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Transcript
Day 1 (Lesson 2.1 Part I) Comparing and Ordering Rational Numbers Lesson Focus: We begin our unit learning about rational numbers and learn to reduce fractions, compare and order fractions, express fractions with a common denominator and convert fractions into decimal form. What are Rational Numbers? They are numbers that can be expressed in the form where a and b are integers and b 0. (They can be written as fractions or decimals.) Examples: 1 3 7 , ,5 ,9,4,0,6.25,0.3 2 4 8 Reducing Fractions: To reduce fractions, find the greatest common factor for the numerator and denominator. 1. 4 10 2. 6 15 3. 24 30 Compare & Order Fractions: We can compare rational numbers by expressing them all as fractions with a common denominator or by expressing them as decimals. a) To compare fractions, express each pair of fractions with the common denominator. To find a common denominator, determine the Lowest Comon Multiple (LCM) of the given denominators. Ex: Which is greater (larger), or ? Step 1: The LCM of 6 and 9 is ________. Step 2: Re-write and as: Step 3: Compare the numerators. Practice: Replace with > or <. 1. 12 8 2 4 2. 84 144 5 12 3. 6 11 5 8 3. List from least to greatest using a common denominator: 1 4 5 1 , , , 3 9 6 2 b) To convert fractions to decimals, divide the numerator by the denominator. (A fraction is a division operation.) Example: 1. means = 0.75 3 5 2. 14 5 c) To convert decimals to fractions, determine the value of denominator by looking at the \ place value of the decimal and put over the division bar. 1. 0.3 = 2. .438 3. 0.024 = Day 2 (Lesson 2.1 Part II) Comparing & Ordering Rational Numbers Review: A. Comparing Fractions A fraction can represent parts of a whole. The shaded part of the diagram shows Compare 3 8 and 2 6 4 8 or 1 2 or 0.5. . Use denominators that are the same. 9 24 8 24 Examples 1. Give the fraction and decimal value for the shaded part of the diagram. , therefore 3 8 2 6 2. What is the opposite of the rational Number? a) a) 3 8 b) 0.2 3 B. Compare the Following Fractions. Which is greater? Replace with > or <. 1. 4 5 7 8 2. 3 4 7 10 C. Arrange from least to greatest (by finding the LCM) 1. 3 2 1 , , 8 7 3 2. 11 3 16 7 , , , 8 2 10 4 New Lesson Focus: Today, we will learn to compare rational numbers using a number line and identify rational numbers between two given rational numbers. A. Match each fraction with a letter on the number line. 4 1 ____ 5 2 5 ____ 4 4 ____ 5 ____ a) Which letter is closest to zero? ____ b) Which fraction is closest to zero? ____ c) Which fraction is smallest? ____ d) Is 5 4 or 4 5 closer to 0? Explain. _____________________________ ___________________________________________________________ B. Match each letter on the number line to one of the following rational numbers. 7 4 – 1 3 ____ –0.3 ____ ____ –2.1 ____ 2 1 5 ____ 0.49 ____ C. Identify the rational number (in decimal and fractional form) between two given rational numbers. 1. 3 4 and 4 5 2. 0.4 and 0.5 D. Find as many integers as you can between 11 7 and ? 3 2 E. Compare and order the following rational numbers. 2.2 , To solve: a) Express your numbers in the same form (decimal or fractional) 5 , 8 9 , 10 0.3 , 9 10 Day 3 (Lesson 2.2) Rational Numbers in Decminal Form Lesson Focus: Today, we will expand our understanding of decimal numbers. We will learn to estimate and calculate decimals and apply operations with rational numbers in decimal form. Why estimate? Estimation can help you work with decimal numbers. For example, you can use estimation to place the decimal point in the correct position in the answer. 16.94 + 3.41 + 81.07 Estimate: 17 + 3 + 80 = 100 Calculation: 101.42 Place the decimal so that the answer is close to 100. 1. Without calculating the answer, place the decimal point in the correct position to make a true statement. a) 149.8 ÷ 0.98 = 15285714 b) 2.7 × 100.9 = 272430 c) 40.6 × 9.61 = 39016600 d) 317 ÷ 99 = 32020202 2. a) Is 349 × 0.9 greater than, less than, or equal to 349? ______________ b) How do you know? ________________________________________ 3. a) You know that 48 ÷ 16 = 3. Without finding the exact answer, tell whether the answer to 48 ÷ 15 is greater than, less than, or equal to 3. __________ b) Explain how you know. ______________________________________ Estimate, then calculate (to the neareast thousandth, if necessary). 1. Adding and Subtracting Rational Numbers in Decimal Form Estimate Calculate a) 0.56 + (–3.14) = __________________ ____________ b) –6.92 + (–8.02) = __________________ ____________ c) 7.82 – 5.37 = __________________ ____________ d) –2.75 – (–4.13) = __________________ ____________ e) 0.594 – (–0.085) = __________________ ____________ 2. Multiplying and Dividing Rational Numbers in Decimal Form Estimate Calculate a) –5.1 × (–9.3)= __________________ ____________ b) –1.68 ÷ (–1.4)= __________________ ____________ c) 35.7 ÷ (–4.2)= __________________ ____________ d) (2.7)(–4.2)= __________________ ____________ e) –8.83 ÷ (–0.33)= __________________ ____________ 3. Calculate: Order of Operations a) –6.2 + (–0.72) ÷ (–1.3 + 0.4) c) –6.2 × (–4.2) – 1.02 ÷ 0.51 b) –2.2 × (–3.2) + (–0.88) × 2.3 Applying Operations with Rational Numbers in Decimal Form For Questions 4 and 5, a) write an expression using rational numbers to represent the problem, then calculate b) write a sentence to answer the problem 4. Camille’s chequing account balance is$135.25. She writes a cheque for the amount of $159.15. What is the balance in her account now? 5. The hottest day in Canada on record was on July 5, 1937, in Midale and Yellowgrass, Saskatchewan, when the temperature peaked at 45 °C. The coldest day in Canada was in Snag, Yukon, at –63 °C. What is the difference in temperature between the hottest day and coldest day in Canada? Day 4 (Lesson 2.3 Part I) Multiplying & Dividing Rational #s. Lesson Focus: Today, we will learn to multiply and divide rational numbers. Recall: Multiplying Integers: Rules: 1. + + = _____ 2. – + = _____ 3. + – = _____ 4. – – = _____ Dividing Integers: Rules: 1) + + = 2) – + = 3) + – = 4) – – = _____ _____ _____ _____ 1 4) 2 1 5) 2 Recall as well: 1) 1 2 2) 1 2 3) 1 2 Muliplying Fractions: Simplying the following expressions. Ensure your answer is in lowest term. When simplifying rationals, it is best to 1. Reduce the fractions first before multiplying 2. Find positive pairs of two negatives 2 6 1. a) 3 7 2. 3 2 4 3 or 64 42 3. 96 60 2 6 4 b) = 3 7 7 16 20 15 4. 30 48 10 Dividing Fractions: Simplying the following expressions. Ensure your answer is in lowest term. When dividing fractions, we 1. Multiply the reciprocalof the divisor. 2. Ensure that our fractions are in the improper form. 6 2 1. 2. 1 3 4. 1 2 6 2 14 5. 12 3 2 – 3 4 2 5 3. 1 6 5 12 25 21 25 4 Working backward: Complete each statement. Show your work. 1) –1 3 8 ____ = 2 1 4 2) ____ 2 3 –3 1 2 Word Problem: 1. Mark has 24 newspapers to deliver. In one apartment building, he delivers them. In the next apartment building, he delivers How many papers does he have left to deliver? 2 3 3 8 of the remaining amount. of 2. John created a painting on a large piece of paper with a length of width of 1 3 4 m. Write an expression in the form a b c 2 5 8 m and a that represents the area of the painting in lowest terms. NOTE: In solving problems with fractions, the word OF means to multiply. Day 5 (Lesson 2.3 Part II) Adding & Subtracting Rational #s. Lesson Focus: Today, we will learn to add and subtract fractions. To add and subtract fractions, we need to 1. Find the Lowest Common Denominator (LCD) 2. Find positive pairs of two negatives 3. When subtracting, add the positives (or tick, tick) Recall: to find the LCD, use the L method. 1 5 3 Example: Find LCD of 3 6 4 Recall as well: we can find positive pairs when there are two negatives 1) 1 2 2) 1 2 1 3) 2 1 4) 2 Simplify: 1) 4. 2 1 3 4 3 10 2 5 – – 2) – 3 4 12 3 1 3) 4 5 1 2 11 5. 15 6 2 3 6. 2 1 9 10 Working backward: Complete each statement. Show your work. 1. 1 1 2 – ____ = 5 2. 6 2 5 ____ = 3 10 1. One January day in Prince George, British Columbia, the temperature read 10.6ْ C at 9:00 a.m. and 2.4 C at 4:00 p.m. a) What was the change in temperature? b) What was the average rate of change in temperature? 2. The Rodriquez family has a monthly income of $6000. They budget 1 4 for rent, expenses? 1 5 for clothing, and 1 10 1 3 for food, for savings. How much money is left for other Day 6 (Lesson 2.3 Part III) Order of Operations with Rationals Lesson Focus: Today, we will learn to simplify rationals using order of operations. Recall: When solving equations, the following order must be taken: In addition, follow these simple keep these orders in mind: 1) B____________________ 2) E____________________ 2. Start inside and work outward. 3) D____________________ 3. Go Left to Right M____________________ 4) A____________________ S____________________ Do the following examples: 1 10 6 4 1) 5 8 3 2 1 3) = 7 5 4 17 27 3 1 2) 5 5 2 2 Day 7 (Lesson 2.4) Determining Square Roots of Rational #s Lesson Focus: Today, we will learn to find square roots of rational numbers. Key Ideas If the side length of a square models a number, the area of the square models the square of the number. If the area of a square models a number, the side length of the square models the square root of the number. A perfect square can be expressed as the product of two equal rational factors. ex. 3.61 = 1.9 x 1.9 1 1 1 4 2 2 The square root of a perfect square can be determined exactly. ex. 2.56 1.6 4 2 9 3 The square root of a non-perfect square determined using a calculator is an approximation. ex. 1.65 1.284523258 Example 1 Determine whether each of the following numbers is a perfect square. Show your work. a) 121 64 b) 1.2 c) 0.9 d) 0.09 Example 2 Evaluate (round to the nearest thousandth where necessary). Show your work a) 2.25 b) 0.34 c) 256 d) 3.61 e) 1225 f) 0.0484 Example 3 A square garden has a side length of 5.2 m. Calculate the area of the garden. Example 4 The area of Mara’s square pumpkin patch is 2.25 m2. She has a square tomato garden with the same area. She wants to determine the dimensions of each garden. Maras solution is shown below. A = s2 2A = s2 2(2.25) = s2 4.5 = s2 4.5 = s 2.12 = s What error did Mara make in her solution? Correct her solution and determine the dimensions of each garden. Example 5 A square lot has an area of 0.5 ha. What are the lot’s dimensions to the nearest metre? Show your work. Hint: 1 ha = 10 000 m2 Rational Numbers Homework Assignments Day 1: Day 2: Day 3: Day 4: Day 5: Day 6: Day 7: Day 8: Day 9: Text page 51 # 4-6, 8-13, 19, 22, 25, 27 (2.1 Part I) Text page 51 # 7, 14-16 (odd letters), 21a, 24, 29 (2.1 Part II) Text page 60 # 4-9 (odd), 10-12, 14, 18, 19, 22, 27 (2.2) Text page 68 # 7-10, 13, 16, 18 (2.3 Part I) Text page 68 # 1, 5, 6, 12, 15, 19 (a, b) (2.3 Part II) Text page 69 # 14, 17, 21, 23, 26 (2.3 Part III) Text page 78 # 1, 2, 7-14(odd), 15, 18, 25 (2.4) Review lesson Unit Test