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lakehead adult education centre 125 south lillie street thunder bay on p7e 2a3 math unit prior learning assessment and recognition plar gwl3o textbook lessons 1 ­ 5 student success nothing less revised january 26 2010 1

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math textbook prior learning assessment and recognition plar lesson 1 integers fractions 2

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operations with integers there are people who suggest that the world would be a better place without negative numbers there would be no debts no below zero temperatures and no need to learn the rules for integers unfortunately few of us would be able to afford to make major purchases like cars or houses without going into debt if there were no temperatures below zero we could not ski skate or make snowpeople and without learning to deal with integers we could not handle these real life situations involving negative numbers the numbers on a thermometer are part of the set of integers this set includes positive numbers such as 1 2 3 and so on it also includes zero and negative numbers such as 1 2 3 and so on you will notice that we have placed brackets around each integer these brackets are not essential but help to avoid confusion in some questions 3

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p. 4

adding integers when adding integers it is usually helpful to think of positive integers as money you have and negative integers as money you owe or debts adding two positive integers together is just like adding whole numbers except that we show the signs in front of the integers for example 3 34 8 17 11 51 when adding two negative numbers together just think of adding two debts the result being an even larger debt to get the answer add the two numbers together ignoring the sign and then put the sign in front of the answer to show that it is negative a debt for example 6 63 14 71 20 134 adding is a bit more difficult when one number is positive and the other is negative remembering to think of positives as money you have and negatives as money you owe or debts makes it easier 4

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p. 5

for example 7 12 since the amount of money you owe 12 is larger than the amount of money you have 7 you will still end up owing money but not as much as the original debt to find the amount you still owed find the difference and then decide on the appropriate sign 12 thus 7 7 12 5 -5 the debt is larger here remember to add two integers when one is positive and the other is negative follow these steps 1 subtract to get the number part of the answer 2 use the sign of the larger integer for the answer here examples 1 15 are a few more examples of how to add two integers work through each one to make sure that you understand what to do 42 57 5

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2 32 18 14 3 55 7 62 4 26 12 14 5 38 19 19 the key on your calculator do not to use it for adding and subtracting on most calculators you can use this key to do questions involving integers to remind yourself of how this key works try pressing 8 and then the several times did you notice how the 8 keeps switching back and forth from 8 meaning 8 to 8 this is the only way we can enter a negative number into a calculator the regular -key can only be used for subtraction it cannot be used to create a negative integer 6

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thus to enter a negative integer we must enter the number first and then press the key for example to get 6 you would press 6 to work out example 3 on the calculator which is 55 7 62 on the previous page we would enter to work out example 5 on the calculator which is 38 19 19 on the previous page we would enter this is already positive there is no need to use try each of the examples in practice 1 for lesson 1 in your student workbook to make sure that you can answer these questions both with and without a calculator then you will be ready to look at how to subtract integers 7

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subtracting integers there are several different ways to show the rule for subtraction in the long run you just have to accept the rule learn it and apply it in this lesson we will try to show you that the rule for subtraction is reasonable and then we will focus on applying it the problems in subtraction occur when you are trying to subtract a negative number we will show you how to figure out answers to subtraction questions by looking at a pattern in a series of questions in these questions the first number will always be 10 while the second number will start at 5 and then decrease to 4 3 and so on down to zero take a minute and go to practice exercise 2 for lesson 1 decide what the answer to each question should be before you continue 8

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10 ­ ­ ­ ­ ­ ­ 5 5 10 10 10 10 10 4 3 2 1 0 6 7 8 9 10 look closely at the pattern of the answers they keep getting one number larger each time it seems that the next answer would have to be 11 if we follow this pattern we can then construct the question 10 ­ 1 11 notice that 11 is also the answer to 10 1 this leads us to the important rule for subtraction to subtract integers add the opposite which means the integer that has the opposite sign for example 5 is the opposite of 5 using this rule if we want to subtract 7 ­ 9 we first change the question to 7 9 and then add as before getting 2 for the answer 9

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the table below shows subtraction questions and the addition questions that result when the rule for subtraction is followed subtraction question 7 21 23 5 38 18 63 120 ­ 12 ­ 6 ­ 7 ­ 18 ­ 17 ­ 9 ­ 15 ­ 6 addition rule applied 7 12 answer 5 15 16 23 55 27 48 126 21 6 23 7 5 18 38 17 18 9 63 15 120 6 you will notice that from the third example down we have not bothered to show the brackets on the answers we have also left off the sign when it is this is a common practice which you may follow if you wish the general guidelines are that you must show the sign if it is negative and you must include the brackets if leaving them out would be confusing for example 6 ­ +3 is not good form when you use your calculator you don t have to remember to add the opposite your calculator will remember that for you to do the question 5 ­ 9 you must enter try each of the examples in practice 3 for lesson 1 in your student workbook to make sure that you can answer these questions both with and without a calculator then you will be ready to look at how to multiply integers 10

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multiplying integers usually the results from multiplying integers are exactly what you would expect 3 x 6 18 3 x 2 6 5 x 3 15 however sometimes confusion exists when multiplying two negative numbers neither nor seems right for the answer as before we will investigate the answer using a pattern go to practice 4 of your student workbook and answer the questions use the pattern you see from your answers to predict how this series of questions will continue before you go on now look at the answers to these questions below and see how the pattern has continued do your answers in practice 4 match these is this the pattern you predicted if not look at your practice exercise again and see if you can figure out the pattern 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 6 5 4 3 2 1 0 1 2 3 24 20 16 12 8 4 0 4 8 12 11

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p. 12

did you notice that the product of two negative numbers seems to be a positive number this leads us to the two rules for the multiplication of integers 1 if the two integers have the same sign both positive or both negative then their product is positive 2 if the two integers have different signs then their product is negative again your calculator already knows these rules just remember to use the key when you enter a negative number for example to multiply 8 x 13 enter multiplying questions can be written in two ways one way uses the x the other way uses brackets 10 x 4 40 10 4 40 rule when multiplying the answer is positive if both signs are the same 11 5 55 11 5 55 rule in this case both 11 and 5 are positive answer is positive since both signs are negative the answer is positive when you multiply the answer is negative if the signs are opposite 11 5 55 11 5 55 practice 5 for lesson 1 in your student workbook 12

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dividing integers the division of integers is a two-step process 1 divide the numbers without worrying about the signs to find the number part of the answer 2 use the rules for multiplication to decide on the correct sign thus 45 9 5 and 36 4 9 these questions can be done on your calculator as follows dividing can also be indicated in two ways one way is to use 10 5 10 5 2 the other way is to write the question as a fraction rule 2 when dividing the answer is positive if both signs are the same 12 3 4 12 4 3 rule when dividing the answer is negative if the signs are opposite 12 4 -3 12 3 -4 go to practice 6 of your student workbook 13

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fractions the number on the top of a fraction is called the numerator the number on the bottom is called the denominator a proper fraction is one where the numerator is smaller than the denominator 5 eg 7 an improper fraction is one where the numerator is larger than the denominator eg 10 3 2 5 a mixed number is one which contains both a whole number and a proper fraction eg 3 to change an improper fraction to a mixed number divide the numerator by the denominator write how many times it goes in as a whole number with the remainder over the denominator as the fraction part example 10 into a mixed number 3 divide 10 by 3 it goes in 3 times with a remainder of 1 10 1 3 therefore 3 3 to change to change a mixed number to an improper fraction multiply the whole number by the denominator and add it to the numerator write the improper fraction with the answer over the original denominator example change 6 3 into an improper fraction 5 multiply the whole number 6 by the denominator 5 the answer is 30 add 30 to the numerator 3 the answer is 33 3 33 6 therefore 5 5 go to practice 7 of your student workbook 14

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p. 15

reducing fractions to lowest terms a fraction is in lowest terms if there is no whole number other than 1 that will divide evenly into both the denominator and numerator examples 3 is in lowest terms because only 1 will divide evenly into both 3 and 4 4 6 is not in lowest terms because 2 will divide evenly into both 6 and 8 8 6 8 62 82 3 4 to change a fraction into lowest terms divide both the numerator and denominator by the largest whole number that will go into both of them evenly example 24 36 24 12 36 12 2 3 although the numerator 24 and denominator 36 can be divided by 2 and 4 and 6 use the largest whole number to reduce this fraction to the lowest terms go to practice 8 of your student workbook 15

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