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Modeling Division of Whole Numbers by Unit Fractions

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Grade Level/Course: Grade 5 Lesson/Unit Plan Name: Modeling Division of Fractions – Division of a Whole Number by a Fraction. Rationale/Lesson Abstract: Before students begin using multiple algorithms for the division of fractions, students should have a conceptual understanding of division of fractions. In this lesson students will understand the concept of division of fractions by using fractions tiles and other models. Timeframe: Dividing a Whole Number by a Unit Fraction - 45 minutes Common Core Standard(s): Grade 5: 5.NF7 Apply and extend previous understandinGs of division to divide unit fractions by whole numbers and whole numbers by unit fractions a. Interpret division of a unit fraction by a non-zero whole number, and compute such Quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the Quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3. b. Interpret division of a whole number by a unit fraction and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the Quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and divison of whole numbers by unit fractions, e.g., by using visual fraction models and eQuations to represent the problem. For example, how much chocolate will each person get if 3 people share ½ lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? Grade 6: Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS1 Interpret and compute quotients of fractions, and solve real-world problems involving division of fractions by fractions, e.g., by using visual fractions 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 ½ 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 ¾ mi and area ½ square mile? Instructional Resources/Materials: - Class sets of Fraction Tiles - Colored pencils and/or highlighters Page 1 of 17 MCC@WCCUSD 02/20/13 Activity/Lesson: A few tips for using Fraction Tiles (or any manipulative): 1. Have a set of Fraction Tiles for each student or each pair of students. 2. Have a system for passing out and collecting the Fraction Tiles. All Fraction Tiles should be in a closed Ziploc bag when passed out and returned in the same manner. 3. Fraction Tiles must remain on the desk and should not be on the floor. Students should handle the fraction tiles with care so that the fraction tiles can be used over and over again. Have students check the floor, at the end of the lesson, before returning the Fraction Tiles. 4. Give students about 2 minutes, prior to the actual lesson, to “play” with fraction tiles. Ask students to sort the Fraction Tiles by the end of the time limit. This will be helpful for selecting the Fraction Tiles for each example. 5. As the teacher, you will need your own set in order to model each problem. Page 2 of 17 MCC@WCCUSD 02/20/13 12 Division of Whole Numbers by Unit Fractions – An Introduction Example 1 “One divided by one-half means “how many halves are in one?”” Pull out the One Fraction Tile. “This length represents one-whole. We need to find how many halves are in one-whole. Find the Fraction Tiles labeled one-half. Place the one-half Fraction Tiles on top of the one-whole Fraction Tile.” (You could also place the one-half Fraction Tiles underneath the one-whole Fraction Tile. This will help students understand how to record their work. It also can help students work with bar models). “How many one-halves are in one-whole?” Count the number of one-halves. “There are 2. Therefore, there are 2 one-halves in one-whole. One divided by one-half equals 2.” Create in your (teacher) notes, on the board, or on chart paper the following: Problem: Think about it as: Build it with Fraction Tiles: Simplify: 1 How many are in 1? 1 1÷ 2 = 2 Have students record the work in their math notebooks or on a graphic organizer. Students should then place the Fraction Tiles back in their organized piles to show they are ready for the next problem. 1 Example 2 1÷ 3 Continue Example 2 using similar questions and language as Example 1. “One divided by one-third means “how many thirds are in one?”” Pull out the One Fraction Tile. “This length represents one-whole. We need to find how many thirds are in one-whole. Find the Fraction Tiles labeled one-third. Place the one-third Fraction Tiles on top of (or underneath) the one-whole Fraction Tile. How many one-thirds are in one-whole?” Count the number of one-thirds. “There are 3. Therefore, there are 3 one-thirds in one-whole. One divided by one-third equals 3.” Continue in Teacher notes, on the board, or on chart paper the following: Problem: Think about it as: Build it with Fraction Tiles: Simplify: 1 1 How many ’s are in 1? 1 1÷ 3 3 1 3 3 = Have students record the work and place the Fraction Tiles back in their organized piles, ready for the next problem. 1 You Try 1 1÷ 4 Continue the You-Tries using similar Questions and language as Example 1 and 2. Students can work in partners to build the division problems and find the quotient. Once partners have agreed upon a correct model and Quotient, students should add the problem to their notes/graphic organizer. Debrief: Have a pair of students come to the board/document camera to share their thinking and how they solved the problem using the Fraction Tiles. Encourage students to use the academic language as addressed in Examples 1 and 2. Page 3 of 17 MCC@WCCUSD 02/20/13 “One divided by one-fourth means “how many fourths are in one?”” Show the One Fraction Tile and the one-fourth Fraction Tiles. “Let’s find how many fourths are in one-whole. Place the one-fourth Fraction Tiles on top of (or underneath) the one-whole Fraction Tile. How many one-fourths are in one-whole?” Count the number of one-fourths. There are 4. Therefore, there are 4 one-fourths in one-whole. One divided by one-fourth equals 4. Problem: Think about it as: Build it with Fraction Tiles: Simplify: 1 1 1 1÷ 1÷ How many are in 1? 4 4 = 4 Record in Teacher notes, on the board, or on chart paper the following and make sure students have correctly recorded the problem in their notes/graphic organizer. Check-in with students and see if they understand the division of fractions done so far. Students can show “thumbs-up” if the understand and are ready for another You-Try, maybe even independently. Students can show “thumbs-sideways” if they think they got it but are not sure; they would like to do another partner You-Try. And students can show “Thumbs-down” if they don’t get it or feel lost; they would like another example by the teacher. Make adjustments to the next few problems, whether the problems are completed as Examples or You- Tries (done in partners or independently) based on student feedback. Continue to record problems in notes/board/chart paper. 1 Example 3 1÷ 5 Predict: Based on our previous examples and you-tries, how many fifths do you think are in 1? Have students show on their fingers. [Take all student answers. Have students explain how they came up with his/her prediction] Let’s build it. Show me with your Fraction Tiles how to build One-whole divided by one-fifth. Give students enough time to use the Fraction Tiles to build the division problem. What Question am I thinking about when I am solving ? [ How many one-fifths are in one-whole?] How many one-fifths are in one-whole? [5] Have students record: Problem: Think about it as: Build it with Fraction Tiles: Simplify: How many are in 1? 1 “One divided by one-fifth means “how many fifths are in one?”” Find the One Fraction Tile, and all of the one-fifth Fraction Tiles. “How many one-fifths are in one-whole?” Count the number of one-fifths. “There are 5. Therefore, there are 5 one-fifths in one-whole. One divided by one-fifth equals 5.” Page 4 of 17 MCC@WCCUSD 02/20/13 Continue recording in Teacher notes/board/chart paper and have students correctly record in his/her notes/graphic organizer. Discussion: What pattern do you notice so far? Have students discuss this Question in partners and share out as a class. [The number of “pieces” that are needed to make the whole is the same as the denominator.] [The Quotient is the same as the denominator of the divisor.] More Examples and You Tries Continue, as needed, dividing 1 by other unit fractions so that students see the relationship between the denominator and the quotient (how many pieces are in the 1 whole).
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