Understanding Radians: A Challenge in Measurement Units
Explore the concept of radians as a unit for measuring angles through challenges involving arc lengths, radius comparisons, and conversions to degrees. Dive into a word search activity to test your knowledge and understanding of radians vs. degrees.
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Week 4 3 problems to work through this week! Answers to weeks 2 and 3 coming next week!
Challenge 1: Radians We can measure distances using several different units. For example, centimetres or inches. Look at the two sides of this ruler.
Challenge 1: Radians We can also measure angles using different units. We usually use degrees, but another unit for measuring angles is radians. Look at these different protractors. We will have a look at how radians work in this challenge . ? ? ? ?
What do you notice about these sectors? What is the same? What is different? In each diagram, the arc length is the same as the radius. There is only 1 angle at which this happens, either 57.3 or 1 radian! This is how a radian is defined.
Challenge 1: Radians 1 radian is much bigger than 1 degree. Much in the same way as 1 mile is longer than 1 kilometre. As we have just seen, 1 radian is a special angle because it s the angle which makes the arc length the same as the radius. (In degrees it s 57.3 which isn t very memorable!) If half a turn is 180 , this is the same as 3.14159265359 radians. Recognise this number? Yes it s pi or . So 180 = radians. So in a full turn, 360 ,we have 2 radians.360 = 2 radians. Mathematicians quite like radians as we really like ! We can either use a little c or rads to show what units we are using. One of the easiest conversions to remember is: 180 = ?? or 180 = ? rads So, we can use fractions of to represent other angles e.g. 90 =? 2rads
Challenge 1: Radians Wordsearch Fill in the blanks to make the statement correct and then find that word in the wordsearch. Z L F B S T N Q R N Question 57.3o = 1 _ _ _ _ _ _ _ X P Z X I G X U A Q Pi radians = _ _ _ hundred and eighty degrees S X Z C X D O I M U Two pi radians = Three hundred and _ _ _ _ _ degrees One _ _ _ _ _ _ _ pi radians = 90o E I O W T F D F U A One _ _ _ _ _ pi radians = 60o P E X Z H A N I Y R _ _ _ thirds pi radians = 120o _ _ _ _ _ quarters pi radians = 135o P I R T R O N E T T ? 2rads = _ _ _ _ _ _ degrees O Q Q H Y M Q H E E _ _ _ _ pi radians = Seven hundred and twenty degrees Three pi radians = _ _ _ _ hundred and forty degrees F I V E T S I U N R One _ _ _ _ _ pi radians = 30o T K D E G R E E I I ? 180rads = one _ _ _ _ _ _ O K O Y D I A A N J Put your answer on the Desmos activity (page 2) .
Challenge 1: Radians The horizontal axis on the graph of ? = ??? ? below has got muddled! Drag the red labels into the correct positions! Complete on the Desmos activity (page 3) .
Challenge 2: The Curious Incident of the Dog In the Sherlock Holmes story Sliver Blaze the question is asked: Is there any other point to which you would wish to draw my attention? To the curious incident of the dog in the night-time The dog did nothing in the night-time That was the curious incident remarked Sherlock Holmes 1, 2, 4, 7, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44, ? What is next in the sequence? Think about what Sherlock remarked. Hint: Do not think about which numbers are included. [Credit: Professor Stewart s Hoard of Mathematical Treasures] Put your answer on the Desmos activity (page 4) .
Challenge 3: Digits! Using the following digits, make 100. Rules: No changing the order of the digits. No subtraction allowed. No concatenation of digits. e.g. you cannot use 2 and 3 to make 23. 1 2 3 4 5 6 7 8 9 = 100 [Credit: Professor Stewart s Hoard of Mathematical Treasures] Put your answer on the Desmos activity (page 5) .
