alignment purposes, a small square is left out at the lower left corner of the tool (X=0 and Y=0). As well as near the endings of the curve a flat section is provided. The graph contour endings are equipped with a pen blocker. Pushpin markers are provided at the top surface.
Detailed description
The formula represented in this tool is a smooth curve only going down, seen fromm X equals 0 to X equals 8. Along the curve small indents are provided to indicate Y values corresponding with X values equal 0,25, 0,50, 1, 2, 3, 4, 5, 6, 7 and 8 respectively.
For convenience place a pushpin in the crossing of the X and Y axis. When positioning the tool, be aware, the additional material covers the X and Y axis. Therefore, align the tool with the two axis by placing the flat sections at the tails with the axis. Draw along the contour of the tool. Rotate the tool 180 degrees clockwise. Align the tool with the axes and draw the second part of the graph.
Photo: An X and Y axis are drawn. The graph is drawn in the first quadrant. The tool has been rotated 180 degrees clockwise so the second part can be created. A pushpin is at position X=0, Y=0.
The inner space of this U-shaped tool is a parabola. It has the contour of the function X to the power of two. The two poles of the U have indented centimetre indications along the outside. Three medium size alignment indents are provided; one in the middle of the bottom side and one in each of the two poles.
Pushpin markers are provided in the top surface of the tool. The contour of the graph goes down and up again. At the two ends of the contour pen blockers are provided. Indented positions along the curve reflect Y values for X equals -2, -1, -0,5, 0, 0,5, 1 and 2.
Photo: A parabola has been drawn at X=0 and Y=0. The tool is positioned to create a second one at X-3 and Y=2.the ruler serves for the alignment.
Steps
Draw an X and Y axis with the ruler. Align the bottom indent with the Y axis and the ones in the poles with the X axis. Draw along the inner contour of the tool.
Lowest value for Y. You may choose any other coordinate as the minimum for the y value. The centimetre indications help to align the tool with the axes or the ruler.
Photo: The GraphGrid frame with a grid of rubber bands and two axes on the TactiPad.
Photo: The GraphGrid frame provided with the X, Y and Z axis
Although the main application for the GraphGrid is often a regular rectangular grid, there are other applications in which one or more rubber bands can be placed diagonally, in addition to the regular horizontal and vertical axes.
3D coordinate system
By adding a third axis which passes through the intersection of the other two axes (the origin) at an angle between 30 and 45 degrees a 3D coordinate system occurs. In here you can create drawings of three-dimensional bodies.
Shape outline
You can use the rubber bands to compose the outlines of 3D bodies as well, such as a cube or pyramid. Again, you can distinguish between different (visible or invisible) line segments with higher and lower running rubber bands. The indents at all four outsides of the GraphGrid allow you to position the rubber bands at any position and angle.
Photo: The GraphGrid frame with a grid of rubber bands and two axes on the TactiPad.
The easiest way to draw a graph is to draw the x- and y-axis first using the ruler and measurement indications on the TactiPad and then place the GraphGrid on the drawing board. The axes, the higher rubber bands, can now coincide with the axes drawn on the paper. You can then easily count and draw the x and y values of the graph along the flexible grid lines.
Because the grid lines are flexible, the graph values can be drawn exactly at the intersections by pulling the bands aside with the pen.
Once you have marked all the coordinates of the graph, you can remove the frame and create the graph by connecting the points. If you want to create multiple graphs in the same coordinate system, it is recommended to finish one graph before starting with the next to avoid confusion between the two sets of coordinates.
Photo: two hands on the TactiPad, one referring to the coordinates, the other drawing the graph
Photo: The GraphGrid frame sits on the drawing board. Along the lower edge of the frame cells are provided with crosses.
The columns of the GraphGrid frame can be considered as bars of a bar chart. The value in each bar is indicated by the number of marked cells, starting from the lower edge of the frame or from an additional rubber band. The lower edge or the rubber band is the X axis. A cell can stand for one or multiple units or for a percentage.
Bar chart for birthday data
This bar chart example presents an overview of when people in a group have their birthday spread over the year. Setup: Each column stands for a month. Each cell stands for one person.
Procedure: Ask who has his/her birthday in January and mark the number of cells accordingly. Do the same for all respective months. Take the GraphGrid away. You can now interpret the data by counting the number of markings in the column.
More complex data
One bar can present multiple data by applying different tactile markings in the same cell. For a male count you can apply a line in the cell from lower left to upper right and for female a line from upper left to lower right. Data that are presented with two colours could follow the same routine. Additional graphical markings can present even more complex data types.
Providing maximum value/top of the bar
To present numeric values you can find the position in the column/bar by counting upwards from the X axis. Assume the maximum value can go as high as 1000, than you might take five cells as the longest bar. A value of 700 can be indicated as a straight line horizontally in the middle of the forth cell up. Only the top of the bar needs to be indicated in order to interprete the data.
Photo: Tactile marks have been placed on the drawing board at gridbox crossing made up by the rubber bands. After connecting them a traditional house shows itself.
Coordinate system
A crossing of two rubber bands can be considered as a coordinate in a grid system. Starting in the upper left corner of the GraphGrid to the right we call the X direction. Going down is the Y direction. Each crossing is identified by two numbers. With this method, the coordinate 1,1 is the lower right hand corner of the top left cell.The GraphGrid frame does not need axes. The cell size is two centimetres.
Steps
Place the frame on the drawing board.
Mark the coordinates as given in the below examples.
Take the frame away and connect the dots with straight lines as instructed with the word ‘to’.
Front face of a traditional house A2,8 to B6,8 to C6,2 to D4,1 to E2,2 to 2,8 (which is A again).
You can add a door or windows by free hand drawing separately or place the GraphGrid once again and mark the respective coordinates.
A cube
The eight corners of a cube are named A To H. The 12 ribs complete the cube. Some of them are marked as ‘dashed’ because they are invisible lines. In coordinates and lines they can be listed as follows:
Bottom face A2,8 to B6,8 to C10,7 dashed to D4,7 and dashed to A.
Top face E2,4 to F6,4 to G10,3 to H4,3 to E.
A to E , B to F, C to G, D dashed H.
Pyramid
Place the GraphGrid on the drawing board.
The floor of a pyramid is similar as of a cube. From the four corners sides go up to the top ‘T”.
The top can be right above the centre of the floor or somewhere else. Create the coordinates and lines A to B to C dashed to D dashed to A.
The top T has coordinate 5,3.
Draw the lines A to T , B to T , C to T, D dashed to T.
Share and surprise; connect the dots
You can create a list of coordinates for someone else or you might receive a list of coordinates from someone else. Provide instructions to mark the coordinates and how they should be connected. In the list is an ‘hidden’ image that will appear only when you follow the instructions carefully.
More details
For more detail in the image you can work with halves like 3.5, 6 etc. The level of detail of the image can be increased by working not only with straight lines. Curved lines could be indicated as ‘cu’’ or ‘cl’. The ‘u’ refers to upper half of a circle. The ‘l’ to the lower half of the circle.
Photo: GraphGrid frame with rubber bands per two centimetres and two crossing axes placed on the TactiPad.
Cells as playing field for games
A playing field for games based on cells such as tic tac toe, battleship etc can be created with the grid boxes that mature by the crossing rubber bands. The ‘game board borders’ can be marked with high rubber bands. The default distance of two centimetres for the cell size is enough, but reconfiguring the rubber bands to a cell size of three centimetres offers more space.
Yahtzee
Keep the scores for Yahtzee by marking the cells of the first column as 1 to 6 and further down as three of a kind, four of a kind, full house, small and large Straight, free choice and Yahtzee. In the first column you can use regular characters or some tactile graphics that are meaningful for you so you can identify the categories. For the first game we use the second column for the scores per category. For the next games use the next columns. The scores one to six are all kept in one cell.
Draw a line along the lefthand side of the cell for score one.
Draw a line along the bottom side of the cell as well for score two. For three a line along the right hand side etc.
For score five and six you can create diagonal lines in the cell.
For a missed category provide a line from one side to the other in the middle of the cell.
Gas-Water-Electricity puzzle
Photo: The three resources gas, water and electricity provided to three houses; some pipe lines are provided.
Ideally we would like to have gas, water and electricity available in our homes.
In this challenge there are these three resources that you have to provide to three houses under the condition that the (pipe) lines may NOT intersect!
Steps
Place the GraphGrid on the drawing board.
Find the cell in the third column and third row and trace the inner contour. This is the gas supply.
Skip three cells down and trace the inner contour of the third cell, which is the water supply.
Skip three more cells down and trace the third cell as the electricity supply.
The three resources are all in the same column.
Skip three columns to the right and create the squares similar as before. The houses are in one column.
Take the GraphGrid away.
Now provide every house with non-intersecting pipelines with all three resources.
The GraphGrid frame is a frame (36 by 28 cm) with a thickness of four millimetres. It needs to be mounted on the TactiPad drawing board like a picture frame. Eight oval holes in the frame fit around the knobs of the TactiPad to hold it in place. Three corners are rounded. The fourth corner is flattened.
Along the inner edge, small curved hooks are placed at a regular interval of 1 centimetre. Along the outer edge of the frame indents are made with a spacing of 1 centimetre as well. Slightly wider indications mark 5 centimetre intervals. They exactly match with the centimetre scale along the edges of the TactiPad.
Photo: detail of GraphGrid with indents at every 5 cm, corresponding with the indents per centimetre at the inner side of the GraphGrid.
The purpose of the indents is to hold rubber bands to have the measurements system of the TactiPad tangible on the entire drawing area in rows and/or columns. Because of the A4 dimensions of the drawing surface (29.7 cm by 21.0 cm), the centimetre scale is not symmetrical along the GraphGrid. Therefore the starting point for horizontal and vertical distances is at the upper left corner of the drawing board and GraphGrid frame when in landscape orientation and the flattened corner of the GraphGrid is placed in the upper right corner.
Note: The frame covers the drawing surface along the edges for one centimetre.
The frame of the GraphGrid will sit or can sit already on top of the drawing board for storage. The additional tools for the GraphGrid have their place in the paper pocket.
Preparation
To begin with, make sure the TactiPad has the landscape orientation with the hinge facing backwards. To position the GraphGrid frame, place the knobs at approximately five centimetres away from the corners. To orient the GraphGrid frame correct, place the flat corner in the upper right corner of the TactiPad. Make sure all eight knobs of the TactiPad are kept in a hole. It is possible that a rubber band coincides with the position of one of the knobs. Therefore, the holes in the GraphGrid around the knobs are extra-long so you can slide the knobs aside for the desired placement of the rubber band.
Initial setup for the rubber bands
The initial setup of the GraphGrid frame has rubber bands every two centimetre in horizontal and vertical direction, so forming a grid. Two rubber bands go around the frame to indicate two axes. Of course, other setups are possible.
Low and high rubber bands – Using differences in height
The framework of the GraphGrid has a thickness of four millimetres. Therefore there is a well noticeable difference in height between the bands that run along the top side (outer) part or the bottom side (inner) part of the frame. The lower rubber bands that are attached to the hooks run alongside the inner part of the GraphGrid and can lay flat on the drawing surface. The rubber bands can be placed horizontally and vertically with a minimum distance of 1 centimetre.
Note: The inner and outer bands can also be placed at an angle of any number of degrees.
Note: The hooks at the inside edge are at every centimetre. Adding rubber bands to each hook could result in a (too) dense grid.
Placing the grid lines and axes
The minimum cell dimensions are one by one centimetre. To form a larger grid, you can place the rubber bands two or more centimetres apart by skipping one or more hooks in opposing sides of the GraphGrid. Additional rubber bands for replacing broken ones or create more grid lines and axes are supplied with the GraphGrid. These are standard, thin rubber bands with a length of 15 centimetres, 6 inches.
Photo: close-up of 3 hooks
The hooks along the inner side of the frame have a curved shape so the rubber bands cannot come loose if they are positioned properly. Placing a rubber band is easiest if you hold the rubber band with two hands, keeping it perpendicular to the frame so you can slide it through the slot to the end of the hook. First of all, you span the rubber band in two opposing hooks, so that it forms a double line between two sides of the GraphGrid. You then take the upper of these two lines and slide it into the next desired hook in both sides of the frame to form the second grid line.
Photo: two hands placing a rubber band in the correct position on the GraphGrid.
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