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Unit 2 Focus standard: 6.NS.A.1: Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions by Fractions Overview: This unit extends students’ work with division of fractions. They will use models and equations to represent problems. Students will be given a division of fraction problem and they must create a story about the problem and solve it. Essential Questions: How does division of fractions relate to multiplication of fractions? How is division of fractions used in the real world? Possible Student Outcomes: The student will: Compute with fractions to determine quotients. Divide with fractions to solve word problems. Use visual models of the procedure/process used to determine quotients. Analyze multiplication and division of fractions to discover the relationship between these two operations and their effect on fractions Vertical Alignment: Key Advances from Previous Grades: o Between grade 5 and grade 6, students grow in their ability to analyze division of fractions. o In grade 5 students perform operations of addition, subtraction and multiplication with whole numbers, fractions and decimals . o In grade 5 students will divide a unit fraction by a whole number and a whole number by a unit fraction. Additional Mathematics: Students will use division of fractions: o in grade 6 these are the culminating standards for extending division to fractions o numerical work with the four basic operations with rational numbers culminates in grade 7 o in grade 8 the four basic operations are expanded to real numbers Theresa’s ideas Seacoast Charter School for Grade 6 Math 2014-2015 Page 1 of 5 Unit 2 Focus standard: 6.NS.A.1: Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions by Fractions Focus Mathematical Practices indicated in bolded italics 1. Make sense of problems and persevere in solving them. Analyze a problem and depict an appropriate way to solve the problem. Consider the best way to solve a problem. Can interpret the meaning of their answer to a given problem. Create a diagram or draw a picture to solve the problem. 2. Reason abstractly and quantitatively Consider the ideas that division of fractions can be represented in more than one way. Decide if your answer connects to the question. 3. Construct Viable Arguments and critique the reasoning of others. Convince some of your class members that your answer is reasonable. Justify your argument using model or equations. 4. Model with Mathematics Draw or model a diagram that represents division of fractions. Analyze an authentic problem and use a nonverbal representation of the problem. Use appropriate manipulatives. 5. Use appropriate tools strategically Use virtual media and visual models to explore division of fractions. Identify the tools that will help you solve the problem. 6. Attend to precision Demonstrate understanding of the mathematical processes required to solve a problems by communicating all of the steps in solving the problem. Label appropriately. Use the correct mathematics vocabulary when discussing problems. 7. Look for and make use of structure. Look at a diagram and recognize the relationship that is represented in each. Compare, reflect and discuss multiple solution methods. Theresa’s ideas Seacoast Charter School for Grade 6 Math 2014-2015 Page 2 of 5 Unit 2 Focus standard: 6.NS.A.1: Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions by Fractions 8. Look for and express regularity in reasoning Pay special attention to details and continually evaluate the reasonableness of answers. Using mathematical principles to help in solving the problem. Notes about the number system concepts measurement concept: This is referring to division of fractions. Reviewing,4 ÷ 3 with this concept means, “How many sets of 3 are in 4?” If you have 4 pints of ice cream to divide among 3 people, how much does each person receive? 1 0 2 3 4 P1 Person 1 Person 2 P2 P3 Person 3 1 Therefore each person gets 13 pints of ice cream. partition concept: Modeling a quotient, using the partitive concept, requires that only the dividend be modeled. The divisor represents the number of equal parts into which the dividend is to be partitioned. Thus, the modeling materials representing the dividend are rearranged, partitioned, or sub-divided into equal groups. The quotient is the number shown in each of the equal groups. Due to the very nature of the partitive concept, the divisor of a quotient must be a whole number ≥ 2. 1 2 1 ÷6 Theresa’s ideas Seacoast Charter School for Grade 6 Math 2014-2015 Page 3 of 5 Unit 2 Focus standard: 6.NS.A.1: Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions by Fractions 1 So 12 can be divided into 6 equal groups by dividing each part in 6 equal pieces. Take 1/6 of each part and add those together. 1 1 + 6 12 = 2 12 + 1 12 = 3 12 = 1 4 1 4 Each group is equal to . If, after some of the materials are rearranged into equal groups, there are materials remaining, the remaining materials should be traded for equivalent smaller pieces and the partitioning continued. If a number, less than the divisor, of the smallest pieces in your model remain after the partitioning has been completed, a fraction may be expressed where the remainder (the remaining number of smallest pieces) is the numerator and the divisor is the denominator. The quotient is the number in each equal set plus this fraction 1 22 ÷ 4 1 2 1 8 1 2 1 5 +8=8 5 So each group of 4 contains a part equal to 8. common denominator algorithm: The common-denominator algorithm is repeated subtraction concept of division. Example : 5 4 ÷ 1 2 = Theresa’s ideas 5 4 Seacoast Charter School for Grade 6 Math 2014-2015 Page 4 of 5 Unit 2 Focus standard: 6.NS.A.1: Apply and Extend Previous Understandings of Multiplication and Division to Divide Fractions by Fractions the blue represents one set of 1 2 which you will subtract from the diagram divide in half subtract 1 2 subtract the second 1 2 Now we have 2 wholes. The blue that is left is 2 parts out of 4 parts (the original one-half divided into 4 parts…see drawing above) which equals 2 out of 4 or Theresa’s ideas 1 2 1 = 2 2 . Seacoast Charter School for Grade 6 Math 2014-2015 Page 5 of 5