How can we represent this problem with a diagram? In Lesson 6.1.3, you made sense of the answers to division problems.  You paid particular attention to the meaning of each part of the division sentence.  In this lesson, you will extend your understanding about dividing to include division of fractions by other fractions.  As you work with your team, recall what you know about the relationship between multiplication and division and keep the following questions in mind. How can we represent this problem with a diagram? Can we represent it in more than one way?

43. Dria is writing a piece of music 43. Dria is writing a piece of music.  She has decided to replace a   note which takes up   of a small section of the music called a measure, with   notes which each take up   of a measure.  Work with your team to use diagrams to help you figure out how many   notes she will need.  Then represent the problem and its solution with a mathematical division sentence and a diagram.  Be prepared to describe your strategies to the class

In other words, 3 ÷ = 3 · 5 = 15. He asked his teammates, 44. Malik was catching up on homework when he noticed that he got the same answer dividing 3 by  as he did when he multiplied 3 by 5. In other words, 3 ÷   = 3 · 5 = 15.  He asked his teammates,  “Is dividing by   always the same as multiplying by 5?”  a.) Liam drew the two diagrams below and wrote down 5 ·   = 1 and 1 ÷   = 5.  “Does it have something to do with the fact that there are 5 one-fifths in each whole, 10 one-fifths in 2 wholes and 15 one-fifths in 3 wholes?” Liam asked. “Or,” asked Malik, “Is it because 1 is   of 5, 2 is   of 10 and 3 is   of 15?” Discuss this with your team.  Do Liam’s and Malik’s explanations both make sense?  Why or why not?  Can you think of any other ways to explain this?  b.) Malik was looking at problem 6-43 and asked, “Does this work when both numbers are fractions?  Can you find how many  ‘s are in   by multiplying   by 8?”  What do you think?  Refer back to the diagram you drew for problem 6-43.  Be ready to explain your ideas.

45. DORA’S DOLLHOUSE, Part 1 Dora is building a dollhouse for her cousin.  She needs several boards that are each   of a foot long.  She went to the store and found that the lumber she needs is sold only in lengths of 8 feet.  She laid a tape measure next to a board and drew the diagram below.  The diagram is also available on the Lesson 6.1.4 Resource Page, “Dora’s Dollhouse” Work with your team to figure out how many of her  -foot boards she can cut from one 8-foot piece of wood.  Be prepared to explain how you got your answer and to show why it makes sense using the diagram.  After she cuts her boards, how much lumber will be left over from the 8 foot piece?  What part of a   foot board is this?  Show how you know. Represent this situation with a division sentence.

46. DORA’S DOLLHOUSE, Part 2 Dora has taken a closer look at her blueprints and figured out that one 8 foot board is exactly   of the length of wood that she needs for her whole project.  What length of wood does she need for her project?  Work with your team to use the second diagram on the Lesson 6.1.4 Resource Page, “Dora’s Dollhouse,” to help you make sense of this question.   Represent this question and its answer with division.  Compare this problem with the question and answer in problem 6-45.  In what ways are these questions and answers the same?  How are they different? 

32. LEARNING LOG How are multiplication and division related?  Include examples and diagrams in your Learning Log that demonstrate the relationship.  Title this entry “Multiplication and Division” and label it with today’s date.

Tonight’s homework is… 6.1.3 Review & Preview, problems # 33-42 Show all work and justify your answers for full credit.