Repeating Decimals

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SWBAT apply long division skills while discovering patterns in multiplication.

Big Idea

Students will reflect on repeating decimals through approximation


10 minutes

The Problem of the Day to start the class will be a review of the lesson from the day before. Students will complete a problem that requires them to convert a fraction into its decimal equivalent and make a statement about whether the decimal is a terminating or a repeating decimal. I will have a volunteer come to the SMART board and complete the steps of conversion for the class to see. The goal is for students to see the problem being solved while checking their own work. Discussion about the problem can include fluency with the steps of the algorithm along with highlighting any differences that students may identify in their work. We can also highlight the decision as to whether the decimal is terminating or repeating and what knowledge they used to make the determination.  Student will be engaging in MP3 here because they will have to explain why they think it is one answer or another.

POD: Convert 2 7/8 into a decimal. Show all of the work in your long division.


25 minutes

Guided Practice/Teacher Modeling:  Duration: 10 minutes
Students will receive the Repeating Decimals worksheet with problems to solve with a partner. Students will work on the first problem of the sheet with a partner. All of their work needs to be done longhand. The division algorithm has to be done several times to find the point of repetition. I want students to continue the algorithm so they can see when the repetition occurs. I want them to see the patterns that show in their work and recognize the point of repetition. A common misconception I have seen is for students to randomly use the repetend to indicate repetition but they are not specific about what is actually repeating. Continuing the algorithm will also allow students to see exactly what is repeating and encourage them to be specific with the bar notation. This will require students to persevere in their problem solving (MP 1). They can check their work with a calculator at the end of the activity. The goal is to practice fluency with the multiplication and the division. Working on these problems will allow students to engage in MP 8, look for and express regularity in repeated reasoning. As students work on the tasks, I can move among the pairs and watch them solve the problems. I want to see that they are comfortable with the algorithm and are not questioning next steps. I can also answer any questions that may arise. If students have questions or are unclear, I can use what I see to help determine any differentiation necessary for the second problem or the next lesson.

Independent Practice with Learning Task:   Duration: 10 minutes
Students will work on the second problem and all of its elements individually. Again, as they work, I can circulate and see how they attack the problem and create the solution. I will also be able to read their extended responses to get an understanding of their thinking. I can ask questions and have students explain their thinking to me. The goal is to identify the regularity and see the pattern of repeated reasoning in their work.

Student  Discussion:  Duration: 5-7 minutes
Once students finish both questions, we will have a whole group discussion about the problem-solving processes used. I want to touch on each segment of both problems to discuss student understanding. Do they understand how answers were derived for each sub question? Were they able to apply their understanding to both questions? I want to make sure that students are not just mimicking work done on the first question to the second question. I want part of the discussion about the individual work to touch on their reasoning. Does each student understand what was done at each segment? Do they have conceptual understanding of not only the relationship between multiplication and division, but understanding of the reasoning used when they answer the segments in the problem?


5 minutes

To wrap up the lesson, I will give the students a problem involving convert a fraction to a decimal and ask them to determine whether that decimal is a terminating or repeating decimal.

Exit Ticket Problem:

Convert 5/6 into a decimal. Does this fraction convert to a terminating decimal or a repeating decimal?