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Unit 10: Shapes and Sizes

10 Introduction

This unit is about shapes and considers how some of their properties (such as area or volume) can be measured. It is a practical unit, so you will need some basic equipment to tackle some of the activities—a ruler, scissors, glue, a piece of thin cardboard (for example, from an empty cereal box), and a sheet of graph paper. [ The word geometry comes from two Greek words— “geo” means earth and “metros’’ means measurement. ]

10.0.1 What to Expect in this Unit

This unit should take around 14 hours to complete. In this unit you will learn about:

  • Writing very small and very large numbers.
  • More on working with exponents.
  • The roots of numbers.
  • Scale diagrams.
  • Math language and notation for shapes.
  • Perimeter and area of simple shapes.
  • Calculating the volume of simple solids.
  • Further appreciate that drawing diagrams or working with models can help solve problems.

You learned about measurement in Unit 5, where mathematical units were introduced. Section 10.1 extends those ideas by considering measuring very small things (viruses, bacteria, or germs) and very large things (galaxies, distances in the universe), and shows how these measurements can be written using scientific notation. It also explains some important mathematical techniques in working with exponents, which you met in Unit 4.

One of the difficulties in considering very large or very small things is in visualizing them. Section 10.2 explores scale diagrams to help with this. These allow you to represent large areas, such as a road network on paper in the form of a map. This also builds on the ideas of ratio from Unit 8.

Section 10.3 introduces some basic properties of shapes, such as triangles, rectangles, and circles, and also introduces new vocabulary relating to these. Sections 10.4 and 10.5 then introduce perimeters and areas of various shapes, including circles, and uses this knowledge to solve a practical problem.

Section 10.6 extends these ideas to volumes: How much will that box hold? How much water do I need for the fish tank?

The study skills in this chapter concentrate on understanding and using new notation and vocabulary, and extending your problem solving strategies. (Remember to add the new vocabulary to your math notebook glossary as you come across it.) One of the problem solving strategies you have already used is drawing a diagram to help you to visualize a problem, and this unit looks at the use of diagrams and models in more detail. Problems can be tackled in a variety of different ways, and deciding which approach is most appropriate in a particular situation is important. To illustrate this, in Section 10.5 we consider which mathematical technique is best in a medical emergency!

Practice Quiz

Check your understanding of geometric shapes and sizes before you start by giving the Unit 10 pre quiz a try, then use the feedback to help you plan your study.

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend most time on. Remember, it doesn’t matter if you get some or even all of the questions wrong—it just indicates how much time you may need for this unit!

Click here for the pre quiz.

10.1 Big and Small

In Unit 5, you looked into for some everyday measurements and considered some everyday problems. However, you may have to deal with much larger and much smaller quantities than you have already—particularly if you are interested in science or technology.

Activity Symbol Activity: Biggest and Smallest Physical Objects

Think of the biggest and smallest physical objects or numbers that you can and consider how you would describe their size to someone else, making brief notes in your math notebook.

Answer Symbol

Answer

There are many different ideas that you may have come up with. Here are a few that we thought of:

Biggest

  • The width of the universe.
  • The distance to a star.
  • The number of grains of sand in the world.

Smallest

  • The size of a cell in your body.
  • A virus.
  • An atom.

A quick check on the Internet gives the width of the observable part of the universe as about 92 billion light years, where a light year is about 9.5 trillion kilometers. So to find the width of the universe in kilometers, you need to multiply 92 billion by 9.5 trillion. How would you do that? [ A trillion is one thousand billion or a million million—which means a number followed by 12 zeros. What is the biggest number you can put in your calculator?]

Well, you may want to use the calculator, but there’s a problem: 92 billion is 92,000,000,000, and while that number will work on this course's online calculator, that is too big a number for many other calculators.

What could you have done if the number were too big for your calculator? You could have worked the calculation out on paper, or multiplied 92 by 9.5 to get 874 and deduced that the distance must therefore be 874 billion trillion kilometers, or you may just have been bewildered by the enormity of the numbers. Clearly, the skills and notation we have used so far are not particularly helpful here. However, there is a way around this that builds on the work you did with exponents in Unit 4.

In Unit 4, you discovered that some large numbers could be written using power notation. For example, 100 is the same as 10 multiplication 10 comma and this can be writ times ten as 10 super two. Here, 10 is known as the base number, and the 2 is the power. Similarly, 1,000 is the same as 10 multiplication 10 prefix multiplication of 10 or 10 super three, and so on.

We can use this idea to help us write very small and very large numbers, but first let's have a recap on powers of 10.

10.1.1 Powers of 10

Activity Symbol Activity: Powers of 10

Write each of the following numbers as a power of 10.

(a) 10,000

Solution Symbol
Answer

(a) 10,000 equals 10 multiplication 10 multiplication 10 prefix multiplication of 10 equals 10 super four

(b) 1,000,000

Solution Symbol
Answer

(b) 1,000,000 equals 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 prefix multiplication of 10 equals 10 super six

(c) 1,000,000,000

Solution Symbol
Answer

(c) 1,000,000,000 equals 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 prefix multiplication of 10 equals 10 super nine

(d) What do you notice about the number of zeros in the original number and the power of 10 in parts (a) through (c)?

Solution Symbol
Answer

(d) As you may have noticed, the number of zeros in the original number is equal to the power of 10.

Now use this idea to help you write each of the following expressions in decimal form, and as a power of 10.

(e) There are about one hundred thousand hairs on an average human head.

Solution Symbol
Answer

(e) One hundred thousand is 100,000, or 10 super five

(f) By 2050, the population of Earth may be about 10 billion people.

Solution Symbol
Answer

(f) Ten billion is 10,000,000,000, or 10 super 10

(g) In 1961, the French poet Raymond Queneau wrote a book called A Hundred Thousand Billion Poems.

Solution Symbol
Answer

(g) One hundred thousand billion is 100,000,000,000,000, or 10 super 14 Queneau’s book contained ten sonnets, each with 14 lines. Each page, containing one sonnet, was cut into 14 strips with one line on each strip, so it was possible to combine lines from different sonnets to form a new sonnet. There are 10 super 14 different ways of making a sonnet in this way. So, the title of the book was correct! [ Can you explain why there were 10 super 14 different sonnets? ]

Video clip iconThis website has a digital copy of the book that allows you to change lines in one sonnet.  The number of sonnets created by visitors to website already is displayed at the bottom of the page.   When we visited the site less than 1 million had been created.

Powers of 10 can be extended to write other numbers in this form. For example, 6,000,000 can be written as six multiplication 1,000,000 comma or six prefix multiplication of 10 super six. In the same way, 6,500,000 can be written as six .5 multiplication 1,000,000 or six .5 prefix multiplication of 10 super six.

When a number is written in this form, where a number between 1 and up to but not including 10 is then multiplied by an integral power of 10, it is said to be in scientific notation or standard form.

So, the scientific notation for 6,500,000 is 6.5 prefix multiplication of 10 super six, not 65 prefix multiplication of 10 super fivebecause 65 does not lie between 1 and 10. This is an important convention to remember.

10.1.2 Scientific Notation

A number written in scientific notation has the following form:

Notice that to write a number in scientific notation, you can start by writing down the number between 1 and up to but not including 10. So for 130,000, this number is 1.3; then multiply this number by 10 repeatedly until you reach the number required.

Each time you multiply by 10, the power of 10 increases by 1. So, 130,000 equals one .3 prefix multiplication of 10 super five because you have to multiply 1.3 by 10 five times to get 130,000.

Activity Symbol Activity: Understanding Numbers in Scientific Notation

Write the following numbers without using powers of 10.

(a) two prefix multiplication of 10 super four

Solution Symbol
Answer

(A) two equation left hand side prefix multiplication of 10 super four equals right hand side two multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals 20,000

(b) three .82 prefix multiplication of 10 super eight

Solution Symbol
Answer

(b) three .82 equation left hand side prefix multiplication of 10 super eight equals right hand side three .82 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals 382,000,000

(c) nine .3567 prefix multiplication of 10 super two

Solution Symbol
Answer

(c) nine .3567 equation left hand side prefix multiplication of 10 super two equals right hand side nine .3567 multiplication 10 multiplication 10 equals 935 .67

Activity Symbol Activity: Writing Numbers in Scientific Notation

Write the following numbers in scientific notation.

(a) 92 billion

Solution Symbol
Answer

(d) 92 billion equals 92,000,000,000 equals nine .2 prefix multiplication of 10 super 10

(b) 400 trillion

Hint Symbol

Comment

1 trillion is a million million.

Solution Symbol
Answer

(b) 400 trillion equals 400,000,000,000,000 equals four prefix multiplication of 10 super 14

(c) 9,500,000,000,000

Solution Symbol
Answer

(c) 9,500,000,000,000 equals nine .5 prefix multiplication of 10 super 12

Which of these numbers is the biggest?

Hint Symbol

Comment

Compare the powers of ten.

Solution Symbol
Answer

Notice how using scientific notation enables you to compare the sizes of numbers quickly. The highest power of the three numbers in this activity is 14, so 400 trillion is the biggest number here.

Video clip iconRelating to very small and very large numbers and what they mean is difficult. You may find that this website which allows you to zoom out from the extremely small scale to the very large scale will start to help with this.

10.1.3 Multiplying and Dividing Numbers with the Same Base Number

How does this notation help in working out the width of the universe?

Consider what happens when you multiply two different powers of 10 together, say 10 squared times and 10 super four.

multiline equation line 1 equation left hand side 10 squared multiplication 10 super four equals right hand side open 10 multiplication 10 close multiplication open 10 multiplication 10 multiplication 10 multiplication 10 close line 2 equation sequence equals 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals one comma 000 comma 000

There are six 10s multiplied together, so we can write this as 10 super six. So, 10 super two equation left hand side prefix multiplication of 10 super four equals right hand side 10 super six

Have you noticed anything about the powers of ten in the answer and the original numbers?

You can get the same result by adding the powers together:

10 squared equation sequence prefix multiplication of 10 super four equals 10 super open two plus four close equals 10 super six

Try working out two cubed prefix multiplication of two super five the “long way” and then by adding the powers together. Do your answers agree? The following rule applies in general: To multiply two numbers with the same base number, add the powers.

Activity Symbol Activity: Multiplying Numbers with the Same Base Number

Work out the following, giving your answer first in power form and then in decimal form.

(a) two squared prefix multiplication of two super four

Solution Symbol
Answer

(a) two squared equation sequence prefix multiplication of two super four equals two super open two plus four close equals two super six equals 64

(b) open three minus close squared multiplication open three minus close cubed

Solution Symbol
Answer

(b) equation sequence open three minus close squared multiplication open three minus close cubed equals open three minus close super two plus three equals open three minus close super five equals two minus times 43

(c) 10 super four prefix multiplication of 10 super three

Solution Symbol
Answer

(c) 10 super four equation sequence prefix multiplication of 10 super three equals 10 super open four plus three close equals 10 super seven equals 10 comma 000 comma 000

Notepad iconActivity: The Width of the Universe

You can now work out the width of the universe!

Remember that the universe is 92 billion light years across and a light year is about 9.5 trillion kilometers.

Solutions icon
Answer

Width of the universe = 92 billion prefix multiplication of 9.5 trillion km.

. First express both numbers in scientific notation: 92 billion equals nine .2 prefix multiplication of 10 super 10 and nine .5 trillion equals nine .5 prefix multiplication of 10 super 12.

So the width = 92 billion multiplication nine .5 trillion km equals nine .2 prefix multiplication of 10 super 10 multiplication nine .5 prefix multiplication of 10 super 12 km = 9.2 multiplication nine .5 prefix multiplication of 10 super 10 prefix multiplication of 10 super 12. Multiplying 9.2 by 9.5 gives 87.4. Then adding the powers gives 10 super 10 equation sequence prefix multiplication of 10 super 12 equals 10 super open 10 plus 12 close equals 10 super 22.

So the calculation is:

nine full stop two multiplication nine full stop five multiplication one zero super one zero postfix multiplication one equation left hand side zero super 12 equals right hand side 87 full stop four multiplication one zero super 22 km
equation left hand side prefix of equals right hand side eight full stop 74 multiplication one zero super one postfix multiplication one zero super 22 km
prefix of equals eight full stop 74 multiplication one zero super open one plus 22 close km
equation left hand side prefix of equals right hand side eight full stop 74 multiplication one zero super 23 km

That means the width of the universe is 8.74 prefix multiplication of 10 super 23 km.

We have a rule for multiplying two numbers with the same base number, but what about division?

Consider 10 super six division 10 squared. This can be written as a fraction, and then canceled, as follows:

equation sequence 10 super six divided by 10 squared equals 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 divided by 10 multiplication 10 equals 10 multiplication 10 multiplication 10 multiplication 10 equals 10 super four

Therefore, equation left hand side 10 super six division 10 squared equals right hand side 10 super four.

This time you can get the same result by subtracting the powers: equation sequence 10 super six division 10 squared equals 10 super open six minus two close equals 10 super four

Try working out three super eight division three cubed the “long way,” and then by subtracting the powers. Are you convinced that the following rule works?

To divide two numbers with the same base number, subtract the powers.

Make a note of these rules about powers with the same base in your math notebook - remembering to highlight that they apply only with the same base number.

Activity Symbol Activity: Dividing Numbers with the Same Base Number

Without using your calculator, work out each of the following.

(a) four super seven division four super five

Solution Symbol
Answer

(a) equation sequence four super seven division four super five equals four super open seven minus five close equals four squared equals four multiplication four equals 16

(b) three super four division three super seven

Solution Symbol
Answer

(b) equation sequence three super four division three super seven equals three super open four minus seven close equals three super negative three equals one divided by three cubed equals one divided by three multiplication three multiplication three equals one divided by 27

(c) two super four division two super two minus

Solution Symbol
Answer

(c) equation sequence two super four division two super two minus equals two super open four minus open two minus close close equals two super four plus two equals two super six equals two multiplication two multiplication two multiplication two multiplication two multiplication two equals 64

(d) 10 super two minus division 10 squared

Solution Symbol
Answer

(d) equation sequence 10 super two minus division 10 squared equals 10 super open two minus minus two close equals 10 super four minus equals one divided by 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by 10 comma 000 equals 0.0001

10.1.4 Zero as a Power

Notepad iconActivity: Zero as a Power

Work out 10 cubed division 10 cubed the “long way” and using the rules that we have just learned about.

Solutions icon
Answer
equation sequence 10 cubed division 10 cubed equals 10 cubed divided by 10 cubed equals 10 multiplication 10 multiplication 10 divided by 10 multiplication 10 multiplication 10 equals one divided by one equals one

The powers rule gives equation sequence 10 cubed division 10 cubed equals 10 super open three minus three close equals 10 super zero. So it follows that 10 super zero equals one. But what does 10 super zero mean? “10 multiplied by itself zero times” does not make sense, so 10 super zero is defined as having the value 1. You can use a similar argument to show that any number to the power zero has a value of 1. For example, two super zero equals one, 3.25 super zero equals one, and open six minus close super zero equals one.

Now let’s take a look at negative exponents or powers.

10.1.5 Negative Exponents

Notepad iconActivity: Further Investigation of Exponents

Work out 10 squared division 10 super six the “long way” and using the rules for powers with the same base.

Solutions icon
Answer
equation sequence 10 squared division 10 super six equals 10 squared divided by 10 super six equals 10 multiplication 10 divided by 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by 10 super four

The rule gives equation sequence 10 squared division 10 super six equals 10 super open two minus six close equals 10 super four minus, so 10 super four minus means one divided by 10 super four.

We read 10 super four minus as “10 to the power negative four.” [ You learned about reciprocals in Unit 7 when dividing fractions. ] Often, the word “power” is left out and we say “10 to the negative four” instead. In the same way, five super two minus is said as “five to the negative two” and means one divided by five squared, or one divided by five multiplication five, which is equivalent to one divided by 25.

Alternatively, you can say that five super two minus is the reciprocal of five squared.

Activity Symbol Activity: Understanding Negative Powers

Without using your calculator, write the following numbers as fractions or whole numbers, as appropriate.

(a) two super three minus

Solution Symbol
Answer

(a) equation sequence two super three minus equals one divided by two cubed equals one divided by two multiplication two multiplication two equals one divided by eight

(b) three super four minus

Solution Symbol
Answer

(b) equation sequence three super four minus equals one divided by three super four equals one divided by three multiplication three multiplication three multiplication three equals one divided by 81

(c) six super zero

Hint symbol
Comment

What does any number to the power 0 equal?

Solution Symbol
Answer

(c) six super zero equals one

(d) 10 super six minus

Solution Symbol
Answer

(d) equation sequence 10 super six minus equals one divided by 10 super six equals one divided by 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by one comma 000 comma 000

Notepad iconActivity: Writing Negative Numbers

Write the following as powers of 10.

(a) 0.01

Hint symbol
Comment

Convert the number to a fraction first, thinking back to place value if you need to.

Solution Symbol
Answer

(a) equation sequence 0.01 equals one divided by 100 equals one divided by 10 squared equals 10 super two minus

(b) 0.0000001

Solution Symbol
Answer

(b) equation sequence 0.0000001 equals one divided by 10 comma 000 comma 000 equals one divided by 10 super seven equals 10 super seven minus

10.1.6 Small numbers and Scientific Notation

In the preceding section you have seen how to write numbers that are less than 1 using powers of ten. So you can now write numbers less than 1 in scientific notation. For example,

equation sequence 0.003 equals three divided by 100 equals three divided by 10 multiplication 10 equals three divided by 10 squared equals three multiplication 10 super two minus

An alternative way to write a number less than 1 in scientific notation is first to write it down as a number between 1 and up to but not including 10; in this case it is 3. Then divide this number by 10 repeatedly until you reach the number required. Each time you divide by 10, the power of 10 reduces by 1. So here you have to divide 3 by 10 twice to get 0.03, so 0.03 equals three prefix multiplication of 10 super two minus.

To convert a number in scientific notation back into decimal form, write down the negative power of 10 as a fraction and then divide the numerator by the denominator. For example:

equation sequence 5.72 multiplication 10 super four minus equals 5.72 divided by 10 super four equals 5.72 divided by 10 comma 000 equals 5.72 division 10 comma 000 equals 0.000 57

Activity Symbol Activity: Understanding Small Numbers in Scientific Notation

Write down the following numbers as decimals.

(a) three multiplication 10 super four minus

Solution Symbol
Answer

(a) equation sequence three multiplication 10 super four minus equals three divided by 10 super four equals three divided by 10 multiplication 10 multiplication 10 multiplication 10 equals three divided by 10 comma 000 equals 0.0003

(b) 7.5 prefix multiplication of 10 super five minus

Solution Symbol
Answer

(b) equation sequence 7.5 multiplication 10 super five minus equals 7.5 divided by 10 super five equals 7.5 divided by 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals 0.000 075

(c) 1.46 prefix multiplication of 10 super three minus

Solution Symbol
Answer

(c) equation sequence 1.46 multiplication 10 super three minus equals 1.46 divided by 10 cubed equals 1.46 divided by 10 multiplication 10 multiplication 10 equals 0.00146

Notepad iconActivity: Writing Small Numbers in Scientific Notation

Write down the following numbers in scientific notation.

(a) 0.000007

Solution Symbol
Answer

(a) 0.000 equation sequence 007 equals seven divided by one comma 000 comma 000 equals seven divided by 10 super six equals seven multiplication 10 super six minus

(b) 0.0742

Solution Symbol
Answer

(b) equation sequence 0.0742 equals 7.42 divided by 100 equals 7.42 divided by 10 squared equals 7.42 multiplication 10 super two minus

(c) 0.0000000098

Solution Symbol
Answer

(c) 0.000 000 009 equation sequence eight equals 9.8 divided by one comma 000 comma 000 comma 000 equals 9.8 divided by 10 super nine equals 9.8 multiplication 10 super nine minus

Before we move on, think about how scientific notation fits in with what you already know about decimals. Earlier, you saw that when you multiplied a number by 10, the decimal point in the number moved one place to the right, and when you divided a number by 10, the decimal point moved one place to the left. So, extending this idea, if you multiply a number by 10 cubed, the decimal place will move three places to the right to make the number bigger. For example, 4.5 prefix multiplication of 10 super three equals four comma 500. Similarly, if you multiply a number by 10 super four minus, which is the same as dividing by 10 super four, the decimal point will move four places to the left to make the number smaller. For example, two multiplication 10 super four minus equals 0.0002.

Videoclip iconIf you want some more practice with scientific notation before you move on to look at how to use scientific notation on a calculator, have a go at this game .

10.1.7 Calculator Exploration: Scientific Notation

In this exploration, you will learn how to use the calculator for scientific notation.

Calculator iconThe calculator can be accessed on the left-hand side bar under Toolkit.

To enter a number that is given in scientific notation into the calculator, use the The location of the 10 to the x button is on the fifth line of buttons on the very left (two buttons left of the button for the number zero). button. You’ll find it here on the calculator.

To input 4.25 multiplication 10 squared, click on the following keys:

Note the parentheses at the end. Before you press Enter, you should see this:

The number in scientific notation appears in the black window. As you type The buttons shown pressed are, multiplication button and power of 10 button, parentheses open in the white window, and these must be closed to complete the expression.

Before investigating what happens if you miss the parentheses, complete this exercise by clicking on the Equals key or pressing Enter. You should see the number converted back into the normal format.

So to convert a number from scientific notation to the normal format, enter it using the The button being shown is the power of 10 key key, and then click on the Equals key or press Enter.

10.1.8 The Case of the Missing Parentheses

Activity Symbol Activity: The Case of the Missing Parentheses

Suppose that you want to find the answer to the following calculation: 6.7 multiplication 10 squared multiplication three

(a) Enter this into the calculator, exactly as it is presented, remembering to close the parentheses after the power (exponent) of 10. Make a note of the answer.

Hint Symbol

Comment

To enter a number in scientific notation, enter the number, then The buttons being shown are the multiplication button and the power of 10 button, then the power of 10.

Solution Symbol

Answer

(a) You should get the answer 2010.

(b) Now repeat the calculation, leaving out the parentheses after the power of 10. What answer do you get this time?

Solution Symbol

Answer

(b) This time, you get 6700000.

Look at the two calculations. In the first one, the calculator has done what you wanted. In the second, you didn’t close the parentheses, so the calculator assumed that the “times 3” was part of the exponent, the power of 10.

So be careful using the The button being shown is the power of 10 key. key. Watch the white window as you enter the calculation and remember to close the parentheses when you have completed the power of 10.

10.1.9 Very Big and Very Small Numbers

The calculator will convert very large numbers (numbers over a million) to scientific notation. Enter the number and click the Equals key or press Enter.

Try this with the number 123456789. Enter it into the calculator and click the Equals key. You should see the number 123456789 in the white window and written in scientific format in the black window.

Similarly, very small numbers (numbers in scientific notation which have a power of 10 that is –6 or smaller) are automatically written in scientific notation. Enter 0.000000123 into the calculator, then click the Equals key or press Enter and see what happens. You should see this:

Working with Numbers in Scientific Notation

Here are activities to practice working with numbers in scientific notation.

Activity Symbol Activity: Adding Very Big Numbers

Add 9.2 multiplication 10 super 10 to 9.5 multiplication 10 super nine by entering both numbers in scientific notation. Give your answer first in normal format, then in scientific notation.

Hint Symbol

Comment

Use the The button being shown is the power of ten key. key, remembering to close the parentheses after you have entered the power of 10.

Solution Symbol

Answer

Here is what you should see on the calculator:

The calculation is shown correctly in the black window. But be careful, because there is only space for part of the answer. The full answer is shown in the white window. Clicking on the Equals key again gives the answer in scientific notation. The calculation is shown as follows:

9.2 multiplication 10 super 10 plus 9.5 multiplication 10 super nine equals 101 comma 500 comma 000 comma 000 equals 1.015 multiplication 10 super 11
Activity Symbol Activity: Multiplying Very Small Numbers

If you multiply two very small numbers, then your answer is even smaller. Try this with the following calculation:

3.3 multiplication 10 super six minus multiplication 4.2 multiplication 10 super five minus

Hint symbol

Comment

Do you remember how to enter negative numbers? Use the The button being shown is the positive or negative value key. key.

Make sure you watch the white window as you enter the calculation. In particular, remember to close the parentheses after you have entered each power of 10.

Solution Symbol

Answer

Once again, the full answer doesn’t show in the black window but is given below. Clicking on the Equals key or pressing Enter will give the answer in scientific notation in the black window.

equation sequence 3.3 multiplication 10 super six minus multiplication 4.2 multiplication 10 super five minus equals 0.0000000001386 equals 1.386 multiplication 10 super one minus times zero

The calculator should show this:

10.1.10 How Many Bees?

Activity Symbol Activity: How Many Bees?

In a country, there are 7,532,000 beehives, and each hive contains about 50,000 bees at the peak of the bee season. Write each of these numbers in scientific notation. Then use the calculator to find the total number of bees in the country.

Hint Symbol

Comment

To write a number in scientific notation, write it as a number between 1 and 10 multiplied by a power of 10.

Solution Symbol

Answer

seven comma 532 comma 000 equals 7.532 multiplication 10 super six and 50 comma 000 equals five multiplication 10 super four.

So the total number of bees in the country is equation left hand side 7.532 multiplication 10 super six multiplication five multiplication 10 super four equals right hand side 3.766 multiplication 10 super 11. That’s a lot of bees—about 380 billion!

To recap:

  • To enter a number in scientific notation, use the The button being shown is the power of ten key. button. Remember to close the parentheses after you have entered the power (exponent) of 10.
  • To convert a very large or a very small number to scientific notation, enter the number into the calculator and click the Equals key (or press Enter).

10.1.11 Decimal Forms of Powers of Ten and Prefixes

[ SI stands for International System of Units, the modern form of the Metric System. ] The table below summarizes the decimal form for powers of 10, together with the SI prefixes and their names for reference:

Number Power SI Prefix Prefix Name
0.000001 10 super six minus µmicro-
0.001 10 super three minus m milli-
0.01 10 super two minus c centi-
0.1 10 super one minus d deci-
1 10 super zero - * - *
10 10 super one da deca-
100 10 squared h hecto-
1,000 10 cubed k kilo-
1,000,000 10 super six M mega-
1,000,000,000 10 super nine G giga-
*The dash sign, ( - ) means there is no prefix for this value.

10.2 Roots

In Unit 4 we started to look at exponents and looked at multiplying numbers by themselves a number of times, as we have been looking at with powers of ten in this unit. The root of a number reverses this process. So if you take the square root of 4, written as Square root of four, the answer is 2, since equation sequence two multiplication two equals four equals two squared.

Since a number can be multiplied by itself any number of times there are also any number of different roots as well.  We will look at this further later in this unit.

You can express square roots using power notation as well. Consider the following: four super one divided by two multiplication four super one divided by two.

Adding the powers gives: equation sequence four super one divided by two multiplication four super one divided by two equals four super one equals four.

This shows that when you multiply four super one divided by two by itself, you get 4. In other words, four super one divided by two is a square root of 4.

This is defined as the positive square root. It may be expressed as:

equation sequence four super one divided by two equals Square root of four equals two

Similarly, 27 super one divided by three means the cube root of 27, i.e., the number which when multiplied by itself three times gives 27. Since three multiplication three multiplication three equals 27, we say that 3 is the cube root of 27 or 27 super one divided by three equals three.

Convince yourself that this is the case by using the rules of multiplying powers with the same base.

A square root of a number, say 16, is a value that when multiplied by itself, gives the number (16 here). So one square root of 16 is 4 because four multiplication four equals 16. But there is another square root of 16 because four minus multiplication four minus equals 16, too. Any positive number has two square roots, a positive one and a negative one.

These square roots can be written down using the square root notation: The square roots of 16 are equation left hand side prefix plus minus of Square root of 16 equals right hand side prefix plus minus of four. The symbol ± is read as “plus or minus.” The Square root of symbol means “the positive square root of … .”

So, we can say that the square roots of 25 are prefix plus minus of five, while Square root of 25 equals five.

10.2.1 Calculator Exploration: Square Roots

In this exploration, you will learn to use the calculator to find square roots.

Calculator IconThe calculator can be accessed on the left-hand side bar under Toolkit.

To use the calculator to find square roots, use the The button being shown is the square root key. key.

Try using your calculator to find Square root of 25.

Enter The buttons shown are the square root button, followed by the number 2, the number 5, parentheses closed, and equals sign keys.

Note that when you press the square root key, the calculator opens parentheses; you must remember to close them when you have completed the number whose square root you are finding.

Before you click the Equals key (but after you have closed the parentheses), you should see this:

Here is the completed calculation:

Activity Symbol Activity: Some Square Root Investigations

(a) Use the calculator to work out Square root of 25, Square root of 49, and Square root of open 25 plus 49 close.

Does equation left hand side Square root of 25 plus Square root of 49 equals right hand side Square root of open 25 plus 49 close?

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Comment

When you enter Square root of open 25 plus 49 close, don’t close the parentheses until you have entered the whole expression under the square root sign, that is, all of 25 plus 49.

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Answer

(a) Square root of 25 equals five and Square root of 49 equals seven, so equation sequence Square root of 25 plus Square root of 49 equals five plus seven equals 12.

However, equation sequence Square root of open 25 plus 49 close equals Square root of 74 equals 8.60 (to 2 decimal places).

Therefore, Square root of 25 plus Square root of 49 is not equal to Square root of open 25 plus 49 close.

So it is important to note that the square root of a sum is not the same as the sum of the individual roots.

(b) Use the calculator to work out Square root of 25, Square root of 49, and Square root of open 25 multiplication 49 close.

Does equation left hand side Square root of 25 multiplication Square root of 49 equals right hand side Square root of open 25 multiplication 49 close?

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Answer

(b) Square root of 25 equals five and Square root of 49 equals seven, so equation sequence Square root of 25 multiplication Square root of 49 equals five multiplication seven equals 35.

Then, equation sequence Square root of open 25 multiplication 49 close equals Square root of 1225 equals 35.

Therefore, equation left hand side Square root of 25 multiplication Square root of 49 equals right hand side Square root of open 25 multiplication 49 close.

The square root of a product is the same as the product of the individual roots.

10.2.2 Repeating Square Roots

Activity Symbol Activity: Repeating Square Roots

Work out Square root of two. Now, click on The buttons shown are the square root button followed by the equals button. to find Square root of Square root of two. Keep repeating this so that you are finding Square root of Square root of Square root of two and Square root of Square root of Square root of Square root of two, and so on. What do you notice happening?

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Comment

Each calculation needs just The buttons shown are the square root button followed by the equals button.. Watch the windows at the top of the calculator to see what is happening.

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Answer

Square root of two equals 1.414 full stop full stop full stop If you keep taking square roots of the answer, the number gets closer and closer to 1. Try starting with a different number, like 37. Does the same happen? What about a number between 0 and 1, such as 0.5?

Activity Symbol Activity: Square Roots of Negative Numbers

What do you think the square root of 16 is? It can’t be 4, because four minus multiplication four minus equals 16, not 16. Use the calculator to find the answer.

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Comment

Enter the square root, then 16 using the ± button.

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Answer

You should get an answer like this:

Notice that the 4i in the black screen is colored green. This is to show you that you have a different type of number. In this context, “i” is shorthand for Square root of super minus times one. This is known as an imaginary number. You will learn more about imaginary numbers if you continue studying mathematics. However, for the moment, it is sufficient to know that the square root of a negative number can’t be found among the ordinary numbers that you work with every day. If you have a hand calculator, you could try finding the square root of a negative number on it; you may find that it displays an error message.

To recap:

  • To find a square root on the calculator, use the The button shown is the square root key. key, remembering to close the parentheses after you have entered the number.

10.2.3 Calculator Exploration: Cube Roots and More

In this exploration, you will use the calculator to find cube roots and other types of roots.

Calculator IconThe calculator can be accessed on the left-hand side bar under Toolkit.

4 is a square root of 16 because four multiplication four equals 16. 3 is a cube root of 27 because three multiplication three multiplication three equals 27, and 2 is a fourth root of 16 because two multiplication two multiplication two multiplication two equals 16.

The type of root is determined by the number of identical factors that make up the product: two identical factors are square roots, three identical factors are cube roots, four identical factors are fourth roots, and so on.

Identifying cube roots and other roots is not easy, so the calculator is very useful. Cube, fourth, fifth, and any other roots are found using the The y-th root button key. The image below shows where to locate this key on the calculator.

Cube roots and higher roots are a bit trickier than square roots to input into the calculator. Check that the cube root of 27 is 3 by entering the following: Press the yth root button followed by 27, a comma, the number 3 and a parentheses closed, then equals..

Before you click the Equals key, you should see this:

In the white window you see sqrt (27, 3), which the calculator uses as shorthand for the cube root of 27. If you find this confusing, then watch the black window, where you can see the correctly formatted cube root. Clicking on the Equals key gives the answer 3.

Let’s do another example for extra practice. See if you can work out the method by entering the following to find root of order four over four.

The button sequence is similar to the one used above, so you should enter the following: Press the yth root button followed by 16, a comma, the number 4 and a parentheses closed, then equals..

Before you click the Equals key, you should see this:

Once again the white window will display special notation, but you can just look at the black window to verify that the calculator is computing the fourth root of 16. Clicking on the Equals key gives the answer 2.

Activity Symbol Activity: Cube Roots and More

(a) Use the calculator to find the cube root of 4,913. Check your answer by cubing it.

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Comment

Take your time entering each calculation, watching the windows to make sure that you are entering everything correctly. Remember that after you use the The button shown is the y-th root key. key, you will need to type in the number you are taking the root of, followed by which root you want to take, separated by a comma.

To check your work, take the answer given by the calculator and raise it to the root as a power using the The button shown is the the y-th power key. key.

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Answer

(a) The cube root of 4,913 is 17. You can check this by finding 17 cubed equals 4913.

Screen shot showing cube root of 4913 equals 17 in the black display window and the white window underneath showing 17. Screen shot showing 17 cubed equals 4913 in the black display window and the white window underneath showing 4913.

(b) Use the calculator to find the fifth root of 7,776. Check your answer by raising it to the power 5.

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Answer

(b) The fifth root of 7,776 is 6. You can check this by finding six super five equals 7776.

Screen shot showing fifth root of 7776 equals 6 in the black display window and the white window underneath showing 6. Screen shot showing 6 to the 5th power equals 7776 in the black display window and the white window underneath showing 7776.

To recap:

  • To find a cube, fourth, fifth, or any other root, use the The button shown is the y-th root key. key. Enter the number you want to find the root of, followed by a comma, and then the root required. Then, close the parentheses, and press the Equals key to see the answer.
  • To check your solution, use the The button shown is the y-th power key. key. Enter the answer you found, and raise it to the root as a power. Then, close the parentheses, and press the equals key to see the answer.

The last two sections have introduced new notation and new ways of manipulating numbers written in scientific notation as well as the roots of numbers. If you have not already done so, look back through these sections and summarize the key points in your math notebook before moving to the next section.

10.3 Scale Diagrams

This section considers how you can use scale diagrams in practical situations. Scale diagrams are particularly useful when you want to describe something accurately, but the life-size version is either too big or too small to do so directly. For example, a map can highlight key features of the land and help organize a trip, or a plan can indicate how to build something to specifications. In a scale diagram, all the measurements are a fraction (or a multiple) of the real measurements. For example, suppose you have a plan for a table, and a scale of 1:50 is used. This means that the real measurements will be 50 times the measurements on the plan. So if a particular length on the plan is 4 cm, the corresponding length on the full-size table would be four cm multiplication 50 equals 200 cm or two m.

To understand how this works in practice, try making a geometrical puzzle from a scale diagram in the activity below. This kind of puzzle is known as a tangram and has been popular in many countries for hundreds of years. It consists of seven geometrical pieces arranged in a square, and the challenge is to arrange all seven pieces to make another shape or picture that is given to you. You will need the tangram for activities in sections 10.4 and 10.5. [ The earliest known book of tangram puzzles was published in China in 1813. ]

You will need to print out the tangram for the following activity. Use the link provided below.

Adobe PDF IconThe following activity requires you to print out a tangram. Use the link below to download and print a copy of the tangram:

View document

Activity Symbol Activity: Making a Tangram Puzzle

A diagram of the tangram puzzle is shown below. Use the similar tangram that you have just printed out.

Note that the image you see on-screen may not appear drawn to the proper scale.

(a) Draw the square outline of the full-size puzzle on a thin cardboard, at a ratio of 1:2. Use your ruler to enlarge the outline of the printed tangram.

Measure the length of the side of the entire puzzle you printed out. What do you need to use for the corresponding lengths on the resized puzzle?

Hint Symbol

Comment

You may find it helpful to place a piece of graph paper onto the cardboard first, so that your lines are straight and the corners square.

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Answer

(a) If you scaled your tangram properly, the length of each side of the tangram will double to 4 inches.

(b) Add the grid of 16 squares to the puzzle by drawing the three evenly spaced horizontal vertical lines. What are the side lengths of the 16 small squares in your scaled diagram?

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Answer

(b) The lengths of the small squares on the new puzzle should be twice the length on the original diagram.

The length of the sides of the small squares in the original puzzle are 0.5 in. The corresponding lengths on the new puzzle are therefore 0.5 in multiplication two equals one in.

Finally, add the thick blue construction lines by matching the points of intersection. Keep your puzzle handy, but don’t cut it up yet—there’s more to be done!

You may have used scale diagrams for do-it-yourself work in your house or garden. If you are planning a new look for a room or a garden, it helps to create a detailed plan so that you can see how things will look. Bathroom suppliers and do-it-yourself stores often provide planning grids and scaled cut-out shapes for items such as bathtubs and cupboards, so you can try different arrangements without moving the real furniture. A scale diagram also helps you to see what will fit and more importantly what won’t, and this can prevent expensive mistakes.

Imagine a planning grid marked in 0.5 cm squares with scale at 1:20. What would each square on the plan represent in real life?

Since the scale is 1:20, the side of each of these squares will represent 0.5 cm prefix multiplication of 20, or 10 cm.

In order to use such a grid, you will need to add the dimensions of the room and the measurements of fixtures like doors, windows, and radiators. For example, suppose you measure your bathroom and find that it is 3.5 meters long and 2.45 meters wide. So that the diagram will fit onto a standard letter-size piece of paper, use either centimeters or millimeters as the units of measurement.

  1. First, convert the real-life measurements into centimeters. The length of the bathroom is 3.5 m equals three .5 multiplication 100 cm equals 350 cm.
  2. The edge of each square on the plan represents 10 cm.
  3. Since 350 division 10 equals 35, the edge of 35 squares will represent 350 cm.
  4. The width of the bathroom is 2.45 m equation left hand side equals right hand side 2.45 multiplication 100 cm equals 245 cm.
  5. Since 245 division 10 equals 24.5, we know that the edge of 24.5 squares will represent 245 cm.

On the planning grid provided separately, draw a rectangle 35 squares long and 24.5 squares wide to mark the boundary of the bathroom.

Adobe PDF Icon Use the link below to download the bathroom planning grid:

View document

Note: when printing, be sure to place your settings on “Scale: None” to ensure that the dimensions print accurately.

Alternatively, rather than counting squares, you can calculate the lengths on the plan. If the scale is 1:20, then you will have to divide both the length and the width of the bathroom by 20 to find the corresponding lengths on the plan. The length of the bathroom will be 350 cm division 20 equals 17 .5 cm, and the width of the bathroom will be 245 cm division 20 equals 12 .25 cm. Check that your rectangle measures 17.5 cm by 12.25 cm!

10.3.1 Designing a Bathroom

In the bathroom described on the previous page, the door is on the shorter wall in the corner, and it is 75 cm wide. A heated towel rail 1 m wide is on the longer wall, 1.25 m from the corner of the bathroom. On the wall opposite the door, there is a window that is 62 cm wide and 91.5 cm from the corner (but at a height of 1.6 m from the floor).

Here is a rough sketch:

What will these measurements be on the plan?

Mark the towel rail and window on the plan. How would you arrange the bathtub, toilet, and basin in the bathroom?

The door is 75 cm wide, and this will be represented by the edge of 75 division 10 equals 7.5 squares, or a length of 75 cm division 20 equals three .75 cm on the plan.

The towel rail is 1 m wide. This is the same as 100 cm, so it will be represented by the edge of 100 division 10 equals 10 squares or a length of 100 cm prefix division of 20 equals five cm on the plan. It is at a distance of 1.25 m or 125 cm from the corner. This length will be represented by the edge of 12.5 squares or by 125 cm division 20 equals six .25 cm.

The window is 62 cm wide, and this will be represented by the edge of 6.2 squares or by a length of 62 cm division 20 equals three .1 cm on the plan. It is 91.5 cm from the corner. This length will be represented by the edge of 9.15 squares or by 91.5 cm prefix division of 20 equals 4.575 cm on the plan. You will have to round these measurements before drawing them on the plan. The diagram shows how the plan should look. (It is not to the scale given above.)

To arrange the bathtub, basin, and toilet, you could cut out shapes drawn to scale and position them on the plan, allowing space around each item. For ease of installation, the toilet should be positioned next to the sewage pipe, but there are several possibilities for the bathtub and basin. If other items such as cupboards are required, you can make scaled shapes for these and add them.

These two examples, the tangram and the bathroom plan, illustrate how diagrams can be used to make practical problems easier to solve. If you are constructing your own scale diagram and have taken a lot of measurements, you may find it clearer to tabulate your results as shown below. Why not try this for a room in your house?

10.4 Geometry

Geometry is a branch of math that, at its basic level, is concerned with describing shapes and space. So it helps to be familiar with basic definitions and vocabulary relating to angles, lines, and shapes. As you study this section, add new definitions to your notes.

Angles and Lines

Angles measure the amount of turning from one position to another. Imagine looking straight ahead and then turning around until you return to your starting position. The angle you have turned through is a full turn. If you turn so that you are facing in the opposite direction, you will have made half of a full turn. If you turn from looking straight ahead to facing either directly to your right or directly to your left, you will have made a quarter turn.

Smaller turns can be described by splitting the full turn, that is a circle, into 360 equal intervals known as degrees. So one full turn is the same as 360 degrees, which is written as 360º. This means that half a turn is 180º and a quarter turn is 90º as shown below. [ Why 360º? We inherited this from the Babylonians! Their counting system was based on 60 and they first used degrees in astronomy. ]

An angle of 90º is known as a right angle.

Lines which are at right angles to each other are said to be perpendicular. On a diagram, right angles are denoted by a small square drawn at the angle. Angles can be measured using an instrument called a protractor.

Parallel Lines

Parallel lines are always the same distance apart and, if you extended them indefinitely, they would never meet. A railway track is an example of a set of two parallel lines. If two lines are parallel, this can be shown on a diagram by drawing an arrow (or double arrow) on each line.

10.4.1 Everyday Shapes

Notepad iconActivity: Everyday shapes

You are probably already familiar with shapes such as triangles, squares, rectangles, and circles from everyday life. Spend a few moments summarizing what you know about these shapes already, including how you recognize them and where you might find them.

Solutions icon
Answer

Triangles have three straight sides and are often used in buildings since they cannot be distorted. For example, they may be used for bracings or supports.

Squares and rectangles both have four straight sides and their angles are all right angles. In addition, the sides of a square are all the same length. These shapes can be seen everywhere—books, tables, tiles, buildings, etc.

Circles can be drawn by specifying a center point and a radius—the distance from the center to the circle’s edge. The radius is always a constant length. The diameter of a circle is the distance from one edge of the circle to the other, passing through the center. The edge of the circle (or the length of this edge) is known as the circumference. Circles are often used where movement is involved (for example, wheels) or where lack of corners is important (for example, cups or bowls). [Why are manhole covers usually rounded? Hint: Think about the possibility of dropping the cover into the hole!]

Don’t worry if you didn't get all these details—you are here to learn! Just take the time to note down any new information to you in your math notebook.

Let's take a more in-depth look triangles and four-sided shapes.

You may have mentioned some other properties of these shapes as well.

Some special kinds of triangles are shown below; the right-angled triangle, isosceles triangle, and equilateral triangle. Sides that are the same length are marked with the same symbol, usually a short line, perpendicular to the side.

A triangle in which all the sides have different lengths is known as a scalene triangle. Some examples are shown below. You will see that a right-angled triangle can also be described as a scalene triangle.

Another important fact is that the angles of a triangle add up to 180º. To illustrate this, if you cut out any triangle and then tear off the angles, you will be able to arrange them to form a straight line. Draw a triangle and have a go yourself.

Shapes that have four straight edges are known as quadrilaterals. Squares and rectangles are special kinds of quadrilaterals.

A quadrilateral that has one set of parallel sides is known as a trapezoid.

If the quadrilateral has two sets of parallel sides, it is called a parallelogram.

When you are describing a geometrical figure, you often need to refer to a particular line or angle on the diagram. This can be done by labeling the diagram with letters. For example, the diagram below shows a triangle ABC, in which the longest side is AB and the angle cap a times cap c hat times cap b is a right angle. cap a times cap c hat times cap b is the angle formed by the lines AC and CB. The point where two lines meet is known as a vertex (the plural is vertices). So A, B, and C are vertices of the triangle.

Note that you can use the shorthand notation “ prefix white up pointing triangle of cap a times cap b times cap c” for “the triangle ABC” if you wish. You may have come across a lot of new math vocabulary in this section—remember to include any in your math notebook.

10.4.2 What Can You See?

Activity Symbol Activity: What Can You See?

Find the tangram puzzle you made in Section 10.3 and label it as shown in the image below.

(a) Which sets of lines appear to be parallel?

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Answer

There are four sets of parallel lines. AB is parallel to DC. This can be written as cap a times cap b parallel to cap d times cap c. Similarly, cap d times cap b parallel to cap h times cap j comma and cap a times cap i parallel to cap e times cap h. Also, AD, GI, and BC are all parallel to each other.

(b) Which lines are perpendicular?

Solution Symbol
Answer

AB is perpendicular to AD and to BC. This can also be written as cap a times cap b up tack cap a times cap d comma and cap a times cap b up tack cap b times cap c