Problems using a square 9-pin geoboard

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1 Problems using a square 9-pin geoboard The 9-pin geoboard can be used with learners of all ages, from making and talking about shapes with younger pupils to working out the intersecting area of two overlapping shapes with A-level students. The issue of working with a concrete manipulative in order to enable learners to develop abstract thinking is of fundamental importance. The 9-pin geoboard is, I believe, pedagogically iconic in this regard. Starting with a 9-pin geoboard and some dotty grid paper (below) the following problems can be offered to pupils: 1) How many different triangles can be made? This problem automatically raises the question about whether the same triangle drawn in a different orientation is considered to be different. The teacher may or may choose not to engage pupils with the concept of congruency when this arises. There can also be an opportunity to engage pupils with the vocabulary of reflection, rotation and possibly translation. Pupils may benefit by having access to tracing paper so they can see for themselves whether two triangles are congruent or not. The further issue about whether they count the three right-angled isosceles triangles, of different areas, as being different might also be discussed; there exists the opportunity to engage with the vocabulary of similarity if the teacher feels this is appropriate. 2) What properties does each triangle have and, therefore, what is the name of each triangle? This issue of discussing the different properties of triangles and aligning names of triangles to these properties is important. Whilst an equilateral triangle cannot be drawn on a 9-dot square grid, or in fact on any size square grid would be a suitable challenge for A-level students to consider. With younger pupils it is important for them to recognise that the shape below, which is often thought to have three equal sides, is in fact isosceles. Again, the use of tracing paper might be used to help pupils arrive at this understanding.

2 3) How many different sized angles are there in the different triangles? Finding a complete set of angles will be a worthy challenge. Pupils can measure angles made with a protractor or name them as acute, right or obtuse by inspection and observation. Interestingly, all angles can be defined or calculated by knowing that the following triangle has angles 90, 63 and 27 (to the nearest 1 ) in the triangle below 4) Measure the distances around each triangle in order to find the perimeter of each. Again bigger grid paper might enable more accurate lengths to be measured. As with the angle work above, pupils actually only need to measure the following three lengths to be able to deduce the perimeter of each triangle:

3 The three defined lengths can also be used to describe perimeters algebraically. E.g. red = a, green = b, blue = c so the perimeter of the triangle above becomes P = 3a + c Placing the triangles in order of size of perimeter is a development of this work. 5) What is the area of each triangle (if the area of a small square is 1 square unit)? Quite often pupils will work out area of shapes by drawing internal dissections and adding together the areas of the resulting shapes. This method works okay for some though not all triangles. A more robust method is to surround a shape with a rectangular (including a square) frame and to calculate the areas of the shapes between the frame and the shape under consideration. 6) Trying to prove all the possible triangles have been found Given that mathematics is ultimately about proof then asking the question How do you know you have found them all? is an important one. Encouraging pupils, therefore, to design a systematic way of ordering the triangles they have made as a way of proving they have found a complete set is similarly important. Asking pupils to use and develop their system to find all the possible triangles on a 16, square dot grid will extend potentially higher attaining pupils. 7) Asking pupils to find all the quadrilaterals are there on a 9-dot grid will be a significant challenge Proving a complete set has been found would challenge A-level students, so for KS2 pupils it will be sufficient to ask them to try to find all 16 and to classify them according to properties. With quadrilaterals the further property of parallel sides comes into play. Below are all 16 solutions.

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5 A B C D F E G H I J L K M N P O

6 Properties of the quadrilaterals on a 9-pin geoboard Shape A is a square because Shape B is a parallelogram because Shape C is a Shape D is a trapezium because Shape E is a Shape F is a symmetrical arrowhead or a kite because Shape G is a Shape H is a kite because Shape I is a Shape J is a Shape K is a rectangle because Shape L is a Shape M is a Shape N is a Shape O is an isosceles trapezium because Shape P is a

7 Shape Name Area Perimeter Lines of Symmetry Order of rotational symmetry A Square 1 4a 4 4 B Parallelogram 1 2(a + b) 0 2 C asymmetrical arrowhead 2 a + 2b + c 0 1 D Trapezium 1.5 4a + b 0 1 E asymmetrical quadrilateral 2.5 3a + b + c 0 1 F Kite 2 2(a + c) 1 1 G Square 4 4b 4 4 H Kite 2 2(a + c) 1 1 I asymmetrical quadrilateral 2 a +2b + c 0 1 J Kite 1 2(b + c) 1 2 K Rectangle 1 6a 2 2 L Parallelogram 2 2a +2c 0 1 M Trapezium 3 5a+ c 0 1 N Square 4 8a 4 4 O Isosceles trapezium 1.5 2a + 3b 1 1 P asymmetrical arrowhead 1.5 3a + b + c 0 1