Interpreting Rational Numbers

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1 Interpreting Rational Numbers Student book pages 4-8 jg5b Relate rational numbers to fractions and integers. You will need Iculator «« Math j Terms rational number A number that can be "Dressed as the quotient t iwo integers; it can De written in fraction, mixed-number, or decimal -orm, or as an integer. apposites ; Two numbers with opposite signs that are the ame distance from 0; for -: "ample, t-2 and -2 are pposites, and +0.5 and -0.5 are opposites. What rational number could represent position A on the number line above? Express the number in at least 2 forms. What rational number could represent position B on the number line above? Express the number in at least 2 forms. A and C are opposites. Determine the value of C, and label its position on the number line. 1 C = or - and D are opposites. Determine the value of D, and label its position on the number line. 0 = or - 1 j- A, B, C, and D are all rational numbers. Determine the value of a rational number between and -. Locate j and on the number line below. 0 Create a new scale on the number line by making 8 equal sections between 0 and 1. is a number between - and. Label the position of Eon the number line above. 2 Lesson 1.1: Interpreting Rational Numbers jpynqht « Nelson Education Ltd.

2 t as a fraction = n u m b f pfjectipris counted number of equal sections u as a decimal = Locate the opposite of E on the number line and label it F, Then determine its value. F = 0 Rachel looked at the thermometer outside. The temperature was between - 18T and -19"C. What might the temperature have been? A. The opposites of -18 and-19 are. and _..What is a rational number between these opposites? JO 40 B. What is a possible value for the rational number located at PI Explain. C. What other form of this rational number could you use to describe PI D. Place this rational number on the line below Math Terms mixed number., I 'Jin*,i' i. 1 M i. Tt r,,» «,inu!. - ; l i mii it improper traction 1 't ii n <r,,vh'.^>' i.i '! it f i I u< > i.' I JMlP-il: I, r r,11 if! i II i H H' 4 % ' ir/hjk HI Go back to the problem in part A. Locate another possible solution to this problem on the number line. Write your new solution in three different forms. Decimal: Mixed number: Improper fraction: Reflecting How do you know that there are always many rationals between two given rationals? Explain. ( opvnqht <o 2010 Nelson Education Ltd Lesson 1.1: Interpreting Rational Numbers

3 ,77Ty?Tfl (,, it.!' '! u. ' ; " Practising 7. Write each number as the quotient of two rational numbers.!l ' i 1 ii-'! f! I r l! '. a) 5,1 5,1 - r ^ ^ b) -4 i Think of the opposite first. f Therefore, -4 ^ = -. c) -.02 Think of the opposite first..02 = = Therefore, -.02 =. Write each number in decimal form. 2 a) 5 g 5-5 M2 -r 5) 5 = b) - 7 Think of the opposite first. 7 \ = 7 r ( - _ >\->+ Therefore, - / ~ = c) 12 IS! = 19 = I jsson t.l: aiterpretinu National Numbers jpynqnt 2U10 Nelson education Ltd.

4 9. Mark integers from - 10 to 2 on the number line below. Then estimate and mark the location of each rational number on the number line. c a) -~ Think of the opposite as a decimal. = is between and - ^ -, so it is between and b) 1 ~ 1 ~ is between and 1 ~ is closer to c) -7.2 Think of the opposite as a decimal. 7.2 is between and 7.2 is closer to -7.2 is between and but it is closer to 12. a) Name three fractions between and. We need a smaller scale than eighths for the denominator. Write and as equivalent fractions with a denominator of 2. 2 _, n r i = Can you name three fractions between these two equivalent fractions? b) How would your answers in part a) help you name three rational numbers between - and -? c) Are your answers in part a) rational numbers? Explain. topvnnht 2010 Nelson fcducation Ltd Lesson 1.1: Interpretina Rational Numbers 5

5 Name: Date; Chapter 1: Rational Numbers Page 6 Subtracting Integers To subtract integers, you can add the opposite. For example, for 10 - (-5), calculate = 15. You can use a number line to subtract integers by using an arrow that starts at the second integer, and then by extending it to the first integer. The length of the arrow is the difference between the second integer and the first integer. If the arrow points to the right, the difference is a positive integer. If the arrow points to the left, the difference is a negative integer. 16. Calculate. a) 6 - (-2) b) - - (-9) c) -8 - (+7) d) 1 - ( r5) e) -4 - (-10) f) 11 -(-1) Multiplying and Dividing Integers The product or quotient of two integers with the same sign is positive. For example, 5x4-20; -5 x ( --4) = 20; 20 ^ 5 = 4; -20 -r- -5 = 4. The product or quotient of two integers with different signs is negative. For example, - 5 x 4 = -20; 5 x (-4) = -20; 20 -J- -5 = -4; = Calculate. a) 16 - (-4) c) -7 x 4 e) (-10) h- (-2) b) -9 x (-) d) 6 - (-12) f) -5 x 6! 6 I Review of Essential Skills Masters Copyright O 2010 Nelson Education Ltd.

6 Name: Date: Chapter 1: Rational Numbers Pages Dividing Decimals To divide decimals, first multiply the dividend and divisor by the same multiple of 10 so that there are no decimals. Then divide as you would divide whole numbers. For example, to divide by 2.15, first multiply both numbers by 1000 to eliminate the decimals. Then divide: (0.645 X 1000) + (2.15 X 1000) Calculate. a) :- 2.5 b) C) s- 1.7 d) e) f) Adding Integers To add integers with the same sign, add and keep the sign. For example, to add -6 + (-9), add = 15. The addends were negative, so the sum is negative: -6 + (-9) = -15. To add integers with different signs, subtract and keep the sign of the larger. For example, to add 12 + (-8), subtract > 8, so the sum is positive: (-8) = 4. You can use a number line to add integers by representing the first integer with an arrow that starts at 0, and the second integer with an arrow that starts at the end of the first arrow. Positive integers are represented by arrows that point to the right. Negative integers are represented by arrows that point to the left. The sum of the integers is the endpoint of the second arrow. 14. Calculate. a) 5 + (-7) b) - + (-15) + 10 c) (-10) + (-) + 5 Opposites The opposite of an integer is the number that is the same distance from 0 on the number line, but in the opposite direction. For example, the opposite of +4 is Write the opposite of each integer, a) 12 b) -7 c) -15 d) 20 Copyright 2010 Nelson Education Ltd.

7 Name; ^,. Date: 1 Chapter 1: Rational Numbers Page 4 Adding and Subtracting Decimals Add or subtract decimals by adding or subtracting place by place..14 For example, to calculate , add the hundredths, the tenths, and the ones: Regroup as necessary. For example, to calculate , you'll need to regroup S?44 ones as 2 ones and 10 tenths and 1 tenth as 10 hundredths: Calculate. a) 2.15 f 4.2 c) 6.75 f.29 e) b) d) f) 9.14 f } Multiplying Decimals Multiply decimals as you would multiply whole numbers. Calculate partial products and then add to determine the product. For example, 2.11 x.4 First multiply 2.11 by 100 to avoid using two decimals: 2.11 X 100 = X.4 - (211 x ) f (211 X 0.4) Then divide by 100 to reverse the earlier multiplication: = Estimate to verify your answer. For example, 2x = 6. The answer must be close to 6, so is reasonable. 12. Calculate. a) 5.1 x 4.4 c) 8.18 x 2.05 e) 10.7 x 6.12 b) 7.42 x.1 d) 2.79 X 4. f) 9.2 X 1.8 ~~1 I Review of Essential Skills Masters CoQvriaht.'run Moi^n c^,.-,..

8 Name: Date: Chapter 1: Rational Numbers Page Adding and Subtracting Fractions with Unlike Denominators To add or subtract fractions, you should try to create equivalent fractions with a common denominator, so you can add the numerators or subtract one numerator from the other. For example, to calculate 5 +, both fractions can be rewritten as equivalents with a denominator of _ 5_ 4 _12 _5_ 12 5 r " 15 5 " Calculate. Multiplying Fractions Multiply fractions by multiplying numerator by numerator and denominator by denominator. For example, x \ 9. Calculate a) x - ' 4» 5 1 b) - x e - x n 4 f) - x - ' 8 5 Dividing Fractions To divide fractions, you can multiply by the reciprocal of the divisor. For example, i ^ 1 _ 4 _ A 1 ~ 5 You can also rename the fractions with like denominators and divide the numerators. For example, -r- \ = j Calculate Copyright 2010 Nelson Education Ltd. Rt'lftfM Sit

9 Chapter 1: Rational Numbers Page 2 Adding and Subtracting Fractions with Like Denominators Add or subtract fractions with like denominators by adding or subtracting the numerators and keeping the denominator. For example, {Q + JQ - JQ and - = \. 5. Calculate a) H - c) e) H ' 5 5 ' 8 8 ' u % 2 ^ 10 5 ^ 5 2 b) d) f) ; ' 9 9 ' Mixed Numbers Write a mixed number as an improper fraction by multiplying the whole number part by the denominator of the fraction and adding that number to the numerator. For example, 2\ = f because 2x4 = 8 and 1 +8 = 9. Write an improper fraction as a mixed number by dividing the numerator by the denominator to obtain the whole number part, and writing the remainder as a fraction. For example, I = if 5 '5 because 7 divided by 5 is 1 with a remainder of Write the mixed numbers as improper fractions, and then add or subtract. a, li + 2f c>9f-7 e> 6 l + 2f b, l-lf d,4 + 5l f ) 8 7T ~ 1 TT 7. Write the improper fractions as mixed numbers, and then add or subtract x 22 7, a) c) 1 e) ' 8 8 ' 9 9 ' x n 7 10 e ' T" T d ) 6 + T. 5 6 f»4 + 4 Review of Essential Skills Masters Convrinht a?nin woi.^r, CA..,-.*-.

10 Name: Date: Chapter 1: Rational Numbers Page 1 Equivalent Fractions and Decimals To write a fraction as a decimal, you can divide the numerator by the denominator. For example, = 4 -H 5 = Write as a decimal. a) -j b) - c)? It is easy to write a decimal as a fraction. Just write the digits after the decimal as the numerator with the place name as the denominator. For example, = j^. Then you can write the fraction in least terms by dividing the numerator and denominator by the same factor: 125 r 125 _ i Write as a fraction. a) 0.6 b) 0.25 c) 0.05 '' Equivalent fractions represent the same number. To write an equivalent fraction, multiply or divide both numerator and denominator by the same factor. For example, you can write 5 with a denominator of 12. Determine the factor to multiply the original denominator by to get the new denominator: x4 = 12, so = f. So, 5 and ^ are equivalent fractions.. Write each fraction as an equivalent fraction with a denominator of , a) - b) c) - d) ' 2 ' 16 ' 4 ' 24 Comparing Fractions and Decimals To compare fractions, write equivalent fractions with a common denominator and compare the numerators. For example, to compare I with 1, rename I as I and compare the numerators. > 2, so i ^ > a I To compare fractions and decimals, express both numbers as fractions or decimals and then compare. 4. Write <, >, or = to make the number sentence true. a ) D b)in c)faf d)0.d^ Copyright 2010 Nelson Education Ltd. Review ot Essential Skiilt Mn<r*rt I t I

Chapter 4 Section 2 Operations on Decimals

Chapter 4 Section 2 Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers.