Type of Knowledge & SBAC Claim Grade 5 Fraction Multiplication and Division Unit - Conceptual Lessons Lesson Title and /Description Suggested Time Frame C, RK- 1, 2, 3 Sharing Vanilla Wafers In this lesson, students will build on their understanding of division as equal shares to use play- doh to divide amounts that do not partition equally. They will understand and describe fractions as division of whole numbers as well as explain the meaning of the numerator and denominator in context. C, RK- 1, 2, 3 Making a Zip Line Students will cut or fold play- doh representing a linear measure into equal parts to understand the relationship between division of whole numbers and fraction as well as to explain the meaning of the numerator and denominator of a fraction. P- 1 RK- 2 C- 1, 3 Practice and problem solving- Using fractions to represent the division of whole numbers with number lines or other models. Painting a Wall- Area and Multiplication of Fractions Students will represent a fraction of a rectangle using play- doh and then show what a fraction of that fraction would represent. Students will connect this representation to a visual model of the area model and use this to multiply fractions. Students will study patterns to generalize a rule for multiplication of fractions. 1-2 class 1 class period 1-2 class 2-3 class P- 1 Practice- multiplying fractions, including mixed numbers 2-3 class RK- 3 Problem solving- Multiply Fractions including finding area 2-3 class C- 1, 3 Multiplying Fractions on a Number Line Students will use a number line to model the multiplication of fractions, seeing a factor as representing one distance and the other factor representing 1-2 class Math Practice embedded 1, 2, 3, 5, 6 1, 2, 3, 5, 6 1, 2, 3, 7, 8 1, 2, 3, 7, 8 1
the fraction of that distance they will travel. Students will study the results to explain when the product of two fractions is greater than, equal to or less than 1. Students will look at patterns with the factors and products to derive the fact that a b c d = ac bc. P- 1, 3 Practice- Game- show the product of two fractions and students decide if product is greater than 1, equal to 1 or less than 1. Practice multiplying fractions 1 class period RK- 2 Problem Solving- using a number line to multiply fractions 2-3 class C- 1 Dividing Fractions I Students will use quotative division to rewrite division of fraction problems as how many are in, and use this definition with an area model to divide a whole number by a fraction. Students will first understand this by seeing how many of a smaller pattern block fit into a larger one and then relating the context to the math behind it. 1-2 class C- 1 C- 1, 3 P- 1 Dividing Fractions II Students will use partitive division and an area model to divide fractions by whole numbers. Students will explain the relationship between the division and corresponding multiplication problem. Dividing Fractions: Noticing Patterns Students will study patterns between each division problem from the previous lessons and a corresponding multiplication sentence that will have the same result. Students will explain that division of fractions is the same as multiplication of the first fraction by the reciprocal of the second fraction. Students will explain this in their own words and then be introduced to the academic term, reciprocal. Practice- dividing whole numbers by fractions and fractions by whole numbers. 1-2 class 1 class period 2 class 1, 2, 3, 7 1, 2, 3, 7 3, 6, 7, 8 2
C- 1, 3 P- 1 Dividing Fractions on a Number Line Students will use a number line to model the division of fractions, using quotative (measurement) division. Students will study patterns with types of problems to summarize what happens to the quotient of two fractions less than one, greater than one or equal to one. Students will study the pattern of the division problems with their related multiplication sentences to explain a procedure for dividing fractions. Practice- Game- show the quotient of two fractions and students decide if quotient is greater than 1, equal to 1 or less than 1. Practice dividing fractions 1-2 class 1 class period RK- 2 Problem Solving- Dividing fractions 2-3 class RK- 2, 3 How Far Can You Jump Students will measure the distance (in fractions of meters) each student can jump and represent this data on a line plot. Students will answer questions comparing distances both with additive and multiplicative comparisons, representing each question with a math sentence and solution applying the four operations with fractions. 1 class period RK- 2 Problem Solving with Line Plots 1-2 class 1, 2, 3, 5, 7, 8 1, 5 Notes: Unit Addresses: NF 3, 4a, 4b, 5a, 5b, 6, 7a, 7b, 7c, MD 2 5 Week Unit 3
Teacher Directions Materials: Play-doh (1 can per person or pair) Plastic Knife or dental floss (1 per person or pair) to cut play-doh Plates on which to place the play-doh (optional)- 1 per person or pair Optional- Vanilla Wafers for students to enact scenarios (will be messy!) In this lesson, students will build on their understanding of division as equal shares to use playdoh to divide amounts that do not partition equally. They will understand and describe fractions as division of whole numbers as well as explain the meaning of the numerator and denominator in context. Explain the scenario to the class: Caiden choose to bring in an approved healthier snack for his birthday, but he did not count out the Vanilla Wafers to pass out before he came. The class task is to figure out how many cookies each student would get based upon different scenarios. Pass out play-doh, plastic knives and plates to each student or pair. Explain that they will model the cookies with the play-doh and then draw a picture to represent each scenario. Give the class 5-7 minutes to complete the first two problems and then bring the class back together. Have a volunteer share #1, and ask if any one solved it differently. Repeat the same process for #2, where you should have a variety of explanations how to solve as well as at least two ways to record (see below). #2-4 3 or 11 3 Possible Method #1: Give each student 1 cookie and then divide remaining cookie into 3 equal parts, so everyone gets 1 whole and one-third. Possible Method #2: Divide all the cookies into 3 equal parts and pass out the one-thirds. Each student gets four one-thirds of a cookie. Have a discussion about what 4 3 means. Help students explain and see it as four one-third pieces and connect this to a drawing/model. Have a class discussion as to what 1 1 means. Help 3 students explain and see it as one whole plus one-third. If students are doing well, give them about 20 minutes to work through scenarios 3-10, encouraging them to build with play-doh, draw a picture and list the answer two ways when applicable. Bring the class back together and choose students to come share and explain their thinking, making sure to always seek an alternate methods, explanations or answers. Part 2 is meant as formative assessment. Numbers 3 and 4 are especially challenging, but will reveal deep understanding, if present! For #1, you should be looking for wording such as each whole (or wafer or cookie) is divided into 3 equal parts and each person gets 5 of those onethirds. For #2, look for language that explains the whole is divided into 5 equal parts and each student gets 3 of those parts. Problems 3 and 4 can have multiple, correct answers. IMP Activity: Sharing Vanilla Wafers 3
Teacher Directions Materials: Play-doh (1 can per person or pair), Plastic Knife and Plate OR Strips of paper Optional- rope and scissors for students to enact scenarios Optional- rulers (1 per person or pair) Students will cut or fold play-doh representing a linear measure into equal parts to understand the relationship between division of whole numbers and fractions as well as to explain the meaning of the numerator and denominator of a fraction. Show the students a piece of rope (or play-doh rolled out like a rope or a strip of paper). Ask how they can measure this length into three equal parts. Let them think silently for a minute and then pass out rope, play-doh or strips of paper and let the students experiment. Allow students to share ideas with their neighbors and then choose students to share strategies with the class. Pass out the activity sheet and have a student read the background aloud. Once students understand the task, allow them to use play-doh, string or strips of paper to solve. Explain that you want them to solve with materials and then represent the solution with a drawing in the space provided. Note: students can use rulers to measure in inches or they can make the whole length change each time and then define the part in terms of the distance the whole represents. Encourage students to write each solution two ways, if possible (mixed number and fraction). Circulate to ensure students are understanding. If you find many students struggling, bring the class back and have volunteers model a few problems. Focus your questions to students on ideas of, How long is the whole? How many parts is the whole divided into? How much does each boy get? Part 2. Students are now directly modeling their work above by placing the solutions on the number line. Have students record the fraction or mixed number above each point as well as which problem # the point represents. Choose a student who has a correctly labeled number line and place this on the document camera so students can check their answers. Once all students have correct answers, give them time to answer the questions. Let them think and work silently for 5 minutes on these questions before discussing answers with a partner. Below are solutions: 1. 3 or1 These are two names for 3 3 3 2. 8 3 or2 2 3 These are two names for 8 3 3. 2 3 These is another name for 2 3 4. 1 These is another name for 1 3 3 Completethe lesson by having students write the summary. Select a few students to read what they wrote. The big idea is that fractions are a way to represent division of whole number- the denominator representing how many equal parts the whole is divided into and the numerator representing how many equal parts each person gets. IMP Activity: Making a Zipline 3
Teacher Directions Materials: Play-doh (1 can per person or pair), Plastic Knife and Plate Colored Pencils or highlighters Students will represent a fraction of a rectangle using play-doh and then show what a fraction of that fraction would represent. Students will connect this representation to a visual model of the area model and use this to multiply fractions. Students will study patterns to generalize a rule for multiplication of fractions. Pass out the activity sheet and give the students two minutes to read and complete the prediction section (we will come back to this later in the lesson.) Pass out play-doh, plastic knives (or dental floss) and a plate to each person or pair of students. Ask the students to form their play-doh into a rectangle or rectangular prism. Ask them to use their knives to show 1. Call on a few students to come show how they represented one-third 3 and how they know the piece is really one-third. Have the students use colored pencils to show one-third on the picture model. Repeat this process for the remaining problems in part 1. It is okay if students represent their fraction differently, as long as it really is the correct fraction of the wall and they can explain. For part 2, however, students will soon find that shading like above will be easier to do with all models (generalizing!). Thebig ideas for part 1 are: 1) 1/3 can llook different ways cuts can be horizontal or vertical or other ways too and 2) we need equal parts if we are going to use the fractional names 2 pieces that are unequal cannot be called halves). Direct the class attention to part 2. Have each student build their wall again and show onethird. Then ask, what if you only painted half of that amount on day 1? Have the students use their play-doh to show one-half of the one-third. Ask, how many equal pieces is the whole divided into (shoud be 6, regardless of which method students used). Ask how many pieces represent what they will paint that day (should be one of the sixths, or 1 6 ). Have students share methods, and if possible, guide the students to agree upon a method where you mark halves on one side and thirds on the other side (as shown below). If other methods seem equally easy, you can agree upon this during a later problem (such as number 4 where other methods get too messy). See example. IMP Activity: Painting a Wall- Area and Fractions 7
1/2 of 1/3 1/3 If students did well with #1, let them continue using play-doh and then drawing a picture to represent the fraction of the wall painted each time. Have students come present their play-doh model and picture for each problem, making sure to ask for alternate ways to build or draw. Discuss the alternate ideas, noting which are the same (commutative property) and which are different and if they still work and what advantages or disadvantages might be. Make sure to support all correct methods, but agree, by the end of #4, to use the model above (or the same model with the dimensions reversed). Part 3 Allow students to continue with play-doh, if they would like, but require them to draw using an area model. (See below for #1). Have the students try #1, come back to have students present and discuss and then repeat this for #2. After this, allow students time to work on their own for about 20 minutes (while you circulate and ask questions) before coming back and having students share their thinking and work. 1/2 a) How many equal parts is the whole divided into? 8 b) What fraction of the whole represents the area or product? 1/8 c) So, 1 4 1 2 = 1 8 1/4 of 1/2 Part 4 Some students may have already recognized a pattern for how to multiply without the area model- great! To help all students see this, have them begin by filling in the table, using the answers for part 3. (To save time, you can pre-fill in the table and put up the correct answers). Use think-pair-share to have students answer the conclusion questions. The goal of this section is for students to use logic in repeated reasoning to see that the product of two fractions can be found by calculating the product of the numerators and the product of the denominators. Additionally, the revisiting the prediction section is intended for students to explain the concept of the multiplication of fractions as finding a part of a part. You should hihglight answers that reveal that the product can be less when multiplying two fractions less than one as you take a fraction of a whole and then only a fraction of that fraction. Note that we will deal with the concept of when the product of two fractions is less than one, equal to one or greater than one in a subsequent lesson. IMP Activity: Painting a Wall- Area and Fractions 8
Teacher Directions Materials: Optional- Number line or meter stick on the floor Students will use a number line to model the multiplication of fractions, seeing a factor as representing one distance and the other factor representing the fraction of that distance they will travel. Students will study the results to explain when the product of two fractions is greater than, equal to or less than 1. Students will look at patterns with the factors and products to derive the fact that a b c d = ac bd. Pass out the activity sheet and have a student read the opening scenario aloud. Give the students a few minutes to complete the prediction section on their own. Let them share ideas with a partner and then select students to share with the class. To check their predictions and help with the concept of the lesson, have two students come up front to model. Have one student try to jump and land on a half-meter. Have another student jump about half that distance and discuss where the second student is in relation to the first. Repeat this for the next two scenarios. Task 1 Ask the student to point to 1 on their number line. Direct the class attention to the sentence 2 frame just above the number line, find of one-half. Have the class chorale reply to explain what letter a means. (They should say, find one-fourth of one-half ). Since they have already found one-half, give them a minute to figure out what one-fourth of that distance would be (there should be marking going on!). Call on a few students to come up and show you how they determined where one-fourth of one-half was and then ask the class what the name for this point is and how they know. They should state 1 8. Label this point on the number line as both 1 8 and under that, 1 4 of 1 2. Follow the same process for problem b and then have students work in pairs to complete the rest of task 1. Bring the class back together to have them share their solutions and explain how they came up with their solutions. Task 2 The students will be following the same method of thinking in task 2, but now finding a fraction of one-fourth. Ask the class to tell you the sentence frame for problem a, and then have them work alone on the remaining problems. Come back together and select students to present answers and methods. Noticing Patters: What Happens to the Product Complete this section together, by showing the students on which problems to focus, reading the sentence and then using think-pair-share to have the class explain what they have noticed and how this relates to the predictions. They should conclude that the product of two fractions less IMP Activity: Multiplying Fractions on the Number Line 4
than 1 results in a product less than 1; the product of a fraction and 1 is the fraction itself and the product of a fraction and a number greater than 1 is a number greater than the fraction. Noticing Patterns: What s the Algorithm? This last section provides an opportunity for students to verify what they may have already discovered using the area model to multiply fractions: that you can multiply the numerators and denominators to find the product. Give the students a few minutes to study the table and then complete the summary. Call on students to read their summary and show the example problem to the class. IMP Activity: Multiplying Fractions on the Number Line 5
Teacher Directions Materials: Play- doh (1 small tub per student) Plastic knife (one pr student) Paper plates (1 per student) Colored Pencils (options) Students will use quotative division to rewrite division of fraction problems as how many are in, and use this definition with an area model to divide a whole number by a fraction. Students will first understand this by seeing how many of a smaller pattern block fit into a larger one and then relating the context to the math behind it. Pass out the activity sheet and have students read the opening question. Have 12 students come up to the front of the class and stand in one long line. Use think- pair- share to have students decide how many 4 s there are in 12. Have a student show how to measure out three fours in the 12 by having a group of four students sit down, then another group of four sit down, then the last group of four sit down. Relate this to repeated subtraction by writing a subtracting equation(s) as students sit down (12-4- 4-4=0 or 12-4=8, 8-4=4, 4-4=0). Explain that you will be using this measurement definition of division to divide whole numbers by fractions today. Pass out the play- doh and have students build a rectangle to represent one whole tray of brownies in #1. Ask students to cut the tray of brownies in half and think about how many 1/2s there are in 1. Allow student to continue using their play- doh and solving the remaining questions. a) Write the problem in words: How many 1 s are there in 1? 2 b) Using a rectangle, show 1 1 2 : *The cut could be horizontal instead of vertical. c) There are 2 halves in 1. (one whole) Show using repeated subtraction. d) Symbols: 1 1 2 = 2. IMP Activity: Dividing Fractions I 4
For #10, this is a great opportunity for students to share their method which may be different that others in the class. Once option is to use inside- outside line. Number students off by 1 s and 2 s. All the ones stand in a long line side by side; all the 2 s line up in a line across from the 1 s so each person has a partner. Students bring their paper to the inside- outsdie line and explain how they solved, then listen to how their partner solved. Then the person at the end of the #1 line goes to the other end of the line and everyone in the 1 s line shifts over one and has a new partner. Rotate 2-3 times, giving everyone a chance to explain how they solves and listen to others explain how they solved. There are many methods, but an equation that could model the problem is: 10) 6 1 4 = 24 or There are 24 one- fourth pieces in 6 pies, but I need 35 pieces; so I need 11 more pieces. Halve of 24 pieces is 12 pieces, which would be 3 more pies that are needed. IMP Activity: Dividing Fractions I 5
Teacher Directions Materials: Colored Pencils or a highlighter (optional) Students will use partitive division and an area model to divide fractions by whole numbers. Students will explain the relationship between the division and corresponding multiplication problem. Pass out the activity sheet and have students read the opening questions. Use think-pair-share to have students discuss why it is hard to say how many 12 s there are in 4 (there is not a whole 12, so it is hard to picture or answer). Then ask question #1, to have students see that it can be easier to sometimes see division as making equal groups (partitive or fair share division) as compared with Dividing Fractions I, where students used measurement or quotative division. Have the students look at the pictures of the cookies and revisit the opening question by asking how 12 kids can equally share the 4 cookies. Use think-pair-share to have students answer this question. Make sure students understand that this is answering the division questions 4 12. We can think of this as 4 shared equally among 12 people. Direct the class attention to #3 and begin with the students writing a sentence to represent the problem using partitive division. 3) For 1 2, let the rectangle represent one whole. 3 a) I have 1 and need to make 2 equal groups. 3 1 group/person b) Shade the rectangle to show what we have. c) Divide the rectangle in to 2 equal groups. 2 nd group/person d) How much of the whole does each person get? 1 6 e) Math: 1 3 2= 1 6 Make sure to always ask the class, How many pieces/parts is the whole divided into, as this will be the most common mistake. Once the class is doing well with this, have them work with a partner on # s 4-6. Call on students to come show their work and explain their thinking. Once you have reviewed # s 4-6, turn to the questions on top of page 3 and model for the class how to go back to # 3 and show the area by labeling the dimensions and the area of a group (see below). Ask the class what IMP Activity: Dividing Fractions II 4
multiplication problem this looks like ( 1 3 1 2 = 1 ) or halve of one-third is the same as onethird shared equally among 2 6 people 1/2 1/3 Have the students do the same for # s 4-6. Once students share so that all students see how to label, give the students a few minutes to silently think about the challenge question (again, it is okay if not everyone sees this yet). Have students complete # s 7-10 on their own and then let them check answers with a partner. IMP Activity: Dividing Fractions II 5
Materials: None Teacher Directions Students will study patterns between each division problem from the previous lessons and a corresponding multiplication sentence that will have the same result. Students will explain that division of fractions is the same as multiplication of the first fraction by the reciprocal of the second fraction. Students will explain this in their own words and then be introduced to the academic term, reciprocal. Explain that over the past few days, students have noticed that there is some relationship between division and multiplication of fractions and today you will investigate this further. Have students take back out their activity sheets from Dividing Fractions I and II. Tell them to find and record the answers on the table provided in the top row of each box. Their task is then to find what the missing number must be to make each multiplication sentence true. (Note: it will also work for you to complete this table and let students study it on the document camera or make copies of the completed table). Give the students a few minutes to look at conclusion questions #1 and 2. Ask for students to share ideas, focusing on the fact that the first number and answers are the same in both rows, but the second number is different. Let students explain how it is different using their own words (the number is flipped, you put a 1 on top or a 1 on the bottom, etc). Explain that we call numbers like this reciprocals and define this for the students. Have the students look at #3 and answer these questions. If students are unsure of how to find reciprocals, spend a few more minutes doing some practice with this. Complete the lesson by having students complete #4 and then use their words to explain how to divide fractions. IMP Activity: Dividing Fractions Noticing Patterns 3
Teacher Directions Materials: Optional- number line on floor marked in quarters from 0-2. Students will use a number line to model the division of fractions, using quotative (measurement) division. Students will study patterns with types of problems to summarize what happens to the quotient of two fractions less than one, greater than one or equal to one. Students will study the pattern of the division problems with their related multiplication sentences to explain a procedure for dividing fractions. Pass out the activity sheet and have a student read the opening scenario aloud. Give the students a few minutes to complete the two questions. Let them share ideas with a partner and then select students to share with the class. To check their predictions and help with the concept in the lesson, have a student come up to a number line marked on the floor to show 0, 1 and 2 marked in quarters and show how many quarter jumps could be made in 2 total meters. Task 1 Direct the students to letter a. Have the class chorale reply what the problem is asking, using measurement division (the sentence frame just above the number line). They should say, How many one-fourths are there in 2? Give them a minute to use their number lines to mark and show this. Encourage the students to label the new marks they create on the number lines. Have the students follow the same process for letters b-g, realizing they may get stuck on e and f (this is okay, as long as they realize it is less than 1). Come back as a class and choose students to come show their thinking to the class. Task 2 The students will be following the same method of thinking in task 2, but now asking how many there are in one-half. Ask the class to tell you the sentence frame for letter a, and then have them work alone on the remaining problems. Come back together and select students to present answers and methods. Noticing Patters: What Happens to the Quotient Complete this section together, by showing the students the problems on which to focus, reading the sentence and then using think-pair-share to have the class explain what they have noticed and how this relates to the predictions. They should conclude that the quotient of a whole number and a fraction less than one is larger than the whole number; the quotient of a fraction and itself is 1, and the quotient of a smaller fraction and a larger fraction is less than 1. Noticing Patterns: What s the Algorithm? This last section provides an opportunity for students to verify what they may have already discovered using the area model to divide fractions- that you can multiply the first fraction by the reciprocal of the second fraction to find the quotient. Give the students a few minutes to study the table and then complete the summary. Call on students to read their summary and show the example problem to the class. IMP Activity: Dividing Fractions on the Number Line 4
Teacher Directions Materials: Place for students to run and jump Starting line for each group Tool to mark landing distance for each group Meter Sticks (1 per group) Students will measure the distance (in fractions of meters) each student can jump and represent this data on a line plot. Students will answer questions comparing distances with multiplicative comparisons, representing each question with a math sentence and solution applying the four operations with fractions. Ask the class if they have ever seen anyone long jump. Model a long jump for the class or show a short video of a long jumper. Ask the class to predict how far they think they can jump. Pass out the activity sheet and review the instructions together. Make sure students understand how to measure in fractions of a meter (each group can decide how precise they would like to be). Put students in groups of 5-6 students (to generate a good amount of data). Have the students assume the following roles: materials manager, a recorder, a starting line observer, and landing observer (they can rotate roles). Have the materials manager come get rope or chalk and a meter stick for their group. Take the class outside (preferrably to grass to avoid injury on the blacktop). Allow students (or lead them in) stretching before jumping. Have each group collect their own data. If a student repeatedly faults (goes over the starting line), set a limit of three tries and then they just do a 2- footed jump (no running start). Give the class 5-10 minutes to collect their data and record it in the table. Bring the class back inside. Have each student copy down their groups data and use this to make a line plot. Once each line plot has been made, have students work on the analysis questions. Allow them to work together, noting that they should have different answers for # s 4-6 and then 8, as they are only using their data point. Allow the students to struggle some, and ask guiding questions to help. Encourage the use of the number line on the line plot to help the students complete the multiplication and division problems. If the class does not know how to calculate average, you will need to model this first with whole numbers and then let them try. Bring up different students to explain their solutions to each problem. IMP Activity: How Far Can You Jump? 3