Directed Numbers Directed Numbers Curriculum Ready ACMNA: 124, 134, 183, 186, 280 www.mathletics.com Numbers come in all sizes and forms. They can be positive or negative, whole numbers, fractions or decimals and rational or irrational. Before you start, investigate these terms and write a brief description about them in the boxes. Positive and negative numbers: Whole numbers, fractions and decimals: Rational and irrational numbers: is Give th Q a go! Calculate the value of: (-2) 3 + {- 2 + [2 # (2 + 2)]} Work through the book for a great way to solve this Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 1 How does it work? Your Turn Directed Numbers Types of numbers Natural numbers are all the positive counting numbers "1, 2, 3, ... , . Integers are all the positive (+) and negative (-) whole numbers (including zero). Circle all the numbers below that belong to the family known as integers. (psst!: some of them need to be simplified by calculating their value first) 1 2 - 10 5 41 2 0 4 5 - 2 - 9 1.19284750219... 4 5 - 5.0 13 -1 4 7 0.42 -16 9.86 - 8 3 36 8 -1 3 -0.3o 10 3 30 Place each of these numbers into their correct group below to classify them. Irrational numbers can’t be written as a fraction Rational numbers can be written as a fraction Integers: the positive and negative whole rational numbers including 0. Put the special types of rational numbers into their correct group -1 Natural numbers: the counting rational numbers (not including 0). 2 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning How does it work? Directed Numbers Measuring instruments and integers There are many applications for directed numbers. Thermometers are a great example because the temperature can go below zero. The temperature at midday was 20 degrees Celsius. During the afternoon, the temperature rose 4 degrees before falling 9 degrees in the late afternoon. (i) Shade in the thermometer showing the late afternoon temperature. Starting at 20cC , temperature changed + 4cC then - 9cC 25o 24o 23o 22o 21o 20o 19o 18o 17o 16o 15 14o 13o 12o 11o 10o 9o 8o 7o 6o 5o 4o 3o 2o 1o 0o Temperature at midday 20cC Temperature by late afternoon 15cC rose =+ and falling =- + 4cC - 9cC Centigrade o C (ii) What directed number gives the overall change in temperature between the two readings? Overall change =+4cC - 9cC =-5cC Digital scales need to use directed numbers to calculate changes in mass. Use this information to calculate the mass of olives purchased by Maria at her local Deli: Maria asked for 500 g of olives. After first weighing 575 g of olives, the Deli owner took out 98 g and then put back 46 g. The directed number sentence for this is: + 575 g - 98 g + 46 g Increase = + and decrease = - ` Maria purchased + 575 g - 98 g + 46 g = 523 g of olives If the first term is positive, we usually don't write in the plus sign. Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 3 How does it work? Your Turn Directed Numbers Measuring instruments and integers 1 Shade in each thermometer below to accurately display these temperatures: (i) 11cC (ii) -5cC (iv) 0cC (v) 8cC above zero (vi) 2cC below zero 20o 20o 20o 20o 20o 20o 15o 15o 15o 15o 15o 15o 10o 10o 10o 10o 10o 10o 5o 5o 5o 5o 5o 5o 0o 0o 0o 0o 0o 0o -5o -5o -5o -5o -5o -5o -10o -10o -10o -10o -10o -10o Centigrade o C 2 (iii) 6cC Centigrade o C Centigrade o C Centigrade o C Centigrade o C This thermometer shows a reading of 12cC at the start. Write the new readings that would be displayed after each of these changes in temperature: 25o 20o 15o (ii) Becomes 8cC cooler cC (iii) Decreases by 13cC cC (iv) Warms up by 9cC cC (v) Rose by 22cC cC (vi) Dropped 28cC cC (vii) Decrease by 7cC then increase by 12cC cC (viii)Warms by 3cC and then a further 9cC cC (ix) Cools 15cC before warming by 19cC cC (x) Changes to 0cC before dropping 4cC cC 5o -5o o -10 Centigrade o C H 4 SERIES TOPIC cC cC 10o 0o 12 (i) Increases by 3cC 30o 4 Centigrade o C Directed Numbers Mathletics Passport © 3P Learning Your Turn Directed Numbers 3 The mass of a glass container with some water in it is shown below: INSTRUM TS AND EN Measuring instruments and integers G MEASURIN How does it work? ..../...../20... TEGERS * IN 0402 grams Write down the new reading on the scales after any one of the following changes occurred. (i) 320 g of water poured into the glass grams (ii) 212 g of water was spilled out of the glass grams (iii) 88 g of flavour syrup was stirred in and then 250 g was drunk grams (iv) A similar glass with the same amount of water was placed on the scales next to this one. grams (v) A cup filled with 216 g of water was emptied into the glass and during the day, 7 g evaporated away because the room was hot. grams Try this tricky one! 4 A glass container holding cold water below was placed on the digital scales after being moved from a 24cC room into a 32cC room. The glass had collected 3 grams of water (through a process called condensation) on the outside for every 2 degree rise in room temperature. What was the original mass of the water filled glass if 24 grams of water was spilt during the move? 60o 55o 40o 30o 32cC 20o 10o 0o Centigrade o C 0583 grams Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 5 How does it work? Directed Numbers Magnitude and direction of integers Integers are positive (+) and negative (-) whole numbers, including zeros. They tell us the size (magnitude) and direction of values associated with change. -1 It is important to first decide what will be the positive direction. +3 • If up ( ) is positive, then the opposite direction down ( ) is negative • If right ( ) is positive, then the opposite direction left ( ) is negative • If hotter is positive (+), then the opposite colder is negative (-) , ER ST , NEGATIVE , CE DU E, COOLER, S EAS OU CR TH DE , BE LO RE INCREASE, W AR ER, ME OV R, R, INK, REVERSE SHR , R, LESS, SLOW ER OSE , CL FURTHER, M E, OR OV E, AB FA POSITIVE , ST WE W, D, G EAST, ROW, F OR H, T WA R R NO CEND, LOWE R , DES UN DE R RA, ASCEND , EXT HI GH E These are some other words usually associated with positive and negative directed numbers Only negative numbers must have the sign written in front. Write down the directed number for each of these statements: (i) 5 degrees below the zero: =-5c The magnitude of the temperature is 5 degrees (ii) 30 degrees above the zero: =+30c 30c = The magnitude of the temperature is 30 degrees The positive sign (+) can be hidden (iii) $12 extra money: The magnitude of the money is $12 =+$12 = $12 The directed number of an amount left after some changes represents the overall change. One day Matt started with no money and was given $40. He spent $14 of it downloading music. What is his overall change of money on this day? The change in money for Matt during this day was: $0 + $40 - $14 = $26 left ` the overall change in money for Matt on this day is +$26 The final directed value = overall change 6 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Your Turn Directed Numbers Magnitude and direction of integers For each of these statements, write down: (i) the directed number that matches it and, (ii) the magnitude of the directed number. a ..../...../20... EGERS * NT 1 The tree in Jamie’s yard grew 3 metres in one year. (i) Directed number: AND DIRE ION OF I CT E NITUD MAG How does it work? (ii) Magnitude of the tree’s growth: The same tree was then trimmed which reduced the height of the tree by 4 metres. (iii) Directed number: b (iv) Magnitude of the reduction in height: Aki walked 250 m West from her starting point. (i) Directed number: (ii) Magnitude of the distance Aki walked: Aki then turned around and walked another 600 m East (iii) Directed number: c (iv) Magnitude of this distance walked by Aki: Sean’s bank account balance earned interest and increased by $4. (i) Directed number: (ii) Magnitude of Sean's bank balance increase: Sean’s bank account balance then changed due to being charged $3 in bank fees. (iii) Directed number: 2 (iv) Magnitude of the change in Sean's bank balance: Think carefully for each of these statements and write down the directed number that matches the overall change. a Pip was given $24 and then spent $15 on a dress the same day. Directed number for the overall change in money that day: b +$9 The temperature initially increased by 6 degrees and then increased by a further 4 degrees. Directed number for the overall change in temperature: c Nigel went down 6 rungs on a ladder and then up 2 rungs. Directed number for the overall change in ladder rungs: d Cameron's coolness rose by 4 points when he played guitar, and a further 20 points when he did maths. Directed number for the overall change in Cameron's coolness after doing both activities: e Shiromee hiked 12 km North, 2 km East and then 19 km South. North is the positive direction. Directed number for the overall change in Shiromee's North-South movement: f Adele's hair grew from 20 cm to 30 cm long at the back. After a hair cut it was only 17 cm long. Directed number for the overall change in Adele's hair length from the initial 20 cm: Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 7 How does it work? Directed Numbers Ascending and descending order When comparing directed numbers, it often helps to arrange them into numerical order. De sce ng ndi e Asc Ascending order = lowest to highest ndi ng Descending order = highest to lowest Don’t be tricked by the magnitude of a number. Which of these two values is higher? 4 2cC - 10cC 2cC - 10cC represents a very cold temperature (below/less than 0cC ) 2cC is a cold temperature, however it is (above/greater than 0cC ) 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 ` 2cC is a higher value than - 10cC -8 -10cC -9 -10 -11 -12 -10 is a bigger number in magnitude than 2, however the negative sign makes it lower in value Centigrade o C Let’s look at an ascending and descending order example. For the numbers -4 , 0 , 14.5 , -6 , 5.2 , 3 , -10 , 5.4 : (i) Arrange them into ascending order -10 , -6 , -4 , 0 , 3 , 5.2 , 5.4 , 14.5 Lowest negative Highest positive (ii) Arrange them into descending order 14.5 , 5.4 , 5.2 , 3 , 0 , -4 , -6 , -10 Highest positive 8 H 4 SERIES TOPIC Lowest negative Directed Numbers Mathletics Passport © 3P Learning Order is opposite to ascending How does it work? Your Turn Directed Numbers 2 3 Circle the word that represents the order of the values in these statements: a Shortest to Tallest Ascending Descending b Longest to shortest Ascending Descending c Closest to farthest Ascending Descending d Warmest to Coolest Ascending Descending e Heaviest to lightest Ascending Descending f Thinnest to widest Ascending Descending ..../...../20. .. G ORDER * IN 1 ASCEN Ascending and descending order END SC NG AND D DI E Arrange the following groups of numbers into ascending order (lowest to highest). a 14 , 0 , 17 , 3 , , , b 21 , 25 , 20 , 19 , 22 , , , , c -10 , -8 , -12 , -4 , -16 , , , , d 2.2 , 2.4 , -2.6 , 0 , -2 , , , , e -1 , 11 , 1 , -1.5 , 1 , 1 5 4 , , , , , Arrange the following groups of numbers into descending order (highest to lowest). a 2 , 16 , 5 , 8 , , , b 40 , 31 , 32 , 38 , 29 , , , , c -19 , -11 , -16 , -18 , -13 , , , , d 1.6 , 1.9 , -1.3 , -1.0 , 1.2 , , , , e 0 , - 1 , -2 1 , -2.1 , 1.2 , -30 4 2 , , , , Directed Numbers Mathletics Passport © 3P Learning , H 4 SERIES TOPIC 9 How does it work? Your Turn Directed Numbers Ascending and descending order Combo Time! 4 Elevator Riding A department store elevator operator started work on the 3rd floor of the 15 storey building. For the first 15 minutes of work, the operator travelled to the following floors in the order written: 12 th floor 8 th floor 13 14 15 Ground floor 10 11 12 2nd floor 7 8 9 4 5 6 15 th floor 1 2 3 4 th floor Ground floor 10 th floor G >|< >|< 5 th floor a If up is positive and down is negative, write numbers that represent the movement of the elevator operator during the first 15 minutes of work. , b , , , , , , , , , Arrange the directed movements into descending order. , c , , , , , During the first 15 minutes, did the operator mostly ascend or descend in the elevator? Explain you answer. 20... ../ ..../... ME * TI C 10 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning * COMB O BO TIME OM COMBO TI ME * How does it work? Directed Numbers The number line Directed numbers can be plotted on a number line to instantly see their order of value. Values decrease (descending) Values increase (ascending) -15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112 1314 15 Lower numbers are further left and higher numbers are further right. Display the numbers -4, -1, 7 on a number line. -9 -8 -7 -6 -5 -4 -3 -2 -1012345 67 8 9 Use the plotted points to compare the values of: (i) -4, -1: -4 is further left than -1, ` -4 1 -1 (ii) -1, 7: -1 is further left than 7, Remember: 1 means 'less than' and 2 means 'greater than' ` -1 1 7 This example requests numbers to be plotted using a given rule. Display all the even positive integers between -5 and 13 on a number line. Even and positive -6 -5 -4 -3 -2 -1012345678910 11 12 13 Write these numbers in descending order. Read the numbers from right to left ( ) ` The numbers in descending order are: 12 , 10 , 8 , 6 , 4 , 2 Half values are plotted by placing the dot half-way between the integers on either side. Display the numbers 1, -2, 4 1 , -5.5, 9, 0 on a number line. 2 -9 -8 -7 -6 -5 -4 -3 -2 -1012345 67 8 9 4 1 is half way between 4 and 5 -5.5 is half way between -5 and -6 2 Write these numbers in ascending order. Read the numbers from left to right ( ) ` The numbers in ascending order are: -5.5 , -2 , 0 , 1 , 4 1 , 9 2 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 11 How does it work? Your Turn Directed Numbers 2 Insert the correct symbol 1 (less than) or 2 (greater than) for each of these. a 6 d -11 g -1 2 j - 1.8 ..../...../20... LINE * ER 1 THE NUMB The number line THE NUMB LINE * ER b 3 -12 c -5 -2 e 8 -8 f -7 1 2 -7 0 h 91 2 i -5.1 -5.8 k -12.5 l 11 1 2 11 1 3 4 1.2 9.9 -21.5 4 List the numbers displayed below in ascending order a -15-14-13-12-11-10-9-8 -7 -6 -5-4 -3 -2 -1012345678910 11 121314 15 , , , , b -9 -8 -7 -6 -5 -4 -3 -2 , 3 -10123456789 , , , List the numbers displayed below in descending order a -18-17-16-15-14-13-12-11-10 -9-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112 , , , , b -9 -8 -7 -6 -5 -4 -3 , 12 H 4 SERIES TOPIC -2 -10123456789 , , Directed Numbers Mathletics Passport © 3P Learning , How does it work? Your Turn Directed Numbers The number line 4 Display the following sets of numbers on a number line. a -5, 8, -1, 3, 6. -9 -8 b -5 -4 -3 -2 -10123456789 -7 -6 -5 -4 -3 -2 -10123456789 -7 -6 -5 -4 -3 -2 -10123456789 -3 -2 -10123456789 - 7, 7.5, 3 1 , 0, 2 1 , 2. 2 2 -9 -8 5 -6 -4 1 , 4, 6 1 , - 1 , 5. 2 2 2 -9 -8 d -7 -8.5, 1.5, -2, 7.5, 5. -9 -8 c 1 = 0.5 2 -7 -6 -5 -4 Display the following sets of numbers on a number line. a The first six odd integers above -8. -9 -8 b -6 -5 -4 -3 -2 -10123456789 All the negative even integers less than 7 but greater than negative 8. -9 -8 c -7 -7 -6 -5 -4 -3 -2 -10123456789 All the multiples of 3 higher than 7 but lower than 18. 0 1 2 3 4 5 6 7 8 9 10 11 1213 14 15 16 17 18 d All the integers that are 3 or 7 whole numbers away from -2. -9 -8 -7 -6 -5 -4 -3 -2 -10123456789 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 13 How does it work? Directed Numbers Addition and subtraction using a number line Starting from zero each time, calculations involving directed numbers can be made using a number line. Negative direction Positive direction Remember Always start from 0 and move using each directed number one at a time. Use a number line to calculate these: (i) 8+4 +4 -15-14-13-12-11-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 101112 131415 Start Starting at +8, 4 to the right finishes at 12 right (+12) Finish ` 8 + 4 = 12 (ii) -6 - 7 -7 -18-17-16-15-14-13-12-11-10 -9-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112 Finish Start Starting at -6, then 7 to the left finishes at 13 left (-13) ` -6 - 7 =-13 (iii) 3 + (-2) (iv) (-2) + 3 Negative numbers are often written inside parentheses -2 -3 -2 -101 2 3 Finish +3 -3 Start Starting at +3 then 2 to the left finishes at 1 right (+1) -2 -101 2 3 Start Finish Starting at -2, 3 to the right finishes at 1 right (+1) ` 3 + (-2) = 1 (v) 10 - 26+4 Same as (iii) but in different order ` (-2) + 3 = 1 -26 +4 -18-17-16-15-14-13-12-11-10 -9-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1112 Finish Starting at +10, then 26 to the left followed by 4 to the right finishes at 12 left (-12) ` 10 - 26 + 4 =-12 14 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Start How does it work? Your Turn Directed Numbers Addition and subtraction using a number line 1 Show the calculation for each of these on a number line and write down the answer. a 3+5 = -9 -8 b -4 -3 -2 -10123456789 -7 -6 -5 -4 -3 -2 -10123456789 -7 -6 -5 -4 -3 -2 -10123456789 -7 -6 -5 -4 -3 -2 -10123456789 7 - 15 = -9 -8 e -5 (-3) - 1 = -9 -8 d -6 (-4) + 8 = -9 -8 c -7 14 - 6 + 3 = 012345678910 11 12 13 14 1516 1718 2 Write down the calculation to get the given answer shown on each of these number lines: a =-12 -15 -14 -13 -12 -11 -10 -9 b -8 -7 -6 -5 -4 -3 -2 -1 01 2 3 -3 -2 -10123456789 -3 -2 -10123456789 =-3 -9 -8 -7 -6 c -5 -4 =4 -9 -8 -7 -6 -5 -4 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 15 How does it work? Your Turn Directed Numbers Addition and subtraction using a number line 3 Show the calculation for each of these on a number line and state the answer. a 1.5 + 2 1 = 2 -9 -8 b -7 -6 -4 -3 -2 -10123456789 Start Finish -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 56 -6 -5 -4 -3 -2 -10123456789 -6 -5 -4 -3 -2 -10123456789 -4 + 10.5 = -9 -8 d -5 (-2.5) - 4.5 = -12 -11 -10 -9 c +2 1 2 4 -7 8 1 - 5.5 = 2 -9 -8 -7 Write down the calculation shown on each of these number lines: a ..../.. * =2 -8 -7 -6 -5 b -3 -2 -10 12 34 .. =-3.5 -9 -8 -7 -6 -5 c -4 -3 -2 -10123456789 -4 -3 -2 -10123456789 -4 -3 -2 -10123456789 = -9 -8 -7 -6 -5 d = -9 16 -4 .../20. MBER LIN NU E -9 D SUBTRA AN C USING ON A TI 4 ADDITIO N Try these trickier ones! -8 H 4 SERIES TOPIC -7 -6 -5 Directed Numbers Mathletics Passport © 3P Learning Where does it work? Directed Numbers The mysterious Dr Thermos! Dr Thermos controls the temperature of his laboratory using large hot or cold spheres. + 1cC Hot Sphere - 1cC Cold Sphere Count the number of each sphere type to find the overall temperature of his laboratory. Calculate the temperature of the laboratory for these sphere combinations: (i) 7 hot spheres and 3 cold spheres ` + 7cC and - 3cC ` laboratory is + 4cC (ii) +7 and -3 combine to make +4 2 hot spheres and 3 cold spheres ` + 2cC and - 3cC ` laboratory is - 1cC +2 and -3 combine to make -1 To change the temperature, Dr Thermos adds or removes spheres. Show two ways Dr Thermos can raise the temperature by + 2cC for the combination of spheres below. 9 hot spheres and 6 cold spheres ` + 9cC and - 6cC ` laboratory is currently + 3cC +9 and -6 combine to make +3 (i) Dr Thermos could add two hot spheres 9 hot spheres +2 hot spheres and 6 cold spheres ` + 11cC and - 6cC ` laboratory is now + 5cC ` a change of + 2cC +11 and -6 combine to make +5 (ii) Dr Thermos could remove two cold spheres 9 hot spheres and 6 cold spheres -2 cold spheres ` + 9cC and - 4cC ` laboratory is now + 5cC ` a change of + 2cC Directed Numbers Mathletics Passport © 3P Learning +9 and -4 combine to make +5 H 4 SERIES TOPIC 17 Your Turn Directed Numbers 1 ..../.. Write down the temperature of Dr Thermos’ laboratory for these sphere combinations: a .../20. .. b Laboratory temperature: cC c Laboratory temperature: cC Laboratory temperature: cC d Laboratory temperature: 2 ! THE MYST E - 1cC Cold Sphere OS + 1cC Hot Sphere cC Write the number and type of hot or cold spheres that would need to be added to the laboratory to cause these changes in temperature. a + 3cC b - 9cC d + 20cC 3 hot spheres c 3 18 E ! TH MYST E The mysterious Dr Thermos! OS S DR.THER OU M I R S DR.THE OU RM RI Where does it work? - 6cC Write the number and type of hot or cold spheres that would need to be removed from the laboratory to cause these changes in temperature. a - 7cC b + 2cC c - 5cC d + 8cC H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Where does it work? Your Turn Directed Numbers The mysterious Dr Thermos! 4 For each of these combinations of hot and cold spheres, explain two different ways Dr Thermos can change the temperature of the laboratory to equal the amount given in the square brackets. a [+4cC] (i) First way: (ii) Second way: b [-2cC] (i) First way: (ii) Second way: Try these trickier ones! 5 If Dr Thermos could only use the spare spheres shown below each given combination, explain two different ways he could change the temperature of the laboratory to equal the amount in the square brackets. You must use all of the spare spheres for at least one of the methods a [+6cC] (i) First way: (ii) Second way: Spare spheres: b Using all the spare spheres [-1cC] (i) First way: (ii) Second way: Spare spheres: Using all the spare spheres Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 19 Where does it work? Your Turn Directed Numbers The mysterious Dr Thermos! Puzzle time! 6 Dr Thermos has a puzzle room at the entrance to his laboratory to keep his experiments secret. The door will open when the display in the puzzle room reads -2cC . The display currently reads + 3cC . • Each hot (+1cC ) or cold (-1cC ) step can be used only once. • Only sideways ( ) or forward ( ) steps can be taken. • Steps in the white section are switched on (added to the puzzle room). • Steps in the grey section are switched off (removed from the puzzle room). (i) Draw a pathway that will allow you to enter the laboratory of Dr Thermos. (ii) See if you can find another path! Finish anywhere along here Start anywhere along here 20 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Where does it work? Directed Numbers Adding and subtracting directed numbers The exercises in Dr Thermos’ laboratory were examples of adding and subtracting directed numbers. Here is what happened: Addition subtraction =+1c C =-1c C When a was added, the temperature went up When a was added, the temperature went down When a was removed (Subtracted), the temperature went down When a was removed (Subtracted), the temperature went up The rules for adding and subtracting represent the same thing. Use the pattern above to calculate these: = +1 and Addition Subtraction = -1 5+ = 5 + (+1) = 5 + 1 = 6 + + = + (Same signs together means plus) 5+ = 5 + (-1) = 5 - 1 = 4 + - = - (Opposite signs together means minus) 5- = 5 - (+1) = 5 - 1 = 4 - + = - (Opposite signs together means minus) 5- = 5 - (-1) = 5 + 1 = 6 - - = + (Same signs together means plus) These rules work for all kinds of directed additions and subtractions Calculate these additions and subtractions: (i) 3 + (- 2) = 3 - 2 =1 + -=- (ii) (-2) + 3 =-2 + 3 Same as (i) but in a different order =1 (iii) 6 - (- 2.5) = 6 + 2.5 = 8.5 - -=+ (iv) (-11) - (-4) =-11 + 4 =- 7 - -=+ (v) 2 1 + (- 3) = 2 1 - 3 + - = 2 2 =- 1 2 (vi) (-12.2) + (-5.2) =-12.2 - 5.2 + - = =-17.4 (vii) 9.5 - (- 8.5) + 6 = 9.5 + 8.5 + 6 = 24 (viii) 4 1 + ^-1h - `-6 2 j = 4 1 - 1 + 6 2 3 3 3 3 = 10 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 21 Directed Numbers ADDING A Adding and subtracting directed numbers 1 2 22 a -5 + 3 b -4 + 7 c -4 - 4 d -11 - 8 e 1 - 16 f 16 - 22 g 6 + (-3) h 30 + (-17) i 0 + (-8) j -4 + (-5) k -10 + (-19) l 13 - (-5) m 20 - (-21) n -18 - (-6) o -15 - (-26) a 4-8+2 b -9 + 6 - 8 c -3 + 4 - 8 - 7 d - 7 + (- 4) + 13 e 2 - (-6) - 7 f 5 + (-1) - (-14) H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning ..../..... * Calculate these additions and subtractions of integers without a calculator. SUBTRAC /20... DIREC NG T TI ND NUMBERS Your Turn ED Where does it work? Where does it work? Your Turn Directed Numbers Adding and subtracting directed numbers Simplify these additions and subtractions: Use the number line on the side if it helps. -12 -11 -10 -9 3 5.5 - 8 b -3 + 2.5 c -2 - 7.5 d -6 1 - 19 2 e 9.5 - 16 f 7.5 + (-18) g 0.5 + (-5.5) h -12 1 - (-3 1 ) 2 2 i 0 - (+8) - (-6) a -2 - 7 + 3.5 b 1 + 0.5 - 6 c 2 + (-3.5) + 1.5 d - 7 1 + ^-9h - `-2 1 j 2 2 e -8.5 - `-6 1 j + 18 2 f -4 + ^-4h - `-4 1 j - 4 1 4 4 -8 a -7 -6 -5 -4 -3 -2 Directed Numbers Mathletics Passport © 3P Learning -1 0 1 2 3 4 5 6 7 8 9 1011 12 4 H 4 SERIES TOPIC 23 Where does it work? Your Turn Directed Numbers Adding and subtracting directed numbers For trickier questions, the calculator can be used. Use the ( – ) button to change between positive and negative when entering numbers into the calculator. Sometimes the button may look like Simplify using the rules -29.3 - (-42.9) + (-19.6) =-29.3 + 42.9 - 19.6 ( – ) 2 . 9 3 + 4 . 2 9 – 1 9 . 6 = Enter into calculator =-6 Use your calculator (only if you need to) for these: 5 24 a 53 - 89 b -43 + 94 c 34 + (-51) d - 25 + (-94) e - 94 - (-28) f -16 + (-87) - (-41) g -2.6 + 10.4 h -0.5 - 6.7 i -10.3 + (-5.6) j 66 - 34.6 + (-24.4) k 1.06 + 4.5 - 9.7 l -0.23 + 1.76 m (- 71.23) + (-52.38) n -1.204 - (-5.093) o 98.23 - (-13.8) -112.3 p -29 1 + 51 3 q -30 1 - `-66 1 j 3 2 r 100 + `-54 1 j - 46 1 2 4 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Where does it work? Directed Numbers Multiplying and dividing directed numbers The rules for multiplying and dividing are similar to adding and subtracting: Multiplication Division Positive # Positive # Negative # Negative # Positive = Negative = Positive = Negative = Positive Negative Negative Positive + + - # # # # +=+ -=+=-=+ (Same signs mean answer is positive) (Opposite signs mean answer is negative) (Opposite signs mean answer is negative) (Same signs mean answer is positive) The exact same rules apply because division is the opposite operation to multiplication! These rules always apply to all multiplication and division calculations. Calculate these multiplications and divisions: -4 # 3 =-12 - # +=- (ii) -14 ' 2 =- 7 -'+=- (iii) 12 # (-1.5) =-18 + # -=- (iv) 24 ' (-4.8) =-5 +'-=- (v) -8 # (-3) = 24 - # -=+ (vi) -46.8 ' (-18) = 2.6 -'-=+ (vii) -3 1 2 - # +=- (viii) -32 ' `-3 1 j = 10 5 -'-=+ (i) #6 =-21 Be careful squaring (or cubing) directed numbers as parentheses affect the question. 2 (x) -3 =-(3 # 3) =-9 (ix) (-3) 2 = (-3) # (-3) = 9 If more than two terms are multiplied or divided, simplify by calculating in order from left to right. Simplify these mixed questions: (i) 6 # (-3) ' 9 (ii) (-2) ' 0.5 # (-3) (iii) (-1) 3 (iv) 8 4 # 5 ' (-2) ' (-11) 5 =-18 ' 9 =-2 Calculate 6 # (-3) first =-4 # (-3) = 12 Calculate (-2) ' 0.5 first + # -=- -'+=- - = (-1) # (-1) # (-1) = 1 # (-1) =-1 = 44 ' (-2) ' (-11) =-22 ' (-11) =2 -=+ Calculate (-1) # (-1) first -'-=+ Calculate 1 # (-1) next +'-=- Calculate 8 4 # 5 first 5 Calculate 44 ' (-2) next + # +=+ +'-=- -'-=+ Directed Numbers Mathletics Passport # -'+=- © 3P Learning H 4 SERIES TOPIC 25 Where does it work? Your Turn Directed Numbers Multiplying and dividing directed numbers 1 2 26 Simplify these without using a calculator: a 3 # (- 7) b -6 # 8 c -36 ' 12 d 48 ' (-4) e -16 # (-1) f -2 # (-2) g -12 ' (-12) h 21 ' (-21) i 3 # (-4) ' 6 Simplify these without using a calculator: a -24 ' 3 ' (-8) b -18 ' (-6) # 4 c -5 # 2 # (-1) # 2 d -5 2 e (-4) 2 f (-3) 3 g -23 h (-2) 2 ' 4 i (-3) 2 # -3 j 63 ' (-2) k 24 '-42 l 92 '-32 # (3) H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Directed Numbers TI MUL PLYI 3 4 Simplify these using a calculator: a 27 # (-4) b -21 # 13 c d -95 ' (-19) e 125 # (-49) f g -53 # 16 h (-391) ' (-23) i -4 # 29 ..../...../20... * Multiplying and dividing directed numbers AND DIVI DIREC NG TE DI NG NUMBERS Your Turn D Where does it work? -162 ' 18 -15 # 15 ' (-3) Use your calculator to simplify these: a -25 # 13 ' (-65) b -500 ' (-5) ' 25 c -552 ' (-23) # (-8) d 21 2 e 12.5 # 25 ' (-0.25) f -1 2 ' (-14) # 5 # (-2) 5 g -4 # 3.6 # 22 ' 7.92 h -3404.7 ' (-3.6) ' 1 1 ' 9.7 2 i 260 ' (-26) # 2 3 ' 0.26 5 j (- 7) 3 k -210 l (-8) 3 o -(-1.5) 5 # (- 6) ' 5 -1024 - 343 m (-9) 4 n -(-6) 3 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 27 What else can you do? Directed Numbers Combining the basic operations Questions that mix multiplying/dividing with adding/subtracting need to be done in a certain order. Multiplication or division operations are ranked the highest and therefore must be completed first. Calculate these combined operations questions: (i) -2 + 3 # 5 -2 + 3 # 5 =-2 + 15 Do this first = 13 Add -2 and 15 35 ' (-5) - 1 =- 7 - 1 Do this first =-8 Subtract 1 from -7 (ii) 35 ' (-5) - 1 If more than one multiply/divide sign, work left to right. Do the same for addition/subtraction. Calculate these combined operations questions: (i) -28 ' 7 # 3 + 6 -28 ' 7 # 3 + 6 =-4 # 3 + 6 Do this first =-12 + 6 =-6 Multiply -12 by 3 Add -12 and 6 (ii) 17 - 4 -(-10) # 2 17 - 4 -(-10) # 2 = 17 - 4 -(-20) Do this first = 13 - (-20) = 33 Subtract 4 from 17 Subtract -20 from 13 ( = 13 + 20 ) (iii) 24 ' 3 + (-5) # 4 24 ' 3 + (-5) # 4 = 8 + (-5) # 4 Do this first = 8 + (-20) =-12 Multiply -5 by 4 Add 8 and -20 (iv) 6 # (-3) - 210 ' 30 # (-2) 6 # (-3) - 210 ' 30 # (-2) =-18 - 210 ' 30 # (-2) Divide -210 by 30 Do this first =-18 - 7 # (-2) Multiply -7 by -2 =-18 + 14 Add -18 and 14 =-4 28 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Your Turn Directed Numbers 2 3 Simplify these combined operations without a calculator: a - 4 + 16 ' 8 b 6 # (-8) -12 c 50 ' (-25) + 10 d -8 - 5 # 4 e 5 + 7 # (-6) f -11 # -2 + 18 g 22 - 39 ' (-3) h -10 - 4 # 15 ..../...../20... * 1 SIC OP BA Combining the basic operations COMBIN G THE IN ATIONS ER What else can you do? Simplify these combined operations without a calculator: a 4 # (-2) + 18 - 11 b -3 # 6 ' 9 + 3 c - 9 + 48 ' (-12) + 13 d 32 ' 4 - 3 # (-4) Use a calculator to simplify these combined operations: a 28 ' (-4) + 13 # 7 b 120 ' (-15) # 24 - 8 c - 25 # 6 - 224 ' (-32) d 30 # (-15) - 35 ' 4 # (-16) Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 29 What else can you do? Directed Numbers Order of operations For more complex calculations involving parentheses and indices, we use a similar convention: P Parentheses I Indices Do these first D M Division A Multiplication S Addition Then do these Subtraction Do these last Simplify these directed number calculations using the correct order of operations: (i) (12 - 8) # 7 Subtract 8 from 12 inside the parentheses (12 - 8) # 7 = 4 # 7 = 28 Multiply 4 and 7 (ii) -21 ' (2 + 5) + 3 -21 ' (2 + 5) + 3 =-21 ' 7 + 3 =-3 + 3 =0 Add 2 and 5 inside the parentheses Divide -21 by 7 Add -3 and 3 (iii) (-4) 2 ' (8 - 6) (-4) 2 ' (8 - 6) = (-4) 2 ' 2 = 16 ' 2 =8 Subtract 6 from 8 inside the parentheses Square -4 Divide 16 by 2 (iv) 15 # (-5 + 3) 3 - 20 15 # (-5 + 3) 3 - 20 = 15 # (-2) 3 - 20 = 15 # (-8) - 20 Add -3 and 5 inside the parentheses Cube -2 Multiply 15 by -8 =-120 - 20 =-140 Subtract 20 from -120 (v) 29 - (13 - 4 # 5) + 32 29 - (13 - 4 # 5) + 32 = 29 - (13 - 20) + 32 = 29 - (- 7) + 32 = 29 - (- 7) + 9 = 36 + 9 = 45 30 H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning Multiply -4 by 5 inside the parentheses Subtract 20 from 13 inside the parentheses Square 3 Subtract -7 from 29 Add 36 and 9 What else can you do? Your Turn Directed Numbers Order of operations 1 2 Simplify these directed number calculations using the correct order of operations without a calculator: a (16 - 10) # 4 b 34 ' (6 + (- 8)) c -5 # (27 ' 9) d (-13 - (-19)) # 3 e -5 # (14 - 9) - 5 f 4 # (36 ' 2) - (-4) g 18 + (16 ' 4 + 10) h (18 ' 2 - (-4)) - 7 Simplify these calculations containing indices using the correct order of operations without a calculator: a (-2) 2 ' (-2) b (- 7 - 5) # (-1) 3 c (8 - 14) 2 - 16 d (33 - 43) ' 8 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 31 Your Turn Directed Numbers TI ERA ONS P O ..../...../20... a 26 ' (14 + 2 # 6) + (-1) 2 b -100 # (12 - 32 ' (-4)) ' 40 c (11 # 23 - 10) ' 39 + 8 d 9 # ((-8) - 8 ' 4 + 16) - 43 e 200 ' (-4) # (-3) ' (16 - 14 # (-1) 3) H 4 SERIES TOPIC * TI ERA ONS OP 32 Try simplify these trickier ones using the correct order of operations: ER OF ORD 3 * Order of operations ER OF ORD What else can you do? Directed Numbers Mathletics Passport © 3P Learning What else can you do? Directed Numbers Order of operations with grouping symbols For questions containing multiple pairs of grouping symbols, complete each pair separately. Grouping symbol names: ( ) = parentheses [ ] = square brackets { } = braces Simplify these combined operations questions containing multiple brackets: (i) (10 # 2) ' (- 7 - 3) Simplify the parentheses (10 # 2) ' (- 7 - 3) = 20 ' (-10) =-2 Divide 20 by -10 10 # 2 = (10 # 2) ' (- 7 - 3) -7 - 3 The same question can be written as a fraction. (ii) numerator = (numerator) ' (denominator) denominator 34 - 3 # 4 22 ' (-2) 34 - 3 # 4 = (34 - 3 # 4) ' (22 ' (-2)) 22 ' (-2) = 22 ' (-11) =-2 Re-write with parentheses Simplify the parentheses Divide 22 by -11 When questions contain brackets within other brackets, work from the inside to the outside. Simplify these combined operations questions containing multiple brackets: (i) 3 # 6(1 - 6) ' 5 @ 3 # 6(1 - 6) ' 5 @ = 3 # 6- 5 ' 5 @ = 3 # -1 =-3 Simplify the parentheses first Simplify the square brackets Multiply 3 by -1 (ii) 30 - "18 - 626 ' (2 2 - 17) @, 30 - "18 - 626 ' (2 2 - 17) @, = 30 - "18 + 626 ' (4 - 17) @, Simplify the parentheses = 30 - "18 + 626 ' (-13) @, = 30 - "18 + - 2 , Simplify the square brackets = 30 - 16 Simplify the braces = 14 Subtract 16 from 30 Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 33 Directed Numbers Simplify these questions: a (16 + 9) + (14 - 7) b (1 - 9) - (3 - 8) c (7 - 11) # (4 + 2) d (18 - 3) ' (-12 - (- 7)) e 46 - 19 5-2 f -2 # 10 48 ' 12 h 8 # 5 - 12 42 ' (-6) = (46 - 19) ' (5 - 2) = 27 ' 3 =9 g 2 34 (50 + 54 ' 9) ' (-24 ' 3) Simplify these questions with multiple grouping symbols: a -12 ' 6(7 + 11) ' 9 @ b 610 + (-14 ' 2) @ # 8 c 6-6 # (13 - (- 7)) @ ' 4 d 5 # 6(22 - 9) # 3 @ H 4 SERIES TOPIC Directed Numbers Mathletics Passport © 3P Learning UP GRO ING 1 H Order of operations with grouping symbols NS WI TIO T A R OF OP ER E D R Your Turn 20... / ..../..... LS * MBO O SY What else can you do? What else can you do? Your Turn Directed Numbers Order of operations with grouping symbols a {[12 + (6 # 8)] ' 3} -25 b (-2) 3 + {- 2 + [2 # (2 + 2)]} e? er m Rememb Give these three tricky ones a go to earn an awesome stamp. (14 - 8) # (-1) 15 - (-3) * AWE 3 ME * SO a 2 E* OM 4 Simplify these: /20... ..../..... * AWES 3 b -4 # {- 72 + (-6) 2 ' [(4 # (-5)) - (-2)]} c -3 # {[(-3 - (-3)) 2 + (-3) 2] ' 3 # (-3)} Directed Numbers Mathletics Passport © 3P Learning H 4 SERIES TOPIC 35 Cheat Sheet Directed Numbers Here is a summary of the important things to remember for directed numbers Types of numbers • Positive and negative numbers are numbers that have both size and direction. Eg: +2 or -3 • Rational numbers can be written as a fraction. They include whole numbers and terminating or recurring decimals. Eg: 8, 2.3, 4.5o , 1 , 4 . 2 • Irrational numbers cannot be written as a fraction. These include decimals that keep going without following a pattern, and roots of numbers that are not perfect squares/cubes etc. Eg: 3 , 1.2938164928..., r. • Integers are whole numbers only. They are rational and have no decimal or fraction parts. Eg: 3, -4, 100. Magnitude and direction of numbers • Directed numbers are positive (+) or negative (-) whole numbers. • They show both the size (magnitude) and direction of values associated with change. • It is important to first decide what will be the positive direction. Ascending and descending order Ascending order = lowest to highest Descending order = highest to lowest Adding and subtracting directed numbers Addition Subtraction + + - + = + - = - + = - - = + (Same signs together means plus) (Opposite signs together means minus) (Opposite signs together means minus) (Same signs together means plus) Multiplying and dividing direct numbers Positive # Positive # Negative # Negative # Multiplication Division Positive = Negative = Positive = Negative = Positive Negative Negative Positive + + - # # # # +=+ -=+=-=+ (Same signs mean answer is positive) (Opposite signs mean answer is negative) (Opposite signs mean answer is negative) (Same signs mean answer is positive) The exact same rules apply because division is the opposite operation to multiplication! Order of operations The word PIDMAS helps to remember the order in which we need to perform calculations. P Parentheses Indices Do these first 36 H 4 SERIES TOPIC I D Division M Multiplication Then do these Directed Numbers Mathletics Passport © 3P Learning A S Addition Subtraction Do these last Directed Numbers www.mathletics.com

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