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Aim ‘Bridge the gap’ Allow slower pace Allow room for individual learning Allow space for diverse ways of learning Algebra can’s and cant’s Can Do simple substitution such as R = 3t + 6. Find R when t is 8 Solve simple linear equations, e.g. 3a + 5 = 17 or 4(2x + 5) = 44 Understand x2, x3, etc ,replace values and find roots Factorise very simple expressions such as 3t + 12 Can’t Substitution of one expression in another, e.g. P =2r, r = K + 6, Find P in terms of K Solve equations with letter both sides e.g. 5x + 8 = 3x + 24 Cope with (3x)2 or (x + 3)2 Solve quadratic equations by factorisation or formula, e.g. x2 – 7x + 12 Solve equations with Solve equations with one division line cross-multiplying such as: such as x+4 =2 x+1=x+3 5 6 8 BRIDGING THE GAP Lecturer Nonworkshop students Workshop students The ‘Algebraic Cuboid’ Advantages: 1) Slightly contrived, but shows a possible practical application 2) Easy to create a gradual approach 3) Visual imagination stimulated 4) Provides a concrete aspect to the abstraction of an expression 5) The idea of ‘design’ is appealing to engineers, and creative 6) Link later to calculus-min, max areas,vols Starting point:- Initially, work out the total length of edges (T) surface area (A) and volume (V) of several given cuboids given dimensions. Build to formulas: V = LWH, T = 4L + 4W + 4H, A = 2LW + 2LH + 2WH Open questions: e.g. a) How many cuboids can you design from a total length of wire of 28 cm? Which has the largest volume? b) You have to design a cuboid box of chocolates with a volume of 180 cm3. How many cuboids can you find? Which has the smallest surface area? Then, proceed to algebraic cuboids: A cuboid is designed with the width twice the length. The height is 8 cm. Find expressions for: a) The total length of the edges b) The volume c) The surface area Now, design a cuboid with the above specifications and a volume of 400 cm3. How large is its surface area? Comment: 1) A touch of ‘openness’ (they can choose their own variable) 2) Visual and tactile -they can actually touch the box and draw the faces if they wish 3) They can see the point of allocating letters ( to model that the length is twice the width) 4) They can see the point of simplifying- to work out the second part 5) The algebra techniques- they practise collecting like termsi.e. 4(2L) + 4L + 4 x 8 = 12L + 32 for total edge-length The associative law 2L x L x 8 = 16L2 for volume Both of the above for surface area 6)More subtly, it helps unconsciously with the substitution problem (Q1) as stated earlier (i.e. V = LWH, L= L, W = 2L, H = 8 But they are looking at it on the box – so-visually and tactilely) The algebraic cuboid idea is very versatile from the tutor’s point of view. You can adapt it to bring out almost any algebraic technique. Also, a quick change in the numbers can lead very quickly to other similar examples of the same type for them to try. In other words, it is very adaptable to the individual- which is what we need here. In the following 3 ‘cuboid’ questions, write down what you think are the algebraic techniques involved and what concepts it helps with. 1) A cuboid is designed with sides in the ratio 3:4:5 a) Find expressions for the total edge-length, surface area and volume b) Find the volume if the surface area is restricted to 4606 cm2 2) A square-based cuboid has a height of 8 cm. a) Find expressions for the total edge-length, surface area and volume b) Find the volume if the surface area must be 450 cm2 3) Design a cuboid with length 4 cm more than the width, height 5 cm more than the width and a surface area of 712 cm2 4) A cuboid is designed on the model: Length: t + 2 Width: t + 2 Height: 4t Find the dimensions if the surface area is 658 cm2 5) 2 cuboids are made from the same total length of wire. The first is square based with a height of 6cm, the second has the same length as the first, but the width is 3 times as large and the height is 2 cm Find the dimensions of both cuboids. 6) A measure of the heating cost of a cuboid shaped room is given as C = 2LW + H2 Suppose the room has width 3 times the length and height twice the length.. If the cost of heating ,C, as above is 250 find the dimensions of the room. Difficulties could encounter Differences in teaching style between a) Workshop and lecture b) Workshop and tutorial c) Workshop and 1-1 sessions Differences in resources/exercises provided between a) Workshop and tutorial b) Workshop and 1-1 sessions Profile of workshop students Style of session Teach at their level of understanding Teach to the same point as lecture Provide graded exercises Lots of student activity Individual help