Grade B Interpret Proportions Understand that x is

Grade B Interpret Proportions Understand that "x is inversely proportional to y" is equivalent to "x is proportional to 1/y", and interpret equations that describe direct and inverse proportion. If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl. org. uk

Key Vocabulary Proportional Inversely Directly Formula Equation Gradient Constant

Understanding x is inversely proportional to y “x is inversely proportional to y” means that as y increases in value, x will decrease. This is the same as saying that ”x is directly proportional to the inverse of y”. The inverse of y is 1/y. 5 cm 10 cm 2 2 cm 20 cm 2 4 cm Both of these rectangles have the same area, 20 cm 2, but different widths and lengths. As you can see as the length increases from 2 cm to 4 cm the width decreases from 10 cm to 5 cm. So the length is inversely proportional to the width. Another example would be time and speed. As speed increases the time taken to get to your destination decreases. Therefore speed is inversely proportional to time.

Understanding x is directly proportional to y If x is directly proportional to y, this means that as y increases x also increases. So if x was to double, y would also double. The graph of y = x shows this relationship. y If y increases, x will also increase. y=x x

Understanding x is directly proportional to y continued… However the gradient does not have to be 1. It could be any number: y y y = 2 x x y y = 3 x x y = 10 x x In all of the graphs above x is directly proportional to y. However they have different gradients (steepness). This is known as the k value (constant) in the equation y = kx. The general formula for when x is directly proportional to y is y = kx where k is the gradient and can be any constant.

Now you try Find the value of k from these graphs: Remember k is the gradient… y (10, 20) Gradient = (y 1 -y 2) ÷ (x 1 -x 2) y (18, 9) k = ? ? ? 2 O y O x (56, 8) y x (20, 5) k = ? ? ? 7 O x k = ? ? ? 1/2 k = ? ? ? 4 O x

Problem Solving and Reasoning a) What is the gradient of the graph? b) What is the equation linking C (cost) and A (number of apples)? Cost in £ Number of apples bought

Problem Solving and Reasoning y O y A x x C O y B O Which of the graphs A, B, C or D show: a) y is directly proportional to x? b) y is inversely proportional to x? x D x The graph of y is directly proportional to x so should be a straight line as y = kx is also a straight line. Therefore it is either A or C y = kx has a y intercept of 0 meaning it crosses the point (0, 0) therefore it must be graph A. The graph of y is inversely proportional to x is the same as that of y is directly proportional to 1/x. The graph of 1/x looks just like graph B therefore B is the answer.

Reason and explain § What does the graph of y is directly proportional to x 2 look like? § What does the graph of y is inversely proportional to x 3 look like? § Does y is inversely proportional to the inverse of x mean the same as y is directly proportional to x?