0 Common Core Mathematics Teacher Resource Book 6
Table of Contents Ready Common Core Program Overview Supporting the Implementation of the Common Core Answering the Demands of the Common Core with Ready The Standards for Mathematical Practice Depth of Knowledge Level Items in Ready Common Core Cognitive Rigor Matrix Using Ready Common Core Teaching with Ready Common Core Instruction Content Emphasis in the Common Core Standards Connecting with the Ready Teacher Toolbox Using i-ready Diagnostic with Ready Common Core Features of Ready Common Core Instruction Supporting Research Correlation Charts Common Core State Standards Coverage by Ready Instruction Interim Assessment Correlations Lesson Plans (with Answers) A6 A7 A8 A9 A0 A A A A6 A8 A0 A A8 A A6 CCSS Emphasis Unit : Ratios and Proportional Relationships Lesson Ratios M CCSS Focus - 6.RP.A. Embedded SMPs -, 6 Lesson Understand Unit Rate M CCSS Focus - 6.RP.A. Embedded SMPs -, 6, 7 Lesson Equivalent Ratios 9 M CCSS Focus - 6.RP.A.a Embedded SMPs -,,,, 7, 8 Lesson Solve Problems with Unit Rate 9 M CCSS Focus - 6.RP.A.b, 6.RP.A.d Embedded SMPs - Lesson Solve Problems with Percent M CCSS Focus - 6.RP.A.c Embedded SMPs -,,,, 7 Unit Interim Assessment M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
CCSS Emphasis Unit : The Number System Lesson 6 Understand Division with Fractions 8 M CCSS Focus - 6.NS.A. Embedded SMPs -, 7, 8 Divide with Fractions 66 M CCSS Focus - 6.NS.A. Embedded SMPs -, 7, 8 Lesson 8 Divide Multi-Digit Numbers 78 S/A CCSS Focus - 6.NS.B. Embedded SMPs -,, 7, 8 Lesson 9 Add and Subtract Decimals 88 S/A CCSS Focus - 6.NS.B. Embedded SMPs -, 6, 7 Lesson 0 Multiply and Divide Decimals 98 S/A CCSS Focus - 6.NS.B. Embedded SMPs -,, 6 8 Lesson Common Factors and Multiples 0 S/A CCSS Focus - 6.NS.B. Embedded SMPs -,, 7 Lesson Understand Positive and Negative Numbers 0 M CCSS Focus - 6.NS.C., 6.NS.C.6a, 6.NS.C.6c Embedded SMPs -,, Lesson Absolute Value and Ordering Numbers 8 M CCSS Focus - 6.NS.C., 6.NS.C.7a, 6.NS.C.7b, 6.NS.C.7c, 6.NS.C.7d Embedded SMPs -, 6 Lesson The Coordinate Plane 8 M CCSS Focus - 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8 Embedded SMPs -,, 7 Unit Interim Assessment Unit : Expressions and Equations Lesson Numerical Expressions with Exponents 7 M CCSS Focus - 6.EE.A. Embedded SMPs - 8 Lesson 6 Algebraic Expressions 67 M CCSS Focus - 6.EE.A.a, 6.EE.A.b, 6.EE.A.c Embedded SMPs -, 6 Equivalent Expressions 79 M CCSS Focus - 6.EE.A., 6.EE.A. Embedded SMPs -, Lesson 8 Understand Solutions to Equations 9 M CCSS Focus - 6.EE.B. Embedded SMPs -, 7 Lesson 9 Solve Equations 99 M CCSS Focus - 6.EE.B.6, 6.EE.B.7 Embedded SMPs -, 7 M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
Unit : Expressions and Equations (continued) CCSS Emphasis Lesson 0 Solve Inequalities M CCSS Focus - 6.EE.B., 6.EE.B.8 Embedded SMPs -, 6, 7 Lesson Dependent and Independent Variables M CCSS Focus - 6.EE.C.9 Embedded SMPs -, 7, 8 Unit Interim Assessment Unit : Geometry Lesson Area of Polygons 6 S/A CCSS Focus - 6.G.A. Embedded SMPs - 7 Lesson Polygons in the Coordinate Plane 6 S/A CCSS Focus - 6.G.A. Embedded SMPs -,,,, 7 Lesson Nets and Surface Area 6 S/A CCSS Focus - 6.G.A. Embedded SMPs - 7, 8 Lesson Volume 68 S/A CCSS Focus - 6.G.A. Embedded SMPs -, Unit Interim Assessment 79 Unit : Statistics and Probability 8 Lesson 6 Understand Statistical Questions 8 S/A CCSS Focus - 6.SP.A. Embedded SMPs -,, 6 Measures of Center and Variability 9 S/A CCSS Focus - 6.SP.A., 6.SP.A. Embedded SMPs -, 7 Lesson 8 Display Data on Dot Plots, Histograms, and Box Plots 0 S/A CCSS Focus - 6.SP.B. Embedded SMPs - 7 Lesson 9 Analyze Numerical Data 7 S/A CCSS Focus - 6.SP.B.a, 6.SP.B.b, 6.SP.B.c, 6.SP.B.d Embedded SMPs - Unit Interim Assessment 8 M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
Focus on Math Concepts Lesson 6 (Student Book pages 7) Understand Division with Fractions Lesson Objectives Understand the meanings of division. Use a model to show division of fractions. Use an understanding of multiplication of fractions to explain division of fractions. Prerequisite Skills Know that multiplication and division are inverse operations. Know that division is either fair sharing (partitive) or repeated subtraction (quotative). Divide with whole numbers. Divide a whole number by a fraction. Model division with manipulatives, diagrams and story contexts. Vocabulary There is no new vocabulary. The Learning Progression In Grade, students divided whole numbers by unit fractions. Students continue this understanding in Grade 6 by using visual models and equations to divide whole numbers by fractions and fractions by fractions to solve word problems. In Grade 7, students will continue their work with fractions to include all rational number operations (positive and negative). Students will build on understanding of number lines developed in Grade 6. Ready Lessons Teacher Toolbox Tools for Instruction Interactive Tutorials Prerequisite Skills Teacher-Toolbox.com 6.NS.A. CCSS Focus 6.NS.A. Interpret... quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 8 9 because of is 8 9. (In general, a b c d ad.) How much chocolate will each person get if people share lb of chocolate equally? How bc many -cup servings are in of a cup of yogurt? How wide is a rectangular strip of land with length mi and area square mi? standards FOR MatheMaticaL Practice: SMP, 7, 8 (see page A9 for full text) 8 L6: Understand Division with Fractions
Part : Introduction Lesson 6 At a Glance Students explore dividing a fraction and a whole number by a fraction. Lesson 6 Part : introduction Understand Division with Fractions Focus on Math concepts ccss 6.ns.a. Step By Step Introduce the Question at the top of the page. Read the description of how to divide a whole number by a unit fraction. Ask students to explain in their own words why 6 is the same as 6. Ask them to use the number line illustration to make the explanation clear. Have students explain a reciprocal. Remind them that the reciprocal is the same as the multiplicative inverse. Discuss the number line in Think and ask a student to explain in his or her own words how the model shows 6. ELL Support Discuss the difference between the phrases dividing by and dividing into. Dividing by, for example, asks how many groups of are in a number, but dividing into fourths means to separate a number into four equal groups i.e., multiplying by a unit fraction. What does it mean to divide a fraction by a fraction? You know how to divide a whole number by a unit fraction. For example, you can think of 6 divided by as how many one-fourths are there in 6? Using a number line, you can divide 6 into fourths and count to see there are fourths in 6. 6 0 6 You also learned that dividing a number by a fraction is the same as multiplying the number by the reciprocal of the fraction. 6 is the same as 6, or think What does dividing a whole number by a fraction mean? Madison cuts a 6-yard length of ribbon into yard pieces. To figure out how many pieces Madison cut, think, How many three-fourths are in 6? You can draw the same number line to represent the 6 yards of ribbon and divide it into fourths. 0 6 You can circle three sections to represent yard pieces. You can see there are eight yard pieces in 6 yards. 6 8 0 6 6 8 L6: Understand Division Division with with Fractions Mathematical Discourse circle the multiplication expression that is the same as the division expression. When you divide a whole number by a fraction less than, the quotient is larger than the dividend. Can you explain why? Students might explain that since the fraction is less than, you would be able to take out more than one group from each whole in the dividend. Ask other students to add to the explanation and to use the same reasoning to solve 0 and explain why the answer makes sense. L6: Understand Division with Fractions 9
Part : Introduction Lesson 6 At a Glance Students explore dividing a fraction by a whole number. Part : introduction Lesson 6 Step By Step Read Think as a class. Emphasize that in the previous problem about Madison and the ribbon, students were shown an example of dividing a whole number by a fraction: 6. This question is about dividing a fraction by a whole number: 6. Discuss the area model with students and have them explain how the first model represents. Then have a student explain how the second model represents dividing the model into 6 parts. Ask students how they know how much is in each group. Be sure students understand that refers to the whole. Ask students to use the model to show why 8. Ask students to explain to their partner why 6 is the same as 6. Have students read and reply to the Reflect directive. Visual Model Folding paper to divide a fraction by a whole number. Materials: rectangular sheet of paper Paul uses of his back yard for gardening. He will divide this gardening section into equal parts for vegetables. Model using paper. Start by folding the paper into fourths and shading. Now fold the paper into thirds to show sections for vegetables. Unfold. Ask, How is the paper divided now? [twelfths] Ask, So how much of the gardening section of Paul s back yard is used for each vegetable? [The shaded portion of each column represents of the whole.] think What does dividing a fraction by a whole number mean? Cory wants to pour of a quart of juice equally into 6 glasses. This means he needs to divide into 6 equal parts. You can represent the problem with an area model. First, you can show the quart of juice. Then, you can draw vertical lines to divide the model into 6 equal parts. 6 glasses 6 8 quart of juice divided equally into 6 glasses means Cory will pour or quart of juice 8 into each glass. 6 is the same as 6. Cory pours 6 of quart of juice into each glass. reflect Use the number line to show and explain why 0 and 0 both equal 0. 0 0 0 Possible explanation: see number line above. you can think of divided by 0 as dividing 0 into equal parts. this is the same as finding of 0 because each half is one of the equal parts. L6: Understand Division with Fractions Real-World Connection Ask students to describe real world situations in which they would have to divide a fraction by a whole number. Ask students to think of a problem or situation that represents 6. Share a situation in which of something is divided into 6 equal groups or shared equally by 6 people. Example: Susan wants to create a picture frame in the shape of a regular hexagon. She has a strip of trim that is of a meter long. If she uses the entire strip, how long would each side be? 60 L6: Understand Division with Fractions
Part : Guided Instruction Lesson 6 At a Glance Students use a number line to divide a fraction by a fraction. Step By Step Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through problem together as a class. As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking. Check to see that students label the number line correctly before they use it to answer questions. Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson. STUDENT MISCONCEPTION ALERT: Students may believe that dividing by is the same as dividing into half. Dividing by one half means how many s are in a quantity, 7 divided by. Dividing in half means to take a quantity and split into two equal parts. 7 divided in half equals. Part : guided instruction explore it explore dividing a fraction by a fraction with the problem below. Kate has yards of fabric to make small flags. Each flag requires yard of fabric. How 6 many flags can Kate make? You need to find out how many 6 s are in. The number lines below are divided into thirds. Label on the top number line to represent yards of fabric. 0 0 L6: Understand Division with Fractions Mathematical Discourse Lesson 6 Each flag requires yard of fabric. Divide the bottom number line into sixths to show 6 how many sixths are in. Look at the bottom number line. How many sixths are there in? 6 How many flags can Kate make? 7 6 8 6 How would you explain dividing a fraction by a fraction using the number line model to a student who was absent from class? Students should explain that you first draw and label the dividend on a number line. Then, draw another number line beneath it and divide it equally into the number of parts that are in the divisor. They should then count and circle how many groups of the divisor there are up to the dividend. Students should model the number lines on the board for the class to see. What is another way you could model this problem? Students might describe an area model or paper folding. Which method for solving fraction division problems do you prefer and why? Students may prefer multiplying by the reciprocal or modeling. Listen for and encourage correct usage of math vocabulary. L6: Understand Division with Fractions 6
Part : Guided Instruction Lesson 6 At a Glance Students read fraction division word problems and solve them in pairs or groups. Step By Step Organize students into pairs or groups. You may choose to work through the first Talk About It problem together as a class. Walk around to each group, assessing how the groups are solving the problem. Use the Mathematical Discourse questions to help support or extend students thinking. Students may need to be reminded what direction horizontal is. Review horizontal and vertical as needed. Direct the group s attention to Try It Another Way. Have a volunteer from each group come to the board to explain the group s solutions to problems and 6. SMP Tip: Students construct arguments using verbal or written explanations accompanied by models. Provide students an opportunity to refine their mathematical communication skills through discussions in which they evaluate their own thinking and the thinking of other students (SMP ). Part : guided instruction talk about it L6: Understand Division with Fractions Mathematical Discourse Why would you use rectangles to model the problem? Using an area model is a strategy for solving fraction division problems. The problem is asking you to divide by s. Why 8 didn t you divide it into thirds? To count by s, you would need to start by 8 dividing it into eighths. How would you explain dividing a fraction by a fraction using a common denominator? Lesson 6 solve the problem below as a group. Kevin has 6 cups of flour. It takes cup of flour to make one cake. How many cakes can 8 Kevin make? 9 You need to find out how many 8 s are in 6. 0 Do you think the number of cakes Kevin can make is greater than or less than 6? Why? greater than 6. it takes less than cup of flour to make one cake. Represent 6 cups with 6 rectangles. rectangles are shown below. Draw more rectangles. Draw horizontal lines to divide each rectangle into eighths. Circle and count groups of in the model. How many did you circle? 8 How many -cups of flour are in 6 cups of flour? 6 8 6 8 6 try it another Way explore dividing by a unit fraction using a common denominator. To solve, write as a fraction with a denominator of and think, How many halves are in ten halves? 0 0. Use the same reasoning to find 8 6. Write 8 6 as a fraction with a denominator of. To solve, think, How many two-thirds are in four-thirds? 6 Write as a fraction with a denominator of 6. 6 To solve 8 6, think, How 6 many four-sixths are in eight-sixths? Students should realize that once they have a common denominator they know the parts are the same size, so they can simply divide numerators. 6 6 L6: Understand Division with Fractions
Part : Guided Practice Lesson 6 At a Glance Students demonstrate their understanding of dividing a fraction by a fraction. Step By Step Discuss each Connect It problem as a class using the discussion points outlined below. Explain: Ask, What does the top number line represent? 6 Ask, What does the bottom number line represent? [twelfths] Ask, What do the circled portions represent? [Groups of in. There are 0.] 6 Analyze: Students may draw an area model or a number line. Remind students that the problem is asking how many groups of s are in. The quotient must be greater than one because there is more than one i n. Justify: Read the problem as a class. Remind students that the problem is asking how many groups of are in. 6 Students should draw a number line or area model to model the problem. SMP Tip: Students need many opportunities to connect and explain the connections between different representations. Encourage students to compare the number line and area models and explain how they help to solve the problem (SMP ). 6 Part : guided Practice connect it talk through these problems as a class, then write your answers below. 7 explain: Look at the model below. Write the division equation that the model represents. Explain how to find the quotient using the model. 0 6 0 6 0 ; the bottom number line is divided into twelfths and shows there are 0 twelfths in 6. 8 analyze: Sam said that equals. Draw a model and use words to explain why 8 Sam s statement is not reasonable. 0 0 9 justify: Show that by using a model. Explain why the answer is greater 6 than the number you started with. L6: Understand Division with Fractions Possible explanation: asks how many fourths are in three halves. there is more than group of fourths in three-halves so the quotient must be greater than. the quotient could not be a fraction less than, like. the 8 model shows that 6. 0 Possible explanation: When you divide a given number by a fraction less than, the quotient is always greater than the given number. 6 Lesson 6 L6: Understand Division with Fractions 6
Part : Common Core Performance Task Lesson 6 At a Glance Students choose a fraction division word problem and solve by writing an expression and drawing a model. Step By Step Direct students to complete Put It Together. As students work, walk around to assess their progress and understanding, answer their questions, and give additional support, if needed. If time permits, have students share their problem with the class and discuss how they solved it. Scoring Rubrics See student facsimile page for possible student answers. A Points Expectations The student wrote a division expression and drew a model to represent the problem. The student only draws the model or the expression, does not include both. 0 The model or expression is incorrect. Part : common core Performance task Put it together 0 Use what you have learned to complete this task. Choose one of the following problems to solve. Circle the problem you choose. Greg made gallon of lemonade and plans to share it equally among friends. How much lemonade will each friend get? Keisha plans to run miles this week. If she runs of a mile each day, how many days will it take her to run miles? Will she be able to run miles in a week? L6: Understand Division with Fractions Lesson 6 a Write a division expression and draw a model to represent the problem. Division expression for the first problem: Division expression for the second problem: Models will vary, but should represent the problem. b Estimate what you think the quotient will be. Will the quotient will greater than or less than the dividend? How do you know? For the first problem: the quotient will be less than because divided by means dividing into parts; each part will be less than. the quotient will be a fraction less than. For the second problem: the quotient will be greater than because divided by means finding how many groups of are in. there are groups of in. is less than so there are more than groups of in. c Use your model to explain how to find the quotient and what the quotient means. For the first problem: or. this means each of greg s friends 6 will get gallon of lemonade. 6 For the second problem: 6. this means it will take keisha 6 days to run miles. yes. she will be able to run miles in a week. 7 B Points Expectations Student uses the model to explain how to find the quotient and what the quotient means. Student does not explain how to find the quotient or what the quotient means. 0 Student provides an incorrect explanation or response. C Points Expectations The student estimates what they think the quotient should be and proves if it will be greater than or less than the dividend. The student does not explain why he/she thinks the quotient would be greater than or less than the dividend. 0 The student estimates incorrectly or gives an incorrect explanation. 6 L6: Understand Division with Fractions
Differentiated Instruction Lesson 6 Intervention Activity Explore fraction division on a number line. Materials: painter s tape, index cards, markers, string Use painter s tape to create a number line on the floor across the classroom. Label the number line through with index cards spaced evenly apart on the number line. Write the problem on the board. On-Level Activity Make a fraction division handbook or poster. Guide students to create a three-page handbook or three-part poster highlighting the steps to solving a fraction division problem. Examples should include dividing a fraction by a whole number, dividing a whole number by a fraction, and dividing a fraction by a fraction. Each example should include an expression, a model, and an explanation for how to solve the problem. Ask students to explain how the number line can be used to solve the problem. Ask students to use painter s tape to divide the line into s. Ask them to explain why is. Next, ask students to model. Students should use string to circle groups of. There will be 7 groups of and left over. The quotient would be 7. Help students to realize that is half of, so they have 7 and g r oup s. Continue to model using the number line for problems such as 0, then 0. Challenge Activity Think about dividing whole numbers by fractions and dividing fractions by whole numbers. Tell students, Yolanda wants to cut -foot pieces from a -foot rope. Ask, How would you record this number sentence? [ 8] What do you know about the quotient? [It will be larger than the dividend.] How many pieces will she cut? [8] Tell students, Now Yolanda wants to cut -foot pieces of rope from a -foot piece of rope. How would you record this number sentence? 8 What do you know about this quotient? [It is smaller than dividend.] Is it possible to cut feet pieces of rope from a foot piece of rope? [No, so the real world answer is 0.] Ask students to think of other real-world situations in which they would use division with fractions. Work with students to determine that their situations make sense. L6: Understand Division with Fractions 6
Develop Skills and Strategies (Student Book pages 8 69) Divide with Fractions Lesson Objectives Solve word problems using division of fractions. Write an equation to solve a problem using division of fractions. Write a story problem that will use division of fractions. Prerequisite Skills Know that multiplication and division are inverse operations. Know that division is either fair sharing (partitive) or repeated subtraction (quotative). Divide with whole numbers. Divide a whole number by a fraction. Model division with manipulatives, diagrams, and story contexts. Vocabulary multiplicative inverse: a number which when multiplied by x yields the multiplicative identity, reciprocal: the multiplicative inverse of a number; with fractions, the numerator and denominator are switched The Learning Progression In Grade, students learn to understand fractions as division and to divide whole numbers by unit fractions. In Lesson 6, students built upon the understanding from Grade using models to show division of fractions. In this lesson, students continue to build upon their knowledge by using visual models and equations to divide whole numbers by fractions, fractions by fractions, and mixed numbers by fractions to solve word problems. In Grade 7, students will continue their work with fractions to include all rational number operations. Ready Lessons Teacher Toolbox Tools for Instruction Interactive Tutorials Prerequisite Skills Teacher-Toolbox.com 6.NS.A. CCSS Focus 6.NS.A. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 8 9 because of 8 9 is. In general, a b c d ad bc. How much chocolate will each person get if people share lb of chocolate equally? How many -cup servings are in of a cup of yogurt? How wide is a rectangular strip of land with length mi and area square mi? standards FOR MatheMaticaL Practice: SMP, 7, 8 (see page A9 for full text) 66
Part : Introduction At a Glance Students read a word problem and explore dividing a whole number by a fraction using a model. Part : introduction Divide with Fractions Develop skills and strategies ccss 6.ns.a. Step By Step Tell students that this page models dividing whole numbers by fractions using a visual model. Have students read the problem at the top of the page. Work through Explore It as a class. Ask students how they determined whether the number of planters would be greater or less than. ELL Support Use the diagram on the page to review the words divisor, dividend, and quotient. Throughout the lesson, use models or manipulatives to demonstrate concepts and processes. Allow students to use the models to demonstrate their learning. SMP Tip: Students map important quantities in the problem to the diagram as a way of understanding dividing with fractions (SMP ). Students need many opportunities to explain the connections between different representations. Have students explain how the model helps them solve the problem. Visual Model Model dividing by a number less than. Draw identical circles on the board. Write. Ask, Is the divisor,, greater than or less than? [less] How many s are in circles? Let a volunteer draw lines in the circles to show fourths. Ask, Which is greater: the number of parts or the number of whole circles? [The number of parts is more.] Write. Ask, Will the quotient be greater than or less than? [greater than ] Then let a volunteer draw a model to illustrate. 8 in the previous lesson, you learned what dividing by fractions means. in this lesson you will divide with fractions to solve problems. take a look at this problem. Charlie is growing vegetables in planters. He has bags of soil and uses of a bag of soil to fill each planter. How many planters can he fill? explore it use the math you already know to solve the problem. Think of the number of planters that Charlie can fill as how many s are in. Will that number be greater than or less than? Explain your reasoning. the number of planters will be greater than. When you divide a given number by a number less than, the answer will be greater than the given number. The model below represents the bags of soil. Draw horizontal lines to divide each bag into thirds. Circle and count groups of in the model. How many did you circle? 6 Why do you circle groups of to represent this problem? each planter holds of a bag of soil, so each circled group fills one planter. you are trying to find how many s are in. How many planters can Charlie fill? 6 Explain how the model helped you solve the problem. you can count the groups of that are in. Mathematical Discourse How does using a model help you solve the problem? Students may answer that drawing the model helps them to see and count the groups. Explain, in your own words, dividing fractions with a model. Responses should discuss drawing the dividend and dividing it into groups of the divisor. Why does it make sense that the quotient is greater than the dividend when you divide with a fraction less than? If the divisor is a fraction greater than, will the quotient be greater or less than the dividend? Responses should show understanding that taking out groups that are less than one whole will mean that there are more groups than the dividend. Dividing by a fraction greater than will result in fewer groups than the dividend. 67
Part : Introduction At a glance Students explore solving a division problem using multiplication. Part : introduction Find out More Step By Step Read Find Out More as a class. Remind students that a fraction and its reciprocal must have a product of. Point out that when you multiply fractions you multiply the numerators, then multiply the denominators, and then simplify if possible. Discuss Reflect. Guide students to think about dividing as being the same as multiplying by the reciprocal (multiplicative inverse). Visual Model Use a model to understand using reciprocals in division. Help students understand why they can solve any division problem by multiplying the dividend by the reciprocal of the divisor. Draw this model on the board for : Explain that the expression can be read as how many groups of are in? Show on the model that contains groups of, and there are groups of in. The total number of groups of in is simply. When you found the number of s that are in, you were dividing. You are solving the problem. You can solve this problem by multiplying. You know that multiplication and division are related. divided by is the same as of, or multiplying by. Think of as. Dividing by is the same as multiplying by. When dividing with unit fractions, you learned that dividing by is the same as multiplying by. Dividing with any fraction works the same way. Dividing by is the same as multiplying by. 6 6 You can solve any division problem using multiplication. To divide by any number, you can multiply by its multiplicative inverse, which is also known as the reciprocal. reflect Explain how you can solve this division problem by using multiplication. 6 Dividing by is the same as multiplying by. so, you can solve 6 divided by by multiplying 6 by. 6 6 8 = 9 Dividing by is the same as multiplying by or. Dividing by is the same as multiplying by. Real-World Connection Ask students to think of everyday places or situations where people might need to divide by fractions. Encourage them to share their ideas with the class. Examples: cooking, making crafts, measuring, building structures 9 68
Part : Modeled Instruction At a Glance Students read a word problem and explore how to divide a whole number by a fraction using a bar picture and by modeling the problem using words and an equation. Step By Step Read the problem at the top of the page as a class. Read and discuss Picture It. Ask, What does the shaded part of the bar show? [how much of the whole bottle Kelly drank, or two fifths] SMP Tip: Students reason abstractly when they analyze a problem and represent it as an equation with a missing factor in order to find a solution (SMP ). Ask students to explain how their equations or models represent the context of the problem. Read and discuss Model It. Walk through each step to be sure students understand how to use the inverse operation. Hands-On Activity Make a model to show how many s are in. Materials: sheet of paper for each student, pencils Tell students they will make a model to show how many s are in. Give each student a sheet of paper. Have them fold it into equal parts and draw lines on the folds to show equal sections. Tell them this will be a model for. Ask, What is the first thing you could do to the model to start showing how many s are in? [Draw lines to divide each of the whole sections into thirds.] What might you do next? [Circle each group of of the thirds and count them.] Let students finish their models. Ask them what the solution is. Suggest they label their model with the equation solved: 6. 60 Part : Modeled instruction read the problem below. then explore how to divide a whole number by a fraction. Kelly drank of the water in her bottle. She drank cups of water. How many total cups of water were in her bottle? Picture it Mathematical Discourse you can draw a picture to understand the problem. The bar represents Kelly s water bottle. You can divide the bar into fifths and shade to represent the amount of water Kelly drank, cups. Model it cups cups cups? cups you can use words and equations to understand the problem. of the total amount of water equals. of the total amount of water equals? To solve a missing factor problem like?, you can divide.? What is another way you could solve the problem regarding Kelly s water bottle on page 60? Responses may include dividing cups in half to find of the amount in the bottle, and then multiplying by to find the total amount in the bottle. How is dividing fractions similar to dividing whole numbers? Listen for responses that indicate dividing a quantity into groups. 69
Part : Guided Instruction At a Glance Students revisit the problem on page 60 to learn how to solve it using the bar picture and the equation model. Step By Step Read Connect It as a class. Be sure to point out that the problems refer to the problem on page 60. For problem, remind students to change to an improper fraction before multiplying by. Multiply the denominator times the whole number, and then add the numerator. The result is the numerator, and the denominator stays the same. In problem, review how to change an improper fraction to a mixed number: Divide the numerator by the denominator to get a whole number, the remainder is the new numerator, and the denominator stays the same. Have students work through Try It on their own. Then discuss with them how they solved the problems. Part : guided instruction connect it now you will solve the problem from the previous page using the picture and model. Look at Picture It on the previous page. Why do you divide the bar into fifths? the problem says kelly drank of the bottle, so you need to show fifths. How can you use Picture It to find out how many cups of water are in the bottle? since cups, you know that is cups. you can multiply by to find the total number of cups of water that were in the bottle. How many total cups of water were in Kelly s bottle? 7 cups. Look at Model It on the previous page. Find. Show your work. cups or 7 cups. 6 Explain how to use multiplication to divide a whole number by a fraction. try it Dividing by a fraction is the same as multiplying by its inverse. Multiply the whole number by the reciprocal of the fraction. use what you just learned about dividing with fractions to solve these problems. show your work on a separate sheet of paper. 7 How many -cup servings are there in cups of juice? 8 servings 8 It takes Emily 9 minutes to bicycle of the way to school. How many minutes does it 0 take Emily to bicycle all the way to school? 0 minutes 6 Try It Solutions 7 Solution: 8 servings; Students solve the problem by using the equation? or a drawing such as cups with lines dividing each cup into halves and circled groups of. ERROR ALERT: Students who wrote 8 servings forgot to multiply by the reciprocal of the divisor. Remind them that they need to find the reciprocal of the divisor before multiplying. 8 Solution: 0 minutes; Students may use the equation 9? They may use a drawing such as a bar 0 divided into tenths with shaded to model the 0 distance she went in 9 minutes. This would show that is equal to minutes; 0 0 minutes. 0 70
At a Glance Part : Modeled Instruction Students read a word problem and explore how to divide a fraction by a fraction using a double number line and by modeling the problem using words and an equation. Step by Step Read the problem at the top of the page as a class. Discuss Picture It: Ask, What does the top number line represent? [the distance divided into fourths] Ask, What does the bottom number line represent? [the distance divided into eighths] Discuss Model It. Ask, What does the question mark in the equation represent? [the number of hurdles Eli jumped over during his -mile run] Part : Modeled instruction read the problem below. then explore how to divide a fraction by a fraction. Eli ran of a mile. Every of a mile, he jumped over a hurdle. There was a final 8 hurdle at the mile mark. How many hurdles did Eli jump over? Picture it you can draw a picture to understand the problem. The top number line shows the distance Eli ran, mile. The bottom number line shows the number of s that are in. 8 0 0 Model it you can use words and equations to understand the problem. Think: How many s are in? 8 Use division to find how many s are in. 8 divided into 8? s 8 equals the number of hurdles 8? 6 Mathematical Discourse Which method for dividing fractions do you prefer? Why? Are there situations when one method may be easier to use than another? Encourage students to support their opinions and to listen to the opinions of others. Point out that there is no correct answer, and that different students may have different preferences. Can you think of another way to describe dividing fractions? Explain. Encourage students to suggest ideas or knowledge on other ways to divide fractions. 7
At a Glance Part : Guided Instruction Students revisit the problem on page 6 and solve it using the double number line and the equation model. Step by Step Read and discuss Connect It as a class. Refer to the problem on page 6. For problem, remind students that they should find the inverse (reciprocal) only of the divisor and not the dividend. Also remind them they should simplify improper fractions to a mixed- or wholenumber answer. Have students work through Try It on their own. Then discuss their answers and solutions. SMP Tip: Give students multiple opportunities to solve and model problems. Students use repeated reasoning to understand algorithms and make generalizations about patterns (SMP 8). Part : guided instruction connect it now you will solve the problem from the previous page using the picture and model. 9 Look at Picture It. Why is the top number line divided into fourths? Why is the bottom number line divided into eighths? the top number line is divided into fourths to mark, the total distance eli 0 Explain how Picture It helps you figure out how many hurdles Eli jumped over. i could count how many s are in. 8 How many hurdles did Eli jump over? Look at Model It. Explain how to use multiplication to find 8. Dividing by 8 is the same as multiplying by its inverse, 8 or 8, so is the 8 same as 8. Evaluate. Show your work. 8 or 6 8 Explain how to divide a fraction by a fraction. to divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction. try it ran. the bottom number line is divided into eighths because eli jumped over a hurdle every eighth of a mile. use what you just learned to solve these problems. show your work on a separate sheet of paper. Keisha cuts a -foot rope into -foot pieces. How many pieces of rope did she cut? 8 6 Jade makes half a liter of lemonade. She pours liter of lemonade into each glass. 0 How many glasses is Jade able to fill? 6 6 Try It Solutions Solution: 8; Students may draw a double number line or may use the standard algorithm to solve the problem. 8 ERROR ALERT: Students who wrote 8 pieces of rope may have multiplied the inverse of both the dividend and the divisor. Remind students that they should only multiply by the inverse (reciprocal) of the divisor. Review Find Out More on page 9 of this lesson to help students understand why multiplying by the reciprocal is mathematically valid. 6 Solution: ; Students may draw a bar picture and shade half of the figure, then draw lines to divide the figure into tenths. parts would be shaded. Or, they may use the standard algorithm. 0 0 7
At a Glance Part : Modeled Instruction Students explore how to divide a mixed number by a fraction using bar pictures and by modeling the problem using words and an equation. Step by Step Read the problem at the top of the page as a class. Discuss Picture It: Ask, How do the shaded bars in the first picture represent pounds of granola? [One whole and more parts are shaded.] Ask, What does each circle in the second picture represent? [Each circle represents one -pound bag of granola.] In Picture It, some students might be confused by the circle that contains part of the whole bar and part of the partial bar. Suggest to students that they think of the entire bar plus bar as one entity. Discuss Model It. Remind students that they must write the mixed number in the dividend as an improper fraction before dividing. 6 Part : Modeled instruction read the problem below. then explore how to divide a mixed number by a fraction. Mari divides pounds of granola into -pound bags for a bake sale. How many bags of granola can she sell? Picture it you can draw a picture to understand the problem. The shaded bars represent pounds of granola. Each circle shows a -pound bag of granola. bag of granola The remainder is half of. Model it you can use words and equations to understand the problem. Think: How many s are in? Use division to find how many s are in. divided into? 9? s equals the number of bags of granola? Mathematical Discourse Mari is selling bags of granola to raise money, so she wants to have as many bags as possible to sell. What else might she think about when dividing up the granola? Listen for responses that show students making connections to personal experiences to make sense of the problem. They might discuss how the size of each bag might make a difference to buyers: Customers might not buy if they think there is not enough granola in a bag. Students might also mention cost to customers or the amount left over after filling the bags. 7
At a Glance Part : Guided Instruction Students revisit the problem on page 6 and solve it using the bar picture and the equation model. Step by Step Discuss Connect It as a class. Point out that Connect It refers to the problem on the previous page. When discussing problem 0, be sure students make the connection between the bar model and the mathematical process for changing a mixed number to an improper fraction. For students having trouble understanding why writing a mixed number as an improper fraction is mathematically valid, ask students to count the total number of shaded squares (9) in the first bar picture and point out that the bars are divided into fifths, so there are 9 fifths altogether. Have students work through Try It on their own. Let volunteers share their solutions and answers with the class. Clear up misconceptions and discuss any questions students may have. Part : guided instruction connect it now you will solve the problem from the previous page using the picture and model. 7 Look at Picture It. Why do you circle groups of to solve this problem? Mari divided the granola into -pound bags. you are trying to find how many s there are in. 8 Count the circles. How many -pound bags of granola can Mari sell? 9 What fraction of a bag would the remaining pound of granola be? Explain your answer. each bag is pounds. the remaining pound of granola is half of, so the remainder is of a bag. 0 Look at the Model It. Explain how you know is equal to. 9 is, which equals. 9 Explain how to use multiplication to evaluate 9. Dividing by is the same as multiplying by its inverse,, so 9 is the same as 9. 9 Evaluate 9. Show your work. 0 0 or Explain how to divide with mixed numbers. Dividing with mixed numbers is just like dividing with fractions. you can write mixed numbers as fractions, then multiply by the reciprocal of the divisor. try it use what you just learned to solve these problems. show your work on a separate sheet of paper. A recipe requires of a cup of water. Kyle has a -cup measuring cup. How much of the measuring cup is filled with water? of the cup How many -cup servings are in cup? 6 servings 6 Try It Solutions Solution: of the cup; Students may draw a model that shows and divide it into fourths. of a cup would be half of the model. Solution: ; Students may write an equation. 6 6 6 ERROR ALERT: Students who wrote transposed the fractions and found. Encourage students 6 who made this error to draw a model to help them visualize the problem. 7
Part : Guided Practice Part : guided Practice Part : guided Practice The student divided the number of gallons of paint used,, by the gallons of paint she bought,. Pair/share How could you justify your answer with a picture? study the student model below. then solve problems 6 8. Student Model Lydia bought gallons of paint and used gallons of paint. What fraction of the paint did she use? Look at how you can show your work using a model. think: What fraction of is? some fraction of equals.? to solve?, divide.? ; 6 0 or Lydia used of the paint she bought. Solution: 7 A marathon is miles long. If people divide up the distance equally, how many miles does each person need to run? Show your work. Solution: 0 6 0 each person will need to run 6 0 miles. Dividing by is the same as multiplying by what number? Pair/share How is this problem different from the others you ve seen in this lesson? Will the answer be less than or greater than? Why? 6 Lexi has planted seeds in of the garden. She used pound of seeds. How many pounds will she use for the entire garden? Show your work. 8 Which of the following problems can be solved by finding? a people equally share of a pizza. How much of the pizza does each person eat? b How many -cup servings of soup are in cups of soup? What kind of picture could represent the expression? ; 6. c D A pie recipe requires pounds of apples. How many apples are needed for pies? A family ate of a -foot sandwich. How much did they eat? Pair/share How did you and your partner decide which fraction is the dividend and which is the divisor? Solution: Lexi will use pound of seeds for the entire garden. 6 Arthur chose a as the correct answer. How did he get that answer? arthur confused the dividend and divisor. the situation he chose could be solved by dividing by. Pair/share Does Arthur s answer make sense? 66 67 At a Glance Students practice solving problems that require dividing a whole number by a fraction, dividing a fraction by a fraction, and dividing a mixed number by a fraction. Step by Step Ask students to solve the problems individually on pages 66 and 67. In the student model at the top of the page, call attention to writing each mixed number as an improper fraction before multiplying. When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. Solutions Ex Using a model of words and an equation is one way for students to show their solution to the problem. 6 Solution: Lexi will use pound of seeds for the 6 entire garden; Students could solve the problem by using an equation. (DOK ) 6 7 Solution: Each person will need to run 6 0 miles; Students could solve the problem by using an equation. 0 6 0 (DOK ) 8 Solution: B; Students must recognize the language as a division problem how many are in? Explain to students why the other two answer choices are not correct: C is not correct because the problem asks how many are needed for, which is a multiplication problem. D is not correct because the problem states of, which is a multiplication problem. (DOK ) 7
Part 6: Common Core Practice Part 6: common core Practice Part 6: common core Practice Solve the problems. Write each expression in the correct column to show whether the quotient is less than, greater than, or equal to. What is the value of the expression 8? A 9 6 B 6 8 C D 9 7 0 6 9 8 8 quotient is less than 0 6 quotient is equal to 9 8 8 quotient is greater than 9 7 Find the expression that does NOT answer the question: What fraction of 8 is? A 8 B 8 C 8 D? 8 Explain the difference between dividing in half and dividing by half using pictures, models, or numbers. Possible answer: Dividing in half means dividing into parts or multiplying by. Dividing by half means finding how many s there are in the number. if you divide in half, you get. if you divide by, you get 8. there are eight s in. The area and one dimension of a piece of land are given. From the list, write the fraction inside each box that represents the second dimension of the piece of land described. 7 7 7 8 7 8 9 9 7 9 7 7 8 9 The area of a rectangular piece of land is square mile. One dimension of this piece of land is 7 8 mile. The area of a piece of land that is in the shape of a triangle is square mile. One dimension of this piece of land is mile. The area of a rectangular piece of land is square mile. One dimension is miles. 6 Write a story to represent the expression 6. Draw a model and use multiplication to show the solution. Explain how the dividend, divisor, and quotient relate to the story. stories will vary. Possible answer: a recipe calls for 6 cups of flour. if the only measuring cup you have is cup, how many times will you have to fill the measuring cup to get 6 cups of flour? 6 6 8 Possible student model: self check Go back and see what you can check off on the Self Check on page. 68 69 At a Glance Students divide by fractions to solve word problems that might appear on a mathematics test. Solutions Solution: D; 6 (DOK ) 8 8 Solution: C; transposed the dividend and divisor. Correct reasoning should be? of 8, or? 8. (DOK ) Solution: See student book page above for solution; Use the area formula for either a rectangle or triangle to find the unknown dimension. (DOK ) Solution: See student book page above for solution; To divide fractions, multiply the first fraction by the reciprocal of the second fraction. (DOK ) See student book page above for possible student explanation. (DOK ) 6 See student book page above for possible student model and explanation. (DOK ) 76