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1 EVALUATING ACADEMIC READINESS FOR APPRENTICESHIP TRAINING For ACCESS TO APPRENTICESHIP MATHEMATICS SKILL OPERATIONS WITH INTEGERS AN ACADEMIC SKILLS MANUAL for The Precision Machining And Tooling Trades This trade group includes the following trades: General Machinist, Tool & Die Maker, Mould Maker, Pattern Maker, and Machine-Tool Builder Integrator Workplace Support Services Branch Ontario Ministry of Training, Colleges and Universities 2011

2 In preparing these Academic Skills Manuals we have used passages, diagrams and questions similar to those an apprentice might find in a text, guide or trade manual. This trade related material is not intended to instruct you in your trade. It is used only to demonstrate how understanding an academic skill will help you find and use the information you need.

3 MATHEMATICS SKILL OPERATIONS WITH INTEGERS An academic skill required for the study of the Precision Machining and Tooling Trades INTRODUCTION Integers include all of the positive whole numbers and all of the negative opposites of the positive whole numbers. The number +5 (read positive five) has a negative opposite 5 (read negative five). Every positive whole number has a negative opposite. On a cold day, the weatherman says it is minus 20 o C, or 20 o. The negative sign indicates that the number is opposite in value to Integers consist of all the positive whole numbers, their negative opposites, and zero. Operations with integers are most often used when doing algebra and using certain gauges. Working with integers provides a different way of thinking about subtraction. Understanding negative numbers on the number line is useful if you have to read a measuring tape both forwards and backwards. This skills manual covers the following operations with integers: Integers on a number line Rational numbers Operations with positive and negative numbers, including addition of integers subtraction of integers removing brackets multiplication of integers and rational numbers division of integers and rational numbers Simplifying equations with integers INTEGERS ON A NUMBER LINE All whole positive and negative numbers, including zero, make up the set of numbers called integers. Integers can be arranged in order as points on a number line. The line extends from zero in both a positive and a negative direction. The points on the line to the right of zero are positive and the points to the left of zero are negative. Mathematics Skills for The Precision Machining & Tooling Trades Operations with Integers Page 1

4 Here is what the number line looks like: Although only the integers from 4 to +4 are shown here, the arrows indicate that the numbers extend endlessly in both directions. The positive sign is not usually written in front of a positive integer. It is assumed that if no sign is shown in front of a number then the number is positive. If the + sign is written in front of an integer, it indicates that the number following the sign is to be added when you simplify the expression. The negative sign is written in front of a negative number every time. Understanding a number line is useful in reading certain gauges. On these gauges there is a zero in the centre of the numbering. When the dial is pointed to the right of the zero, the numbers are positive. When the dial is pointed to the left, the numbers are negative. RATIONAL NUMBERS All integers (whole numbers) fractions and decimal numbers, are called rational numbers. 1. Between each integer on the number line is an infinite number of fractional numbers. \$ For example, the fractions along with countless other fractions and decimals, can all be found as points on a number line between any two integers. 2. Any fraction can also be written as a decimal and so any decimal number has a corresponding point on the number line. \$ For example, the decimal number 4.5 (which can also be expressed as the mixed number 4 2 ) can be found on the number line half way between 4 and There is an endless number of fractions so we only show a few if we are drawing a number line that includes fractional or decimal numbers. Operations With Integers Page 2

8 The rules for subtracting integers can be used to find the distance between two points on the number line. The number to the left on the number line is subtracted from the number to the right to find the distance between them To find the distance between two integers or two rational numbers on the number line, you could start from the point that is furthest to the left and count the number of spaces to the second point To find the distance between and 2 on the number line, we subtract the number on the left, which is, from the number on the right, which is 2. The problem is written like this: We change the sign of the number to be subtracted, which is 3, to its opposite, which is 3. The minus sign changes to an addition sign. The problem is now written: (We use addition rule # 1.) = 5 This gives us the same answer as when we counted the number of points between the two integers. Using the rules of subtraction to find the distance between two points on the number line is helpful when the points indicate decimal numbers. It is difficult to count the distance between the points and 6.7, so we use the rules for subtracting positive and negative numbers. We subtract the number furthest to the left, which is -5.8 from the number on the right, 6.7. To add decimals, write them so the decimal points fall under each other Write the question like this: 6.7 (5.8) Follow the rules for subtracting. The number to be subtracted, which is - 5.8, changes to its opposite, which is 5.8 and the minus sign changes to an addition sign The distance from to 6.7 is 12.5 units Operations With Integers Page 6

10 Example: 4 (+5) the and signs become a sign = 4 5 = 9 Example: = -6 2 = 8 Example: Simplify: the and signs become a sign = First remove the brackets following the three rules that determine signs. = Next simplify using the rules for addition of integers. = 11 Multiplication of Integers and Rational Numbers There are several ways to indicate multiplication when working with rational numbers. Multiplication is indicated in the following ways: 6 x 5 6 (5) 6 (5) Multiplication of integers is usually indicated by the third example. The two numbers to be multiplied are written next to each other, with the second number enclosed in brackets. If there are operations to be done inside the brackets, these are done first, before the answers are multiplied. If there are two sets of numbers within brackets to be multiplied, then the sign is usually shown to clearly indicate the calculations to be done. In the complex example the sign is used to indicate multiplication. Multiplication is done in the usual way but the sign of the answer must be considered. Operations With Integers Page 8

11 To determine the sign of the answer to a multiplication question follow these rules: 1. When two positive numbers are multiplied, the answer is positive. 2. When two negative numbers are multiplied, the answer is positive. 3. When a positive and a negative number are multiplied, the answer is negative. To Multiply Integers Example: When two positive integers are multiplied, the answer is positive. 9 (8) = 72 Example: When two negative numbers are multiplied, the answer is positive. 4 (6) = 24 Example: When a positive number and a negative number are multiplied, the answer is negative. If the first number is negative, it is often also enclosed in brackets: (7) (5) = 35 Example: When a positive number and a negative number are multiplied, the answer is negative. 12 (2) = 24 The rules for multiplying integers also apply to all rational numbers. Example: 1.5 (3.3) = 4.95 Division of Integers and Rational Numbers Division of integers is usually indicated algebraically in an equation by writing the number as a fraction, with the number you are dividing into (the dividend) written on top and the number you are dividing by (the divisor) written on the bottom. So 9 is written: in an algebraic expression. In a complicated formula, the division sign, like the x sign, might still used. Operations With Integers Page 9

13 Determine the correct sign for each answer, depending on the rules for that operation. You also determine the resulting sign of the number when any brackets are removed. Example: Simplify = = 5 Example: Simplify = = 27 Example: Simplify = 5 x (4 80) = 5(76) = 380 Operations With Integers Page 11

15 Simplify the following equations. You will need to use all the different rules given on operations with integers. Answers are on the last page of the skills manual. 1. 5(8 +4) + (36 9) (2) (2) + (8)(7 + 5) (6) (-5) 4. (6)(5) [8 + (3)] + 4(7 1) Operations With Integers Page 13

16 ANSWER PAGE 1. 5(8 +4) + (36 9) (2) = 5(4) + (4) + 2 = = (2) + (8)( 7 + 5) 12 = 8 + (8)( 2) 12 = = (6) (5) = = (6)(5) [8 + (3)] + 4(7 1) = (11) + 4(6) = = 11 Operations With Integers Page 14

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