Terrifying Trapeziums (They re not really that scary!) Question 1) a) Look at the trapezium below. Draw how the trapezium can be made into a rectangle that would have the same area (be careful this is a little trickier than we have seen before). b) Make sure you are correct by finding the area of the trapezium by counting the squares and compare it to the area of rectangle (assume each square has an area of 1cm 2 ) c) If your answers were not the same in part b) change your rectangle and try again until you get a rectangle the same answer. Question 2) a) What is the length of the top of the trapezium? (remember each square is 1cm 2 ) b) What is the length of the bottom of the trapezium? c) What is the length of your rectangle? d) What do you notice about the length of the rectangle compared to the length of the top and bottom of the trapezium?
To find the area of the trapezium this is what we are actually doing 1) Cut each triangle at each end of the trapezium in half as shown below by the red lines 2) Move the triangle cut off each end so that it fills the missing part of the rectangle. The trapezium is now a rectangle.
Since we cut half of each triangle off the ends the LENGTH of the rectangle is HALF WAY in between the length of the top and the length of the bottom of the trapezium. The next problem is how do we find half way between two numbers? Question 3) What is half way between the following pairs of numbers? a) 2 and 4 b) 6 and 16 c) 7 and 10 Easy, right? The last one was a little trickier but not horrible, but hang on these are nice whole numbers. Any builder will tell you they don t get to work with whole numbers very often. So how are we going to find half way in between harder number? Let s try some harder ones. Question 4) What is half way between the following pairs of numbers? a) 3.2 and 4.8 b) 7.9 and 13.5 c) 7.2 and 6.5 d) 10.94 and 54.53 That s making it hard isn t it! But it s simple really, we just have to realise what we are doing when we find the middle of two numbers. Question 5) Can you think of ways to find the middle of a set of numbers? (Hint we do it all the time in statistics)
A cricketer made the following scores; 180, 0, 0, 0, 0, 60, 0 0, 40 and 20 Question 6) a) Did the cricketer have a good season? Why or why not? b) How would you describe this cricketer s batting performance to another person? Would you only talk about the total runs they scored? A second cricketer had the following scores; 30, 40, 0, 70, 50, 35, 45, 30 Question 7) a) Who was the better cricketer? Why b) Is comparing total runs scored the only way to compare batter? Is it the best way? c) What is a better way to talk about a batter s performance? Averages! If you didn t quite get there it s ok, but average is one of the best ways to describe a batter s performance, as it gives you an idea of how many runs a batter it likely to make each week. It doesn t mean they will make that amount on any given weekend just that in the long term that is how many they make each week. When you find an average of a group of numbers you are finding the MIDDLE NUMBER of the group, exactly what we are trying to do with our top and bottom length of trapeziums.
So most importantly how do we find an average of numbers? We ADD them all together and DIVIDE by how many numbers there are. EXAMPLE Find the average of the following numbers 1, 5, 7, 4, 8 ANSWER Step 1) Add them together 1+5+7+4+8 = 25 Step 2) Divide by how many numbers there are So the average is 5. Question 8) Find the average of each pair of numbers from question 4, (you ll find it much easier now!) So let s recap for a second. To find the area of a trapezium we need to find the average of the top and bottom lengths of the trapezium and multiply it by the height. Question 9) Why do we need to find the average of the top and bottom lengths of the trapezium? (look back to the explanation after question 2 if needed)
Question 10) Look at the trapezium below. The top is labelled a the bottom b and the height h Write down the formula for calculating the area of a trapezium. Question 11) Explain why this is the formula for the area of a trapezium. Question 12) Find the area of each of these trapeziums a)
b) c) d)
e) f*) A trapezium has a base length 3 times the length of the top and twice the size of the height. i) If the base is 6m long what is the area? ii) If the base is 18cm long what is the area? iii) It the TOP is 4m long what is the area? iv) If the base is n meters long what is the area in terms of n? v) If the base is 4y meters long what is the area in terms of y?