Dividing Fractions by Fractions

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Objective

SWBAT: • Demonstrate the relationship between multiplication and division involving fractions. • Use visual models to divide a fraction by a fraction. • Develop strategies for dividing a fraction by a fraction.

Big Idea

What do you get when you divide ¼ by ½? What does your answer actually mean? Students continue building their skills and strategies in order to divide fractions by fractions.

Do Now

10 minutes

See the Do Now video that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want student to review dividing whole numbers by fractions, which we worked on in the previous lesson.  Some students may draw a picture, while other students may use the algorithm to find the answer.

For each problem pick 1-2 students (who I observed during the do now) to come to the document camera to share and explain their work.  I ask students if they agree or disagree with this work and why.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

If students do not mention the picture, the algorithm, or the common denominators method I will present it as a possible solution and ask students what they think.  See my “Do Now Problem 1 Strategies” and “Do Now Problem 2 Strategies”.  I want students to be familiar with multiple strategies.

Problems

20 minutes

Notes:

  • For this lesson, each student needs a fraction kit to help model problems.  Example: Fraction Kit
  • These problems can be very challenging for students.  This is to be expected!  I explain to students that they are not only learning how to divide with fractions, but also how to create visual models.  It will take time, but they can do it!

I have volunteers pass out the fraction kits.  We work through these problems together.  For problem 1a and 1b, we read the situation and create a model of the gallons.  We use the fraction kit to help us model what is going on.  See my Problem 1a and 1b Fraction Kit pictures for more details.  Then we use the information to draw the tables.  In the packet there is only one table provided, which is intentional.  If more than one table is painted I want students to draw it.  As we are working I am creating a table on chart paper that displays our division and multiplication number sentences.  This will be displayed on the board so students can use it as a reference throughout the lesson.  See “Teacher Created Visual” to see what the chart will look like at the end of the Problems section.

Then we move on to problems 1c, 1d, and 1e.  These problems are more complicated because students have to deal with a remainder.  In each problem there is leftover paint that will only cover part of the table.  Students are engaging in MP1: Make sense of problems and persevere in solving them, MP2: Reason abstractly and quantitatively, and MP4: Model with mathematics.

For problem 1c, we create the model with our fraction kit (see Problem 1c Fraction Kit).  I ask, “How many ½ gallons fit into one-fourth?”  If students struggle, I ask them if it is more than 1 or less than 1.  By looking at the fraction kit model, it is clear that one group of ½ gallon will fit into ¼ gallon.  If students don’t answer, I use the kit to show that ¼ gallon takes up ½ of the ½ piece, so ¼ divided by ½ is ½.  We can use the multiplication sentence ½ times ½ equals ¼ to support our answer.  In drawing the tables, covering the entire table would take ½ gallon (or 2/4 gallon) of paint.  Since we only have ¼ of a gallon, it will cover ½ of the table (See Problem 1c 1d 1e drawing tables). 

We repeat the process with problems 1d and 1e (See my fraction kit and drawing tables examples).  For problem 1e, a common misconception is that 2/3 of a gallon will cover 1 1/6 tables, rather than 1 1/3 tables.  Students who think the answer is 1 1/6 are most likely looking at their fraction kit model and seeing that the gap between 2/3 and ½ can be filled by using a 1/6 piece.  Instead, students need to think of ½ gallon as the whole, since it takes ½ gallon to cover a whole table.  See “Problem 1e Fraction Kit” to the steps I use with students.  In “Problem 1c 1d 1e Drawing Tables” I show how drawing the tables forces students to realize that it takes ½ of a gallon (or 3/6 gal) to cover the first table.  Then the students only have 1/6 of gallon of paint left.  If it takes ½ gallon (or 3/6 gallon) to cover the whole table, than 1/6 of a gallon will cover one-third of the table.  The multiplication problem 4/3 x ½ = 4/6, or 2/3 confirms our answer.

I ask students if their strategies (flip and multiply, common denominators) work for problem 1d and 1e.  I have 2 students come to the document camera to show us that each strategy will produce the same answers that we came up with using the fraction kits and table drawings.  

Painting More Tables

20 minutes

Students work on this section independently.  I remind them if they get stuck they should use their fraction kit, previous problems, and classmates as resources.  If they continue to be stuck, they can raise their hand to ask me a question.  I Post A Key so students can check their work as they go.

If students are struggling, I may ask them some of the following questions:

  • What division problem is going on?  How do you know?
  • How much paint does it take to cover an entire table?  How much paint does Avery have?
  • How can you use your fraction kit to create a model?
  • How can you draw the tables?  How much paint is leftover? 

If students successfully complete their work, they move on to work on the challenge problems.

Closure and Ticket to Go

10 minutes

For Closure I ask students how they figured out problem 2c.  Students participate in a Think Pair Share.  I call on students to share out their thinking.  If a student says that the paint covered 1 ½ tables, I disagree out loud and declare that the answer is 1 1/8 tables.  I want students to tell me that 2/8 of a gallon (or ¼) covers one table and then we are left with 1/8 of a gallon of paint.  If it takes 2/8 of a gallon to cover the whole table, then 1/8 of a gallon will cover ½ of the table.  I use my fraction kit to demonstrate this under the document camera.  I ask students to share a multiplication number sentence that confirms the answer is 1 ½.  Lastly I ask students whether the flip and multiply or the common denominators strategies would produce the same answer.  My goal is that students can connect the visual models with the algorithms. 

I pass out the Ticket to Go.  Then I pass out the HW Dividing Fractions by Fractions.