Maths Week 2021

Survivor Series/Kia Mōrehurehu

Monday Level 5

Questions

What to do for students

1 You can work with one or two others. Teams can be different each day.

2 Do the tasks and write any working you did, along with your answers, in the spaces provided (or where your teacher says).

3 Your teacher will tell you how you can get the answers to the questions and/or have your work checked.

4 When you have finished each day, your teacher will give you a word or words from a proverb.

5 At the end of the week, put the words together in the right order and you will be able to find the complete proverb! Your teacher may ask you to explain what the proverb means.

6 Good luck.

Task 1 – numbers in te reo Māori

The following chart gives numbers in te reo Māori.

Look at the chart carefully and note the patterns in the way the names are built up from 10 onwards.

Work out what each of the numbers in the following calculations is, do each calculation, and write the answer in te reo Māori.

Question Answer

(a) whitu + toru

(b) whā x wa

(c) tekau mā waru – rua

(d) ono tekau ma whā + rua tekau ma iwa

(e) toru tekau ma rua + waru x tekau mā ono

Task 2 - Roman numerals

The picture shows the Roman Emperor, Julius Caesar, who was born in the year 100 BC.

(a) How many years ago was 100 BC?

You may have seen places where numbers have been written in Roman numerals. The Roman numeral system uses letters for numbers as shown in the table below. To successfully complete what follows, you will need to know these by heart.

Letter Meaning

I 1

V 5

X 10

L 50

C 100

D 500

M 1000

They make other numbers by combining the letters. The following table shows this for the numbers 1 – 10.

Number Symbol

2 II

3 III

4 IV

5 V

6 VI

7 VII

8 VIII

9 IX

10 X

To start with, letters are written after one another. But after three of the same letter occur together after another letter, the system changes, and the letter being written goes in front of the next letter. So after III (ie “3”) we have IV, which means “1 before 5”, ie 4, then V (for “5”) and then the system continues. So VI means “1 after 5”, that is 6, then we have VII (for “7”), VIII (for 8), then IX (meaning one before 10, ie 9) and so on. Here are some more examples. Make sure that you understand how each numeral is written.

Symbol Meaning Number IL 1 before 50 49 XL X before 50 40 LX 10 after 50 60 XC 10 before 100 90 CX 10 after 100 110 XIV 4 after 10 14 XLI 10 before 50 and 1 after 41 LXI 10 after 50 and 1 after that 61 XCIV 10 before 100 and 4 after that 94 XCVI 10 before 100 and 6 after that 96

Now see how you do with Roman numerals.

(b) Complete the following table. Hindu-Arabic is the system of numerals we normally use.

Hindu-Arabic 1 2 3 4 5 6 7 8 9 10 Roman

Hindu-Arabic 11 12 13 14 15 16 17 18 19 20 Roman

Hindu-Arabic 31 42 54 67 79 84 107 258 563 987 Roman

(c) Now see if you can write these Roman numerals as Hindu-Arabic numerals.

Roman Hindu-Arabic Roman Hindu-Arabic XD CCCL XDI CCCV DL MI IM XM CX MXVI CCLV MXXIV ICC MMCVI XCC MMMLIX

(d) The photo shows the famous Trevi fountain in Rome. It was built in the year 1732 AD. Convert 1732 into a Roman numeral.

(e) The Darlington Courthouse in Sydney has this Roman numeral above its main entrance: MDCCCLXXXVIII. What number does it show? What will this number relate to?

(f) When Sarah was learning about Roman numerals, she wrote one as LLI. Her teacher told her that it wasn’t a proper Roman numeral.

(i) What number do you think Sarah intended to write in Roman numerals?

(ii) How should she have written this number?

(iii) Using the symbol X, try writing a number incorrectly in Roman numerals (you can include some other letters as well).

(g) Now see what it would have been like if you had been doing maths using Roman numerals!

(i) Here are some numbers to be added. Do the calculations (without using a calculator!).

4 2 2 0 5 + 1 9 + 5 8

(ii) Here are the same calculations in (a) but written using Roman numerals. Try to do the calculations now.

X L I I C C L V + I X X + L V I I I (h) Can you give two reasons why using Hindu-Arabic numerals makes the calculations a lot easier?

(i) Give some places where you could see Roman numerals being used today.

Task 3 – Computer numerals (binary)

The Hindu-Arabic numbers that you are familiar with are based on 10 (which is why the number system is called a decimal system). It used the ten digits 0, 1, 2, 3, . . .9. In a Hindu-Arabic number, digits have place value based on powers of 10, as shown in the following example for 7358.

Digit 7 3 5 8 Value thousands hundreds tens ones (units) 1000 = 103 100 = 102 10 = 101 1

Computers store data as numbers in a special form called binary. That’s because the numbers use just two digits – 0 and 1. The circuits in a computer's processor are made up of billions of transistors. A transistor is a tiny switch that is activated by the electronic signals it receives. The digits 1 and 0 used in binary reflect the on and off states a transistor can have (the only two possibilities).

In a binary number, digits have place value similarly to a decimal system but based on powers of 2. The following table shows this for the binary number 1101.

Digit 1 0 1 1 Value 23 22 21 1 Hindu-Arabic 8 4 2 1

So 1011 in binary means 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1 = 11 (eleven as a Hindu-Arabic numeral).

Similarly, the binary number 101001 means 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 1 ie 32 + 8 + + 1 = 41

(a) Write the numbers 1 – 20 as binary numerals.

(b) Write these binary numerals as Hindu-Arabic numerals (i) 110011 (ii) 11010101

(c) Write these Hindu-Arabic numerals as binary numerals.

47 100 234 Task 4 – Hexadecimal numbers

You will have seen that binary numbers become quite large very quickly as the numbers get bigger. For example, the number 859 is 110111011 in binary.

Computer system designers and programmers make things easier for themselves by having a more human-friendly representation of binary-coded values. They use a number system based on 16 (although the computers themselves still work in binary). Numerals in this system are called hexadecimal. Hexadecimal numbers use the 16 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. A is used for 10, B for 11, C for 12, D for 13, E for 14 and F for 15. Hexadecimal numbers have place values based on powers of 16.

The hexadecimal number 2E means 2 x 161 + 14 = 32 + 14 = 46

The hexadecimal number B05 means 11 x 162 + 0 x 16 + 5 = 11 x 256 + 5 = 2821

(a) Write the hexadecimal number 3AE as a Hindu-Arabic number.

(b) Write the Hindu-Arabic numbers 25, 67, 143 and 458 as hexadecimal numbers.