VIEWS: 11 PAGES: 14 POSTED ON: 6/14/2012
Using Cabri to explore the properties of triangles 1 Constructing triangles Use Cabri’s construction tools to construct: A scalene acute triangle A scalene obtuse triangle A right angled triangle An isosceles triangle An equilateral triangle Label all the angles and side lengths. Save your constructions to file. Print off at least 1 construction. Consider what mathematical skills and knowledge you needed to complete this task. 2 The angle sum of a triangle. Use Cabri to construct the proof for the angle sum of the triangle. Save and print off your work. What happens when you ‘drag’ the various points? Consider the implications of this for teaching and learning in the primary school. Does this method constitute a formal proof? 3 The exterior angle of a triangle. Use Cabri to construct the proof for the exterior angle of a triangle. Save and print off your work. What happens when you ‘drag’ the various points? Consider the implications of this for teaching and learning in the primary school. Does this method constitute a formal proof? Cabri Investigations FML 1 Using Cabri to explore Pythagoras’ Theorem 1 Pythagoras’ Theorem Use Cabri to construct the squares on the sides of a right angled triangle. Test the hypothesis: The square on the hypotenuse is equal to the sum of the squares on the other two sides. Test this for several triangles. Does it seem to be true? Consider the implications of using Cabri for exploratory geometry work in the primary school. 2 Exploring Pythagoras’ Theorem further Above you have tested the theorem. Use Cabri to explore: What shapes other than squares does the theorem hold for? Record your findings and produce some printouts to illustrate them. Can you prove your findings? 3 Extension Does the theorem only hold for right angled triangles? Test the squares on other types of triangles. Record and attempt to justify your findings. Cabri Investigations FML 2 Right Angled Triangles Use Cabri to investigate the proprieties of right angled triangles. Hints: Consider triangles whose side lengths are whole numbers. Consider triangles whose side lengths are not whole numbers. Consider the ratio of the sides. You may use additional software packages to support presentation and analysis of your results. Cabri Investigations FML 3 Using Cabri to explore similarity Use Cabri to investigate triangles with different scale factors of enlargement including fractional and negative. Use a centre of enlargement outside the figure. In what way is performing an enlargement on Cabri different to the standard computer drawing package? Explore changes in area under enlargement. What happens when you use Cabri’s dynamic facility to move the centre of enlargement? Investigate Cabri’s transformation tools. Consider their use for supporting ideas related to transformations. Make notes on your observations. Cabri Investigations FML 4 Angles in the same segment Major segment chord Minor segment P O O B A We define angle APB as the angle in the segment (in this case the major segment). Investigate angles the same major segment. What do you notice? Investigate the angles in the minor segment. What do you notice? Can you prove any findings? Cabri Investigations FML 5 Quadrilaterals in circles investigation Construct a quadrilateral inside a circle. Move the vertices around the circumference of the circle. Note what happens to the angles of the quadrilateral. What else can you notice as you move the vertices around the circumference? Record your findings as a succinct statement. Can you prove your statement using pencil and paper methods? Alternate segments investigation Construct a triangle inside a circle. Draw a line segment to produce a tangent to the circle which touches one of the vertices of the triangle. Move the vertices of the triangle around the circumference of the circle and note what happens to the angles. Will your observation always be true? Record your findings as a succinct statement. Can you prove your statement using pencil and paper methods? Cabri Investigations FML 6 Using Cabri to explore quadrilaterals Use Cabri to find as many different quadrilaterals as possible with: Zero sides of equal length 2 sides of equal length 3 sides of equal length 4 sides of equal length No angles of equal size 2 angles of equal size 3 angles of equal size 4 angles of equal size Zero pairs of parallel sides 1 pair of parallel sides 2 pairs of parallel sides In each case above, write down the name of the quadrilateral(s) found. Can you now classify quadrilaterals according to the 3 properties investigated? Page 23 Cabri Investigations FML 7 More Cabri Investigations 1 Pentagon Puzzle Can you construct a regular pentagon? Your completed pentagon should not ‘come apart’ when pulled. 2 Perpendicular Bisectors *this is a diagram only – perpendicular bisectors are approximate This is a quadrilateral together with the perpendicular bisectors of the 4 sides. * Can you find a quadrilateral which has 4 perpendicular bisectors crossing at one point? Find as many answers as you can. Try to find out what is special about the quadrilaterals which work, and why. Explore the perpendicular bisectors of pentagons, hexagons or other polygons. Cabri Investigations FML 8 3 Reflection Reflect the triangle in the mirror line. What shapes can you make by dragging the triangle? What shapes can’t you make? Explain your results. 4 Equal Sides, Equal Angles This pentagon has equal sides but the angles are not all equal. This hexagon has equal angles but not equal sides. Find ways to generate polygons which have equal sides or equal angle but not both. Cabri Investigations FML 9 5 Hexagon hallucinations You might like to try this investigation on paper first. Draw an irregular hexagon. Join up the mid-points of the sides. Join up the mid-points again. Continue the procedure. What do you notice? Explore this idea using Cabri. Can you generalise what you ‘see’? Can you prove what you see using Cabri or anther method? 6 The golden ratio The golden ratio can be defined as: A division of a line into 2 segments such that the ratio of the larger segment to the smaller segment is equal to the ratio of the whole line to the larger segment. A P B AP AB So AP : PB = AB : AP which is the same as PB AP Use Cabri to investigate if you can construct the golden ratio and hence calculate the value of the golden ratio (known as Phi Ф). Cabri Investigations FML 10 7 Centres of Triangles A This is a the circumcircle of a triangle: 1 Use cabri to create the circumcircle of a triangle. 2 It is said that the centre of the circumcircle, which is called the circumcentre, is equidistant from the 3 vertices of the triangle. Test the veracity of this statement by constructing the circumcircles of triangles. B This is the incircle of a triangle: 1 Use Cabri to construct the incircle of a triangle. 2 It is said that the centre of the incircle, which is known as the incentre, is equidistant from the sides of the triangle. Test the veracity of this statement by constructing the incircles of triangles. Cabri Investigations FML 11 7 Centres of Triangles (cont.) C A median of a triangle is defined as a line joining a vertex to the midpoint of the opposite side. 1 Investigate the properties of the medians for a particular triangle. 2 What do you notice? 3 Test your ideas for other triangles. 4 Can you generalise your hypotheses. 5 Can you prove your hypotheses using geometric theorems? D The altitude of a triangle is defined as a line drawn from a vertex perpendicular to the opposite side. 1 Investigate the properties of altitudes for a range of triangles. 2 Attempt to generalise and prove your findings. Cabri Investigations FML 12 8 Arithmetic Composition The image below is based on a painting by Theo Van Doesbury (1883 – 1931). The problems based on this image and given below are adapted from: Sharp, J (2004) Art Inspires Geometry Mathematics in School, January 2004 Please note, the image below has been reproduced in MSWord so will not be exactly to scale. How the image is constructed The composition above is based on drawing a tilted square in the isosceles triangle formed by bisecting the square whose base is on the diagonal and whose other vertices are on the sides of the original square. Another horizontal square is created whose side is half the side of the original one and the process is repeated. (See next page for investigation questions.) Cabri Investigations FML 13 8 Arithmetic Composition (cont.) Starting with a unit square, what is the side of the resultant square? What proportion is this of the diagonal? What is the relative area of two successive squares? Doesburg only paints four squares. Suppose the construction is carried on infinitely. What proportion of the original square is the area of the tilted squares? Consider other geometric compositions and investigate them. Cabri Investigations FML 14