# In the money

'Seconds to make up your mind! - Have fun with this hypothetical problem of sequences'

A long lost relative wants to share their fortune with you! There are 4 different options about how to receive your money in yearly installments. Each one starts with a different amount that changes each year according to a given rule. You are given 2 minutes to make your choice only! Having made your choice, investigate the 4 options to look at their pro's and cons and the eventual wisdom of your choice. Have a bit of fun and learn about different types of sequences and how they grow!

## Activity

Here you will find all the details you need about the activity. Instructions for students, downloads etc..... This will look different for different activities.

### Aims

The main aim of the activity is to introduce geometric sequences in comparison to arithmetic ones. Using this problem as a starting point the aim is to look at the structure of the two different types of sequences and try to generalise about them. Students will generate terms and calculate sums.

### Resources

The problem and tasks are outlined in the   In the money activity sheet. There is also a partially completed   In the money spread sheet to work with. Teachers can read more in the teacher's notes at the bottom of the page

### Description

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

• Consider the initial question and make an instinctive choice.
• Using a spreadsheet or otherwise generate the terms of the different sequences for a given period of time and conclude on the wisdom of the decision.
• Make graphs to model the situation.
• Work on the arithmetic sequences and generate some terms.
• Work on the geometric sequences, generate some terms, work on some problems and begin to generalise.

### Winner !!!

You receive a phone call that says you are the winner of a mystery lottery and that you can choose one of 4 ways in which to be paid your winnings. You have only 30 seconds to decide!

Option 1- \$3000 in the 1st year, \$3250 in the 2nd, \$3500 in the 3rd and so on with the amount received increasing by \$250 each year.

Option 2- \$500 in the 1st year, \$1000 in the 2nd, \$1500 in the 3rd and so on with the amount received increasing by \$500 each year.

Option 3- \$1 in the 1st year, \$3 in the 2nd, \$9 in the 3rd and so on with the amount received multiplying by 3 each year.

Option 4- \$50 in the 1st year, \$100 in the 2nd, \$200 in the 3rd and so on with the amount received multiplying by 2 each year.

a)– you have 30 seconds to make your choice and record it

b)– Having made your choice, construct an excel spreadsheet that will monitor the amounts received for these four options over a 25 year period. Include a column for each option that shows the ‘total’ or ‘sum’ of money received after 5, 10, 15, 20 and 25 years.

c)– Use the spreadsheet to produce two graphs; one that shows how the annual amount compares with the four options and a second to show how the sum compares.

d)- Discuss the wisdom of your choice!

### Identifying Arithmetic Sequences

Once again consider the four options and the sequences that are generated by them (the column that shows annual amount)

1. Which two of the four options produce arithmetic sequences?
1. For each of these, fill in the table below
 Option U1 d U1 Sn U25 S25

### Defining Geometric Sequences

Refer now to the other two options (not dealt with above)

1. Describe these sequences using words
1. Fill in the table below for the other two options
 Option U1 U2 U7 U10
1. Now for both options fill in the table of expressions below
 Express Option Option U2 in terms of U1 U7 in terms of U1 U10 in terms of U1 Un in terms of U1

### Formulae

Use your formula booklet to find the formula for the nth term of a Geometric sequence and compare it to yours

Write down the formula, define the terms and explain what each part means

### 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 collection 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....

## Teacher Notes

Here is a section specifically for teachers about how to run the activity.

Winning money generally provides an engaging context quickly and the 30 second decision piles pressure on the intuitive skills! Even amongst those that opt for the exponential options there is surprise at the speed and size of the growth of their fortunes!! The context also provides an obvious need for both term and sum. Using Excel invites students to use the formulas that they ultimately aim to derive! The analysis brings sequences and functions together and makes a valuable link! Students are also asked to define variables and make different expressions using them which invites them to think about about the relationships between the different terms of the sequence and to generalise. Its a fun activity that introduces the structure of geometric sequences well.

### How

The following is some practical advice about how the activity might be run.

#### Resources provided

There is a written worksheet that clearly describes all the stages of the activity.

#### Resources needed

This activity is really well aided by the use of a spreadsheet which, of course, can also be done on a GDC.

#### Time needs

This activity can generally be done in 1 hour, depending on the usual variables. It may be useful to follow it with a more formal lesson on the topic so references can be made to it.

#### Starting and finishing

• Its important not to give too much away at the start of this activity. Present the problem either with the worksheet or by other means and ask for a quick decision from students. The point of this is to challenge our intuition.
• constructing the spreadsheet columns for these different options really brings out the desire and need for formulae which is a great moment worth dwelling on.
• graphing the sequences is a good way of looking their different properties and eventually linking to other topics. It is also a good opportunity to make the distinction between sequences and functions.
• the second part of the activity is about distinguishing between the different types of sequences and is more worksheet based. The questions are posed in a way that hopefully prompts students to think carefully about the relationship between different terms of sequences and the fundamental differences between the different types of sequences.
• students should probably do this part individually, but the teacher may expect to give a good deal of input here.

#### Records

The worksheet offers a good record of this activity and use of computers would let students incorporate some of the numbers from the spreadsheet in to their records as well.

### What

• This paragraph talks about the the sorts of things to expect and watch out for during this activity and the possibilities that exist within it to change or extend the task.
• Students like to make intuitive judgments in Mathematics and so it is fun to give them the chance to do so and hear them begin to add some reason to their decisions. Having collected the decisions, there is an opportunity for an excellent discussion that encourages some of that reasoning.
• considering formulae for spreadsheets or lists brings a range of problems to the core that can be interesting too. The main issue is the difference between the formulae for nth term and for the sum. In what year did you receive the most money?' and 'in what year did you have the most money in total?' are two different questions and its nice that these issues turn up by themselves.
• The geometric sequences quickly move in to the use of scientific notation, which is good practise!
• the second part of the exercise has students working more with algebraic variables and this transition needs a little oiling. Some of this is about getting used to the notation, but mostly it is about the slow transition to the general case.

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