Proportionality to powers In many situations, one variable may be directly proportional to a power of the other variable. For example, the kinetic energy of an object is proportional to the square of its speed. This means that if the speed of an object doubles, its kinetic energy will be four times greater. If the speed of the object trebles its kinetic energy will be nine times greater. When the object is at rest it will have no kinetic energy. Teacher notes As in a directly proportional relationship, when the speed is 0 the kinetic energy is also 0.

Proportionality to powers The kinetic energy of an object is proportional to the square of its speed. What will happen to the energy of the object if it quadruples its speed? What will happen to the energy of the object if it halves its speed? Can you draw a graph to illustrate the relationship between the kinetic energy of the object and its speed? What do you notice about the shape of the graph? Teacher notes If the object quadruples its speed, its energy will be 16 times bigger. If the object halves its speed, its energy will quarter. Getting students to draw a graph to illustrate the relationship between the kinetic energy of the object and the speed of the object will allow them to see how the relationship works. If you wish, you could use the graph from slide 20 to see how the students’ graph matches up. In order to draw an accurate graph, you will need to provide the students with a constant and values for either the kinetic energy of an object or its speed.

Equations and square proportion If one quantity, y is directly proportional to the square of another quantity, x, we can link them to each other with the symbol  by writing: y  x2 We can also link these variables with the equation: . y = kx2 In this equation, k is called the constant of proportionality. Teacher notes Stress that x and y are variables, while k is a constant number. Photo credit: © Michael J Thompson, Shutterstock.com By rearranging the equation we can see that k = . y x2 This means that the ratio between y and x2 is constant.

Equations and square proportion One quantity, b, is in direct proportion to a2. a b 1 2 3 27 4 5 10 3 12 48 75 300 By writing an equation in a and b, or otherwise, can you complete this table? To find the value of the constant, k, square the value a then find what this has been multiplied by to give you b. b is directly proportional to the square of a, so: b = ka2. When a = 3, b = 27. This means that: 27 = k32 9 27 Rearranging the formula shows: k = = 3

Equations and square proportion One quantity, b, is in direct proportion to a2. a b 1 2 16 3 64 81 5.5 4 4.5 4 36 121 By writing an equation in a and b, or otherwise, can you complete this table? What value will b have when a is 7? What value of a will make b equal to 144? What value of a is needed to make b greater than 100? Teacher notes Discuss how we can use the given values to find the constant of proportionality, k and write equations linking the two variables. b is proportional a² so, b = ka². When a = 2, b = 16, so 16 = 4k. Rearranging the formula shows that k = 16 ÷ 4 = 4. We can write b = 4a². Using inverse operations establish that a =b/2. Use this formula to complete the table. The missing numbers are: a = 1, b = 4, a = 3, b = 36, a = 4, b = 64, a = 4.5, b = 81, a = 5.5, b = 121. When a = 7, b = 4 × 7² = 4 × 49 = 196. When b = 144, a = 144/2 = 12/2 = 6 . When b = 100, a = 100/2 = 5. When a > 5, b > 100. So, any value over 5 for a will make b greater than 100.

Equation and square proportion Teacher notes This open-ended activity has set numbers for a and b. The nature of the activity allows different permutations of a and b to be put together to calculate different constants, k. While some of the combinations will produce complex values for k, students should be able to work with the numbers given.

Using proportionality to write formulae Russian dolls, known as matryoshka, fit inside each other. Each of the dolls is the same shape, but they all have different heights. How are the surface areas and the heights of the dolls related? The surface area, S, of each doll is directly proportional to the square of its height, h, meaning that the dolls are mathematically similar. Teacher notes Discuss the fact that if two three dimensional shapes are similar, any corresponding areas in the shapes will be directly proportional to the square of any corresponding heights. Photo credit: © Andriano, Shutterstock.com We can write this as: S  h2 or S = kh2.

Using proportionality to write formulae The largest matryoshka doll is 11 cm high and has a surface area of 193.6 cm2. Write a formula in terms of S and h. We can find k by substituting the known values of the largest doll into S = kh2. 193.6 = 121k k = 193.6 ÷ 121 Photo credit: © Andriano, Shutterstock.com k = 1.6 The formula linking the surface area and height is therefore: S = 1.6h2

Using proportionality to write formulae The surface area of each doll is directly proportional to the square of its height, making the dolls mathematically similar. Use the formula S = 1.6h2 to complete the questions below. Teacher notes This slide should give the students a chance to practise substituting into formulae when given specific information and also the students’ ability to rearrange a given formula in order to find a specific missing quantity. By substituting into the formula for the first question, we can see that: S = 1.6h2, S = 1.6 × 72 = 78.4 cm2 By substituting into the formula and then rearranging it in the second question, we can see that: S = 1.6h2, 3.6 = 1.6 × h2, h² = 3.6 ÷ 1.6 = 2.25, h = √2.25 = 1.5 cm. Photo credit: © cristi180884, Shutterstock.com One of the dolls is 7 cm tall. What is its surface area? The smallest doll in the set has a surface area of 3.6 cm2. What is its height?

Inverse proportionality to powers In some cases, one variable can be inversely proportional to a power of the other variable. For example, the electrical resistance, R, of a metre of wire is inversely proportional to the square of its diameter, d. We can write this relationship as: R  . d² 1 Photo credit: © Galushko Sergey, Shutterstock.com Alternatively, we can express this relationship by writing it as an equation using a constant of proportionality. d² k R =

Using proportionality to write formulae The electrical resistance of a metre of wire with a diameter of 2 mm is 1.2 ohms (unit of electrical resistance). Write a formula linking the electrical resistance of the wire, R, to its diameter, d. d² k Substitute the given values into: R = . 2² k 1.2 = 4 k Teacher notes Note that here we have worked out k for diameters that are given in mm. This is fine, so long as we continue to calculate using diameters in millimetres. Photo credit: © Christopher Sykes, Shutterstock.com 1.2 = k = 4 × 1.2 = 4.8 4.8 d² R =

Using proportionality to write formulae The electrical resistance, R, of a metre of wire is inversely proportional to the square of its diameter, d. What is the electrical resistance of the wire when the diameter is 5 mm? What diameter of wire would have an electrical resistance of 0.3 ohms? Teacher notes This slide should give the students a chance to practise substituting into formulae when given specific information and also the students’ ability to rearrange a given formula in order to find a specific missing quantity. By substituting into the formula for the first question, we can see that: R = 4.8/d² = 4.8 ÷ 5² = 4.8 ÷ 25 = 0.192 ohms. By substituting into the formula and then rearranging it in the second question, we can see that: R = 4.8/d² d² = 4.8 ÷ R = 4.8 ÷ 0.3 = 16 d = √16 = 4 mm. Photo credit: © foto, Shutterstock.com

Using inverse proportionality

Graphs of proportional relationships When trying to find the relationship between two variables, it is often useful to construct a table of values for the variables and use these to plot a graph. If y  xn, four different shaped graphs are possible: n = 1 n > 1 Teacher notes Discuss what type of proportion is shown by each graph. When n = 1, y is directly proportional to x. When n > 1, y is directly proportional to x2, x3 or any other positive power of x. When 0 < n < 1, y is directly proportional to any nth root of x. When n < 0, y is inversely proportional to x or any power of x. 0 < n < 1 n < 0

Graphs of proportional relationships Teacher notes This activity can be used to introduce students to the different graph types that proportional relationships produce. The activity allows you to vary the value of k and note the basic properties of each graph. When y  x, i) the graph is a straight line that passes through the origin; ii) as x increases by a fixed amount, y increases by a fixed amount, iii) the value of k, the constant of proportionality is given by the gradient of the graph. When y  1/x, i) the graph is a curved line that does not pass through the origin. It never touches the x- or y-axis; ii) as x increases by a fixed amount, y decreases in decreasing amounts; iii) the greater the value of k, the steeper the graph. When y  x2; i) the graph is a curved line that passes through the origin; ii) as x increases by a fixed amount, y increases by increasing amounts. This can be investigated by using the pen tool to show how many units y increases by for each unit in the x direction; iii) the greater the value of k, the steeper the graph. If y varies directly with any power of x greater than 1, a similar shaped graph will be produced. Compare this graph with those showing repeated proportional change by an amount greater than 1 in the presentation ‘Graphs of non-linear functions’. When y  √x; i) the graph is a curved line that passes through the origin; ii) as x increases by a fixed amount, y increases by decreasing amounts. This can be investigated by using the pen tool to show how many units y increases by for each unit in the x direction; iii) the greater the value of k, the steeper the graph. Remind pupils that x can be written as x½. If y varies directly with a fractional power of x (between 0 and 1), a similar shaped graph will be produced. Compare this graph with graphs showing repeated proportional change by an amount between 0 and 1 in the presentation ‘Graphs of non-linear functions’.

Identify the graph

Complete the table of values Teacher notes Ask pupils to fill in the missing values in the table by finding the constant of proportionality for the given relationship or otherwise.

Using proportionality to solve problems Drinks manufacturers make different sized products that are mathematically similar. The surface area of each can is directly proportional to the square of its height. How much raw material is used in the manufacture of each small can? Aluminium costs £13.45 for a 9 m² sheet and cans are manufactured in bulk. The minimum order is 10 000 units. Teacher notes Work out the constant, k, using the information from the larger can: S = kh², k = S ÷ h² = 371.8 ÷ 13² = 2.2 Work out the surface area of the smaller can: S = kh² = 2.2 × 9² = 178.2 cm². Work out how much raw material would be used in one batch of each sized can: 10 000 x 178.2 = 1 782 000 (small cans), 10 000 x 371.8 = 3 718 000 (large cans) Work out how many whole sheets of aluminium you need for each batch: 1 782 000 cm² / 90 000 cm² = 19.8 or 20 full sheets (small cans) 3 718 000 / 90 000 cm² = 41.3 or 42 full sheets (large cans). Work out the cost of aluminium for one batch of each size of can: 20 × £13.45 = £269 42 × £13.45 = £564.90 As an extension task, you may wish to get students to imagine they are working for the company producing this soft drink. How much do they need to sell each can for in order to make a profit? Compare the selling price with the prices of competitors. Is this realistic? You could make this task as all-encompassing as you wish. Photo credit: © Roman Sigaev, Shutterstock.com How many aluminium sheets are needed for one order of each of the two types of can? How much money does it cost to make 10 000 units?

Using proportionality to solve problems Some manufacturers make concentrated products to save on packaging. The surface area is proportional to the square of the width. The smaller carton has a surface area of 500.926 cm². Find the value of the constant, k. As a percentage, how much more packaging does the large carton use? A shelf is 2 m long by 30 cm deep. How many more small cartons will fit on a shelf than large cartons? Teacher notes The surface area is proportional to the square of the width of the carton. Therefore, S = kh². Rearrange this to find k: k = S ÷ h² = 500.926 ÷ 7.3² = 9.4 This means that the surface area of the larger carton is: S = 9.4h² = 9.4 × 12.4² = 1445.344 cm². Find percentage increase in packaging used 1445.344 ÷ 500.926 × 100 = 288.5% increase in packaging Find how many of each sized carton can fit on a shelf. Small carton: 200 cm ÷ 7.3 cm (width) = 27 whole cartons 30 ÷ 7.3 (length) = 4 whole cartons 27 × 4 = 108 on a shelf Large carton: 200 ÷ 12.4 = 16 whole cartons 30 ÷ 12.4 = 2 whole cartons 16 × 2 = 32 on a shelf You can fit 108 small cartons on the shelf, but only 32 large cartons. Therefore, you can fit 108 – 32 = 76 more of the small cartons on a shelf.

Investigating relationships The following table contains data taken from five schools in England in 2009. Teacher notes Students need to take a pair of data sets and investigate if there is a relationship between them e.g. overall absence and persistent absence, overall absence and average GCSE points, persistent absence and average GCSE points. To investigate the relationships they need to set the data out in some kind of ascending order by one data set, and decide which relationship is most likely – square proportion, inverse proportion etc. When they have decided which one to try, they need to take a pair of values and work out k. Then apply the constant to the next pair to see if there is a close match. They need to do this to all 5 pairs of data and record their results. They then need to try a different proportional relationship using the same set of data to see in they get a closer match. Can you find any relationships amongst the data? What kind of relationships can you find? Show your working.