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Level Five > Number and Algebra

# Playing with Money (From the Other Point of View)

Keywords:
Purpose:

In this unit we look at exchange rates for five countries. We see (i) how much various foreign currency amounts are worth in New Zealand dollars; (ii) how to make a currency converter; (iii) how to approximate these conversions; and (v) how to use the exchange rates to find foreign currency values in other foreign currencies.

Achievement Objectives:

Achievement Objective: Use rates and ratios.

Specific Learning Outcomes:

explain what exchange rates are

convert money amounts accurately from New Zealand dollars to another currency

convert money amounts from New Zealand dollars to another currency

convert money amounts from one country's currency to another country's currency

graph currency fluctuations and interpret the graphs

Description of mathematics:

This unit is about proportion and ratio. We use exchange rates, the relation between various currencies, to practice finding proportionate values.

Proportion is something that many of us find very difficult to understand. However, it is something that we find in many areas of daily life and it is essential in mathematics itself. Apart from currencies, proportion is necessary if you have to transfer from metric to imperial units; if you have to make a scale drawing or model; if you have to change a recipe for 4 people to a recipe for 6; and if you have to make several pieces of clothing from a pattern.

It is important for students to master proportion and ratio before they move on to algebra as it is indispensable in solving equations.

In this unit we mainly look at the corresponding values of currency amounts in different countries. To find out more about how this is done, you should look at Content Tutorial "Breafast in Sydney".

Prior to teaching this unit, we suggest that you work with students on the unit Playing with Money (From the New Zealand Point of View), Level 5.

In preparation for session 3, you will need to get the students to find exchange rates for a number of countries well ahead of the day that you use that session's material.

Required Resource Materials:
Television news (to get exchange rates) though these could be obtained from newspapers or the web
Calculator
Activity:

The above screen appears every night on TV One's news. Clearly this contains different numbers to the ones used in Playing with Money (From the New Zealand Point of View), Level 5. You can find the screen on the web by using the URL: http://www.tvnz.co.nz/view/news_sub_cat_skin/news_business_index_group. Click on ‘Markets ’. We will use the information here to begin a unit on proportion.

What does it all mean? Starting at the top this screen is about exchange rates, that is, the rates at which currency is exchanged. More specifically, the table tells us how much in given overseas currencies one New Zealand dollar is worth. So we can see that one New Zealand dollar is worth

0.7007 United States dollars
0.9211 Australian dollars
0.3882 British pounds
0.5710 Euros
77.19 Japanese yens

The burgundy triangles (pointing down) indicate that the New Zealand dollar has gone down against the given currencies since the last news; the pointing up triangles show that the New Zealand dollar has gained in value against the foreign currency; the ‘NC' means that there has been no change.

In session 2, the students will produce currency converters for each of the currencies here to convert from a foreign currency into New Zealand dollars. We suggest four methods of currency conversion here but you should encourage them think about how this might be done. And you might especially let them think about what amounts they will break for the rows and columns of the converter if they are using Methods 1 or 2.

Method 1: A table. You can set up a table like the one below. In the table you put the exchange value of the currency you are converting to. For instance, in the position marked XXX you will put the New Zealand equivalent of, say, US \$2.20.

 US \$0.00 \$0.10 \$0.20 \$0.30 \$0.40 \$0.50 \$0.60 \$0.70 \$0.80 0.90 \$1 \$2 XXX \$3 \$4 \$5 \$6 \$7 \$8 \$9 \$10

Make sure that the students find quick ways to fill in the table. You won't expect them to use the exchange rate for each one. For instance, the \$US4.40 cell can be filled by simply doubling the entry in the \$US2.20 cell. Encourage them to use as many short cuts as they can think of.

Method 2: A similar table can be completed more quickly (and maybe more accurately) by using an Excel spreadsheet. Once a column has been started the rest can be filled in by ‘dragging down'.

Method 3: Let them discover an algebraic equation for the conversion so that when they put in the foreign value for x, say, they get out the, New Zealand value y.

Method 4: They could write a program for a computer or graphics calculator.

You can find out about the exchange rates for other currencies (such as the ones that you will need to find for sessions 3 and 4) by looking in your daily newspaper or the web. If you do a search on exchange rates' you will find a number of sites. However, for the purposes of session 3 it would be good to get exchange rates in terms of New Zealand dollars. You will probably need to have the students get the information for session 3 at least 24 hours ahead of time.

In session 4 you will be working with approximations. It is important to discuss what a good approximation is. Presumably it will be good if it is reasonably accurate and if it can be calculated in the head. Are there other criteria?

Some background to this unit can be found in the Content Tutorial "Breakfast in Sydney". This might be particularly useful towards the end in Session 5.

### Teaching Sequence

#### Session 1

In this session, we look at exchange rates and begin to change currency from a foreign currency to its New Zealand dollar equivalent.

1. We start this unit in the same way that we started the unit Playing with Money (From the New Zealand Point of View), Level 5. This should act as a reminder about exchange rates. If you are giving this lesson immediately after that unit then there is no need to use this introduction.
2. Show the class a copy of the TV One News exchange rate table. (If possible, use one that you have got from the web from the night before.) Leave it where they can see it. Ask:
What is this? What does it tell us?
What denominations are there in the (i) United States currency (dollars and cents); (ii) Australian currency (dollars and cents); (iii) British currency (pounds and new pence); (iv) Euros (euros and cents); and (v) Japanese currency (yens)?
What is another name for the British currency? (Sterling)
The symbol for dollars is S with a line through. What is the symbol for pounds? What is the symbol for new pence? (They may need to look this up.)
Which countries use the Euro? (There are many here. Perhaps you could start a list on a notice board and add to it when someone in the class finds another European country that uses the Euro. The complete list is Belgium, Germany, Greece, Spain, France, Ireland, Italy, Luxembourg, The Netherlands, Austria, Portugal, Slovenia and Finland.)
The symbol for dollars is S with a line through. What is the symbol for Euros? (They may need to look this up.)
What is the symbol for yens?
What do the burgundy triangles tell us?
What was the value of the New Zealand dollar yesterday in (i) United States currency (70.07 cents) (ii) Australian currency (92.11 cents) (iii) British currency (38.82 pence); (iv) Euros (57.10 cents ); Japanese currency (77.19 yens there is no smaller currency unit)?

You should feel free to add any questions that you think might be interesting or fit into work in other subjects. For instance you might like to ask students to point out on a map where the various countries are. You might like to ask what the capitals of those countries are.
3. Now get into some harder calculations. Spread the following question around the class (not just to the first person who puts up their hand.) You might ask more than one student to answer the same question. If there is a difference of opinion this can lead to a worthwhile discussion. Make sure that the students use the correct units. Ask:
What is \$NZ 100 worth in US dollars and cents? (\$70.07) How did you work that out?
What is \$NZ 100 worth in Australian dollars and cents? (\$92.11) How did you work that out?
What is \$NZ 200 worth in British currency? (£77 and 64 pence) How did you work that out?
What is \$NZ 200 worth in Euros? (€114.20) How did you work that out?
What is \$NZ 300 worth in yens? (¥231.57) How did you work that out?
4. But let's turn that round.
If you were in the States and buying something for \$US 140.14, a pair of shoes, say, what would be the equivalent cost in New Zealand dollars? (\$200)
If you were in the States and buying something for \$US 175.18 what would be the equivalent cost in New Zealand dollars? (\$250) What might you be buying?
If you were in the States and buying something for \$US 280.28 what would be the equivalent cost in New Zealand dollars? (\$400) What might you be buying?
If you were in the States and buying something for \$US 490.49 what would be the equivalent cost in New Zealand dollars? (\$700) What might you be buying?
If you were in the States and buying something for \$US 665.67, a pair of shoes, say, what would be the equivalent cost in New Zealand dollars? (\$950) What might you be buying?
5. Now let the class go into their groups and each produce four similar questions for each of the other four countries on the TV One News screen. They should provide the answers for each of their questions. You may need to provide help especially in the early stages. Not all students will immediately realise how to make the conversions. (But one way to do this is to cheat and do the question backwards. That is, find out the US value of some New Zealand dollar value and then ask what the US dollar value is in New Zealand dollars.)
6. Before the next session, have all of these questions photocopied for the other groups to work on.

#### Session 2

In this session, the students will produce five currency converters that will convert from a foreign currency to New Zealand dollars.

1. Let the class recall what you did in the last session. Get them to see that it would be easier if they had a device to calculate the New Zealand value of amounts in the other currencies. Discuss how this might be done.
2. Let them work in groups to produce their currency converters for all five currencies. On these converters you might read off, say, \$US 5 on a row and 60 US cents on a column, and the cell in that row and column will contain the New Zealand dollar equivalent of \$US 5.60. As this is a fairly time consuming task, you might get each group to do a currency converter for only one currency.
3. You really don't want them to do the 100 individual multiplications that are necessary to fill all the cells of the currency converter. Get them to think how adding, or multiplying by 2, for example, might be easier than working every value from scratch.
4. Get groups who worked on the same currency to check the entries in their currency converters.
5. Photocopy enough currency converters so that all students can have a copy of each currency converter.
6. Get them to use their currency converters to calculate 10 or so of the questions from the last session. Many of these will be outside the range of the converters so additional arithmetic will be needed.
7. If there is a dispute over the answers, the group that made up the question and the group that got a different answer should discuss where the error lies. You should adjudicate in each case.
8. You might also like to let them go onto the web and pick out goods that are for sale overseas. They can then calculate the New Zealand equivalent of their prices. What are good buys on the web?

#### Session 3

In this session, students will make up a TV screen for a foreign television company. The screen will show the exchange rate for the other five countries we have been involved with.

1. You will need to prepare somethings ahead of this session. First you will need to have discussed with the class what country's television station they would like to work for. Then they will need to find 5 countries that they think would be of interest to the viewers in the chosen country. For example, for Pacific Island countries you might expect the students to find the exchange rates (in terms of New Zealand dollars) of two or three neighbouring countries plus two or three of New Zealand, Australia, and the United States.
2. Make sure that the exchange rates are in terms of New Zealand dollars. (To do this you should probably use a site of a New Zealand bank.)
3. Let's suppose that one student chooses the mythical country Utopia. Working alone, the student needs to find (i) the exchange rate of Utopia relative to one New Zealand dollar; (ii) the exchange rates of four other countries with respect to one New Zealand dollar; (iii) convert these to five exchange rates relative to one unit of Utopia's currency; (iv) design their own exchange rate screen; and (v) produce the final TV screen on A4 cardboard or as a screen in powerpoint or another programme.
4. Some students will need much more support than others with step (iii) above.
5. When complete, let students show their exchange rate screens and explain how they produced them.
6. Display on the wall or suitable notice board, the screens that the students think are the best.

#### Session 4

In case there are no calculating devices handy, it's often useful to make approximations. Here the students calculate approximate cross currency values.

1. Talk about the fact that calculating devices will not always be handy.
So how will you know if you a shopkeeper has charged you about the right amount?
Talk about the use of approximations. Discuss what would be ‘good' approximations for each currency.
2. Discuss roughly how much of each currency you would get for 100 Utopian currency units.
3. Then give them about 8 Utopian currency unit amounts to approximately convert into each of the other five currencies on their TV exchange rate screen.
4. Make sure that the students know about rounding.
How accurate are the approximations?
Are they good enough?
Would it make any difference if you were buying a car or a cake?
6. Use some actual values to highlight the points above.
7. Get the class to compare actual values against approximate values.
What is the error in the approximation (as a percentage)?
Is there a better approximation?

#### Session 5

Here we see that how to use proportions to convert between currencies where we don't know the actual value of the relative exchange rates.

1. Return to the TV One exchange rate screen. The task today is to convert between pairs of countries on that screen given only the information on the screen. In other words the students will have to work out for themselves from the screen how many US dollars, say are equivalent to one pound sterling.
2. In a class discussion, ask how many US dollars you would get for one pound sterling.
How would you use proportions to get this amount?

3. Let the students work in pairs to produce answers to the following questions:
What is the US equivalent of € 25?
What is the US equivalent of ¥ 2,500?
What is the British pound equivalent of € 325?
What is the Japanese equivalent of € 25?
What is the Japanese equivalent of \$ 25 Australian?
What is the US equivalent of \$ 338 Australian?
What is the Australian equivalent of £ 25?
What is the British pound equivalent of ¥ 90,673?
What is the British pound equivalent of \$ 984 Australian?
What is the US equivalent of £ 473?
4. You might want to make up extra questions for yourself or get the groups to make up questions that they can ask of the other groups. You could make this more interesting by putting the questions in a context. Such as:
An Ipod costs \$US 193. What would that cost in Japanese yen?
It might be even more interesting to find out what certain items cost in different countries and then discuss which would be the cheaper country to buy them in.

## Playing with Money (From the New Zealand Point of View)

In this unit we look at exchange rates for five countries. We see (i) how much various New Zealand dollar amounts are worth in these other currencies; (ii) how to make a currency converter; (iii) how to approximate these conversions; (iv) how currencies’ values change; and (v) how to use the exchange rates to find foreign currency values in New Zealand dollars.

## Ratios

In this unit we use the theme of drinks to develop an understanding of ratios. It is suitable for students working at Level 5 of the curriculum or Stage 8 of the Number Framework. The unit provides a sequence of demonstrations and questions to help guide students' learning. It includes activities that require students to convert between the metric units for capacity, and to use their knowledge of fraction, decimal and percentage conversions. Students use proportional thinking to explore, compare and combine ratios. The unit concludes with a session on rates (a subset of ratios).

## Drive

This unit uses one of the digital learning objects, Drive: easy problems, to support students as they investigate linear relationships. The context of driving illustrates the relationship between time, distance and speed. The students use the information from one situation to establish the relationship between the variables and apply it to new numbers. Information is given in a table and on a graph. The students’ answers are shown on the graph. It is suitable for students working at stage 7 of the number framework. It includes a sequence of problems and questions that can be used by the teacher when working with a group of students on the learning object, and ideas for independent student work.

This is a level 4 number, algebra, statistics activity and level 5 number activity from the Figure It Out series. It relates to Stage 7 and 8 of the Number Framework.

## The Right Gear

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.