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Division of Fractions (1 of 3) Missy Bristol 1

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2 This project is funded by the American Federation of Teachers.

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3 *1 st Time Users of 21 st Century Lesson: Click HERE for a detailed description of our project.HERE 21 st Century Lessons – Teacher Preparation Spend AT LEAST 30 minutes studying the Lesson Overview, Teacher Notes on each slide, and accompanying worksheets. Set up your projector and test this PowerPoint file to make sure all animations, media, etc. work properly. Please do the following as you prepare to deliver this lesson: Feel free to customize this file to match the language and routines in your classroom.

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4 Lesson Objective Students Will Be Able To… interpret the rationality of quotients and apply this skill to “real-world” problems. Lesson DescriptionThis lesson is the first of 3 regarding the concept of division of fractions and standard CCSS.Math.Content.6.NS.A.1. Lesson 1 supports students in making general sense of division and applying these generalities to fractional division. It seeks to take the comfort students have with whole numbers into fractions. The second lesson will focus more on the visual model aspect of the standard. Lesson 3 takes students from the visual to numerical and efficiency with fractional division calculations. Each of the 3 lessons culminates with word problems in order to ground the math in reality for students. CCSS.Math.Content.6.NS.A.1 Lesson Overview (1 of 4)

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5 Lesson VocabularyDividend, Divisor, Quotient MaterialsClasswork notes should be a double-sided copy. Homework sheet is single-sided. Summary “key to leave” copies needed. Common Core State Standard (To be broken into a 3-part lesson, lesson 1 focuses on interpreting.) CCSS.Math.Content.6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? Lesson Overview (2 of 4)

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6 ScaffoldingThis is the into lesson and proceeds in a manner that is concrete and paced for all learners. The homework problems escalate in difficulty, so accessible to all. EnrichmentThis lesson focuses on rationality of solutions. Advanced learners will be tempted to solve. Tell these learners that the following 2 lessons will give them more of an opportunity to expand critically on what they know about fractional division. Solving is not important for this lesson’s objective. Online Resources for Absent Students As this lesson is narrow in focus, I caution teachers in directing students to resources that might go more into solving fractional division. A quick run through with a hard copy of the slides is a better option for students that miss class. Lesson Overview (3 of 4)

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7 Lesson Overview (4 of 4) Before and AfterThe standard is trisected. Lesson 1 focuses on interpretation and rationality. Lesson 2 deals with solving using visual models. Lesson 3 solves numerically and with equations. All lessons embed the objective into word problems. Topic BackgroundDivision of fractions has been a source of consternation for teachers and students alike. Most often, the “trick” of flipping the second fraction and multiplying has been taught and then memorized by students. This, however, is not an algorithm. In fact it does not teach the concept of division, but the idea of the reciprocal nature of multiplication and division…a separate skill. Review slide 8 for more about the thinking and research that went into the creation of this series of lessons on fractional division.

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*Please Note* - Not the traditional approach to division of fractions Agenda 8 Division of fractions is a difficult concept for many, students and teachers alike, to understand. The “flip and multiply” method that many of us were taught is functional but not necessarily logical for those trying to dig into mathematical meaning. The approach taken in this lesson is based on the fact that all fractional operations (addition, subtraction, multiplication, and division) can be computed using equivalent ratios (common denominators.) This approach was piloted in my 7 th grade class. Instead of being asked to memorize four different ways to approach fractional operations, students were told to create equivalent ratios no matter the operation required. Two periods of instruction were devoted to “re- teaching” these skills that continued to hinder students even after three years of practice. Students were given mixed-operation pre and post quizzes that were identical except for different numbers used. For the post quiz, students were directed to use only the “equivalent ratios method.” The pre mean score was 57.5% while the post was 78.6%. The number of perfect scores increased 212.5%.

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Warm Up OBJECTIVE: Students Will Be Able To… interpret the rationality of quotients and apply this skill to “real-world” problems. Agenda 9 We learned division long ago, so we know that 10 ÷ 2 = 5. Still, can you explain, in words, what dividing 10 by 2 means? Explain in as many different ways as you can think of. ex) 10 is cut into pieces that are 2 in length ex) The number of times 2 fits into 10 ex) How many quantities of 2 a quantity of 10 can hold

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Agenda and Objectives 1) Warm Up 2) Launch 1 4) Explore 5) Summary 10 Learning Objective: Students Will Be Able To… interpret the rationality of quotients and apply this skill to “real-world” problems. Language Objective: SWBAT define and identify Dividend, Divisor, and Quotient as they relate to division of fractional quantities. independent independent and partner “key to leave” check 3) Launch 2 independent and partner

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Lesson Vocabulary and Language Objectives Agenda 11 WordDefinitionExample/Symbol Dividend A number to be divided Divisor The number a dividend is divided by Quotient The number that results from division, solution of division = 5

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12 Launch 1 – Division: Larger Dividend Agenda OnceTwice Concept 1 When the dividend is larger than the divisor, the quotient is always greater than 1. DividendDivisor Quotient

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*On the lines below, write what you think Concept 1 means in your own words. If needed, you can use the number example, 4 ÷ 2, to help in your explanation. Please use either the words "fits into" or "holds" in your explanation. Once finished, share your explanation with a table partner. (Everyone must share) 5 Minutes of Work Time ! 13 Student Work (SW #1) Click on the timer for internet timer window to open! Agenda

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14 Launch 2 – Division, smaller dividend Won’t Fit Concept 2 When the dividend is smaller than the divisor, the quotient is always less than 1. DividendDivisor Quotient Agenda

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*On the lines below, write what you think Concept 1 means in your own words. If needed, you can use the number example, 2 ÷ 4, to help in your explanation. Please use either the words "fits into" or "holds" in your explanation. Once finished, share your explanation with a table partner. (Everyone must share) 5 Minutes of Work Time ! 15 Student Work (SW #2) Click on the timer for internet timer window to open! Agenda

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16 (SW #3) Don’t freak out if not whole numbers… … the concepts still apply to fractions and decimals !! DividendDivisor DividendDivisor Greater than/Less than Concept 1 or 2 (write out) Greater than/Less than Concept 1 or 2 (write out) Agenda

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17 Explore – Rational Quotients Circle the larger number (Dividend or Divisor). Then, write greater than or less than, you may use inequality symbols ( ). #1 is answered for you. 1) Less than < 1 2)1 3)1 4)1 5)1 6)1 7)1 8)1 9)1 10)1 11)1 12)1 Greater than > > > > > > < < < < > Agenda

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18 An Italian sausage is 8 inches long. How many pieces of sausage can be cut from the 8- inch piece of sausage if each piece is to be Summary & Key to Leave Name 1)Explain whether the quotient to the problem below is greater than or less than 1. Use the words “Dividend” and “Divisor” in your explanation. 3) A hotdog is 7 ½ inches long. How many pieces can be cut from the hotdog if each piece is to be two-thirds of an inch? 2) How many halves, (1/2), are there in an eighth, (1/8)? Problems 2 & 3 involve division to solve: Place a box around the dividend and circle the divisor. Write a numerical expression or equation. Then write whether the quotients will be greater than or less than 1. Agenda

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Back to Overview 19 1 st Time Users of 21 st Century Lessons Welcome to 21 st Century Lessons! We are a non-profit organization that is funded through an AFT (American Federation of Teachers) Innovation Grant. Our mission is to increase student achievement by providing teachers with free world-class lessons that can be taught via an LCD projector and a computer. 21 st Century Lessons are extremely comprehensive; we include everything from warm–ups and assessments, to scaffolding for English language learners and special education students. The lessons are designed into coherent units that are completely aligned with the Common Core State Standards, and utilize research-based best practices to help you improve your students’ math abilities. Additionally, all of our lessons are completely modifiable so you can adapt them if you like. Description of 21 st Century Lessons: Next Slide

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20 1 st Time Users of 21 st Century Lessons The lesson that you are currently looking at is part of a unit that teaches the following Common Core Standards: CCSS.Math.Content.6.NS.A.1 Standards for This Unit Back to OverviewNext Slide

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21 1 st Time Users of 21 st Century Lessons In order to properly use 21 st Century Lessons you will need to possess or arrange the following things: Required: PowerPoint for P.C. (any version should work) Note: Certain capabilities in the PowerPoint Lessons are not compatible with PowerPoint for Mac, leading to some loss of functionality for Mac PowerPoint users. An LCD projector Pre-arranged student groups of 2 – (Many lessons utilize student pairings. Pairs should be seated close by and be ready to work together at a moment’s notice. Scissors – at least 1 for every pair Requirements to teach 21 st Century Lessons: Back to OverviewNext Slide

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22 1 st Time Users of 21 st Century Lessons Computer speakers that can amplify sound throughout the entire class “Calling Sticks” – a class set of popsicle sticks with a student’s name on each one A remote control or wireless presenter tool– to be able to advance the PowerPoint slides from anywhere in your classroom Personalize PowerPoints by substituting any names and pictures of children we included in the PowerPoint with names and pictures of your own students. Since many lessons utilize short, partner-processing activities, you will want a pre- established technique for efficiently getting your students’ attention. (“hands- up”, Count from “5” to “0” etc.) Project onto a whiteboard so you or your students can solve problems by hand. (Lessons often have a digital option for showing how to solve a problem, but you may feel it is more effective to show the work by hand on a whiteboard.) Internet connectivity – without the internet you may not have full functionality for some lessons. Strongly Suggested to teach 21 st Century Lessons: Back to OverviewNext Slide

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23 1 st Time Users of 21 st Century Lessons We suggest spending 30-45 minutes reviewing a lesson before teaching it. In order to review the lesson run the PowerPoint in “Slideshow “- Presenters View and advance to the “Lesson Overview” slide. By clicking on the various tabs this slide will provide you with a lot of valuable information. It is not necessary to read through each tab in order to teach the lesson, but we encourage you to figure out which tabs are most useful for you. Note: All of our lessons are designed to be taught during a 45-55 minute class. If your class is shorter than this you will have to decide which sections to condense/remove. If your class is longer we suggest incorporating some of the “challenge” questions if available. Lesson Preparation (Slide 1 of 2) Back to OverviewNext Slide

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24 1 st Time Users of 21 st Century Lessons After reviewing the overview slide, click your way through the PowerPoint. As you go, make sure to read the presenter note section beneath each slide. The note section is divided into two sections: “In-Class Notes” and “Preparation Notes.” The In-Class Notes are designed to be concise, bulleted information that you can use “on the fly” as you teach the lesson. Included in In-Class Notes are: a) a suggested time frame for the lesson, so you can determine whether you want to speed up, slow down, or skip an activity, b) key questions and points that you may want to bring up with your students to get at the heart of the content, and c) answers to any questions being presented on the slide. The Preparation Notes use a narrative form to explain how we envision the activity shown on the slide to be delivered as well as the rationale for the activity and any insight that we may have. Lesson Preparation (Slide 2 of 2) Back to OverviewNext Slide

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25 1 st Time Users of 21 st Century Lessons There are several features which have been incorporated into our PowerPoint lessons to help make lessons run more smoothly as well as to give you access to additional resources during the lesson should you want them. These features include: Agenda Shortcuts – On the agenda slide, click on any section title and you will advance to that section. Click the agenda button on any slide to return to the agenda. Action Buttons – On certain slides words will appear on the chalk or erasers at the bottom of the chalkboard. These action buttons give you access to optional resources while you teach. The most common action buttons are: Scaffolding – gives on-screen hints or help for that slide Answers – reveals answers to questions on that slide Challenge – brings up a challenge questions for students Agenda – will return you to the agenda at the beginning of the lesson Features built into each PowerPoint lesson Back to OverviewNext Slide

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The goal of 21 st Century Lessons is simple: We want to assist teachers, particularly in urban and turnaround schools, by bringing together teams of exemplary educators to develop units of high-quality, model lessons. These lessons are intended to: Support an increase in student achievement; Engage teachers and students; Align to the National Common Core Standards and the Massachusetts curriculum frameworks; Embed best teaching practices, such as differentiated instruction; Incorporate high-quality multi-media and design (e.g., PowerPoint); Be delivered by exemplary teachers for videotaping to be used for professional development and other teacher training activities; Be available, along with videos and supporting materials, to teachers free of charge via the Internet. Serve as the basis of high-quality, teacher-led professional development, including mentoring between experienced and novice teachers. 21 st Century Lessons The goal… 26

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Directors: Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee Ted Chambers - Co-director of 21st Century Lessons Tracy Young - Staffing Director of 21st Century Lessons Leslie Ryan Miller - Director of the Boston Public Schools Office of Teacher Development and Advancement Emily Berman- Curriculum Director (Social Studies) of 21st Century Lessons Carla Zils – Curriculum Director (Math) of 21 st Century Lessons Brian Connor – Technology Coordinator 21 st Century Lessons The people… 27

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