# Half My Age

'How many times will you be half the age of one of your parents? A look at modelling with functions'

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! This activity invites the simple modelling of this situation/problem starting with linear functions and moving on to things less familiar.

### Aims

The aim here is to refresh memories on how functions (linear to start with) are used to model real situations and then used to solve problems. The activity involves exploring unfamiliar functions as well in the latter stages.

### Resources

The three separate parts to this investigation are outlined below and available in printed format here Half my age 1, Half my age 2, Half my age 3, Teachers can read more about this activity before running it here with these Half my age teachers notes

Here a some example graphs that might be produced.

#### Half my Age 1

- 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?
- Consider that both the man and his mother share a birthday. Use the following variables;
- y
_{1}is the mans age, x is the number of years since his birth, it should follow then that y_{1}= x. - Draw a set of axes with the x-axis going from -30 to 30 and the y – axis going from 0 to 50. Make sure your graph is accurately and neatly drawn, then draw the line y = x on it.
- Now consider that y
_{2}= his mothers age and x is still the number of years since his birth. What is the expression that links y_{2 }and x now? Plot this expression on the graph. - Identify the point at which y
_{2}= 2 x y_{1} - Can you now solve this problem algebraically?
- What effect would it have on the problem if the mother and sin did not share a birthday? What considerations would you have to make

#### Half my Age 2

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?
- For this problem, lets consider that a persons age in years remains the same from one birthday to another. This can be shown on a graph by a ‘step’. For example, for the first year of his life the man is 0 and this is represented by a horizontal line that runs along the line y = 0 between x = 0 and x= 1. At x = 1 a next line starts and runs along the line y = 1 until x = 2.
- Draw a set of axes with the x-axis going from 0 to 25 and the y – axis going from 0 to 50. Make sure your graph is accurately and neatly drawn, then draw the line y = x on it.
- Consider that both the man was born on 1
^{st}January and that his mother was born on the first of April. The draw a step graph for each person on this axis. - Using this graph, how many occasions can you identify when the mans age in years is half of his Mothers age in years?

#### Half my Age 3

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?
- Consider again that the man and his mother share a birthday. Let the man’s age be a and the mothers age b
- Examine the pattern created by y = a/b by looking at various values of a and b. Having done this, decide on appropriate scales and plot some points on the graph.
- What are the features of the graph and what is its equation?
- What would be the implications for this problem if the man and his Mother did not share a birthday?

### Description

Here follows an outline if what the task is. If students are not reading this page then the teacher will need to show and give this overview.

- Pose the problem as written above or on the worksheets (or indeed changed to suit your class).
- There follows a discussion/debate where both answers yes and no are argued so that students experience conjecturing and counter conjecturing.
- Consider how functions could be used to model the situation and help solve the problem.
- Move to defining and drawing these functions by using one or all of the three tasks in the worksheet.
- Finish by asking for modified responses to the initial problem.

### I did it my way!

As a practising maths teacher I know that most of us like to give activities our own little twist and do them 'our way'. It would be great to add a little collect of 'twists' from users. You can either add your twist to the comments section below or e-mail them directly to me at jamesn@inthinking.co.uk In time some of these twists may appear here....