1. MATHEMATICS Working with Negative Numbers

2. The aim of this powerpoint is to teach you how to compare, order and undertake calculations involving negative numbers EITHER Take notes as you go along, include some examples and write down any questions and your answers (which you can mark as you go along) OR At the end of the powerpoint, printout the notes called Calc-8

3. Place Value Negative numbers are worth less than 0 The negative sign appears on the LEFT hand side of the number value i.e. In 4 – 6 + 8 + 5 – 3 the negatives are – 6 and – 3 A good example of negative numbers in real-life is thinking about temperatures. Negative temperatures (very cold) are below zero!

4. Number Scale Here is part of an integer number scale… Here is a lower part of the same integer number scale… Negative numbers can also be decimals! 2 3 4 5 6 7 8 9 – 3 – 2 – 1 0 1 2 3 4 – 2 –1.5 – 1 –0.5 0 0.5 1 1.5

5. Ordering and Comparing When you are asked to put numbers in order remember to look at their signs NOT just their digit value If you are asked to put the smallest number first you are looking for the coldest temperature (i.e. the negative value with the largest digits) and you work up to the smallest negative before ordering the positive values…

6. Example Put these numbers in order starting with the biggest and going down to the smallest… 1, – 6, 7, 2, – 9, 8, 0, 3, – 2 HINT: Biggest equates to hottest so look at the positive numbers first and think of it as hottest to coldest temperatures! ANSWER: 8, 7, 3, 2, 1, 0, – 2, – 6, – 9 Hottest = Most positive Coldest = Most negative

7. Calculating with Negatives There are certain rules you can learn but on a basic level, the easiest ways of adding or subtracting with negative numbers is imagining them on a number line or thinking of them in terms of temperatures! Remember that finding the difference (a subtraction) means finding the distance between two numbers! Negative numbers may have (but not always) brackets written around them so you know they are negative!

8. Using a number line… Here is part of an integer number scale… It can help us solve questions like: What is (– 2) + 4? What is 3 – 5? What is 4 – (– 2)? – 3 – 2 – 1 0 1 2 3 4

9. Example 1 What is (– 2) + 4? This can be interpreted as saying ‘If the temperature is at – 2°C and it rises by 4 degrees, what is the new temperature?’ Answer is 2°C (or + 2°C) – 3 – 2 – 1 0 1 2 3 4

10. Example 2 What is 3 – 5? This can be interpreted as saying ‘If the temperature is at 3°C and it falls 5 degrees, what is the new temperature?’ Answer is –2°C – 3 – 2 – 1 0 1 2 3 4

11. Example 3 What is 4 – (– 2)? This can be interpreted as saying ‘What is the difference in temperature between 4°C and –2°C?’ Counting the distance (dots) = 6°C – 3 – 2 – 1 0 1 2 3 4

12. Adding/Subtracting Rules If we were unable to use a number line, then we would need to follow these rules: Two signs ‘side by side’ with nothing (except perhaps brackets) in-between them can be changed for one sign: Collect all the positive numbers together; Collect all the negative numbers together; Write them as positive total negative total and work out the overall answer. SAME signs  plus DIFFERENT signs  minus

13. Examples Let’s use our previous 3 examples… What is (– 2) + 4? What is 3 – 5? What is 4 – (– 2)? – 2 + 4 rearrange with positives first: +4 – 2 = +2 3 – 5 shows we have 3 positives and 5 negatives. 3 positives cancel 3 negatives leaving 2 negatives so… 3 – 5 = – 2 4 – – 2 replace the ‘– –’ with a positive sign so 4 + 2 = 6

14. Two More Examples – 8 + 3 – 7 – 5 + 12 = Positives: + 3 + 12 = +15 Negatives: – 8 – 7 – 5 = – 20 So we have: 15 – 20 = – 5 (There are 5 more negatives in -20 than positives in 15) Subtract -13 from -8? i.e. – 8 – – 13 = – 8 + 13 = 5 (There are 5 more positives in +13 than negatives in -8) – 8 + 3 – 7 – 5 + 12

15. Practice Work out the answers to the following calculations before clicking to check your answers on the next slide. 1)What is the total of: –7, 4, –2 and 4? 2)What is the difference between 6 and –3? 3)Subtract negative 6 away from negative 9. 4)Subtract positive 5 away from negative 7. 5)7 + 8 – 11 + 1 =

16. Answers 1)–7 + 4 –2 + 4  8 – 9 = – 1 2)6 – –3 = 6 + 3 = 9 3)– 9 – –6 = –9 + 6 = –3 4)– 7 – +5 = –7 –5 = –12 5)7 + 8 – 11 + 1 = 16 – 11 = 5 Now work through the MyMaths lesson (and then the online homework) called Negative Numbers 1 found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=negatives/negatives1&taskID=1069 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=negatives/negatives1OH&taskID=1069 http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=negatives/negatives1&taskID=1069 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=negatives/negatives1OH&taskID=1069

17. Multiplying and Dividing with Negatives When multiplying or dividing PAIRS of values… Multiply/divide the numbers then… If the signs are the same, the answer will be positive If the signs are different, the answer will be negative Let’s look at why… [pos x pos = pos]4 × 6 = 24 (4 lots of 6) [pos x neg = neg]4 × -6 = -24 (4 lots of -6) [neg x neg = pos] – 4 × – 6 = – –24 = +24 or 24 (negate 4 lots of -6)

18. More Examples –54  How many -9’s are there in -54?  6 –9 –36 ÷ 3  Divide –36 into 3 equal groups of what?  –12 20 ÷ –5  How many –5’s are there in 20?  –4 –4 × –2 × –5  +8 × –5 = –40 (Note there is an ODD number of negative values (not just a pair) being multiplied together so their answer is negative).

19. Practice Work out the answers to the following calculations before clicking to check your answers on the next slide. 1)What are 7 lots of –9? 2)What is –24 divided by –4? 3)What is the product of –5 and –7? 4)How many lots of –3 are there in 21? 5)7 × (–4) × 2 = ?

20. Answers 1) What are 7 lots of –9?7 × –9 = –63 2) What is –24 divided by –4?–24 ÷ –4 = 6 3) What is the product of –5 and –7?–5 × –7 = 35 4) How many lots of –3 are there in 21? 21 ÷ –3 = –7 5) 7 × (–4) × 2 = ? –28 × 2 = –56

21. What next? If you haven’t made any notes or copied any examples, questions and answers out during this presentation, print out the notes called Calc8. Read through them and make sure you answer any questions. Work through the MyMaths lesson (and then the online homework) called Negative Numbers 2 found at: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=negatives/negatives2&taskID=1068 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=negatives/negatives2OH&taskID=1068 http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=negatives/negatives2&taskID=1068 http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=negatives/negatives2OH&taskID=1068 Save and complete the worksheet called Negs-S1.xlsx Now move on to the Calc9-BIDMAS powerpoint