MATHEMATICS

Grade 9

NUMBERS

Module 1

NUMBERS – WHERE DO THEY COME FROM?

Numbers – where do they come from?

CLASS WORK

1 Our name for the set of Natural numbers is N, and we write it: N = { 1 ; 2 ; 3 ; . . . }

1.1 Will the answer always be a natural number if you add any two natural numbers? How will you convince someone that it is always the case?

1.2 Multiply any two natural numbers. Is the answer always also a natural number?

1.3 Now subtract any natural number from any other natural number. Describe all the sorts of answers you can expect. Try to write down why this happens.

2 To deal with the answers you got in 1.3, we have to extend the number system to include zero and negative numbers – we call them, with the natural numbers, the integers. They are called Z and this is one way to write them down: Z = { 0 ; ±1 ; ±2 ; ±3 ; . . . }

2.1 Complete the following definitions by writing down what has to be inside the brackets:

3 Is the answer always another integer when you divide any integer by any other integer (except zero)?To allow for these answers we have to extend the number system to the rational numbers:

3.1 Q (rational numbers) is the set of all the numbers which can be written in the form abab size 12{ { {a} over {b} } } {} where a and b are integers as long as b is not zero. Explain very clearly why b is not allowed to be zero.

4 Q` (irrational numbers) is the set of numbers which cannot be written as a common fraction, and are therefore not in Q. Putting Q en Q` together gives the set called R, the real numbers.

4.1 Write down what you think is in the set R` . They are called non-real numbers.

end of CLASS WORK

Quipu is an Inca word meaning a string (or set of strings) with knots in it. This system was used for remembering things, mainly numbers. It was used widely in the ancient world; not only in South America. At its simplest, it was just one string with each knot representing one item. In more advanced systems, more strings were used, often of different colours; sometimes a system of place-values was used.

HOMEWORK ASSIGNMENT

1. What is the importance of having a symbol for zero? Think about all the things we’ll be unable to do if we didn’t have a zero.

2 Find out what we call the set of numbers we get when putting R and R` together. Can you say more about them?

3 Design your own set of number symbols like those in table 1. Show how any number can be written in your system. Now think up new symbols for + and – and × and , and then make up a few sums to show how your system works.

end of HOMEWORK ASSIGNMENT

ENRICHMENT ASSIGNMENT

Let’s check out the rational numbers

Is 3,013 a rational number? Yes! Look at this bit of magic:

3,013 = 31+13100031+131000 size 12{ { {3} over {1} } + { {"13"} over {"1000"} } } {} = 30001000+13100030001000+131000 size 12{ { {"3000"} over {"1000"} } + { {"13"} over {"1000"} } } {} = 3000+1310003000+131000 size 12{ { {"3000"+"13"} over {"1000"} } } {} = 3013100030131000 size 12{ { {"3013"} over {"1000"} } } {}

It is easy to write it down straightaway. Explain the method carefully.

Figure 1

4 Only terminating and repeating decimal fractions can be written in the form abab size 12{ { {a} over {b} } } {}.

4.1 Here are some irrational numbers (check them out on your calculator):

 22 size 12{ sqrt {2} } {}113113 size 12{ nroot { size 8{3} } {"11"} } {} 3,030030003000030…

4.2 These are NOT irrational – explain why not: 227227 size 12{ { {"22"} over {7} } } {}; 2525 size 12{ sqrt {"25"} } {}; 273273 size 12{ nroot { size 8{3} } {"27"} } {}

Figure 2

end of ENRICHMENT ASSIGNMENT

Working accurately

CLASS ASSIGNMENT

1 With every question, simplify the numbers, if necessary, and then place each number in its best position on the given number line.

Figure 3

Figure 4

end of CLASS ASSIGNMENT

ENRICHMENT ASSIGNMENT

Inequalities – translating words into maths

1 The number line tells us something very important: If a number lies to the left of another number, it must be the smaller one. A number to the right of another is the bigger.

For example (keep the number line in mind) 4,5 is to the left of 10, so 4,5 must be smaller than 10. Mathematically: 4,5 < 10.

What about numbers that are equal to each other? Surely 6  3 and 44 size 12{ sqrt {4} } {} !

So: 6  3 = 44 size 12{ sqrt {4} } {}.

1.1 Use < or > or = between the numbers in the following pairs, without swopping the numbers around:

5,6 and 5,7; 3+9 and 4×3; –1 and –2; 3 and –3 273273 size 12{ nroot { size 8{3} } {"27"} } {} and 1515 size 12{ sqrt {"15"} } {}

2 We use the same signs when working with variables (like x and y, etc.). .

For example, if we want to mention all the numbers larger than 3, then we use an x to stand for all those numbers (of course there are infinitely many of them: 3,1 and 3,2 and 3,34 and 6 and 8 and 808 and 1 000 000 etc). So we say: x > 3.

2.1 Use the variable y and write inequalities for the following descriptions:

All the numbers larger than –13,4 All the numbers smaller than or equal to π

3 We extend the idea further:

It works best if you write numbers in the order in which they appear on the number line: the smaller number on the left and the bigger one on the right. Then you simply choose between either < or ≤.

3.1 Now you and a friend must each give three descriptions in words. Then write the mathematical inequalities for one another’s descriptions.

Inequalities – graphical representations

3.1 Again make your own diagrams.

end of ENRICHMENT ASSIGNMENT

GROUP ASSIGNMENT

1 CALCULATORS ARE NOW FORBIDDEN – DON’T DO ANY SUMS. ESTIMATE THE ANSWERS AS WELL AS YOU CAN AND FILL IN YOUR ESTIMATED ANSWERS. This assignment is the same as before – only you have to draw your own suitable number line for the numbers. First work alone, then the group must decide on the best answer. Fill this answer in on the group’s number line. This group effort is then handed in for marking.

1.1 –8 ; 12 ; 5–11 ; 4 + 0 – 412412 size 12{4 left ( { {1} over {2} } right )} {} ; 369+124+2211+1369+124+2211+1 size 12{ { {"36"} over {9} } + { {"12"} over {4} } + { {"22"} over {"11"} } +1} {} ; 814814 size 12{ nroot { size 8{4} } {"81"} } {} ; 4+94+9 size 12{ sqrt {4} + sqrt {9} } {} ; 6+16+1 size 12{ sqrt {6} +1} {} ; 273273 size 12{ nroot { size 8{3} } {"27"} } {}

1.2 2.5 – ½ ; 1313 size 12{ { {1} over {3} } } {} ; 113113 size 12{1 { {1} over {3} } } {} ; 56−2656−26 size 12{ { {5} over {6} } - { {2} over {6} } } {} ; 0,5 ; 0,05 ; 0,005

1.3 3 ; 3,5 ; 3,14 ; 22  7 ; 355  113 ; 

end of GROUP ASSIGNMENT

CLASS WORK

1 Of course one can write any number in many ways:

1.1 Is 1  3 equal to 1,3˙1,3˙ size 12{1, { dot {3}}} {}? What about 1,33˙1,33˙ size 12{1,3 { dot {3}}} {}? And 1,33 of 1,333 of 1,3?

1.2 Is 55 size 12{ sqrt {5} } {} the same as 2,2? Or 2,24? Or 2,236? Or 2,2361? Or maybe 2,2360? Discuss.

1.3 Is 3 and 3,5 and 3,14 and 22 ÷ 7 and 355 ÷ 113 the same as  ? Make a decision.

2 We can’t always write 3,1415926535897932384626 . . . when we want to use. Why not?

If I have to write down exactly what  is, then I must write  ! The others in question 1.3 are only approximately equal to. But when I have to use  in a calculation to get an answer, then I have to be able to round off properly.

This is π rounded off to different degrees of accuracy:

1 decimal place: 3,1

2 decimal places: 3,14

3 decimal places: 3,142

4 decimal places: 3,1416

5 decimal places: 3,14159

6 decimal places: 3,141593

3 Simplify and round off the following values, accurate to the number of decimal places given in the brackets.

3.1 3,1  3 (2)

3.2 2 × 22 size 12{ sqrt {2} } {} 2)

3.3 5 ×  (2)

3.4 4,5 × 77 size 12{ sqrt {7} } {} (0)

3.5 1,000008 + 25  10000 (1)

end of CLASS WORK

How many seconds in a century?

CLASS WORK

1.1 How many hours are there in 17 weeks? 24 × 7 × 17 = 2 856 hours

1.2 How many minutes in a week? 60 × 24 × 7 = 10 080 minutes

1.3 Is it just as easy to calculate how many hours there are in 135 months? Discuss the question in a group and decide which questions have to be answered before the answer can be calculated.

1.4 How many years are there in 173 months? 173  12 = 14,4166 6˙6˙ size 12{ { dot {6}}} {} ≈ 14,42 years

2 Why do we multiply in question 1.1 and 1.2, and divide in question 1.4?

3 How many seconds in a century? It may take a while to get to the answer! How will you know that you can trust your answer?

4.1 There are one thousand metres in a kilometre, so we can say that one metre equals 0,001 kilometres. One metre = 1  1000 kilometres or 1 m = 11000km11000km size 12{ { {1} over {"1000"} } ital "km"} {}

4.2 There are one thousand millimetres in a metre: 1 mm = 11000×1000km11000×1000km size 12{ { {1} over {"1000" times "1000"} } ital "km"} {} = 0,000 001 km

4.3 There are one thousand micrometres in a millimetre: 1 μm = 0,000 000 001 km. (μ is a Greek letter – mu.)

5 Just as we can write very large numbers more conveniently in scientific notation, we also write very small numbers in scientific notation. Below are a few examples of each. Make sure that you can convert ordinary numbers to scientific notation, and vice versa. Calculators also use a sort of scientific notation. They differ, and so you have to make yourself familiar with the way your calculator handles very large and very small numbers.

5.1 1 μm = 0,000 000 001 km So: 1 μm = 1,0 × 10^–9 km

5.2 On a typical lightweight bed sheet, there might be about three threads per millimetre, both across and lengthwise. If a sheet for a double bed measured two metres square, that would mean 6,0 × 10³ threads across plus another 6,0 × 10³ threads lengthwise. That gives us 1,2 × 10⁴ threads, each about two metres long. Calculate how many kilometres of thread it took to make the sheet. Tonight, measure your pillowslip and do the same calculation for it.

5.3 A typical raindrop might contain about 1 × 10^–5 litres of water. In parts of South Africa the annual rainfall is about 1 metre. On one hectare that means about 1 × 10¹² raindrops per year. On a largish city that could mean about 6 × 10¹⁶ raindrops per year, or about 1 × 10⁷ drops for every man, woman and child on Earth. How many litres each is that?

5.4 Calculate: (give answers in scientific notation)

5.4.1 3,501×10−59,5×10−8+4,3×10−113,501×10−59,5×10−8+4,3×10−11 size 12{ { {3,"501" times "10" rSup { size 8{ - 5} } } over {9,5 times "10" rSup { size 8{ - 8} } } } +4,3 times "10" rSup { size 8{ - "11"} } } {}

5.4.2 3,5×106+1,4×10−173,5×106−1,4×10−173,5×106+1,4×10−173,5×106−1,4×10−17 size 12{ { {3,5 times "10" rSup { size 8{6} } +1,4 times "10" rSup { size 8{ - "17"} } } over {3,5 times "10" rSup { size 8{6} } - 1,4 times "10" rSup { size 8{ - "17"} } } } } {}

end of CLASS WORK

We use prefixes, mostly from Latin and Greek, to make names for units of measurement. For example, the standard unit of length is the metre. When we want to speak of ten metres, we can say one decametre; one hundred metres is a hectometre and, of course, one thousand metres is a kilometre. One tenth of a metre is a decimetre; one hundredth of a metre is a centimetre and one thousandth is a millimetre. There are other prefixes – see how many you can track down.

Your computer pals will be able to confirm, I hope, that in computers a “kilobyte” is really 1024 “bytes”. Now, why is it 1024 bytes and not 1000 bytes? The answer lies in the fact that computers work in the binary system and not in the decimal system like people. Try to find the answer yourself.

Assessment

Memorandum

CLASS WORK

1.1 Yes, any informal “proof” is acceptable.

1.2 As 1.1

1.3 Zero and negative numbers make an appearance. The explanation is not important – only the thinking that the learner does.

2.1 N₀ = {0 ; 1 ; 2 ; . . . } and Z = { . . . –3 ; –2 ; –1 ; 0 ; 1 ; 2 ; 3 ; . . .}

3. Here are the fractions. Explain carefully that integers can also be written as fractions – in fact it is quite often a useful technique.

4.1 Not everyone will be able to cope with this. R` gives the answers that are obtained when square roots of negative numbers (inter alia) is taken.

TASK

2. Point out to learners that zero is missing from the table.

HOMEWORK ASSIGNMENT

1. Zero is needed because:

The principle behind place values is totally dependent on having a symbol for zero.

It separates positive and negative numbers.

It symbolises “nothing”.

Algebraically it is defined as: a + (–a)

2. Complex numbers – don’t expect too much.

If one uses the symbol i for −1−1 size 12{ sqrt { - 1} } {}, then we can represent non–real numbers as follows:−12−12 size 12{ sqrt { - "12"} } {} = 2−32−3 size 12{2 sqrt { - 3} } {} = 23×−123×−1 size 12{2 sqrt {3 times - 1} } {} = 23−123−1 size 12{2 sqrt {3} sqrt { - 1} } {} = 23i23i size 12{2 sqrt {3} `i} {}

3 + 5i and 2,5 – 16i are examples of non–real numbers, and each consists of two parts: a real part and a non-real part. The most important consequences of this are that one must be careful when doing arithmetic calculations, and that these numbers cannot be arranged in ascending order!

3. Any reasonable answer can be accepted. This might be a good opportunity to have learners evaluating each other’s number systems.

ENRICHMENT ASSIGNMENT

If there is time, one can go through this work, particularly with a strong group.

4.1 Non-repeating; although 3,030030003000030… has a pattern, it does not repeat.

4.2 Emphasise that the first one is NOT equal to π. The two others must be simplified properly.

4.3.1 1553100015531000 size 12{ { {"1553"} over {"1000"} } } {}

4.3.2 51905190 size 12{ { {"51"} over {"90"} } } {}

4.3.3 3031199930311999 size 12{ { {"30311"} over {"999"} } } {}

4.3.4 24039902403990 size 12{ { {"2403"} over {"990"} } } {}

CLASS ASSIGNMENT

The aim of this exercise is to familiarise learners with unsimplified values, so that they can learn to estimate. It is very important that they mentally simplify correctly so that they can start guessing the magnitudes. Then the values have to be arranged in at least the correct order. If the spaces in between are in reasonable proportion, that is a bonus. This shows the order:

1.1 0,00 ; 1 ; 2 ; 3,0 ; 4 ; 5,0000 ; 5+2 ; 6 ; 9–1

1.2 –4 ; –3 ; –1 ; 3–3 ; 2 ; 5

1.3 1515 size 12{ { {1} over {5} } } {} ; 1414 size 12{ { {1} over {4} } } {} ; 0.666 ; 2323 size 12{ { {2} over {3} } } {} ; 2222 size 12{ { {2} over {2} } } {} ; 1,000 ; 0,2+1 ; 1.75

1.4 32−1232−12 size 12{ { {3} over {2} } - "12"} {} ; −8+75−8+75 size 12{ - 8+ { {7} over {5} } } {} ; –5,5 ; −52−52 size 12{ - { {5} over {2} } } {} ; –2,5 ; 5,55 ; 142142 size 12{ { {"14"} over {2} } } {}

1.5 −9−9 size 12{ - sqrt {9} } {} ; −4−4 size 12{ - sqrt {4} } {} ; 00 size 12{ sqrt {0} } {} ; 11 size 12{ sqrt {1} } {} ; 9494 size 12{ sqrt { { {9} over {4} } } } {} ; 162162 size 12{ { { sqrt {"16"} } over {2} } } {} ; 99 size 12{ sqrt {9} } {} ; 3636 size 12{ sqrt {"36"} } {}–1

1.6 44 size 12{ sqrt {4} } {} ; 66 size 12{ sqrt {6} } {} ; 99 size 12{ sqrt {9} } {} ; 1616 size 12{ sqrt {"16"} } {} ; 2525 size 12{ sqrt {"25"} } {} ; 2020 size 12{ sqrt {"20"} } {}+1 ; 3232 size 12{ sqrt {"32"} } {} ; 3636 size 12{ sqrt {"36"} } {}

ENRICHMENT ASSIGNMENT

1.1 5,6 < 5,7; 3+9 = 4×3; –1 > –2; 3 > –3; 273273 size 12{ nroot { size 8{3} } {"27"} } {} < 1515 size 12{ sqrt {"15"} } {}

2.1 y > –13,4

Figure 6

GROUP ASSIGNMENT

These are the simplified values in the original order:

1.1 –8 ; 12 ; –6 ; 2 ; 10 ; 3 ; 5 ; 3,44… ; 3

1.2 2 ; 0,3… ; 1,3… ; 0,5 ; 0,5 ; 0,05 ; 0,005

1.3 3 ; 3,5 ; 3,14 ; 3,142857… ; 3,1415929… ; 3,1415926… (the last one is π)

These are the same values in the correct order:

1.1 –8 ; –6 ; 2 ; 3 ; 3,44… ; 5 ; 10 ; 12

1.2 0,005 ; 0,05 ; 0,3… ; 0,5 ; 1,3… ; 2

1.3 3 ; 3,14 ; π ; 3,1415929… ; 3,142857…

CLASS WORK

This exercise has been designed to give learners a feeling for the consequences of rounding (approximated answers). They often put complete unthinking faith in their calculators’ answers.

1.1 Note the notation as well as the number of decimal places.

1.2 Again, notation as well as number of decimal places.

1.3 Emphasise once again that an approximation to π is not equal to π.

Discuss the meaning of the term “approximately equal to”.

3. Answers: 1,03 ; 2,83 ; 15,71 ; 12 ; 1,0 (the zero must be there).

CLASS WORK

Learners often have difficulties with conversions – you might have to supply lots of help and guidance.

1.3 The months don’t have the same number of days; simply multiplying will not give the best answer. Find out which months are meant and don’t forget leap years!

1.4 Why division? Help them develop strategies.

3. Similar problems to 1.3. The answer can be approximated. Explain why this acceptable. This problem will motivate them to appreciate the advantages of scientific notation: ≈ 3 157 056 000 seconds.

5.1 9,1 × 10²⁸

5.2 24 km

5.3 100 litres

Figure 7