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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 1 2 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: How many are in 1? 1 1÷ Simplify: = 2 1 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. Example 2 1÷ 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: 1 3 How many ’s are in 1? Build it with Fraction Tiles: 1 1 Simplify: 1÷ 1 3 = 3 Have students record the work and place the Fraction Tiles back in their organized piles, ready for the next problem. 3 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: 1÷ 1 4 Build it with Fraction Tiles: How many are in 1? 1÷ 1 Simplify: 1 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. Example 3 1÷ 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: How many are in 1? 1 Simplify: “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). All problems should be recorded in Teacher notes/board/chart paper and students should correctly record in his/her notes/graphic organizer. ReAsk students about the patterns they see in the relationship between the dividend, divisor and quotient. Continue to discuss this throughout the lesson. Page 5 of 17 MCC@WCCUSD 02/20/13 Teacher & Student Notes Should look like this: Problem: Think about it as: Build it with Fraction Tiles: How many are in 1? 1 1 How many are in 1? 1 How many are in 1? 1 How many are in 1? 1 How many are in 1? 1 How many are in 1? 1 How many are in 1? Page 6 of 17 Simplify: MCC@WCCUSD 02/20/13 Division of Whole Numbers by Unit Fractions We will use what we learned previously and apply our knowledge of dividing one-‐whole by a unit fraction to dividing any whole number by a unit fraction Note: Since we will be working with whole numbers other than one, students may not have Fraction Tile kits with enough pieces to independently build the next set of Examples and You-‐Tries. This would be a good time to begin drawing the models in your notes and not use the Fraction Tiles for every problem. If students still need the experience of building the division problem, then only the students who need the Fraction Tiles should use the kits (and they will need more then one kit). The Fraction Tiles can still be used at the board for the teacher to explain the problems or to have students share his/her thinking about a problem under the document camera. Fraction Tile kits can also be used with small groups of students that still need the support of building the model. Example 4 Three divided by one-‐half means “how many halves are in three?” How many one-‐halves are in three-‐ wholes? Draw the model. (We will begin by using a model similar to the Fraction Tiles and begin to use other models in the Examples and You-‐Tries that follow). Since we have three-‐wholes, we will draw a rectangle that is three units in length. How many halves were in one-‐whole? [2] Under each one-‐unit rectangle, draw a rectangle to show each one-‐half piece. You should have a drawing like the one below: 1 1 1 2 1 To find the quotient of , count the number of one-‐halves. (to emphasize this step in our drawing, you can circle each half. However, this may be something that is skipped once students understand the concept. Circling will help students when dividing by a non-‐unit fraction like two-‐thirds ) There are 6 halves in three. Therefore, there are 6 one-‐halves in three-‐wholes. Three divided by one-‐half equals 6. Problem: Think about it as: How many Simplify: There are 6 ’s are in 3? ’s in 3. Model: 1 1 1 Page 7 of 17 MCC@WCCUSD 02/20/13 Example 5 Problem: Think about it as: 2÷ There are 10 How many ’s are in 2? 1 5 Simplify: Model: 2 1 1 1 5 For the Fraction Bar Model: Draw a bar that represents 2. Show that 2 means 2-‐wholes. Now, each 1-‐whole can be drawn to show 5 fifths are in 1. Count the number of fifths in 2. There are 10 fifths in 2. (In this model each fifth is labeled with a and each one is circled in order to count the number of fifths.) ’s in 2. 1 ∴2 ÷ = 10 5 Another Model: 1 6 2 3 5 7 8 10 9 4 For the Circle Model: Draw 2 circles to represent 2-‐wholes.. Each circle should be divided into fifths. Counts the number of fifths in 2. There are 10 fifths in 2. (In this model each circle is drawn to show 5-‐fifths. Then, each one-‐fifth section is labeled with a number in order to c ount the number of fifths.) Example 6 Josh has 3 oranges. He cut each orange into 1 slices. How many slices does Josh have? 4 Problem: Think about it as: Simplify: There are 12 How many ’s are in 3? ∴3÷ ’s in 3 1 = 12 4 Model: Another Model: 1 2 5 6 9 10 3 4 7 8 11 12 Each square is cut into pieces. There are 4 ’s in each circle. There are 12 ’s in 3 circles. Count each piece. Page 8 of 17 MCC@WCCUSD 02/20/13 1 3 You Try 2 Maria is planning to sew some hats. If the fabric if 5 feet long and she needs feet fabric for each hat, how many hats can she make with the fabric? Problem: Think about it as: 0 Simplify: How many are in 5? There are 15 s in 5. 1 ∴5 ÷ = 15 3 Model: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 “Cut” the 5 ft piece of fabric into 15 pieces. There are 15 1 2 3 4 ft pieces. 5 Another Model: 1 1 1 1 1 You Try 3 Problem: Think about it as: Model: There are 8 1 Simplify: How many are in 2? 1 Page 9 of 17 ’s in 2. Another Model: MCC@WCCUSD 02/20/13 ft You Try 4 Lisa eats 1 cup of raisins everyday, which is one serving. If she buys a package that has 4 cups of 3 raisins, how many servings of raisins does she have? Problem: Think about it as: Model: 1 1 1 1 Simplify There are 12 How many ’s are in 4? ’s in 4. 2 1 3 Page 10 of 17 Another Model: 5 4 6 8 7 9 MCC@WCCUSD 02/20/13 10 11 12 Extension: Divide a Whole Number by a Non-‐Unit Fraction Example 7 Problem: Think about it as: How many Simplify: There are 5 ’s are in 2? ’s in 2. Model: 2 1 1 Example 8 Five divided by two-‐thirds means “how many groups of two-‐thirds are in five?” How many two-‐thirds are in five-‐wholes? Draw five-‐wholes. Next, we need to decide how many pieces to break each whole into. Since we want to know how many two-‐thirds we have, we are working with thirds (two one-‐thirds) – 3 is the denominator which tells us how many pieces make a whole. How many one-‐thirds were in one-‐whole? [3] Under each one-‐unit rectangle, draw a rectangle to show each one-‐ third piece. To find the quotient, we will find how many groups of two-‐thirds we have in five-‐wholes. Circle groups of two one-‐thirds. We have 7 whole groups of two-‐thirds. What about the “extra” third? Since we are looking for groups of two-‐thirds, two-‐thirds becomes the “whole”. We have one-‐third, which is half of the group we need (one-‐third is half of two-‐thirds ) There are 7 ½ groups of two-‐thirds in five. Therefore, five divided by two-‐thirds is 7 ½. Problem: Think about it as: How many Simplify: There are ’s are in 5? ’s in 5. Model: 1 1 1 2 1 3 4 1 5 Page 11 of 17 1 6 7 MCC@WCCUSD 02/20/13 Assessment: Determine whether the given model represents the division problem. Explain why the division problem does or does not represent the model. 1. Yes No Explain: 2. Yes No Explain: -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Write a story context for the given division problem. Then draw a model and solve. 3. Page 12 of 17 MCC@WCCUSD 02/20/13 Assessment: Sample Responses Determine whether the given model represents the division problem. Explain why the division problem does or does not represent the model. 1. No Yes Explain: No, the model does not represent the division problem. There should be 6-wholes and each separated into 6 pieces so that it models “six divided by one-sixth”. This model represents “two divided by onesixth”. 2. No Yes Explain: Yes, the model does represent the division problem. There are 2wholes and each separated into 10 pieces, which is one-tenth of the whole. “2 divided by one-tenth is 20”. -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Write a story context for the given division problem. Then draw a model and solve. 3. [Answers may vary] John has three cups of trail mix to share on the hike. If he puts ½ cup of trail mix in each baggie, how many baggies of trail mix will he have? 1 5 2 3 4 6 I drew 3 circles to represent the three cups of trail mix, then divided each into ½. I counted each half circle and there are 6, therefore, I know that 3 divided by ½ is 6. John will have 6 baggies of trail mix to share with ½ cup trail mix in each baggie. Page 13 of 17 MCC@WCCUSD 02/20/13 Warm-Up y CST/CAHSEE: 5NS2.3/5.NF1 Which expression(s) below can be used to simplify: Review: 4NS3.2/ 4NBT.6 Divide. a. b. c. d. x Current: 5NS2.5/5NF4a Other: 4NS3.2/ 4NBT.6 Write a story problem for: Draw a picture to show: 4• 1 4 348 ÷ 4 n Page 14 of 17 MCC@WCCUSD 02/20/13 Warm-‐Up Debrief: Quadrant 1: Review Divide. Story Context for 348 divided by 4: Method 1: Traditional Algorithm Quadrant 4: Other Method 2 : Using the familiar facts of 4 to determine how many groups of 4 are in 348. There are 348 students going on a field trip. If there are 4 buses, how many students will be on each bus? (Other story context are possible) 25 25 87 25 12 Quadrant 2: CST Which expression(s) below can be used to simplify: Method 1 : Subtracting Whole Numbers and subtracting fractions. a. Method 2: Convert to Improper fractions, and then get common denominators. b. Yes, you get this expression when you decompose the whole numbers and the fractions. (see Method 1) No, The m ixed numbers are not converted to improper fractions correctly. (see Method 2 ) c. d. Yes, correctly converts the fractions to get a common denominator. (see Method 1) Yes, correctly converts to improper fractions and gets common denominators. (see Method 2 ) Page 15 of 17 MCC@WCCUSD 02/20/13 Quadrant 3: Current Draw a picture to show: Method 1: Shows a Bar the has 4 pieces, each ¼ in length. Method 2: Shows a circle the has 4 pieces, each ¼ of the whole circle. Page 16 of 17 MCC@WCCUSD 02/20/13 Problem Think/Ask: Model: Algorithm:
