This unit requires students to apply simple exponential functions in everyday situations.
In this unit students build on their knowledge of exponents to investigate situations where exponential change occurs. Typically, those situations involve a fixed ratio between consecutive members of a geometric sequence. The ratio is the base of the exponential function, and that ratio is applied 'n' times to find the nth member of the sequence.
For example, suppose $100 is invested at an interest rate of 5% per annum. At the end of year 1 an additional $5 in interest will be earned. The total amount in the bank is $105. The ratio between 105 and 100 is 1.05, which is the base of the exponential relationship between number of years and amount in the bank. By the end of the fifth year the base, 1.05, will have been applied five times so the amount will be 100 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05. This can be simplified to 100 x (1.05)5. In general, the amount of money after year n is y = 100(1.05)n.
In general, exponential functions are of the form y = a(b)x though transformations can also be applied (translations and reflections). A represents the quantity of term zero and b is the base. Most situations in this unit are about discrete relationships in which integral values for one variable relate exponentially to integral values of the other variable. Exponential functions are generally continuous meaning the variables are free to take up any value in the domain of the function, e.g. temperature of a forest fire as time increases.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
The context for this unit can be adapted to suit the interests and experiences of your students. Select contexts that involve doubling of quantities, the population of animals, percentage growth, etc. that are relevant to your students. Perhaps you could generate a list of contexts with your class.
Te reo Māori kupu such as ōrau (percent), taupūtanga (exponential), and whārite (equation) could be introduced in this unit and used throughout other mathematical learning.
You might start the table like this:
Year | Worth |
0 | 1 |
1 | 2 |
3 | ? |
… | … |
Slide two shows how a graph of the relationship between year number and Natasha’s worth is created. The video uses Desmos as the platform but it is very important for the students to create the graph manually using graph paper. Look for your students to:
Slides three and four allow you to discuss different types of functions, that are many to one relationships, between the values of variables. The functions shown look similar in equation form, particularly y=x2and y= 2x.
In the next part of the lesson we use an analogy from Professor David Suzuki, a world leader in sustainable ecology. You will easily find a video of a talk by David online. The idea is for students to understand the implication of exponential growth of the world population for a sustainable future.
The final slide shows the trend over time (60 minutes) in the percentage of the food in the dish that the bacteria consume. Students should note the implications of exponential growth, the rapid change in percentage occurs over a short duration of minutes. You might look up graphs of human population growth to show a similar pattern over time.
In this session students explore exponential functions where the base is one half. In that case the number of units of the dependent variable is multiplied by one half for each increase of one unit in the independent variable. The result is a repeated halving process. Exponential decay is used in real life to calculate the amount of a drug in a person’s bloodstream and to date the age of historical artefacts.
Time (hours) | 0 | 1 | 2 | 3 | 4 | 5 |
Amount of drug (mg) | 100 | 50 | 25 | 12.5 | 6.25 | 3.125 |
Imagine we graphed these pairs of numbers as co-ordinates. What do you think the shape of the graph would be?
Students might believe that at some time the curve will cross the x-axis. Let them graph more points to establish that the curve converges to the x-axis but does not cross it. Introduce the idea of an asymptote which is a line in this case. The distance between the curve and line approaches zero as the values of the independent variable change. In this context, the amount of drug in the bloodstream approaches zero as the number of hours since injection increases.
Since the scales are different it may be difficult for students to spot that the point (0, 1) is common to all three curves. You may need to work on lessons from the Level 5 unit, Exponent Power, to help students see why b0=1, for all values of b.
So far students have explored exponential functions with bases of two, and one half, though that was extended in Lesson Two. In this lesson students experiment the effect of changing the base. They do so through two contexts; Lock combinations and management trees.
Part One: Combination Locks
After slide two ask:
How can you work out the chance of entering the correct code by luck? (Need to find all the possible permutations)
What are some ways to find all the possible codes? (Create a systematic list, a tree diagram, etc.)
Number of places | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Number of possible codes | 3 | 9 | 27 | 81 | 243 | 729 | 2187 |
Number of places | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Number of possible codes | 4 | 16 | 64 | 256 | 1024 | 4056 | 16384 |
Imagine we graphed this relationship on the same graph as you used previously. What would happen? (Do the students realise that the vertical scale will not be nearly high enough?)
How much higher does the vertical scale need to be? (about eight times higher)
You might investigate the effect of changing the number of digits on the growth curve for the number of possible codes, graph the curves using digital technology, and write equations or curves that pass through the points.
Part Two: Management Structures
Layers of management | 0 | 1 | 2 | 3 | 4 | 5 |
New employees | 1 | 3 | 9 | 27 | ? | ? |
How many new employees will be added in layers four, five and six?
How is situation similar and different to the number of the bacteria story?
Slide Ten shows a graph of the function and highlights the point (0,1), i.e. No layers but one manager. The curve shows the pattern of a path through the points. Note that the context is discrete, i.e. involves whole numbers of people, not continuous.
Slide Eleven shows the curves for ratios of 1:3 and 1:4 on the same graph. Students should note that the difference between the number of new employees grows dramatically as the number of layers increases.
More sophistication occurs when students recognise that the power immediately below 10 000 is significant. The sum of numbers of people before that power will all be managers. For example, look at the powers of five:
However, there will need to be another 6 875 employees to reach 10 000. They will need managers at a ratio of 1:5. 6 875 ÷ 5 = 1375 extra managers. The total number of managers needed will be 1 + 5 + 25 + 125 + 625 + 1375 = 2 156.
Here is a table of the number of managers needed:
Ratio | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Number of Managers | 9095 | 4427 | 2841 | 2156 | 1926 | 1486 | 1323 | 1203 | 1111 |
Up until now students have worked with exponential growth where the base is a whole number, and the simple case of one half. In this lesson they extend their thinking to realistic situations in which the base is non-integral, e.g. Compound interest rate of 10%.
Students might appreciate that the calculation for five years is 1000 x (1.1)5. Can they generalise that the amount for any year can be calculated as 1000 x (1.1)n?
Slide Six sets up an investigation of the effect of different interest rates. Obviously higher rates produce higher returns that the difference in returns become exaggerated by the end of ten years. Some students might use digital graphing tools to make the comparison efficient.
Slide Seven shows a graph of the relationship between year and amount for interest rates of 5%, 10% and 20%. As the rate doubles the interest amount after ten years more than doubles.
Assessment
PowerPoint Five provides two assessment problems. Sierpinski’s triangle involves an iterative (repeated algorithm) process. The number of black triangles relates to generation through powers of three.
Generation | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Number of black triangles | 1 | 3 | 9 | 27 | 81 | 243 | 729 |
Measles is an infectious disease and the transmission rate is between 1:12 and 1:18, meaning that each infected person spreads the infection to between 12 and 18 people. Luckily the death rate is relatively low at 1-2 people per thousand. An epidemic of measles in an un-immunised population would spread rapidly since the base of the exponential function is between 12 and 18. In six generations measles would theoretically infect over 12 million people, more than twice our current population. The normal infectious period for measles is about 10 days. According to those numbers, in 60 days, the whole population of New Zealand could be infected. About 0.2% of the 5 million people would die, that’s 10 000 people.
Generation | New Infected | Total |
0 | 1 | 1 |
1 | 15 | 16 |
2 | 225 | 241 |
3 | 3375 | 3616 |
4 | 50625 | 54241 |
5 | 759375 | 813616 |
6 | 11390625 | 12204241 |
It is important to realise that the reality of an epidemic would hopefully not follow a mathematical model to the extent of infecting the entire population. The model assumes that every infected person has unrestricted interaction with uninfected people. As a city became more highly infected, many infected people would only have interactions with other infected people, lowering the overall rate of transmission. Also, in the case of an epidemic, efforts would be made to reduce the rate of transmission by encouraging people (whether infected or not) to avoid traveling or interacting with crowds.
Dear families and whānau,
Recently, we have been learning to apply simple exponential functions in everyday situations. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/powerful-growth at 11:36pm on the 26th February 2024