1. Division of Fractions (3 of 3) Missy Bristol 1

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

3. 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.

4. 4 Lesson Objective Students Will Be Able To… divide fractions using the common denominator method. Lesson DescriptionThis lesson is the third of 3 regarding the concept of division of fractions and standard CCSS.Math.Content.6.NS.A.1. Lesson 1 supported students in making general sense of division and applying these generalities to fractional division. It sought to take the comfort students have with whole numbers into fractions. The second lesson will focus on the visual modeling aspect of the standard. Lesson 3 introduces the common denominators approach to division of fractions. CCSS.Math.Content.6.NS.A.1 Lesson Overview (1 of 4)

5. 5 Lesson Vocabulary Recognition, Translation, Rationality, Determine, Execution 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 3 focuses on solving equations.) 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)

6. 6 ScaffoldingClasswork problems involve scaffolding and difficulty escalation. Word problems are performed in partners or small groups and can be differentiated by skill level. HW escalates in complexity. EnrichmentThe word problems provide opportunity for students with advanced skills to assume leadership roles in partner/group pairings. Online Resources for Absent Students I recommend a quick run through with a hard copy of the slides as the best option for students that miss class. This approach to dividing fractions is not the widely taught methodology. Lesson Overview (3 of 4)

7. 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 9 for more about the thinking and research that went into the creation of this series of lessons on fractional division.

8. *Please Note* - Not the traditional approach to division of fractions (Lesson 3) 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” 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%.

9. Warm Up How many halves, (1/2), are there in an eighth, (1/ Agenda 9 We know that 5 wholes cannot “fit into” (8/9) so our answer is less than 1. Attempt to solve for the quotient using the visual fraction models given. If you encounter trouble, record/explain what is giving you difficulty. Learning Objective: Students Will Be Able To… divide fractions using the common denominator method.

10. Agenda and Objective 1) Warm Up 2) Vocabulary 4) Explore (3) 5) Assessment 10 Learning Objective: Students Will Be Able To… divide fractions using the common denominator method. Independent Teacher guided Independent, partners, group “Key to leave” check Teacher guided 3) Launch

11. Lesson Vocabulary /Language Objectives Words Recognition Translation Rationality Determine Execution We will encounter these words later in the lesson. You do not need to write definitions. These words will help you with the process of solving word problems. Agenda 11

12. Launch 1A – Warm Up Revisited Agenda 12 Remember all whole numbers can be placed over a denominator of 1. Why? Common Denominators 9 x 1 = 8 x 1 = = 9 x 1 = 9 x 5 8 45 A common denominator/multiple is a number that both of our denominators can “fit into.” It is helpful to find the least/lowest, but any common multiple can be used. In this problem, 9 and 1 “fit into” 9…which is the least common denominator. 18 is also a common multiple but not the least. Recall… Why is (45/9) an equivalent fraction to (5/1)?

13. Launch 1B – Get to the Division!! Agenda 13 Divide from Left to Right * Since any number (including a fraction) divided by 1 is itself…our quotient is… And with practice you can move quickly… …the denominators “cancel” out (become 1)

14. Explore 1 – The Division Process 14 Agenda Equation Common Denominators DivisionQuotient Fast Track Fill in the missing numbers as indicated by the blank boxes. If the entire space is blank then you must supply all work necessary. “Fast Track” means that you may skip this step (optional).

15. Explore 2 – Tying all Division Skills Together 15 Agenda Rosie is running a track race with five teammates. The course is 4 and (1/8) miles long. If all 6 team members want to run equal distances, how many miles will each be running? 1 Can you recognize the operation needed? (Add, Sub, Mult, or Div) 2 Recognition Translation Can you translate this into a numerical equation? 3 4 Rationality Can you hypothesize what a rational solution might look like before actually solving? (will it be greater or less than 1) 5 Determine Can you determine a method of solving the equation? Execution Can you execute the accurate solution using skills learned? If you can, prove it ! Division Less than 1 Common Den. more efficient than visual models Take that!

16. Rohan is building a table and needs to purchase lumber. He has 8 and 2/3 feet of hard oak wood but determines that he needs 12 and 1/4 feet so he needs to buy the rest. Rohan will use 2/5 of the total 12 and 1/4 feet to make the legs of the table. This will be a tripod table so there will only be 3 legs. a)How many feet of wood does Rohan purchase? b)How many feet of wood is used to make the legs? (all together) a)How long is each individual table leg? b)How many feet of wood are used on the rest of the table? (if legs not included) 16 Explore 3 – Fraction nitty gritty Agenda

17. Name:__________ Key to Leave a)How many feet of wood does Rohan purchase? b)How many feet of wood are used to make the legs? (all together) c)How long is each individual table leg? d)How many feet of wood are used on the rest of the table? (if legs not included) Review your work from Explore 3 and compare your work to the solutions provided. At any point where your work is different or at any point that you don’t understand the solution process, explain either your confusion or what you did incorrectly. If your work matches exactly, write “I had that excatly.” (Be honest to help me understand where we stand). I should see much writing in the spaces on this page. Assessment 17 Agenda

18. Back to Overview 18 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

19. 19 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: Standards for This Unit Back to OverviewNext Slide

20. 20 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

21. 21 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

22. 22 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

23. 23 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

24. 24 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

25. 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… 25

26. 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… 26