There are two essential questions to introduce here: 1) Do the signs of fraction quotients follow the same rules as integer quotients? 2) What is the relationship between multiplication and division?
The latter question is not explored in great depth here. It is assumed that students explored this in the 6th grade. They most likely need a refresher so one is provided here. Many of my students will be working with fractions during an intervention block. This is when they will get the more in depth conceptual framework for multiplying and dividing fractions. Despite this, I think it is worth a reminder of how the two operations are related. A simple pattern is presented so students can see how the multiplicative inverse will help when dividing fractions (MP8). It will be helpful to draw a bar model for each related division and multiplication problem so students have a visual model. When discussing these problems I might say as an example, 12 divided by 3 is equivalent to 12 groups of 1/3 while pointed to the bar model for this problem.
Students will be working on these eight problems under careful scrutiny. Some notation issues are addressed here as well. Also students may be seeing fraction division written as a complete fraction for the first time. I will be treating these as division problems and will not present simplifying these using a common factor. The focus here will be to use the multiplicative inverse to solve a division problem. I love dividing by a common denominator, but will not spend much time on that in this lesson. However, if a students realizes this or a common factor can be done - I will encourage it.
The independent practice is designed solely for fluency. The last 4 problems involve the order of operations. I am most excited about students reaching the extension problem. There are four multiplication equations involving signed fractions. There are also two rate problems. There is often not a lot of time for the extensions in a class period but I really hope to get here.
With the rates, students get their first opportunity (this year at least) of finding a unit rate involving two fractions. We will explore this fully in the next unit, but now is a great time to introduce it. Students may struggle with the unit rate in terms of miles per hour. As students work I will look to make sure that they do not convert 15 minutes to 15/100 of an hour. This is a mistake I see all too often. It can quickly be corrected by asking "How many minutes are in an hour"? This is also an excellent time for a bar model or double number line (MP5) showing four units of 15 minutes with the value 2 1/4 miles.
The exit ticket assess the fluency of signed fraction division. Only 1 of the 4 problems appear as a complex fraction. Three of the four provide students with the opportunity to simplify the factors before multiplying. This is optional; I will take any form of a correct answer.
Students need to answer at least 3 questions successfully to show mastery of this lesson.