Teachers Notes

On his 20th birthday a man realizes that he is now exactly ½ the age of his Mother who is 40. This man has always been curious with numbers and patterns and immediately asks himself if this has ever happened before or if indeed it will ever happen again?

What conclusion did he reach?

Background

So this actually happened to me (different numbers) and I argued with my 11 year old nephew that it could only happen once, safe in my sound mathematical knowledge that our ageing rates were the same and that as a consequence the ration of our ages would constantly change and get closer to 1 without ever reaching it. Using the refreshing metal agility of a student unbound by these constraints my nephew was able to point out to me the limits of this argument when considering the way we describe our ages as integers and how in fact it would happen twice. He went on to explain that the duration of each period of time for which it was true would depend on the two birthdays in question but the sum of the two durations would be 1 year! I was forced to eat humble pie and reflect on what had just happened. Ever since I have enjoyed throwing this problem at students so that they may have the same experiences - some playing my role and some my nephew's.

The problem is easily accessible and is a nice example of how the starting points of a problem can undermine or redefine the mathematics we use to solve it. Equally its good way to remind students how linear functions (and others) can be used to model situations and solve problems. Below there is a possible running order for the activity and an outline of some of the benefits and sticking points involved

Running order

Students are in groups for this activity and are asked to discuss and debate this briefly before volunteering answers.

Upon volunteering answers, hopefully a debate ensues at which point students are asked to consider how they could best present their argument. What form of words or diagrams might be most appropriate? Students are invited to try out their theories

This discussion is then followed with specific, differentiated worksheets that ask students to explore the problem using a variety of functions.

The following important issues will hopefully arise;

• statement of conditions - are we considering exact age down to seconds or simply age in years for the duration of that year.
• What is the starting point of the problem (ie valid domain and thus range for the investigation)

The choice of graph is dependent on the answers above; the graphs that might be considered are

• linear functions for the age of the parent and child leading to the conclusion that when the y-coordinate of one is twice the y-coordinate of the other then the condition is satisfied. This assumes exact age and is easiest done assuming parent and child share a birthday. It is subsequently interesting to consider what happens if they do not! This can lead to the solving of simultaneous equations, where 2 x one function equals the other
• Step functions. When considering a person’s age in years we consider that their age increase by one year on each of their birthdays and that they stay that age for one year. This can be represented by a step on a graph, where the point is closed at the beginning of the year and open at the next, so that our graph remains a function. If step graphs are drawn for each of the people, then students should find 2 occasions when the age of one is twice the age of the other.
• Exponential functions? A third possible function to imagine here is a single function of the ratio of the two peoples ages eg y = a/b where a and b are the two ages. This function is then solved for y = ½ or Y = 2 depending on the way it is set up. This allows the exploration of asymptotes.

Sticking Points

What's a linear function?

Its possible that some students will need further reminding of what is meant by the term linear function and its general form y=mx+c (or equivalent). This is a good and important opportunity to do so and dwell on the modelling aspect.

Axes and scales

This can be a major sticking point and students are still occasionally expected to draw graphs and axes from nothing, making these relevant issues. Pre-prepared axes could be used in order to concentrate the lesson on using the functions. Alternatively, its important to allow enough time for these issues to be discovered and discussed. This is particularly tricky with the step function.