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Math 7 Warm-Up↑ 11/29/16 Divide each of the following problems. Be sure to place the appropriate sign on each problem. 1) −54 −6 = 2) If s = -4 and t = -6, then 𝑠+𝑡 5 = 3) The hot air balloon Midnight Express changes -2,700 feet of altitude in 135 minutes. What is the rate of change in feet per minute? 4) During the past week, Mrs. Thorne recorded the following amounts in her checkbook: $150, -$75, -$15, and -$32. Write and evaluate an expression to find the average of these amounts. Math 7 The Number System: 7.NS.2 Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers. 7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. 7.EE.3 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Mathematical Practices: 1 – Make sense of problems and persevere in solving them. 3 – Construct viable arguments and critique the reasoning of others. 5 – Use appropriate tools strategically. 6- Attending to precision. 7 – Look for and make use of structure. Objective: Know what is a rational number and be able to graph rational numbers on a number line. A. What is a rational number? A rational number is any number that can be expressed as a division of two integers as long as the denominator of the fraction is not equal to zero. 𝑎 A rational is of the form Examples: 1= , 9= , 𝑏 , 𝑏 ≠ 0. 2 , − 2.3 = 9 √25 = , Other examples of rational numbers: Examples of numbers that are not rational: , 5 14 8 = −8= , B. Placing Rational Number on the Number Line To place a rational number follow these steps: 1) Find the two whole numbers the rational is in between. 2) Divide the space in between the two whole numbers by the denominator of the rational number. (Need 1 fewer lines than the denominator). 3) Count over the numerator and place a dot at the location where the rational number should be placed and label it correctly. Example 1: Example 2: 3 Graph the rational 1 on the given number line. 4 3 Between which two whole numbers does 1 lie? 4 How many sections do we need? How many lines is that? 3 Draw a dot at 1 . 4 Example 3: 5 Graph the rational −1 on the given number line. 8 (*Remember that negative numbers decrease in value as the digits get larger). 5 Between which two whole numbers does −1 lie? 8 How many sections do we need? How many lines is that? 5 Draw a dot at −1 . 8 You Try! Example 4: 2 Graph the rational −4 on the given number line. 5 2 Between which two whole numbers does −4 lie? 5 How many sections do we need? How many lines is that? 2 Draw a dot at −4 . 5 Example 5: 4 Graph the rational 6 on the given number line. 7 4 Between which two whole numbers does 6 lie? 7 How many sections do we need? How many lines is that? 4 Draw a dot at 6 . 7 Homework p. 167 Review Problems 1 – 6 & p. 168 Reflection Closure: What is a rational number? Which of the following are rationals? Explain your reasoning: 0.2, 𝜋, − 4, 4 √36 , √13 , 2 5 0. 6̅, -7.2, Math 7 Warm-Up↑ 11/30/16 3 1) Graph the rational −1 on the given number line. 8 3 Between which two whole numbers does −1 lie? 8 How many sections do we need? How many lines is that? 3 Draw a dot at −1 . 8 2) What is the definition of a rational number? 3) Give three examples of numbers which are rational using your definition above in question 2. 1) 2) 3) Math 7 The Number System: 7.NS.2 Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers. 7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. 7.NS.2d Converting rational numbers to decimals using long division and knowing that a rational number either terminates or repeats. 7.EE.3 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Mathematical Practices: 1 – Make sense of problems and persevere in solving them. 3 – Construct viable arguments and critique the reasoning of others. 5 – Use appropriate tools strategically. 6- Attending to precision. 7 – Look for and make use of structure. 8 – Look for and express regularity in repeated reasoning Objective: Know that a rational number can be expressed as a repeating or terminating decimal and be able to accurately represent a rational as a decimal using long division. Writing Rationals As Terminating or Repeating Decimals All rational numbers can either be expressed as a terminating or repeating decimals. Terminating decimals are decimals that stop or which have a zero that repeats. Repeating decimals have one or more digits that repeat over and over again. Repeating decimals can be represented using bar notation over the first digit or set of digits that repeat. First divide the numerator by the denominator. Example 1: 1 3 = Example 2: 2 9 = Example 3: 2 3 8= Example 4: 5 4 6= Example 5: 74 100 Example 6: −3 = 7 20 = You Try! Example 7: −6 Example 8: Example 9: 7 9 1 2= = −1 40 = Which denominators do you think cause decimals to repeat? Why? Which denominators do you think cause decimals to terminate? Why? Closure: All rationals can be expressed as what two kinds of numbers? Homework: p. 179 – 180 (2, 6, 8, 10, 11, 15, 16, 18, 19, 22) Math 7 Warm-Up↑ 11/16/15 Express each rational number as a terminating or repeating decimal without a calculator. 1) 1 8 = 8 2) −2 = 9 If you finish #1 and #2, try these two problems with your calculator. 3) 5 3 16 4) −7 = 8 11 = Math 7 The Number System: 7.NS.2 Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers. 7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. 7.NS.2d Converting rational numbers to decimals using long division and knowing that a rational number either terminates or repeats. 7.EE.3 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Mathematical Practices: 1 – Make sense of problems and persevere in solving them. 3 – Construct viable arguments and critique the reasoning of others. 5 – Use appropriate tools strategically. 6- Attending to precision. 7 – Look for and make use of structure. 8 – Look for and express regularity in repeated reasoning Objective: Be able to represent any terminating or repeating decimal as a fraction in simplest form using the concept of place value. A. Writing Terminating Decimals as Rational numbers. By using the last place value a number terminates in we can write all terminating decimals as rational numbers. 0 0 2 -3 . . . . 7 1 9 1 0 5 0 4 Complete the table below. Write the fractions in simplest form. Words Decimal Fraction 0.7 seven tenths 0.19 Two and one hundred five thousandths 2.105 -3.04 Example 1: -0.2 = Example 2: 0.04 = Example 3: -0.875 = B. Writing Terminating Decimals as Rationals Every repeating number can also be written as a fraction with a denominator of 9, 99, 999, 9999, etc. Use the place value of the final digit that repeats the first time minus 1 as the denominator. Then simplify your answer if possible. Example 4: 0. 5̅ = ̅̅̅̅ = Example 5: 0. 09 ̅̅̅̅ = Example 6: −1. 27 ̅̅̅̅̅ = Example 7: 3. 123 You Try! Turn each of these mixed examples of both terminating and repeating decimals into fractions (rationals) in simplest form: Do at least examples 8 & 9. Example 8: 0.8 = Example 9: 0. 6̅ = Example 10: 0.34 = ̅̅̅̅ = Example 12: 0. 72 Homework: p. 181 - 182 (24, 25, 26, 27, 28, 29, 30, 35, 36, 37 ) Name ________________________________________________ 11/30/16 Ticket out the Door Closure: What is the main math concept that you can use to change a decimal into a fraction? _____________________________________________________________ _____________________________________________________________ Give an example of a repeating or terminating decimals show how you would convert each into a fraction. Terminating Example: ______________ = Repeating Example: ______________ =