KS3 Mathematics N2 Negative numbers The aim of this unit is to teach pupils to: Order, add, subtract, multiply and divide positive and negative numbers. Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp 48-51. N2 Negative numbers

N2 Negative numbers Contents A N2.1 Ordering integers A N2.2 Adding and subtracting integers A N2.3 Using negative numbers in context A N2.4 Multiplying and dividing integers

Introducing integers A positive or negative whole number, including zero, is called an integer. For example, –3 is an integer. –3 is read as ‘negative three’. This can also be written as –3 or (–3). It is 3 less than 0. Introduce the key words highlighted in orange. Distinguish between ‘minus three’ which means ‘take away three’ and ‘negative three’ which means ‘three less than zero’. In the first case, the – sign is being used as an operation and in the second case the – sign is being used to tell us that the number is negative. We could write a + sign in front of positive numbers although this is not usually necessary. 0 – 3 = –3 Or in words, ‘zero minus three equals negative three’

Integers on a number line Positive and negative integers can be shown on a number line. –8 –3 Negative integers Positive integers We can use the number line to compare integers. Discuss what we mean when we use the terms ‘greater than’ and ‘less than’ with negative numbers. We say that –3 is ‘greater than’ –8 because it is further along the line in the positive direction. For example, –3 > –8 –3 ‘is greater than’ –8

Ordering negative numbers We can also use a number line to help us write integers in order. Write the integers –2, 8, 2, –6, –9 and 5 in order from smallest to largest. Look at the position of the integers on the number line: –9 –6 –2 2 5 8 Together, order the numbers on the number line. So, the integers in order are: –9, –6, –2, 2, 5, and 8

Ordered Paths The object of the activity is to select a starting number on the far left and to find a path from left to right. You must always move into a hexagon containing a smaller integer. Ask pupils to come to the board to show possible paths.

N2.2 Adding and subtracting integers Contents N2 Negative numbers A N2.1 Ordering integers A N2.2 Adding and subtracting integers A N2.3 Using negative numbers in context A N2.4 Multiplying and dividing integers

Mid-points Explain to pupils that we wish to find the number that is exactly half-way between the other two. Reveal all three numbers as an example. Generate a new example and reveal the two end-points. Ask pupils to justify their answers. Vary the activity by revealing the mid-point and one of the end-points. As an extension include negative decimals. Links: N1 Place value, ordering and rounding - ordering decimals. D3 Representing and interpreting data – calculating the mean

Adding integers We can use a number line to help us add positive and negative integers. –2 + 5 = = 3 -2 3 How can we use the number line to work out –2 + 5? Start at –2 (click to highlight the –2 on the number line) and count forwards 5. On a number line we move to the right for forwards (in a positive direction) and to the left for backwards (in a negative direction). Explain that the answer can be positive or negative depending on the starting point. It is important to stress this fact because when pupils learn the rules for multiplying and dividing negative numbers they often confuse these rules for the rules for addition and subtraction. To add a positive integer we move forwards up the number line.

–3 + –4 = = –7 Adding integers We can use a number line to help us add positive and negative integers. –3 + –4 = = –7 -7 -3 How can we use the number line to work out –3 + -4? Explain that when we add a negative number we have to move backwards down the number line. The answer can be positive or negative depending on the starting point. Stress that adding a negative number is equivalent to subtracting the positive value of that number. To add a negative integer we move backwards down the number line. –3 + –4 is the same as –3 – 4

Ordered addition square This addition square can be used to draw the pupils’ attention to the number patterns produced when adding positive and negative integers in order. Start by working out the positive number additions 3 + 0, 3 + 1, 3 + 2 and 3 + 3. Ask pupils the answer to 3 + –1, using the number pattern in the table. Fill in the rest of the row to 3 + –3. Repeat in the same order for the next three rows. For the last three rows fill in the first answer by looking at the pattern in the first column and continue along the row. Before revealing each answer ensure that the question corresponding to that cell has been clearly stated. For example, What is 1 + –2? Pupils may then look for number patterns to deduce the answer. Once the table is complete, ask pupils to use the table to answer some given questions. For example, Use the table to work out –2 + 1.

Mixed addition square Work out the numbers in the empty squares by adding and subtracting. Ask pupils why we do not need to label the sides of the square ‘first number’ and ‘second number’ (it makes no difference if we add 3 + 7 or 7 + 3).

5 – 8 = = –3 Subtracting integers We can use a number line to help us subtract positive and negative integers. 5 – 8 = = –3 -3 5 How can we use the number line to work out 5 – 8? Start at 5 (click to make the 5 orange) and count backwards 8. To subtract a positive integer we move backwards down the number line.

3 – –6 = = 9 Subtracting integers We can use a number line to help us subtract positive and negative integers. 3 – –6 = = 9 3 9 How can we use the number line to work out 3 – –6? Many pupils may find this difficult to understand. Explain what is happening. When we subtract 6 we move backwards down the number line, so, when we subtract -6 we we need to move forwards up the number line. “3 – –6 is equivalent to 3 + 6” Yow may wish to give further examples of double negatives being equivalent to positives. To subtract a negative integer we move forwards up the number line. 3 – –6 is the same as 3 + 6

–4 – –7 = = 3 Subtracting integers We can use a number line to help us subtract positive and negative integers. –4 – –7 = = 3 -4 3 Here is another example of subtracting a negative number. Again, explain carefully that subtracting a negative number is equivalent to adding. In this example, –4 is the starting point and then we move 7 forwards up the number line. Ask pupils to tell you the equivalent calculation before revealing it on the board. To subtract a negative integers we move forwards up the number line. –4 – –7 is the same as –4 + 7

Using a number line Use the interactive number line to demonstrate a variety of addition and subtraction calculations. Ask volunteers to come to the board to demonstrate solutions to problems using the number line. Include problems with several small numbers such as 3 – 7 + 1 – –5.

Ordered subtraction square This ordered subtraction square can be used to draw the pupils’ attention to the number patterns produced when subtracting positive and negative integers in order. Point out that we are subtracting the integers across the top from the integers down the side. Clicking on a cell will reveal the number inside it. Start by revealing the cells on the far right and moving in order along the row to the left. Alternatively, make the activity more difficult by revealing the cells in random order. Once the table is complete, ask pupils to use it to answer some given questions. For example, Use the table to work out –1 – –3.

Mixed subtraction square Work out the numbers in the empty squares by adding and subtracting.

Complete this table This exercise may be completed orally as a class. Alternatively, give pupils a time limit to complete the task in writing before going through the answers.

Integer cards - addition and subtraction Ask volunteers to come to the board to show a solution for each calculation. Tell them that each example must include at least one negative integer card. Discuss all possible solutions for each calculation and ask pupils to tell you any patterns or relationships that they notice.

Magic Square Explain that in a magic square, each row, column and diagonal adds up to the same number. Start by finding the ‘magic total’, that is the sum of each row, column and diagonal. Ask pupils to find the values of the integers in the empty cells. Click on the empty cells to reveal the solution to the magic square. Click on the reset button for a new puzzle.

Chequered sums Explain that the numbers in the white squares are equal to the sum of the numbers in the four coloured cells touching them. Explain clearly how to work out the answer in one of the squares. Now ask pupils for the number which goes into each square. Pupils may give their answers using individual white boards or by putting hands up. Make the activity more difficult by hiding integers in the coloured cells.

Integer circle sums Select a volunteer to come to the board and drag numbers into the intersections so the the total inside each circle is the same. Members of the class should call out suggestions. Simplify the problem by revealing the circle sum.

Adding and subtracting integers summary To add a positive integer we move forwards up the number line. To add a negative integer we move backwards down the number line. a + –b is the same as a – b. To subtract a positive integer we move backwards down the number line. Stress that adding a negative integer is the same as subtracting and that subtracting a negative integer is the same as adding. Unlike multiplying and dividing integers, when adding and subtracting integers the answer is positive or negative depending on the start number and how much we move up or down the number line. To subtract a negative integer we move forwards up the number line. a – –b is the same as a + b.

N2.3 Using negative numbers in context Contents N2 Negative numbers A N2.1 Ordering integers A N2.2 Adding and subtracting integers A N2.3 Using negative numbers in context A N2.4 Multiplying and dividing integers

Negative numbers in context There are many real life situations which use negative numbers. For example, Balance -£34.52 Bank balances Temperature Measurements taken below sea level Games with negative scores. Discuss each context briefly, asking pupils for examples or supplying your own. Ask pupils if they can think of any other situations in which negative numbers may be used.

Sea level Use this activity to introduce negative integers in the context of sea level. Ask questions based on the diagram and drag the images to illustrate the answers. For example, The crab craws 12m up the sand bank. What is his new position? How many metres does sea gull have to move downwards to catch the fish?

Temperatures Use the slider to point to a variety of temperatures and ask pupils to identify them. Ask pupils to answer questions mentally and illustrate the answers using the slider. The temperature is -1°C. It rises by 5°C. What is the new temperature? The temperature is -13°C. It rises by 17°C. What is the new temperature? The temperature is -4°C. It falls by 7°C. What is the new temperature? The temperature is 12°C. It falls by 20°C. What is the new temperature? The temperature is 6°C. It rises by 10°C and then falls by 18°C. What is the new temperature? The temperature is -3°C. It falls by 6°C and then rises by 13°C. What is the new temperature? Links: S7 Measures - Reading scales

Ordering temperatures Ask pulpits to put the temperatures in order from coldest to hottest. Extend the activity by including decimals. Link: N1 Place value, ordering and rounding – ordering decimals

Comparing temperatures Look at this map of temperatures around Europe on a day in January. The questions below may be answered orally as a group warm up activity or may be set as a written exercise. Use the vertical thermometer at the side to illustrate answers. What is the coldest place on the map? What is the warmest place on the map? Which city is 4°C warmer than Glasgow? Which city is 8°C warmer than Bucharest? Which city is 10°C cooler than Istanbul? Which city is 18°C cooler than Nice? What is the difference in temperature between Seville and Istanbul? What is the difference in temperature between Moscow and Glasgow? What is the difference in temperature between Warsaw and Nice? Write the names of the cities in order from coldest to warmest.

N2.4 Multiplying and dividing integers Contents N2 Negative numbers A N2.1 Ordering integers A N2.2 Adding and subtracting integers A N2.3 Using negative numbers in context A N2.4 Multiplying and dividing integers

Multiplying and dividing integers –3 + –3 + –3 + –3 + –3 = –15 –3 –3 –3 –3 –3 –15 –12 –9 –6 –3 5 × –3 = –15 Ask the class to work out –3 + –3 + –3 + –3 + –3. The answer is –15, as we can see on this number line. How do we usually add together the same number many times? (We multiply) Another way to write –3 + –3 + –3 + –3 + –3 is 5 × –3. So, 5 × –3 = -15. In fact, when we multiply a negative number by a positive number, the result is always a negative number. Reveal the rule on the board. A positive number × a negative number = a negative number

Multiplying and dividing negative numbers –7 × 3 = = 3 × –7 = –21 –7 –7 –7 –21 –14 –7 Now, what is –7 × 3? We know that when we multiply it does not matter what order we put the numbers in so –7 × 3 is exactly the same as 3 × –7. This is called the commutative law. It is not necessary for pupils to know the name of the law at this stage as long as they are familiar with the idea. Again, we can see this illustrated on a number line. The next rule we need to remember is that when we multiply a positive number by a negative number, the result is always a negative number. A negative number × a positive number = a negative number

Multiplying and dividing negative numbers –4 × –6 = 24 – –6 – –6 – –6 – –6 6 12 18 24 Explain that multiplying a negative number by another negative number is like subtracting a negative number. Subtracting a negative number is equivalent to adding the positive value of that number. Multiplying –6 by –4 is equivalent to subtracting –6 four times which is equivalent to adding 6 four times. Multiplying by a negative number has the effect of changing the sign of whatever it is multiplying. We have seen that multiplying a positive number by a negative number, in whatever order, makes the answer negative. Well, multiplying a negative number by another negative number makes the answer positive. –4 × –6 = 24 A negative number × a negative number = a positive number

Ordered multiplication square Start by working out the positive number multiplications 3 × 0, 3 × 1, 3 × 2 and 3 × 3. Ask pupils the answer to 3 × –1, using the number pattern in the table, or using the rules from the previous slide, and fill in the rest of the row to 3 × –3. Repeat in the same order for the next 2 rows. Establish that 0 × any number is 0 and fill in this row accordingly. For the last three rows fill in the first answer by looking at the pattern in the first column and continue along the row. Before revealing each answer ensure that the question corresponding to that cell has been clearly stated. For example, What is 1 × –2? Pupils may then look for number patterns to deduce the answer or use the rules on the previous slide. Once the table is complete, ask pupils to point out any patterns they notice. Shade the negative numbers blue and shade the positive numbers green. Ask pupils to use the table to answer some given questions. For example, Use the table to work out –2 × –1.

Rules for multiplying and dividing When multiplying negative numbers remember: + × = – + × = – + × = – + × = Dividing is the inverse operation to multiplying. When we are dividing negative numbers similar rules apply: These rules have been drawn graphically to make it easier for pupils to spot the pattern. As each rule appear read it as, for example, A positive number multiplied by a positive number always equals a positive number. Remind pupils of the meaning of ‘inverse operation’ – one ‘undoes’ the other. For example, if 4 × –3 = –12, then –12 ÷ –3 must equal 4. Tell pupils that easiest way to remember these rules is that when we multiply together (or divide) two numbers with different signs (a positive number times a negative number or a negative number times a positive number) the answer will always be negative. If we multiply together (or divide) two numbers with a different sign (a positive number times a positive number or a negative number times a negative number) the answer will always be positive. Encourage pupils to first work out whether their answers will be positive or negative and then multiply or divide. Ask pupils to write down rules for multiplying (or dividing) three numbers. For example, negative × positive × negative = positive and negative × negative × negative = negative. + ÷ = – + ÷ = – + ÷ = – + ÷ =

Multiplying and dividing integers Complete the following: –3 × 8 = –24 –36 ÷ = –4 9 42 ÷ = –6 –7 ÷ –90 = –6 540 × –8 = 96 –12 –7 × = 175 –25 For each example ask pupils what sign the missing number will have and then what the number is. Links: Mental methods – multiplication and division Equations – solving equations 47 × = 141 3 –4 × –5 × –8 = –160 –72 ÷ –6 = 12 3 × –8 ÷ = 1.5 –16

Using a calculator We can enter negative numbers into a calculator by using the sign change key: (–) For example: –417 ÷ –0.6 can be entered as: (–) 4 1 7 ÷ . 6 = The answer will be displayed as 695. Ask pupils to locate the sign change key on their calculator. Note that on some non-scientific calculators this key may be shown as +/– and must be pressed after the number is entered to make it negative. Discuss ways that we can check that the answer given by the calculator is correct. First check that the sign is correct. In this example, we are dividing a negative number by another negative number and so the answer must be positive. 417 ÷ 0.6 is approximately equal to 420 ÷ 0.6 = 420 ÷ 6 × 10 = 700. Set pupils a variety of problems involving negative numbers including negative decimals. As an extension include problems involving negative fractions. Links: Written and calculator methods – Using a calculator Always make sure that answers given by a calculator are sensible.

Mixed multiplication square Work out the numbers in the empty squares by multiplying and dividing.

Mixed division square Work out the numbers in the empty squares by multiplying and dividing.

Integer cards – multiplication and division Challenge pupils to find every possible solution for each calculation using the given set of integer cards. Establish that if the answer to the calculation is negative the signs of the numbers we are multiplying or dividing must be different (one positive and one negative). If the answers to the calculations is positive the signs of the numbers we are multiplying or dividing must be the same (both positive or both negative).

Number spiral –2 –10 –8 ×2 ÷ –5 –4 2 –11 –7 + 4 × –1 – 5 –15 3 –16 –2 –16 –2 + 16 Work towards the centre of the spiral to reveal the picture of a snail blinking. × 5 + 8 –3 6 ÷ –2