Week 3: Everyday patterns and formulas

Introduction

Introduction

Patterns occur everywhere in art, nature, science and especially mathematics. Being able to recognise, describe and use these patterns is an important skill to have when tackling a variety of different problems and in understanding the world, for example, predicting the effect on fish numbers for a given amount harvested from the sea. This course explores some of these patterns, ranging from ancient number patterns to the latest mathematical research. It also looks at useful practical applications that are relevant to everyday life. Your study of this subject is spread over two weeks, and includes plenty of activities for you to do to gain confidence with these ideas.

In the following video Maria introduces you to Week 3:

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Transcript

Maria Townsend:

I can’t see any clear patterns in this piece of artwork at the Open University but if I look at the brick wall below it’s a very different situation. This is made from rows of rectangles with each row overlapping the next by half a brick. And spotting the pattern means I can describe it concisely to other people.

In maths we’re always looking out for patterns and this week you’ll find out how useful they can be.

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After studying this unit, you should be able to:

• recognise patterns in a variety of different situations
• use word formulas to help solve a problem
• write your own word formulas to help solve a problem.

1 Exploring patterns and processes

Patterns occur in many different ways in everyday life. They include how petals are arranged on a flower and the repeating notes of bird song. These patterns give us a powerful tool in understanding the world and universe. Many centuries ago ancient civilisations recognised the yearly pattern of changing sunset and sunrise and used this to build places of worship, such as Stonehenge in the UK. Today, patterns in the progress of the sun and moon are used to work out accurate heights and times of tides. This course does not look at anything so complicated, but this example does show that patterns are important in many different ways.

The first section starts by looking at a do it yourself (DIY) problem using tiles. See how you get on with describing the pattern in this first activity.

Activity 1 Tile pattern

Timing: Allow approximately 5 minutes

Suppose that somebody is tiling a bathroom, and the last row of square tiles is going to be a decorative border made up of blank tiles and patterned tiles, as shown in Figure 1 below:

Figure 1 Tiles

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This is a picture of 11 square tiles arranged next to each other in a line. The tiles have a either bridge shape or a smile shape on them, or they are blank. They are arranged in a repeating pattern of four tiles in the following order: bridge, blank, smile, blank.

Figure 1 Tiles

A friend has offered to help with the job. How would you describe the pattern so that they laid the tiles correctly? Click on ‘reveal comment’ if you would like a hint.

Comment

There is no one correct answer to how to tackle this description, so just get stuck in! Think about what might be the best way to describe this to somebody who didn’t have the pattern in front of them.

Answer

There are lots of ways you could have tackled this. For example, you might say that you will need some blank tiles and some patterned tiles with the ‘bridges’ on. Start with a bridge tile, and then put a blank tile next to it. Take another bridge tile, but turn it around so that the bridge is upside down, like a smile, and put it next to the blank tile in the same line. Then put another blank tile next to the smile tile. This is the pattern: bridge, blank, smile, blank, bridge, blank, smile …

Or maybe you numbered the tiles, something like this:

1. bridge
2. blank
3. smile

The pattern would then be:

1,2,3,2,1,2,3,2 etc.

The decorative border is an example of a type of geometric pattern that has many applications in art, crafts and design. You may have something similar in your own home. If you have, see if you can describe this as some extra practice.

The next section looks at a number pattern that has fascinated people for many centuries.

1.1 Pascal’s triangle

Numbering the tiles in Activity 1 gave a number pattern that described a geometric pattern. This next example is a number pattern that appeared in China and Persia more than 700 years ago, but is still used by students in mathematical and statistical problems today – it even appears in chemistry. It is known as ‘Pascal’s triangle’ after the French mathematician Blaise Pascal who studied the properties of this triangle. The first part of Pascal’s triangle is shown in Figure 2:

Figure 2 Pascal's triangle

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This is a triangular array of numbers arranged as follows: Row 1 1 Row 2 1 1 Row 3 1 2 1 Row 4 1 3 3 1 Row 5 1 4 6 4 1 Row 6 1 5 10 10 5 1

Figure 2 Pascal's triangle

You can continue the triangle indefinitely by following the pattern. So how is it created?

If you look at the triangle you can see that each row of numbers starts and ends with the number 1. Now look at the numbers in the second to last line (1 4 6 4 1) and the last line of the triangle. Look carefully at them both - can you see a way that you can make the numbers in the last line from the line above? If you add each pair of numbers in the second to last line, this gives you numbers in the last line.

Starting from the left-hand side the pairs of numbers are 1 and 4, 4 and 6, 6 and 4, and 4 and 1. So:

This process, plus adding a 1 at end of the line, generates the next row of the triangle. Have a go at creating the next line for yourself in this next activity.

Activity 2 Pascal’s triangle – next line

Timing: Allow approximately 5 minutes

Create the next line of Pascal’s triangle by adding the pairs of numbers in the last line. Remember to add the 1s to the end of the rows when you are done.

Answer

Starting from the left-hand side the pairs of numbers are 1 and 5, 5 and 10, 10 and 10, 10 and 5, and lastly 5 and 1.

Adding these pairs gives us:

So the next line in the triangle should look like the bottom one in Figure 3:

Figure 3 How to generate row 7 of Pascal’s triangle from row 6

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This diagram shows how to generate row 7 of Pascal’s triangle from row 6, by adding adjacent numbers together. For example, 1 + 5 = 6; 5 + 10 = 15, 10 + 10 = 20, and so on. So row 7 becomes 1 6 15 20 15 6 1.

Figure 3 How to generate row 7 of Pascal’s triangle from row 6

You can watch a larger version of Pascal’s triangle being built in this video:

Now you’ve seen how to build Pascal’s triangle by adding pairs of numbers it’s time to see if there are more patterns hiding in this number triangle in the next section.

1.2 Pascal’s triangle – a closer look

As well as the patterns within the triangle being interesting in their own right, they are used in maths and science. If you continue your study at university level in these areas you may well come across Pascal’s triangle and will already be prepared for its delights. Also, having a go at spotting the patterns now will give you a chance to practise your pattern-spotting skills and you may well be surprised by how many there are!

Activity 3 Identifying patterns

Timing: Allow approximately 10 minutes

Study this six-row version of Pascal’s triangle and note down any patterns that you can spot.

Click on ‘reveal comment’ for tips on how to get started.

Figure 4 A six-row version of Pascal’s triangle

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This is a triangular array of numbers arranged as follows. Row 1: 1; row 2: 1 1; row 3: 1 2 1; row 4: 1 3 3 1; row 5: 1 4 6 4 1; row 6 1 5 10 10 5 1

Figure 4 A six-row version of Pascal’s triangle

Comment

Play around with the numbers to find patterns. Try looking down the diagonals and adding the numbers in each row.

Answer

• Each row starts and ends with a 1.
• The counting numbers, 1, 2, 3, 4, 5, are in the diagonal rows next to the sides.
• In the next diagonal row, there is the sequence 1, 3, 6, 10. These numbers are known as the ‘triangular numbers’ because they create triangular patterns as shown in Figure 5:

Figure 5 Triangular numbers

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This diagram shows the first four triangular numbers, 1, 3, 6 and 10, as triangular patterns of dots. So 3 is represented by one dot on the first row and two dots on the second; 6 has an extra three dots on the third row and 10 has an extra four dots on the fourth row.

Figure 5 Triangular numbers

• Pascal’s triangle is symmetrical, too. If you draw a vertical line down through 1, 2, 6, one side of the triangle is a mirror image of the other.
• The sums of the first six rows are 1, 2, 4, 8, 16, 32. The total for the next row is double the total for the current row.

These numbers can also be written as 1, 2¹, 2², 2³, 2⁴ and 2⁵.

The pattern in the powers of 2 shown in Figure 6 suggests that the first total, 1, might be written as 2⁰. Check on your calculator to see if this is correct. So it can also be said that the sums of each row are square numbers and these can be generated by raising 2 to the row number minus 1. Figure 6 may help you to see this more clearly than a written description:

Figure 6 Powers in Pascal's triangle

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This is a diagram of the first five rows of Pascal’s triangle with a vertical dotted line passing down the middle of the triangle through the numbers 1, 2 and 6. This line is marked as the axis of symmetry because the numbers on one side of the line are reflected on the other side. On the left-hand side of the triangle, arrows point down the diagonal lines of the triangle, highlighting the number 1s on the edge of the triangle, the counting numbers, 1, 2, 3, … on the next diagonal line and the triangular numbers, 1, 3, 6, 10, … on the next diagonal. On the right-hand side, arrows point along the rows of the triangle to show the sum of the numbers in each row as follows. Row 2: 1 + 1 = 2 = 21. Row 3: 1 + 2 + 1 = 4 = 22. Row 4: 1 + 3 + 3 + 1 = 8 = 23.

Figure 6 Powers in Pascal's triangle

Well done if you spotted some or all of those patterns. There are many more than those looked at here, so this is just a flavour of what can be found in Pascal’s triangle.

What do the two examples have to do with mathematics? Well, recognising patterns in shapes, sets of numbers, processes or problems, and noticing what is the same and what is different about situations, often makes a task easier to solve. You saw how recognising the tiling pattern in Activity 1 made it easier to remember and describe, and by using the number patterns in Pascal’s triangle, you could work out the sum of each row without adding the individual numbers.

If you can spot a pattern and then describe what happens in general, this can lead to a rule or formula. If you can prove that this rule will always work, it can be used elsewhere. For example, if you can work out the general process for calculating a quarterly electricity bill and then give these instructions to a computer, many electricity bills can be generated, printed and sent out in just a few minutes. It may also help you to understand your bill!

In the next section you will start to look at how this process of looking for relationships can be used to write rules.

2 Looking for relationships

This section considers the relationship between quantities in two practical situations and shows you how to describe these relationships by writing down a general rule or a word formula. You can think of these as a set of mathematical instructions. A good way to start is to use an everyday example.

Suppose you are planning a visit abroad. Your map marks all the distances between places in kilometres rather than miles, which you are used to dealing with. How can you work out what these distances are in miles? The first type of information you will need is how long one kilometre is, measured in miles. If you want to, you could use the internet to verify that 1 km is equivalent to approximately 0.6214 miles.

The next step is to work out how, using maths operations, you would convert kilometres into miles. You may be able to see how to convert from kilometres into miles immediately from your study of weeks 1 and 2 of this course – but, if not, try to visualise a few simple examples. The diagram in Figure 7 should help with this, as it shows how kilometres relate to miles:

Figure 7 Converting miles to kilometres

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This shows a line marked off in 1 km lengths. Above each kilometre, 0.6214 miles is written to show that each kilometre is equivalent to 0.6214 miles.

Figure 7 Converting miles to kilometres

So if the distance is 3 km, you will have 0.6214 miles three times; if the distance is 10 km, you will have 0.6214 miles 10 times; if the distance is 250 km, you will have 0.6214 miles 250 times, and so on.

You can write this last example down mathematically as:

(to the nearest whole number)

Notice that in each of the examples above, the process for calculating the number of miles was the same: multiply the number of kilometres by 0.6214 (the conversion factor). This technique will work for any distance, and so the following word formula can be written:

Note that what the word formula works out is stated first – this is the standard way of presenting any formula.

You can use this formula to convert any distance in kilometres into miles. For example, suppose you wanted to convert 500 km, using the formula and replacing ‘distance in kilometres’ with 500 gives:

So 500 km, rounded to the nearest whole number, is equivalent to 311 miles.

In this example, you used the formula by replacing the phrase ‘distance in kilometres’ on the right side of the equation by the corresponding value, 500. This is known mathematically as substituting the value into the formula.

Have a go at this yourself in our next activity.

Now that you have used a word formula, you can move on to finding your own word formula in an activity with a similar idea behind it. Remember that drawing diagrams can help you with spotting relationships and patterns!

2.1 Finding a word formula

Instead of looking for a way to show the relationship between miles and kilometres in a word formula, the next activity asks you to find a word formula to help work out the value of one currency in another. This is very similar to the previous example, as the focus is again on how much of one quantity is represented by another. The activity takes you step-by-step through the process of writing your own word formula.

(Again, remember that drawing diagrams can help you to spot relationships and patterns)

These examples illustrate how a word formula can be used to summarise a mathematical process such as converting units of length or currencies. Once a formula has been derived, it can then be used in other situations, both for calculations by hand or by computer – for example, for currency transactions in a bank. You will be able to practise writing your own formulas next week.

Now let's look at using formulas in some more real-life examples.

3 Real-life examples of using formulas

The previous section considered how a formula could be built and then how it could be used. This section considers some more complicated formulas that have already been developed and are used in a variety of different situations – such as cooking, health care, business and archaeology. These examples illustrate some of the broad applications of maths and how mathematical relationships can be used in making decisions. As you work through these examples, you might consider where else maths could be used in each of these topics.

When you are trying to solve a real-life problem mathematically, you often use formulas that have already been developed, so it is important to understand how to apply them. This section will help you do that.

3.1 Formulas in cooking

The time to cook a fresh chicken depends on its mass (commonly called weight in everyday language), as given by the following formula:

Roughly how long will a chicken with a mass of 2.2 kg take to cook?

To use the formula, you need to substitute the mass of the chicken into the right side of the formula and then work out the resulting calculation. However, the formula asks for the mass in grams, so the first step is to convert 2.2 kilograms (kg) into grams (g).

Since there are 1000 g in 1 kg,

Substituting 2200 for the ‘mass in grams’ in the formula gives:

Following the BEDMAS rules, the multiplication and division must be carried out first, which gives:

(If you are unsure about what BEDMAS is, you can find out more in Succeed with maths – part 1.)

So the cooking time would be about 125 minutes, or 2 hours and 5 minutes.

Now use the same formula for a different sized chicken in the next activity. Remember to check carefully what units of mass are required.

The next example is an application from health care.

3.2 Formulas in health care

The body mass index (BMI) is sometimes used to help determine whether an adult is underweight or overweight. It is calculated as follows:

Although care needs to be taken in interpreting the results – for example, the formula isn’t appropriate for children, old people or those with a very muscular physique – a BMI of less than 20 suggests the person is underweight and a BMI of over 25 suggests the person is overweight.

In this formula, the units have been included in the expression on the right-hand side of the equals sign. It is important to change any measurements into these units before you substitute the values into the formula. (No units have been included for the BMI on the left-hand side, in line with current health care practice.) This is true for any formula that you will come across, so it is very important to remember.

This time, instead of giving you an example first it is straight into an activity to apply the formula.

The next section example introduces a bit more complexity in a formula, but all the same principles apply that you have already used. These are, converting to required units before substituting into a formula and following BEDMAS as necessary.

3.3 Formulas in business

One of the advantages of identifying the general features of a calculation and then describing it mathematically is that the formula can then be used in a computer to work out different calculations quickly and efficiently. Many utility suppliers (gas, water, electricity, telephone) have rates based on a fixed daily charge and a further charge based on how much you have used during the billing period.

For example, a mobile phone network charged £20 per month for 300 minutes (or less) of phone calls. Extra calls above the 300 minutes were charged at 40p per minute.

Assuming that more than 300 minutes were used per month, the formula for the total monthly cost can be expressed by the following word formula:

For simplicity data usage or texting are not included.

In this next activity you’ll unpick how this formula relates to the information given about the way in which bill is worked out. This insight will help when moving onto building your own formulas.

The formula says that the fixed charge is £20, and then each minute in excess of the 300 minutes allowed costs 40p. How this information relates to our formula can be shown by including some extra notes with it, as shown in Figure 8:

Figure 8 Annotated word formula

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This diagram shows the formula: Total monthly cost in pounds equals 20, then open parenthesis, the total number of minutes used minus 300, close parenthesis, multiplied by 0.4. A bubble points to the 20 and says ‘This is the monthly charge of £20.’ A bubble points to the bracket and says ‘The expression in the brackets works out the extra minutes that have been used.’ A bubble points to the 0.4 and says ‘This is the cost of each extra minute in £. 40p = £0.4.’

Figure 8 Annotated word formula

Now you know how the formula works, and has been put together (or derived), try applying it in this next activity.

If you have a mobile phone with a similar deal, you could work out a formula that fits yours and use it to check your bills – and get some more practice at the same time. It might also help you to decide if a different deal may suit you better.

Moving on now from everyday formula to those used in academic study, the next section looks at how one set of formulas are used in archaeology to work out the approximate height of some of our ancestors.

3.4 Formulas in archaeology

Footprints from prehistoric human civilisations around the world have been found preserved in either sand or volcanic ash. From these tracks, it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running. This can be done by using three formulas:

Note that no units have been included in these formulas – so it is important to make sure that you use the same units, such as centimetres, throughout the calculation.

The formula for relative stride length is used to establish if the person was walking or running. If the value of the relative stride length is less than 2, the person was probably walking; if the value is greater than 2.9, the person was probably running. Between these two values, it is difficult to be sure if the person was running or walking. The expected value for relative stride length is usually between 0 and 5.

Notice that the relative stride lengths quoted have no units. The stride length and the hip height are both measured in cm, and when one is divided by the other, these units cancel each other out, just like cancelling numbers when dealing with fractions. If you go on to study physics or other sciences, you will find many examples like this.

Before trying the next activity, if you would like to see one of these formulas being used in practice, have a look at this short video.

This was a slightly more complicated problem than our previous examples as it required a step in the middle to calculate some extra information needed. So, well done for having a go!

3.5 Combining formulas

There is a quicker way to the answer which requires two of the formula to be combined to give a new formula for relative stride length.

The formula for relative stride length was:

It is known that:

This means that it is possible to substitute the formula for hip height in the formula for relative stride length. This then gives:

This new formula could be used directly with the information that was given in the question. This version would also make more sense in the field, as it contains the two pieces of information that would be known on site.

In the final activity for this week you’ll consider why it is important to always think about whether the answer you have obtained looks sensible. This is something that you should always ask yourself. It can act as a prompt to check over your working or logic in any problem.

Well done, you’ve just completed the last activity for Week 3! You’ve just got one final section to look at before moving on to the Week 3 quiz. This summarises the top tips for using any formula that you've covered here.

4 Tips for using a formula

The following tips for using any formula you come across will be useful, both in next week’s study and any further study at university level you may undertake, particularly if you are interested in maths, science or technology. You might find it useful to make a note of these now, or bookmark them to refer to later:

• Check that the formula can be applied to your particular problem.
• Check what values you need to substitute and also what units these should be measured in. Convert the measurements if necessary before substituting.
• Make a rough estimate for the answer.Substitute the values into the formula carefully.
• Use BEDMAS to work out the resulting calculation, step by step. Explain your steps carefully using words like ‘substituting’ and ‘converting’.
• Check that your answer seems reasonable, both practically and from the estimate. Round your answer appropriately.

Remember your concluding sentence should both answer the question and include units.

5 This week's quiz

Use the tips as you are completing the questions in this week’s quiz.

6 Summary

Congratulations for making it to the end of another week. You may not have realised it, but you have now started on groundwork that will lead you to taking your first steps with algebra. This is a fundamental tool of maths and is used in many different subject areas and other, possibly unexpected subjects. In fact anywhere you want to work out an unknown value from numbers that you already know will probably need some algebra. This is just the start of your journey though, and this course takes you through it step-by-step to help you build your confidence. What you should now feel more confident in doing is:

• recognising patterns in a variety of different situations
• using word formulas to help solve a problem
• writing your own word formulas to help solve a problem.

You can now go to Week 4.

Acknowledgements

This course was written by Hilary Holmes and Maria Townsend.

Except for third party materials and otherwise stated (see FAQs), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence.

Every effort has been made to contact copyright owners. If any have been inadvertently overlooked, the publishers will be pleased to make the necessary arrangements at the first opportunity.

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Week 4.