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.
The shape of all curves represent the curves of the lenses.
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.
Detailed description
The inner space of this U-shaped tool has the contour of the formula 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 near the bottom; one in the middle 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.
SinTang; combination tool for the sine, cosine and tangent graph
Global description
The SinTang tool combines graphs for two formulas; the sin(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 is a to the left shifted 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 centimeters. 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
To use the tangent part of the tool it has to be rotated for 90 degrees 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 value of the graph is 4 cntimetres.
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.
Sine graph
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 position. The distance between the two marks is eight centimetres.
Align the two 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 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 Y axis.
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 unstill you approach the Y axis.
Parabola manual
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.
Draw an X and Y axis.
Align the bottom indent with the Y axis and the ones in the poles with the X axis.
Draw along the contour of the tool.
You may choose any other coordinate as the minimum for the y value. The centimetre indications help to align the tool with the axes.
Lens manual
Photo: for the application the tool is rotated 90 degrees. The median light beam is drawn. A convex lens has been drawn. The tool is positioned to create the mirror.
Draw a line with the ruler to represent the median light beam.
Align the tool on the line.
Draw the inner contour for the mirror, the convex or the concave lens.
You can add additional light beams to show light breaking and reflection effects.
Hyperbole manual
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.
Draw an X and Y axis with the ruler
Be aware, the effect of the additional material is also that the actual X and Y axis are covered by the tool.
Align the tool with the two axis by placing the flat sections at the tails with the axis.
For convenience you can place a pushpin in the crossing of the X and Y axis.
Draw along the contour of the tool.
Rotate the tool 180 degrees clockwise. Align the tool with the axis and draw the second part of the graph.
Preparation
The default setup 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.
However, before mounting the GraphGrid frame on the TactiPad make sure all rubber bands are provided according your preference.
Low and high rubber bands – Using differences in height
The framework of the GraphGrid has a thickness of four millimeters. Therefore there is a well noticable 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 centimeter.
The bands can also be placed at an anglle of any number of degrees.
Note: The hooks at the inside edge are at every centimetre. Adding rubber bands to each hook will result in a (too) denced grid.
Placing the grid lines and axes
The hooks and indents are positioned every centimeter along the frame, so the minimum cell dimensions are one by one centimeter. To form a larger grid, you can place the rubber bands two or more centimeters apart by skipping one or more hooks.
Spare rubber bands
Additional rubber bands for more grid lines and axes are supplied with the GraphGrid. These are standard, thin rubber bands with a length of 12 to 15 centimetres.
To keep the rubber bands in place, the hooks have such a shape that the rubber bands will not come loose if they are positioned properly in the frame. Pplacing 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.
The higher rubber bands are stretched around the outside of the GraphGrid and fall naturally into the indents.
Getting started
To begin with, make sure the TactiPad has the landscape orientation with the hinge facing backwards.
To hold the GraphGrid frame,, place the knobs at aproximately five centimeters away from the corners.
To position the GraphGrid frame correct, place the flat corner in the upper right corner of the TactiPad.
It is possible that of one of the rubber bands 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.
The TactiForma set consists of the following tools: 7 squares, 7 triangles, 1 rectangular L-shape hook, 5 polygon tools plus magnetic foot, 2 spur wheels, 4 ellipses.
Selecting a tool
The square tool, the equilateral triangle tool, the ellipse tool and the regular polygon tool come in different sizes. Knowing about the width of one, two or three of your fingers or the length of one finger, makes it easier to measure distances or finding the right tool without (extensive) counting the measurement indications.
Although the tools are recognizable by their shapes and the names of the tools clarify their purposes, it is highly recommended to read the descriptions and manuals because the tools contain many more applications than you will notice at first sight. Especially the regular polygon deserves reading the detailed description and manual while the usage differs from the rest.
Shape or tool
To avoid confusion while reading the texts where the name of the tool and the shape that is created is used, the words shape or tool will be added as much as possible.
Terminologies
We introduce some of our own terminologies that are applied throughout the pages such as finger fitters, finger entrance, small, medium and large sized indents for measuring and alignment.
The frame that forms the square is one centimetre wide. Two finger fitters are provided in opposing sides. Two opposite diagonal corners are rounded. The others are sharp. At each corner a pushpin marker is located. Along the outside and inside a small indent is provided at every centimetre. In case the length of the side is odd numbered, there is in additional indent at the halfway position.
When you position the square somewhere on the TactiPad in any orientation and then draw along the inner contour, ou create your first square. Alignment for horizontal or vertical oriented shapes. Aligning the tool with the ruler that is placed on two opposing TactiPad knobs, will make sure the aligned side is horizontal or vertical, relative to the measurement indications of the TactiPad.
Other shapes
With the square tool you can create many more shapes such as a rectangle, triangle, parallelogram, trapezium and also 3D figures like cube or pyramid.
Rectangle 1
Choose the square tool that has the longest side of the rectangle that you want to create. Draw the bottom line. Determine the width (is height) of the rectangle and draw the left and right line from the same position on to the bottom line. Move the tool downwards so the top side is almost on the line endings of the two (vertical) sides. For convenience place a pushpin in the positions along the two short sides and draw a line from here to the bottom line. Move the tool towards you until the top side hits the two pushpins. Draw the top line and remove the tool.
Rectangle 2
For larger rectangles alignment with the ruler works well. Place the ruler on two knobs. Place the tool against the ruler and draw the bottom line and the left side. Move the tool to the right for as long as you want the length of the bottom line of the rectangle. Draw the rest of the bottom line and also the right side of the rectangle. Connect the two line endings of the sides with the ruler.
Photo: Square against ruler, which is placed on two knobs, moved to the right after drawing the bottom line and left side.
45 triangle – 1/1/sqrt2 triangle
By drawing two connected full sides of the square and the diagonal line between the two line endings will create a rectangle triangle, also known as a 1/1/sqrt2 triangle. Placing pushpins in two diagonal positioned corners can help to draw the diagonal line.
Right corner triangle
By drawing only parts of two connected sides from the tool and connecting the line endings, will result in a right corner triangle.
Triangle with sides 1/2/SQrt5
Draw a line from a halfway position into a corner. Continue with a line along the full side. Remove the tool. Connect the two line endings. This is also a right corner triangle with specific proportions for the length of the sides: 1/2/SQrt5.
Isosceles triangle
Calculating the surface of an isosceles triangle created with a square tool is not too complicated. Draw a line as the base line for the triangle and check the halfway position on this line. Mark or place a pushpin in exact the same position in the opposite side. Connect this top corner position with each of the two line endings of the base line. The surface is the multiplication of the length of the base line and the length of the side of the tool that is used, divided by two.
Irregular triangle
By drawing – a part of – a side and then marking or placing a pushpin along one of the three other sides and providing lines from this position on to both line endings will show an irregular triangle.
Parallelogram
Draw the bottom side of the square.
Place a pushpin two or three centimetres to the right in the bottom line.
Move the tool to the right until it hits the pushpin. Make sure the tool is still aligned with the bottom line.
Place a pushpin at the desired height position at the inner left side of the tool. Mark or place a pushpin in the right side on the same height as well.
Move the tool downwards until it hits the pushpins. Draw the full top side of the square/parallelogram.
Remove the tool and connect the lines.
Trapezium
Draw the bottom line of the square. Use a smaller square tool as before. Align the bottom line with the one that was created already. Draw the top line of the smaller square. Connect the line endings between the top and bottom line. When placing the halfway position of the smaller square at the halfway position of the larger square, the trapezium will be symmetrical.
Cube 1
Draw a horizontal line on the drawing board and leave the ruler sitting on the knobs.
Align the square tool with the ruler. This square is the front face of the cube.
Draw a second square, the rear face, a little to the right and a little upwards. Make sure the bottom lines of the squares run parallel. Therefor move the ruler up a little.
Connect each of the four corners with the ones closest of the second square.
As the above is a start, you have to be aware that some of the lines are invisible. They should be dashed as in the below example
Cube 2, top right view
The ‘top right view’ of the cube shows the front face, the right face and the top face of the cube. Each corner is given a character indication. The bottom face of the cube has corners A, B, C and D. The top surface has corners E, F, G and H. Corner E is above corner A. Please note, there are also other naming conventions.
Steps
Step 1, front face: Draw the inner contour of the square. The lower left corner of the front face square is called A, the one at the lower right B, going straight up from here is F and going left is corner E.
Step 2, rear face: Move the square tool a little to the right and a little up (*). Make sure the lower side is parallel to the lower side of the front face.
The rear face has corner D in the lower left and going counter clockwise C, G and H.
The square of the rear face is partly covered by the front face. So two of the sides of this square have to be dashed: C-D and D-H.
Step 3, connecting corners: Four lines to connect the corners of the two squares complete the cube. Only the line A-D has to be dashed, the others are solid.
(*) Little to the right and a little up
About the amount for ‘a little to the right and a little upwards’ can be discussed. A rule of thumb is to imagine a line under an angle of 30 degrees starting in the A corner. Set out a distance of 60/70% of the side length along this line. The hypotenuse of the 30triangle can also help here.
Pyramid (symmetric)
For a start take the eight centimetres square and equilateral triangle with the same side length. The base of this pyramid is drawn as a parallelogram. The four side faces are triangles. The front face is equilateral.
Draw a parallelogram where the top side is moved upwards and to the right with the rule of thumb in mind; an angle of 30 degrees and a length of 60% of the length of the side.
Dash the left hand side and the top side of the parallelogram because they are invisible.
Place the triangle aligned with the bottom line of the parallelogram where the corners match. Draw the two remaining sides of the triangle where they meet in the top corner T.
The corners of the base (floor) of the pyramid are named A, to D, counter clock wise.
To complete the pyramid, provide a solid line from C to T and a dashed line from C to T.
Taller pyramid
The top of the pyramid can be placed in many positions and the figure will still be a pyramid, but not symmetrical. To keep it as such, the top T has to stay on the altitude line that arises from the crossing of the diagonals; the lines A-C and B-D, the centre of the parallelogram. To set the top T, use the ruler that goes across the TactiPad, in equal positions of the frame and also meeting the centre position of the parallelogram.
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