Math 9 Unit 1 Rational Numbers Outcomes PDF

# Math 9 Unit 1 Rational Numbers Outcomes ## Student Notes ### General Outcome * Develop number sense. ### Specific Outcomes 1. Demonstrate an understanding of rational numbers by: * Comparing and ordering rational numbers * Solving problems that involve arithmetic operations on rationa...

# Math 9 Unit 1 Rational Numbers Outcomes ## Student Notes ### General Outcome * Develop number sense. ### Specific Outcomes 1. Demonstrate an understanding of rational numbers by: * Comparing and ordering rational numbers * Solving problems that involve arithmetic operations on rational numbers. [C, CN, PS, R, T, V] [ICT: P2-3.4] 2. Explain and apply the order of operations, including exponents, with and without technology. [PS, T] [ICT: P2-3.4] # MATH 9 ## Rational Numbers ### 1.1 Integer Review **Integer:** a positive or negative whole number, including zero, and is represented by *I*. **Integers include**: * the Natural number set, *N*, (also known as the counting numbers) * the opposite of the counting numbers * and the number * the set notation is *I* = { __ } The numbers -5 and 5 are called opposites. Other examples of opposite integers are: * __ and __ * as well as __ and __ Mathematically, we refer to opposite numbers as being the __ . **Example 1** Determine the additive inverse of the following: * a) -12 * b) 50 * c) -1001 * d) 5634 **The Integer Number Line** <br> The number line shows a series of consecutive integers from -6 to 6. * **Smallest:** -6 * **Negative Integers:** -5, -4, -3, -2, -1 * **Zero:** 0 * **Positive Integers:** 1, 2, 3, 4, 5, 6 * **Largest:** 6 The arrow at the end of the number line indicates that the numbers go on forever. The text below the number line indicates that the value of the integers are getting larger when moving to the right and smaller when moving to the left. ### Uses of Integers Integers are often in everyday life. Here are some examples: * __ * __ * __ <br> **Example 2** Put these integers in the following order: * a) **Descending order**: -9, -8, 4, -12, 0, 11, -3 * b) **Ascending order**: 4, 4, 3, -15, 7, -1, 5 <br> **Example 3** Compare the following integers and use `<` or `>` to write a true statement. * a) 3 __ 14 * b) 10 __ -25 * c) -7 __ -8 * d) -37 __ -31 <br> **Example 4** Write the integer(s) that is(are): * a) 5 less than 0 * b) 2 less than -4 * c) Between -2 and 2 * d) 3 units to the left of -4 * e) 2 units to the right of -3 * f) Greater than -10 and less than -6 ### Adding Integers While there are various methods used for integer addition, one common tool is the number line. <br> The number line goes from -10 to 10. **Example 5** Calculate: * a) (+3) + (-5) = * b) (-4) + (+6) = * c) (-3) + (-2) = * d) (+2) + (-7) = * e) (-5) + (+2) = * f) (-7) + (-9) = ### Subtracting Integers * To subtract integers, keep in mind the phrase "add the opposite." * Change the subtraction sign to addition, and then change the sign on the following number. * By changing it to an addition statement, the strategies for integer addition can be used. **Example 6** Rewrite each subtraction sentence as an addition sentence, and determine the answer for the following: * a) (+3) - (-5) = * b) (-4) - (+6) = * c) (-3) - (-2) = * d) (+4) - (+2) = * e) (+2) - (-7) = * f) (-5) - (+2) = * g) (-8) - (-1) = * h) (+3) - (+5) = **Example 7** * a) Jimmy starts at the bottom of the stairs at school. First he climbs 6 stairs, then he descends 3 stairs because he forgot his pencil. He then remembers he can borrow one from his teacher, so he climbs 11 more stairs to the top of the staircase. How many stairs total are in the staircase? * b) Mr. Brown has $150 in the bank. He wrote cheques for $80, $40 and $20. He then deposits a cheque for $110. What is his final bank balance? * c) The temperature outside at midnight is 0°C. The temperature rises 9°C by 6 am. It then falls 12°C by 3 pm. Finally it rises 5°C by 7 pm. What is the final temperature at 7 pm? ### Multiplying Integers There are three rules to follow when multiplying integers. 1. The product of two positive integers is always positive. * 3 x 6 = 18 * 5 x 4 = 20 * 8 x 7 = 56 2. The product of two negative integers is always positive. * (-3) x (-6) = 18 * (-5) x (-4) = 20 * (-8) x (-7) = 56 3. The product of a positive integer and a negative integer is always negative. * 3 x (-6) = -18 * 5 x (-4) = -20 * 8 x (-7) = -56 When multiplying by three or more integers, there are two rules to get the correct sign on the answer. 1. If there is an even number of negative signs, the product will be positive. * (-2) x 5 x (-1) x (-4) x 3 x (-2) = 240 ... four negative signs → positive product 2. If there is an odd number of negative signs, the product will be negative. * (-2) x 5 x (-4) x 3 x (-2) = -240 ... three negative signs → negative product You can write the product of integers without the use of the x sign. * 5 x (-4) can also be written as (5)(-4), (+5)(-4), or 5(-4). **Example 8** Calculate the following: * a) (+3)(-5) = * b) (-4)(+6) = * c) (-3)(-6) = * d) (-4)(-7) = * e) (+2)(-7) = * f) (-5)(+2) = **Example 9** Predict the sign of each product: * a) (-4) (+7) (-8) (-3) (+6) (-1) = __ or + * b) (-4) (+7) (-8) (+3) (+6) (-1) = __ or + **Example 10** Using the integers 0, -3, -1, 1, -2, 4 ... * a) Which two integers produce the greatest product? * b) Which two integers produce the lowest product? **Example 11** A hot air balloon rose from the ground at 5 m/s for 10 seconds. It then descended at 2 m/s for 6 seconds. How high off the ground is the balloon? ### Dividing Integers To divide integers, use the same sign rules that apply to multiplication. * The quotient of two integers with the same sign is positive. * 15 ÷ 5 = 3 * (-15) ÷ (-5) = 3 * The quotient of two integers with the different signs is negative. * 15 ÷ (-5) = -3 * (-15) ÷ 5 = -3 **Example 12** Find each quotient: * a) 24 ÷ (-6) = * b) (-18) ÷ (-9) = * c) (-30) ÷ 5 = * d) 35 ÷ (-7) = * e) (-36) ÷ (-4) = * f) (48) ÷ 6 = **Example 13** Complete each division statement: * a) 25 ÷ __ = 5 * b) __ ÷ (-9) = 10 * c) (-63) ÷ __ = -7 * d) __ ÷ (-4) = -11 When dividing three or more integers, there are two rules to get the correct sign on the answer. 1. If there is an even number of negative signs, the quotient will be positive. * (-240) ÷ (-2) ÷ 3 ÷ (-5) ÷ (-2) = 4 ... four negative signs → positive quotient 2. If there is an odd number of negative signs, the quotient will be negative. * (-240) ÷ (-2) ÷ 3 ÷ (-5) ÷ 2 = -4 ... three negative signs → negative quotient **Example 14** Determine if the quotient is positive or negative. * a) (-150) ÷ (-5) × 3 × (-2) = __ * b) (-1200) ÷ (-5) ÷ (-3) × (-2) = __ **Example 15** Use two of the five integers and write a division fact with each quotient. * -2 * 3 *12 * -1 * 4 * a) A quotient of -2 * b) The greatest quotient * c) The least quotient * d) A quotient between -5 and -10 **Example 16** Karisa made withdrawals of $14 from her bank account. She withdrew a total of $98. How many withdrawals did she make? ### 1.2 Fractions: Part I **Fraction:** a part of a whole **Slice a pizza and you will have fractions.** * The top number (**numerator**) tells you how many slices you have. * The bottom number (**denominator**) tells you how many pieces the pizza was cut into. <br> The image shows three pizzas, each divided into different numbers of slices. * **Pizza 1:** 1/2 (One-Half) * **Pizza 2:** 1/4 (One-Quarter) * **Pizza 3:** 3/8 (Three-Eighths) <br> **Equivalent Fractions:** fractions that have equal value but are expressed in different forms. If a pizza was cut into 2 pieces and you ate one piece, you would have eaten the same amount if the pizza was: * Cut into 4 pieces and you and 2 pieces (2/4) * Cut into 8 pieces and you and 4 pieces (4/8) <br> **Equivalent fractions are multiples of each other** ``` 1/2 x 2/2 = 2/4 2/4 x 2/2 = 4/8 ``` <br> **To calculate equivalent fractions, multiply the numerator and denominator by the same value.** ``` 1/2 x 2/2 = 2/4 ``` <br> **Example 1** Determine three equivalent fractions for the following. ``` 2/5 x 2/2 = 4/10 2/5 x 3/3 = 6/15 2/5 x 4/4 = 8/20 ``` **Example 2** Find the missing value which will make each pair of fractions equal. * a) 1/3 = __ / 12 * b) 3/4 = __ / 18 * c) 5/5 = 36 / __ * d) 7/8 = 28/ __ <br> **Reduced Fraction:** a fraction expressed in its simplest form. **Method 1:** Divide the numerator and denominator of the fraction until you can't go any further. **Example 3** Reduce the following: ``` 24 ÷ 12 = 2 108 ÷ 12 = 9 ``` Therefore: 24/108 = 2/9 <br> **Method 2:** Divide both the numerator and denominator by the Greatest Common Factor. **Example 4** Reduce the following. * a) 20/36 * b) 54/66 * c) 30/75 * d) 72/96 ### Converting Mixed Fractions to Improper Fractions using the MAD Method 1. **Multiply** ... the denominator by the whole number 2. **Add** ... the product of step 1 to the numerator 3. **Divide** ... the sum of step 2 by the denominator <br> **Example 5** Convert the following mixed fractions to improper fractions. * 1) 6 1/2 = * 2) 9 3/4 = * 3) 9 5/7 = * 4) 3 1/10 = * 5) 7 3/5 = * 6) 2 2/3 = * 7) 9 3/8 = * 8) 5 2/9 = ### Converting Improper Fractions to Mixed Fractions 1. Divide the numerator by the denominator. 2. Use the remainder as the numerator of the fractional part of the mixed number. 3. The denominator stays the same. 4. Reduce the fraction to lowest terms, if possible. **Example 6** Convert the following improper fractions to mixed fractions. * 1) 18/5 = * 2) 17/6 = * 3) 22/4 = * 4) 57/8 = * 5) 27/4 = * 6) 53/9 = * 7) 37/10 = * 8) 32/6 = ### Adding and Subtracting Fractions In order to add or subtract fractions, the fractions must have (or be converted to) a common denominator. **Final answers must be written in simplest form (reduced) and may be converted to a mixed fraction, if possible.** **Example 7** Determine the sum/difference in simplest form: * a) 2/5 + 1/2 = * b) 2/3 + 3/4 = * c) 5/6 + 5/8 = * d) 3/8 - 1/4 = * e) 4/5 - 1/4 = * f) 7/9 - 5/6 = ### Multiplying Fractions 1. Reduce the fractions if possible by finding the GCF. 2. Multiply the numerators (top parts) together. 3. Multiply the denominators (bottom parts) together. 4. Reduce the product to lowest terms if necessary. **Example 8** Determine the product in simplest form. * a) 2/4 x 18/22 = * b) 6/25 x 27/18 = * c) 8/18 x 24/26 = * d) 12/24 x 12/45 = ### Dividing Fractions 1. Change the division sign to multiplication. 2. Write the reciprocal of the divisor (flip the second fraction). 3. Proceed as if it were a multiplication statement. **Example 9** Determine the product in simplest form. * a) 3/4 ÷ 6/5 = * b) 3/8 ÷ 2/5 = * c) 7/8 ÷ 1/4 = * d) 4/5 ÷ 1/10 = * e) 2 ÷ 4/3 = * f) 6 ÷ 3/5 = ### 1.3 Fractions: Part II **Example 1** Determine the sum/difference in simplest form. * a) 3 1/3 + 1 1/2 = * b) 1 1/2 + 2 1/6 = * c) 3 2/5 + 2 1/10 = * d) 2 1/2 - 1 2/9 = * e) 2 1/5 - 1 7/8 = * f) 4 1/3 - 2 1/5 = * g) 5 1/6 + 2 2/3 - 3 1/2 = * h) 2 1/3 - 1 2/4 + 1 1/5 = **Example 2** Determine the product/quotient in simplest form. * a) 2 2/9 x 1 1/6 x 1 1/2 = * b) 4 3/4 x 2 1/5 x 1 1/2 = * c) 3 2/3 x 5 1/4 + 1 1/10 = * d) 1 3/5 ÷ (1 1/2 + 1 1/2) = **Example 3** 3 Determine the simplest form. * (5 2/5 - 3 2/5) x 3 2/4 = ### 1.4 Converting Fractions and Decimals **Example 1** In each part of the circle, write its decimal value. The circles are divided into several equal parts with each part labeled with a fraction. **Example 2** Convert each fraction to its decimal equivalent, use a repeating bar where necessary. * a) 1/2 = * b) 1/3 = * c) 2/3 = * d) 1/4 = * e) 2/4 = * f) 1/4 = * g) 1/5 = * h) 2/5 = * i) 3/5 = * j) 4/5 = * k) 1/8 = * l) 2/8 = * m) 3/8 = * n) 4/8 = * o) 5/8 = * p) 6/8 = * q) 7/8 = * r) 1/10 = * s) 2/10 = * t) 3/10 = * u) 4/10 = * v) 5/10 = * w) 6/10 = * x) 7/10 = * y) 8/10 = * z) 9/10 = **Example 3** Convert each decimal to its fraction equivalent(s), using denominators of 2, 3, 4, 5, 8, and 10. * a) 0.1 = * b) 0.125 = * c) 0.2 = * d) 0.25 = * e) 0.3 = * f) 0.3 = * g) 0.375 = * h) 0.4 = * i) 0.5 = * j) 0.6 = * k) 0.625 = * l) 0.6 = * m) 0.7 = * n) 0.75 = * o) 0.8 = * p) 0.875 = * q) 0.9 = ### 1.5 Comparing and Ordering Rational Numbers **Example 1** Which fraction is greater ... 2/3 or 4/7 ? * **Method 1:** * 2/3 = __/7 * **Method 2:** * 2/3 = __ / __ * 4/7 = __ / __ **Example 2** Which fraction is greater ... 3/4 or 2/4 ? * **Method 1:** * **Method 2:** **Example 3** Which fraction is greater ... 5/6 or 7/8 ? * **Method 1:** * **Method 2:** * 5/6 = __ / __ * 7/8 = __ / __ **Example 4** Compare each pair of fractions using <, >, or =. * a) 2/9 __ 1/12 * b) 5/6 __ 2/9 * c) 1/2 __ 2/5 * d) 2/3 __ 3/5 * e) 2/5 __ 3/4 * f) -1/3 __ 5/6 * g) 2/3 __ 11/5 * h) 1/4 __ -1/5 * i) 5/7 __ 16/5 **Example 5** Which is greater ... 7/10 or 0.65? * 7/10 = __ * 0.65 = __ **Example 6** Which is greater ... 2/5 or 3.4? * 2/5 = __ * 3.4 = __ **Example 7** Which is greater ... -5 3/8 or -5.4? * -5 3/8 = __ * -5.4 = __ **Example 8** Which is greater ... 9/√25 or 0.6? * 9/√25 = __ * 0.6 = __ **Example 9** Compare the expressions using <, >, or =. * a) 0.6 __ 2/3 * b) 49/16 __ 1.75 * c) 0.875 __ 4/5 * d) 81/4 __ - 13/8 * e) -4.7 __ - 3 2/5 * f) 1/64 __ 3/300 **Example 10** Write each rational number by the letter representing it on the number line. * a) 3/2 * b) -0.7 * c) 1/4 * d) -1 * e) √121/25 <br> The number line shows numbers from -3 to +3. * A = -3 * B = -2 * C = -1 * D = 0 * E = 1 * F = 2 * G = 3 **Example 11** Compare 1 1/8, -1 2/3, -0.1, 1.9, 36/16, and -1 ... and arrange in ascending order. ### 1.6 Solving Rational Equations and Applications **Example 1** Solve the following, express as a fraction in simplest form. * a) 5/3 + (-2.5+5) - (1/2 ÷ 1/5) × 1.25 = * b) 11/15 + 0.2 x (-1.3) = * c) (1/4 + 0.5)² + (0.125 * 8) = **Example 2** Solve the following. * a) The river fishing season in Bodo is 210 days long. When the season is 3/5 over, how many days are left? * b) Leon has $45 and spends 2/3 of it on a jacket. If he spends 2/5 of the money he has left at the movies, what amount does he have remaining after buying the jacket and going to the movies? * c) Joe owns 240 shares of stock valued at $5 per share. If the price of a share dropped by $1.50 per share, determine the ending value of his investment. * d) Halley's comet travels around the Sun approximately once every 78 years. The first record of its appearance was in 467 BC. How many times will the comet have appeared between 467 BC and 2000 AD? * e) In Tracey's recipe for apple fritters, she uses three-quarters of a cup of milk to make 24 fritters. If she wants to make 6 fritters, how much milk will she need? * f) If it takes a construction crew one hour to repair 3/5 km of highway, how long will it take for them to repair 2 1/3 km of highway? * g) It was a very hot day at the office, so the staff was very thirsty. The volume in the water cooler decreased by an average of 0.45 L per hour. If the tank had 3.6 L water at the beginning of the day, after how many hours would the tank be empty? * h) Advertising posters for a school play will be drawn on poster board. A large piece of poster board measure 3.0 m by 1.2 m, and the size of each poster will be 60 cm by 80 cm. What is the greatest number of posters that can be cut from one piece of poster board?