Activating Prior Knowledge –

Slides:



Advertisements
Similar presentations
RATIONAL AND IRRATIONAL NUMBERS
Advertisements

Rational Numbers and Decimals
Fractions and Decimals 5.2. Writing a Fraction as a decimal Divide the numerator by the denominator.
To Start: 10 Points.
Converting Repeating Decimals to Fractions
Equivalent Forms of Rational Numbers
Chapter 5 Lesson 2 Rational Numbers Pgs
3.2 Rational Numbers. Rational Number Any number that can be written in the form of a fraction. a and b have to be integers.
Math 021.  An equation is defined as two algebraic expressions separated by an = sign.  The solution to an equation is a number that when substituted.
Integrated Mathematics Real Numbers. Rational Numbers Examples of Rational Numbers.
Tie to LO Activating Prior Knowledge – Notes
HOW DO WE CLASSIFY AND USE REAL NUMBERS? 0-2: Real Numbers.
Repeating decimals – How can they be written as fractions? is a rational number.
Graph the following equations. 1.y = x x + 2y = 4 Activating Prior Knowledge – Notes Tie to LO M4:LSN 20 Every Line is a Graph of a Linear Equation.
Number: Recurring Decimals Definition and Simple Conversion to Fraction Form. By I Porter.
Lesson 5.3 The rational numbers. Rational numbers – set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not.
Repeating Decimal to Fraction Conversion
Activating Prior Knowledge
Main Idea and New Vocabulary Key Concept: Rational Numbers
Unit 2. Day 10..
Activating Prior Knowledge
Angles Associated with Parallel Lines
Warm-up Open MSG to next clear page and title it “Solving Multi-Step Equations by Clearing the Fractions” Next, solve the following: Find the LCM of the.
Lesson 7.4e Repeating Decimals
Converting Repeating Decimals to Fractions
Lesson 2 MGSE8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for.
Activating Prior Knowledge –
Activating Prior Knowledge –
Activating Prior Knowledge –
Activating Prior Knowledge
Activating Prior Knowledge
Informal Proofs of Properties of Dilations
Activating Prior Knowledge – Handout
Lesson 1: 1.1 Solving One-Variable Two-Step Equations
3.2 Rational Numbers.
Activating Prior Knowledge – Notes
Unit 2. Day 14..
Activating Prior Knowledge- Exploratory Challenge
12 Systems of Linear Equations and Inequalities.
First Consequences of FTS
Sequencing Reflections and Translations
Unit 2. Day 13..
Activating Prior Knowledge – Module Page 61
Activating Prior Knowledge
Activating Prior Knowledge -Simplify each expression.
Tie to LO Activating Prior Knowledge – 1. y – x = x + 3y = 6
Activating Prior Knowledge
Tie to LO Activating Prior Knowledge – Paper
Activating Prior Knowledge – Notes
What is the difference between simplifying and solving?
Activating Prior Knowledge –
7 Chapter Rational Numbers as Decimals and Percent
Tie to LO Activating Prior Knowledge – 1. y – 2x = x + 3y = 6
Activating Prior Knowledge – Notes
Exercise Use long division to find the quotient. 180 ÷ 15.
Activating Prior Knowledge – Simplify each expression.
Main Idea and New Vocabulary Key Concept: Rational Numbers
Activating Prior Knowledge-
Properties of Dilations
Activating Prior Knowledge – Notes
Do Now Solve. 1. –8p – 8 = d – 5 = x + 24 = 60 4.
Bell Ringer Solve the following: 1. ) 7(4 – t) = -84 2
Lesson 2 MGSE8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for.
Activating Prior Knowledge – Notes
Lesson 6 Ratio’s and Proportions
Module 7 Lesson 6 March 21, 2019 Math Log #39.
Multiplying and Dividing Rational Numbers
Math Log #40  .
Subtracting Rational Numbers Unit 2 Lesson 3
Math 9 Honors Section 1.1 Fractions and Decimals
Presentation transcript:

Activating Prior Knowledge – Find the decimal expansion of the following: 1. 9 15 2. 7 8 0.6 .875 3. 5 9 4. 11 16 0.555… .6875 Tie to LO

Objective: Today, we will show that every real number with a repeating decimal expansion is a rational number

Concept Development CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Concept Development In lesson 8 we learned that every rational number, i.e. fraction, has a decimal expansion with a repeating pattern. But is the converse true? That is, can every decimal expansion with an infinite repeating pattern, be written as a fraction? Let’s start observing the effect of multiplying decimals by powers of 10. Consider for example, the finite decimal 1.2345678. If we multiply by 10 5 we get: CFU

Concept Development: CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Concept Development: Example 1: There is a fraction with an infinite decimal expansion of 0. 81 . Find the fraction. Let 𝑥=0. 81 =0.818181818… Any ideas where to begin? Let’s try multiplying 𝑥=0. 81 by powers of 10. 𝑥=0.818181818… 10𝑥=8.18181818… 100𝑥=81.8181818… 1000𝑥=818.181818… CFU

Concept Development: CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Concept Development: Example 1: Let’s take a closer look at 100𝑥=81.8181818… Do you notice anything? So the infinite repeating decimal 0.818181818…= 9 11 100𝑥=81+0.8181818… Since 𝑥=0.818181818… That means… 100𝑥=81+𝑥 −𝑥 −𝑥 99𝑥=81 99 99 𝑥= 81 99 = 9 11 CFU

Concept Development: Do any of these seem to be helpful? CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 52 Concept Development: Example 2: Could it be that 2.13 8 is also a fraction? Let’s try the same trick and multiply by powers of 10. 𝑥=2.138888… Do any of these seem to be helpful? 10𝑥=21.38888… 100𝑥=213.8888 1000𝑥=2138.8888… What if I asked a separate question: is 0.8888… the decimal expansion of a fraction? CFU

Concept Development: CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Concept Development: Example 2: Could it be that 2.13 8 is also a fraction? Since we know 0.8888… is the decimal expansion of some fraction, let’s let 0.888…= 𝑎 𝑏 . Now are any of the previous powers of 10 helpful? We now know that 100𝑥=213+.8888…, so we can write it as 100𝑥=213+ 𝑎 𝑏 . Let’s do some side work: Let’s let 𝑦=0. 8 10𝑦=8. 8 10𝑦=8+0. 8 10𝑦=8+𝑦 Substitution property 9𝑦=8 Subtract y on both sides 𝑦= 8 9 Divide both sides by 9 CFU

Concept Development: CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Concept Development: Example 2: Could it be that 2.13 8 is also a fraction? Now that we know 0. 8 = 8 9 , we can write the equation: 100𝑥=213+ 8 9 100𝑥= 213∙9 9 + 8 9 100𝑥= 213∙9+8 9 100𝑥= 1925 9 𝑥= 1925 900 ÷25 ÷25 𝑥= 77 36 CFU

Independent Practice CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Independent Practice Complete Exercises 1 – 2 in your module. (4 minutes) Exercise 1: CFU

Independent Practice CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 51 Independent Practice Exercise 2: CFU

Independent Practice CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 53 Independent Practice Complete exercises 3 – 4 in your module. (6 min) CFU

Independent Practice CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 53 Independent Practice CFU

Module 7 LSN 10: Converting Repeating Decimals to Fractions Pg. 54 Lesson Summary CFU

Closure – What did you learn? Why is it important? Module 7 LSN 10: Converting Repeating Decimals to Fractions Notes Closure – What did you learn? Why is it important? How does multiplying a decimal by a power of 10 help to determine the fraction that would produce it? Homework: Problem Set, page 54 – 55, problems 1 – 8 all. CFU