Unit 4, Lesson 12: Fractional Lengths
GRADE 6 MATHEMATICSUnit 4, Lesson 12: Fractional LengthsLesson GoalsRequired Materials Use multiplication and division to solve measurement problems involving fractional geometry toolkitslengths. Recognize multiplicative comparison situations (i.e. questions such as “how manytimes as much as is ?”) as division problems. Use equations, diagrams, and division algorithm to reason about multiplicativecomparison problems that involve fractions.12.1: Number Talk: Multiplication Strategies (5 minutes)Setup: 2 minutes of quiet think time, followed by a whole-class discussion.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths1
GRADE 6 MATHEMATICSStudent task statementPossible responsesFind the product mentally.Anticipated misconceptionsWhen multiplying, students mayonly multiply the tens digits andmultiply the ones digits and add themto get 136. Ask these students toestimate an answer for the problemand consider whether their answermakes sense.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths2
GRADE 6 MATHEMATICS12.2: How Many Would It Take? (Part 1) (15 minutes)Setup:Students in groups of 2. 5 minutes of quiet work time, 5 minutes of partner collaboration and discussion. Access to geometry toolkits.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths3
GRADE 6 MATHEMATICSStudent task statementPossible responses1. Jada was using square stickers with a side length ofspine isinch to decorate the spine of a photo album. Theinches long. If she laid the stickers side by side without gaps or overlaps, how many stickersdid she use to cover the length of the spine?2. How many1. 14 stickers2. 18 binder clips-inch binder clips, laid side by side, make a length ofinches?3. It takes exactly 26 paper clips laid end to end to make a length ofinches.3.a. Less than 1 inch and morethan inch.b.a. Estimate the length of each paper clip.b. Calculate the length of each paper clip. Show your reasoning.inchAnticipated misconceptionsIf students struggle to represent thesituations mathematically, suggest thatthey draw diagrams to represent thesituations. They could start withsketches of the objects and then movetoward other simpler or more abstractrepresentations as they make bettersense of the problems.The last question requires students todivide a mixed number by a wholenumber. If students are unsure how todo so, remind them that, in an earlierlesson, we saw that dividing by a wholeOpen Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths4
GRADE 6 MATHEMATICSnumber has the same outcome asmultiplying by its reciprocal.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths5
GRADE 6 MATHEMATICSAre you ready for more?Lin has a work of art that is1. If each paper clip isinches byPossible Responsesinches. She wants to frame it with large paper clips laid end to end.inch long, how many paper clips would she need? Show your reasoning and besure to think about potential gaps and overlaps. Consider making a sketch that shows how the paperclips could be arranged.2. How many paper clips are needed if the paper clips are spacedinch apart? Describe the arrangementof the paper clips at the corners of the frame.Answers vary. Sample response:1. 38 paper clips. One side of the paper is 20inches long., soLin can fit 11 paper clips alongthe side with a gap of inchsince. If thepaper clips are centered alongthe 20-inch length, there will beinch of gap on either side. The other side is 14 incheslong., so Lin can fit8 paper clips along the sidewith no gap at all. At each corner of the paper,two paper clips will meet. If thepaper clip has a width that isabout inch (to fit in thegap left by the 11 paper clipsalong the longer side), thenthere will be no gap or overlap.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths6
GRADE 6 MATHEMATICS Altogether, Lin will need 38paper clips.2. 34 paper clips. If space is putbetween the paper clips, thenfewer paper clips will be needed. Ifa gap of inch is between thepaper clips, then each paper clipcould have inch of space oneither end so that the paper clipand its space takes up 2 inches.Then there are 7 paper clips alongthe sides of length 14 inches of theframe, and there are 10 paper clipsalong the sides of length 20 inches.There is a gap ofinch betweenthe end of the paper clip and theend of the frame.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths7
GRADE 6 MATHEMATICS12.3: How Many Times as Tall or as Far? (15 minutes)Setup:Students in groups of 4. 1–2 minutes of quiet time for each question in the first problem, 2 minutes to compare diagrams and equationsin groups, followed by a brief whole-class discussion. 8–10 minutes to complete the activity, either independently or collaboratively withtheir group.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths8
GRADE 6 MATHEMATICSStudent task statementPossible responses1. A second-grade student is 4 feet tall. Her teacher isa. How many times as tall as the student is theteacher?feet tall.b. What fraction of the teacher’s height is thestudent’s height?1.a. The teacher isas the student.b. The student is2.3. Write a division expression that can help answer each of the following questions. Then answer thequestion. If you get stuck, draw a diagram.miles on Monday andmiles on Tuesday. How many times her Monday’sa. 15b.3.ora. Tuesday's distance istimesMonday's distance.distance was her Tuesday’s distance?b. A cyclist planned to rideas tall as theteacher.2. Find each quotient. Show your reasoning and check your answer.a.b.a. A runner rantimes as tallmiles but only managed to travelmiles. What fraction of his plannedtrip did he travel?b. The cyclist traveledof hisplanned trip.Anticipated misconceptionsIf students have trouble drawing andusing a diagram to compare lengths,ask them to revisit the Fractions ofRopes activity (in Lesson 7) and use thediagrams there as examples. Suggestthat they try drawing a diagram ongraph paper, as the grid could supportOpen Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths9
GRADE 6 MATHEMATICSthem in drawing and making sense ofthe fractional lengths.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths10
GRADE 6 MATHEMATICS12.4: Comparing Paper Rolls (Optional, 15 minutes)Setup:Students in groups of 4. Ask students to keep their materials closed. Display the image of the paper rolls for all to see. 1–2 minutes tonotice and wonder, followed by a brief whole-class discussion. 7–8 minutes of quiet work time.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths11
GRADE 6 MATHEMATICSStudent task statementPossible responsesThe photo shows a situation that involves fractions.1.a. About(oror 2.5) times.b. About(or 0.4) times.2. Answers vary. Possible equations: 1. Use the photo to help you complete the following statements. Explain or show your reasoning for thesecond statement. a. The length of the long paper roll is about times the length of the short paper roll. b. The length of the short paper roll is about times the length of the long paper roll. 2. If the length of the long paper roll isinches, what is the length of each short paper roll?Use the information you have about the paper rolls to write a multiplication equation or a divisionequation for the question. Note that.3. Answer the question. If you get stuck, draw a diagram.3.(or equivalent) inches.Reasoning varies.Anticipated misconceptionsStudents might estimate therelationships between the lengths ofrolls by rounding too much. Forexample, they might say that the lengthOpen Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths12
GRADE 6 MATHEMATICSof the shorter roll isthe length of thelonger roll, or that the longer roll istwice as long as the shorter roll. If thishappens, ask students to take a closerlook and make a more precise estimate.Suggest that they divide the larger rollinto smaller segments, each of whichmatches the length of the shorter rolls.Lesson Synthesis (5 minutes)How do we interpret and solve problems involving fractional lengths, such as: “How many-inch paper clips, laid end to end, are in a length ofinches?” How do we compare twofractional lengths and find out how many times one is as long as the other?12.5: Building A Fence (Cool-down, 5 minutes)Setup: None.Student task statementPossible responsesA builder was building a fence. In the morning, he worked forof an hour. In the afternoon, he worked forof an hour. How many times as long as in the morning did he work in the afternoon?Write a division equation to represent this situation, then answer the question. Show your reasoning. If you getstuck, you can draw a diagram.Open Up Resources(openupresources.org)Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths(orafternoon, he worked). In theas long as hedid in the morning.13
did in the morning. GRADE 6 MATHEMATICS Open Up Resources (openupresources.org) Unit 4: Dividing Fractions, Lesson 12: Fractional Lengths 13. Title: Condensed Lesson Plan Author: Open Up Re
4 Step Phonics Quiz Scores Step 1 Step 2 Step 3 Step 4 Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 . Zoo zoo Zoo zoo Yoyo yoyo Yoyo yoyo You you You you
Abstract: The aim of this paper, to provide the detail survey of historical and recent developments of fractional calculus. Recently fractional calculus has been attracted much attention since it plays an important role in many . Fractional calculus has been 300 years old history, the development of fractional calculus is mainly focused on .
iii UNIT 1 Lesson 1 I’m studying in California. 1 Lesson 2 Do you have anything to declare? 5 Lesson 3 From One Culture to Another 8 UNIT 2 Lesson 1 You changed, didn’t you? 13 Lesson 2 Do you remember . . . ? 17 Lesson 3 Women’s Work 20 UNIT 3 Lesson 1 We could have an international fall festival! 25 Lesson 2 You are cordially invited. 29 Lesson 3 Fall Foods 32 UNIT 4 Lesson 1 Excuses .
Participant's Workbook Financial Management for Managers Institute of Child Nutrition iii Table of Contents Introduction Intro—1 Lesson 1: Financial Management Lesson 1—1 Lesson 2: Production Records Lesson 2—1 Lesson 3: Forecasting Lesson 3—1 Lesson 4: Menu Item Costs Lesson 4—1 Lesson 5: Product Screening Lesson 5—1 Lesson 6: Inventory Control Lesson 6—1
Fourier transform In this section, we generalize the encryption system presented in section 2 of Ref. [14] using fractional Fourier operators, such as the FrFT (Appendix A), fractional traslation (Appendix B), fractional convolution (Appendix C) and fractional correlation (Appendix C).
Oct 10, 2014 · LAPLACE TRANSFORM OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS SONG LIANG, RANCHAO WU, LIPING CHEN Abstract. In this article, we show that Laplace transform can be applied to fractional system. To this end, solutions of linear fractional-order equations are rst derived by a direct method, w
Ito formula for the two-parameter fractional Brownian motion 1 Itô formula for the two-parameter fractional Brownian motion . to study the problem of stochastic calculus for two-parameter Gaussian processes. The canonical example of such processes is the the fractional . time, including a Tanaka formula. Section 4 describes the extension .
228 Fractional Calculus: Integral and Differential Equations of Fractional Order We now observe that an alternative definition of fractional derivative, orig-inally introduced by Caputo [19], [27] in the late sixties and adopted by Ca-puto and Mainardi [28] in the framework of the theory of Linear Viscoelasticity
