Tangrams. - An Introduction to Constraint Based Geometry -

Transcription

1 - An Introduction to Constraint Based Geometry - Chinese Legend A servant of a Chinese Emperor was carrying a very expensive square ceramic tray and when he tripped and fell it was shattered into seven pieces (called tans). He was not able to arrange the pieces back into the proper shape but he did realise that there were many other shapes that could be built from the pieces. The word Tangram was first used by Thomas Hill in 1848 in his book Geometrical Puzzle for the Youth. Tangram Construction Definitions A polygon is a simple, plane (i.e. 2D) figure bounded by line segments. A triangle is a three-sided polygon. An isosceles triangle has at least two sides of equal length. A right-angled triangle has one right-angle. A quadrilateral is a four-sided polygon. A parallelogram is a quadrilateral with two pairs of opposite sides parallel. A rectangle is a parallelogram that has at least one right-angle. A rhombus is a parallelogram with at least one pair of adjacent sides equal. A square is a rectangle with at least one pair of adjacent sides equal. Notes on the Definitions These are not dictionary style definitions. Dictionaries describe a shape. For example, dictionaries claim that a rectangle has opposite sides equal and has four right-angles. Mathematical definitions establish the minimum amount of information that is needed to identify a shape. If we know a shape is a parallelogram, it must have opposite sides equal and if one vertex of a parallelogram is a right-angle, the other three vertices have to be right-angles. Copyright 2007, Hartley Hyde Page 1 of 12

2 Classifying Shapes We can classify shapes using Venn Diagrams. Each circle represents a clearly defined idea and it is possible to classify triangles and quadrilaterals as belonging to a particular circle of the diagram. When a circle is completely enclosed by another circle we say that the inner circle is a subset of the outer circle. For example, we say that E I. Thus equilateral triangles are special cases of isosceles triangles. Rectangles and rhombuses are both special kinds of parallelogram. A parallelogram is also a special type of trapezium but for some reason, the definition seldom mentions this. If two circles overlap, we call the bit that belongs to both circles the intersection set. We sometimes give an idea that belongs to both categories its own special name. For example, a square is a special kind of rectangle and a special kind of rhombus. We say that S = Re Rh. In the Venn Diagram shown right, the universe of discourse is the set of all triangles (T), (i.e. U = T). E = {equilateral triangles} I = {isosceles triangles} R = {right-angled triangles} S = {scalene triangles} = I c In this lesson we are particularly interested in those triangles that are both isosceles and right-angled I R. In the Venn Diagram shown left, the universe of discourse is the set of all quadrilaterals (Q), (i.e. U = Q). T = {trapeziums} P = {parallelograms} Re = {Rectangles} Rh = {Rhombuses} S = {Squares} = Re Rh K = {Kites and Arrowheads} Copyright 2007, Hartley Hyde Page 2 of 12

3 A Legend about an Isosceles Right-Angled Triangle Pythagoras and his followers, the Mathematioi, lived at a school at Croton in Calabria, Italy. They had known for some time that the triangle shown here presented a difficult problem. When we find the length of the hypotenuse (h) we write: h 2 = (Pythagoras) = = 2 h = (4 sig. fig.) However, the Pythagoreans knew nothing about numbers like 2. They believed that all numbers could be written as fractions. One of their group, Hippasus, had recently proved that it is impossible to find a fraction that you can square and get the number 2. He had discovered irrational numbers. As concepts of number were central to their religious beliefs, the Pythagoreans felt threatened by this heresy. They were particularly concerned that profane people, who were not part of their group, would find out. Legend has it that Hippasus was about to go public. He caught a boat for Athens but during the voyage he was stabbed and thrown overboard to ensure his silence the first mathematically inspired murder. The ClassPad Geometry Measurement Bar Switch on your ClassPad and tap on the Geometry icon G. If your screen has drawings from a previous investigation, check if you need to save them before you clear them from the screen. From the File Menu tap New and then tap OK. From the Draw Menu choose Line Segment. Tap on two points on the screen and they will be automatically labelled A and B. Tap on the arrow u at the far right of the tool bar to Go around the corner to the Measurement Bar. Near the left end of the Tool Bar is a pull down menu of measuring instruments. Copyright 2007, Hartley Hyde Page 3 of 12

4 The Measurement Menu The selection of measuring instruments available from the pull down menu will depend on the object or objects that are highlighted. In the window is the actual measurement. To the right is a tick and then an arrow taking us back to the start of the tool bar. We will consider just line segments here, but you will have different choices if you use the Selection Arrow G to select points or shapes before you measure. Measurement Choices for a Line Segment x Q Shows the length of the line segment. In this case, 3.37 units. Shows the Slope or Gradient of the line segment. Shows the angle the line makes with the x-axis O Shows the equation of the line You can use this option to give lines names. For any of these choices, the contents of the measurement window can be edited and if the tick is tapped, the line will be constrained to fit the new information until the tick is tapped again. Measurement Choices for Two Line Segments Q > e x Shows the acute angle between the segments. Shows the obtuse angle between the segments. Shows if the line segments are congruent. Shows the distance between the segments. And again, we can edit the window contents and use the tick to constrain the line segments. For example, using the congruency measurement, as shown on the screen, we can force two line segments to have the same length by tapping the tick. Copyright 2007, Hartley Hyde Page 4 of 12

5 Defining a Triangle The size and shape of a triangle determines six quantities, three sides and three angles. However, we know that under certain conditions we only need to specify three of these to fix the size and shape of a triangle. And of course, the conditions are the four conditions of congruency SSS, SAS, ASA and RHS. An obvious way to construct an isosceles right-angled triangle is to constrain the lengths of two equal sides and a right-angle; like this. From File choose New. From the Draw Menu choose Line Segment. Tap the screen in two places and you have a line segment A and B. Tap B and a third point and you have line segment BC and then all you have to do is join AC. You have a triangle but it probably doesn t look like either a right-angled or an isosceles triangle. From the View Menu choose Select. Use the Selection Arrow G to highlight AB. Tap on the arrow u at the far right of the tool bar to Go around the corner to the Measurement Bar. Choose the Length Measurement tool x, overwrite the measurement in the window with the number 1 and then tap the tick s. Nothing appears to happen! ClassPad simply accepts this first measurement as your definition of unit length. Now we want to select BC instead of AB. When you finish a selection, get into the habit of always tapping in free space (right away from any drawing) so as to clear all previous selections before you start another. Copyright 2007, Hartley Hyde Page 5 of 12

6 So having tapped in free space, select BC. Overwrite whatever measurement appears in the window with the number 1 and then tap the tick s. Now you will see the line BC change to have a length the same as AB. Now select both AB and BC. From the Measurement Menu select the Acute Angle tool Q. The angle between AB and BC will appear in the window. Overwrite the angle with the number 90 and tap the tick s. And now we have an isosceles right-angled triangle. When making Tangrams we also need to consider how easily the resulting shapes can be manipulated. Try moving the triangle by selecting a vertex and then tugging. Try moving the triangle by selecting a side and then pushing. Now select side AB and go to the Measurement Bar. Choose the Slope Tool Q. The slope of the line AB will appear in the window. Overwrite with a 1 and tap the tick s. The triangle should turn so that the line AB slopes at 45. Now try the same thing using side AC. At this point, my version of the ClassPad system gives up. It can t handle all the calculations needed to turn the triangle using the slope of the hypotenuse. So, although we have a perfectly good way to form a triangle, we also need a shape that is easily moved. The SAS method goes on the back burner this time. Here is a better way. Sorry, but you needed to know why I was being so fussy. I won t do it again. Well not this lesson anyway. From the File Menu select New. Copyright 2007, Hartley Hyde Page 6 of 12

7 A better Way Both the SSS and the ASA method seem to work smoothly on my ClassPad. I also found it much easier to construct tans using the integer grid. From the View Menu check the box marked Integer Grid. If there are axes showing, toggle the axes until they disappear. Your screen should now look like the first example. From the Draw Menu choose Line Segment. Draw the three lines AB, BC and AC exactly as shown in the second screen. From the View Menu choose Select. Tap on the arrow u at the far right of the tool bar to Go around the corner to the Measurement Bar. Choose the Length Measurement tool x and the number 8 will appear in the window. Tap the tick s and the length of AB will be fixed at 8 units. Highlight the line segments AC and AB. The menu offering should automatically change to Acute Angle and 45 should appear in the window. Tap the tick s. Highlight the line segments BC and AB. The number 45 should appear in the window. Just tap the tick s. This triangle is now finished. From the View Menu, turn off the Integer grid. Play with the triangle for a while. You should find it easier to control than the version we made before. When you have finished, turn on the grid and park it back where you made it. Remember: After using the Selection Arrow G, get into the habit of clicking in free space to deselect any highlighted constructs. Copyright 2007, Hartley Hyde Page 7 of 12

8 Another Large Isosceles Right-Angled Triangle From the Draw Menu choose Line Segment. Draw the triangle DEF as shown. Using the Selection Arrow G highlight the line segment DE and tap the tick s. Constrain the two 45 base angles as before. Highlight one of the sides and slide the new triangle up next to the first. Because you have left the integer grid on, it should snap into place. Building Quadrilaterals Triangles are particularly rigid shapes. Notice how most metal towers and cranes are made up of triangles. Quadrilaterals are not like that. They very easily flex to give a variety of shapes. The necessary conditions for a quadrilateral to be a parallelogram are: opposite sides parallel; opposite sides equal in length; opposite angles equal; diagonals bisect each other; one pair of opposite sides is equal and parallel. However, even though these conditions determine that a quadrilateral is a parallelogram, they do not necessarily fix its shape. Opposite angles may be equal but that does not determine the size. Opposite sides may be equal but that does not determine the shape. The strategy used here for both the square and the parallelogram is to constrain all the sides to the lengths defined by the grid and then to constrain one angle. Copyright 2007, Hartley Hyde Page 8 of 12

9 Building a Parallelogram From the Draw Menu choose Line Segment. Draw the parallelogram GHIJ as shown. Using the Selection Arrow G highlight the line segment HI and tap the tick s. Do the same thing for sides IJ, JG and GH. This constrains each of the sides of the parallelogram but once we switch off the grid it will be free to slop around. It needs an angle fixed. Select HI and HG. Constrain the angle HGJ to 135 by tapping the tick s. Slide the parallelogram into place. See if you can build triangle KLM on your own. Building a Square From the Draw Menu choose Line Segment. Draw the square NOPQ as shown. Using the Selection Arrow G highlight the line segment OP and tap the tick s. Do the same thing for sides PQ, QN and NO. This constrains each of the sides of the square but once we switch off the grid it will be free to slop around as a rhombus. It also needs an angle constrained. Select OP and PQ. Constrain the angle OPQ to 90 by tapping the tick s. Copyright 2007, Hartley Hyde Page 9 of 12

10 Finishing the Tangram There are two isosceles right-angled triangles left to draw. As before you need to constrain each hypotenuse to the size it has on the grid and you need to constrain each of the base angles to 45. When you have finished the triangles slide them into place to finish the Tangram Square. From the View Manu, switch off the Integer Grid Save your work as Tangram Take your ClassPad to your teacher. Checkpoint Hints on Building Tangram Pictures Although the tans slide easily enough, the difficult part is rotating them to the right orientation. The best procedure is to: check that none of the sides has been constrained to a particular slope; if there are slope constraints, remove them by un-checking the tick, i.e. un-tap the tick ; make sure you don t remove any constraints that are determining the shape or size of your tans; select a side that you wish to orientate; choose the slope measuring tool; use the slope guide at right to get the right slope; tap the tick s; once the tan has rotated, un-tap the tick immediately ; You may choose to lock the tan in place by constraining some points. Obviously you can use slopes in between the ones given here, but these are the ones that you will use most of the time. For some constructs you may need to Zoom Out to allow room to work, then Zoom to Fit when you finish. Copyright 2007, Hartley Hyde Page 10 of 12

11 Playtime Use your tangram set to build each of the following constructs. When you finish, save your work as a new file. Take your ClassPad to your teacher to sign off. Get on with the next one. The Running Man The Cat Checkpoint The Reindeer Hold the ClassPad sideways Checkpoint Checkpoint Copyright 2007, Hartley Hyde Page 11 of 12

12 Checkpoints The purpose of this lesson is to familiarize students with using some of the Constraint Geometry concepts. The actual construction is very repetitive. If it were otherwise it would be fairly stressful for teachers and students trying to track down where and why figures were falling apart. The important part is the play section. You can obtain extra examples from use of material from this site is free for non-profit use and education. Copyright 2007, Hartley Hyde Page 12 of 12