Activity 15.8Slicing SquaresGive students a worksheet with four squares in a row, each approximately 3 cm on a side. Have them shade in the same fraction in each square using vertical dividing line. You can use the context of a garden or farm. For example, slice each square in fourths and shade three-fourths as in Figure 15.20. Next, tell students to slice each square into equal-sized horizontal slices. Each square must be partitioned differently, using from one to eight slices. For each sliced square, they record an equations showing the equivalent fractions. Have them examine their equations and drawings to look for any patterns. You can repeat this with four more squares and different fractions.What product tells how many parts are shaded?What product tells how many parts in the whole?Notice that the same factor is used for both part and whole
Give students an equation expressing an equivalence between two fraction but with one of the numbers missing and ask them to draw a picture to solve. Here are four different examples:5/3 = _/62/3 = 6/_8/12 = _/39/12 = 3/_The missing number can be either a numerator or a denominator. Furthermore, the missing number can either be larger or smaller that the corresponding part of the equivalent fraction. (All four possibilities are represented in the examples.) The examples shown involve simple whole-number multiples between equivalent fractions. Next, consider pairs such as 6/8 = _/12 or 9/12 = 6/_. In these equivalences, one denominator or numerator is not a whole number multiple of the other.
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4th GradeWord Problems and Fractions Laura Chambless RESA Consultant www.protopage.com/lchambless
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CCSS and Gaps What are your gaps in curriculum?1. Review CCSS for Fractions2. Think about your resources3. Think about your teaching – Highlight anything your resources covers well in YELLOW. – Highlight any part of the standard you would like more clarification on in BLUE.
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Learning TargetExtend understanding of fractionequivalence and ordering. 4.NF.1, 4.NF.2Build fractions from unit fractions byapplying and extending previousunderstandings of operations onwhole numbers. 4.NF.3, 4.NF.4
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FractionsWhat conceptual understanding do students need? 1. Begin with simple contextual tasks. 2. Connect the meaning of fraction computation with whole number computation. 3. Let estimation and informal methods play a big role in the development of strategies. 4. Explore each of the operations using models.Van De Walle Book: Number Sense and Fraction Algorithms
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Fraction Word Problem40 students joined the soccer club. 5/8 of the students were boys. How many girls joined the soccer club? Draw a picture and solve it.1. 2 min. working problem on own2. 5 min. sharing with group3. Class discussionFound at: http://www.mathplayground.com/wpdatabase/Fractions1_3.htm
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Problem Solving with Bar Diagrams1. Understand: Identify what is known and what is unknown. Draw the bar diagram to promote comprehension and demonstrates understanding. (Situation vs. Solution Equation)2. Plan: Decide how you will solve the problem (find the unknown). Analyze the bar diagram to find a solution plan.3. Solve: Execute the plan. Use the bar diagram to solve.4. Evaluate: Assess reasonableness using estimation or substitution. Substitute the solution for the unknown in the bar diagram.
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Bar DiagramsWatch Introduction Videohttp://www.mhschool.com/math/com mon/pd_video/mathconnects_bardi agram_p1/index.htmlhttp://www.mhschool.com/math/com mon/pd_video/mathconnects_bardi agram_p2/index.html
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Practice Bar DiagramsTo: Rani earned $128 mowing lawns and $73 babysitting. How much money did Rani earn?With: Jin had $67 in his pocket after he bought a radio controlled car. He went to the store with $142. How Much did Jin spend on the car?By: There are 9 puffy stickers. There are 3 times as many plain stickers as puffy stickers. How many plain stickers are there?You pick 2 more to do by yourself. Share with partnerDraw Your Way to Problem Solving Success Handout, Robyn Silbey
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Thinking Blockshttp://www.mathplayground.com/think ingblocks.html Explore the site When done exploring go to my Protopage and look at your grade level math tab.
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Fractions Stand and ShareMake a list of what you know and any connections you have about the fraction ¼.
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Representations (Part 2 video, 5:16)Set Purpose of video: List why representations are important in the classroom. •Representations are mathematics content representing mathematical ideas is a practice that students need to learn. •Representations provide tools for working on mathematics and contribute to the development of new mathematical knowledge. •Representations support communication about mathematics. •Using multiple representations can help develop understanding and support the diverse needs of students. From: Dev-TE@M session 2
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Examining Representations (Part 3 & 4 Video 1:48/2:15)Set Purpose of videos: listen to the set up of your task and example.1. Examining Representations of ¾ with a partner (10 min)2. Whole group discussion3. Review math notes From: Dev-TE@M session 2
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Making Connections (Part 6 video, 2:22)Set Purpose of video: think about our discussion of ¾, what connection types did we use?Have you ever used connections for the different math representations in your classroom? From: Dev-TE@M session 2
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Benefit of Representations (Part 4 video, 2:17)Set Purpose of video: Did you benefit from our discussions, and how will your students benefit from class discussions?1. As you listen , list benefits for students2. Compare list with partner From: Dev-TE@M session 3
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Definition of Fractions1. Make a list of what you would like to have in a definition of a fraction2. Partner up and compare lists3. Group discussion From: Dev-TE@M session 3
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Definition of a Fraction (Part 5 and 6 videos, 11:48/4:27)Set Purpose of video: What are some key parts in creating a definition of a fraction that you will use in your room?– Give handout of working definitionArticle: Definitions and Defining in Mathematics and Mathematics Teaching by: Bass and Ball From: Dev-TE@M session 3
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Definition Of Fractions• Identify the whole• Make d equal parts• Write 1/d to show one of the equal parts• If you have d of 1/d, then you have the whole• If you have n of 1/d, then you have n/d• n and d are whole numbers• d does not equal 0Dev-TE@M • School of Education • University of Michigan • (734)408-4461 • dev-team@umich.edu For review only - Please do notcirculate or cite without permission
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Fractions Fraction ActivityPaper Strips Fraction Kit: 1, ½, 1/4 , 1/8, 1/16Add to Fraction Kit: 1/3, 1/6, 1/12Add to Fraction Kit: 1/5, 1/10 Compare/Add/Subtract/with Strips READ and DO:4.NF.3a, 4.NF.3b, 4.NF.3cPlay Greater Than, Less Than, Equal• Prove with Fraction Strips
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Ordering Fractions Order Fractions 8/6, 2/5, 8/10, 1/12How did you figure out what order they went in?
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Fractions Prove with Fraction StripsNumber Line: (Benchmarks) 0, ½, 1Compare (>/<): same numerator or same denominatorEquivalent Fractions: Same Name Frame
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Strategies for Comparing FractionsKey points• The following practices are helpful when analyzing students’ work on tasks:• Anticipate the strategies and representations students may use.• Identify the strategies students did use. If the student used a different strategy than predicted, consider if is it a fitting choice.• If the strategy is unfamiliar, explore whether or not the strategy is mathematically valid.• Identify questions to ask the student about her/his strategy or new problems to pose that would further reveal her/his understanding. From: Dev-TE@M session 9
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Strategies for Comparing FractionsMath Notes: Strategies for Comparing FractionsWhich strategies do you use in your classroom? From: Dev-TE@M session 9
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Fraction On A Number LineWriting about Fractions: Draw a number line. Place 3/6 and 7/12 on the number line. Compare the two fractions- why did put them where you did?
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Key Ideas About the Number LineWhat were some intentional talk moves others used to explain their number line? (Part 5 video, 5:26)Set purpose of video: Listen to the detail that is given in explaining how to construct a number line. From: Dev-TE@M session 4
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Conventions Of A Number LineDev-TE@M • School of Education • University of Michigan • (734) 408-4461 •dev-team@umich.edu For review only - Please do not circulate or cite withoutpermission From: Dev-TE@M session 4
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Talking Through A Number Line1. Understand the problem.2. Think about which representation you are going to use.3. Describe your thinking process while constructing the number line.4. Sum up the solution that proved your answer.Model Example: 3/10 & 6/8
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Fraction On A Number LineUsing a number line, compare 5/6 and 3/8 and tell which one is greater . Have a partner listen to you as you construct the fractions and find the answer.
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Student ErrorsWhat value should be written where the arrow is pointing? What would kids write? Session 4-6: Analyzing students’ errors when labeling marked points on the number line- see slides From: Dev-TE@M session 4
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Student ErrorsKey points When determining how to respond to a student, it can be helpful to consider:• What question(s) could be asked to learn more about the student’s thinking?• What key mathematical idea(s) might be raised with the student?
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Narrating a Representation• Make clear the mathematical problem or context.• Describe how a particular representation is useful for this problem.• Construct the representation and use it to solve the task while describing and giving meaning to each step.• Summarize what the representation has helped to do. From: Dev-TE@M session 5
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Number Lines (Part 2 video, 1:21)Set purpose of video: listen to directions and practice narrating on the number line. Partner Work Compare ¾ and 4/3 From: Dev-TE@M session 5
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Number Lines (Part 3 video, A 3:32/C 1:29/ E :28)Set purpose for video: Where are the problems when narrating the number line? (Part 5 video, 4:24) Set purpose for video: review narration (Part 6 video, 1:53) Set purpose for video: What fractions do you use for examples From: Dev-TE@M session 5
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Add/Subtract Fractions with Unlike Denominators Developing Equivalent Fractions• Slicing Squares Van de Walle book: pg. 304-305 3 x = 3 x 4 = 4 3 x 3 x = = 4 4
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Developing Equivalent FractionsMissing-Number Equivalencies Van de Walle book: pg. 304-305 5 2 6 = = 3 6 3
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Methods for Generating andExplaining Equivalent FractionsMath Notes: Methods for Generating and Explaining Equivalent Fractions Pair Share1. Partner 1: Reads - Reasoning about equivalent fractions using an area model2. Partner 2: Reads - Reasoning about equivalent fractions using a number line3. One minute report4. Report on how your model was different than your partners. From: Dev-TE@M session 9
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FractionsMultiply a fraction by a whole number READ and DO: 4.NF.4a, 4.NF.4b• Work as a group• Use Fraction strips to show answers 4 x 1/3 ¼ x 12• What connection can you make to multiplication? What other representations can you use? Can you use a number line?
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Multiple a Fraction by a Whole Number 4 x 1/3 (4 groups of 1/3) = 4/3 = 1 1/3I want 4 ribbons each at 1/3 of a yard. How much ribbon will I need to purchase? 1/3 2/3 3/3 4/3 ¼ x 12 (1/4 of 12) = 3I have 12 cookies and want each of my friends to have ¼ of them. How many cookies will each friend get?
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MOPLS http://mi.learnport.org Search: MOPLS Math (navigate by using top tabs)Look at Concepts Tab– Introduction– Math Behind the Math– Misconceptions– Tasks & Strategies
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Fractions OnlineCheck out some sites on my 4th grade math Protopage
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Learning TargetExtend understanding of fractionequivalence and ordering. 4.NF.1, 4.NF.2Build fractions from unit fractions byapplying and extending previousunderstandings of operations onwhole numbers. 4.NF.3, 4.NF.4
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Closer ActivityList something you learn about story problems and fractions today.
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Thanks for a great day Please contact me if you have any questions or would like more information.
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