SWBAT write any fraction as a terminating or non-terminating decimal using long division

Students use the long division algorithm to write fractions as decimals and use the remainder to distinguish between a terminating and repeating decimal.

10 minutes

This lesson is mainly about the mechanics of long division as applied to converting a fraction to a decimal value. Students have already worked on long division of decimals in their 6th grade course, so this is an extension of that work. That being said, this lesson is presented in a direct instruction format.

I'll try not to overwhelm the students with too many problems. Long division can be tedious and students don't always love practicing it. It will be necessary to keep the pace moving and keep up the encouragement. I found simply pulling out whiteboards will make them want to practice a lot. Also, despite it not being the most glamorous topic, I have found that students confidence in their math abilities goes way up when they have conquered long division!

There is still an essential question to answer: How can you determine if the decimal representation of a rational number will terminate in zeros or repeat?

We then will discuss the terms terminating, repeating, decimals and repetend. As opposed to just presenting the information about these terms in the resource I will ask students for examples. Many will remember 0.33333...... etc or 0.666666....... Then we can define it and discuss the correct bar notation for a repeating section. The term repetend will also be repeated. This is a somewhat strange sounding word so I will ask the students to repeat it after me a few times.

Next I will ask students what it means to terminate? Many of the students will know this means to end, so a terminating decimal is one that ends.

I will remind students that we want to look for clues as to when a decimal value will repeat or terminate.

Then we go through the long division steps. I like using the mnemonic dad, mom, sister, brother as a way to remember to divide, then multiply, then subtract, then bring down. Their 6th grade teacher taught his division unit around Halloween so he preferred to say "Dracula must suck blood."

We will then work through for examples. I will give students a chance to first set up the long division "house". I have not left a lot of room in the resource for showing work so I will provide my students with whiteboards or ask them to use notebook paper.

Many errors in long division come from simple mistakes in computation. The other errors come from improper alignment. To mitigate the alignment problem, I will encourage students to practice long division using graph paper or the grid side of their whiteboards. Only 1 digit may be placed in each square.

At the end of this section I will ask students to look for what is different about the quotients in example 1 compared to example 2. They will note that example 1 has a decimal that terminates and example 2 has a decimal that repeats. I'll then repeat the essential question and have kids do a brief **turn-and-talk. **I will listen for answers in these conversations. One is that the remainder repeats on the repeating decimals, the other is the divisor is 3. The terminating decimal ends up with a remainder of zero and the divisor is 8.

At this point we can conclude that when a remainder value repeats we will have a repeating decimal otherwise the decimal will terminate. The answers about the divisors will be considered, but then I'll ask students if they think 4 problems is enough evidence. Hopefully, they say NO! I'll say we'll be on the look out for more evidence as we work. This is **MP8**.

15 minutes

I have included 6 addition problems similar to the four examples from the previous section. Students will probably need a bit more practice. Students may get confused by the negative symbols. I'll remind them to ignore them until they have finished the long division. There are two mixed numbers in this set. I prefer converting the fraction value without converting to an improper fraction. I will let students decide which method they prefer, but I'll ask them to solve GP5 by making an improper fraction; GP6 is up to them.

I'll do another **turn-and-talk **around the essential question. Students will have more evidence of the divisor of 8 resulting in a terminating decimal. Some may even suggest that divisors that are multiples of 8 terminate. Others may notice that denominators of 6, 9, and 12 repeat. They may conclude that divisors that are multiples of 3 repeat. If they come to this conclusion, I may say what about 3/3 or 18/9. This will then make them refine their answers even more.

20 minutes

Now students are on their own to solve 8 problems. The last 4 problems ask students to place values on a number line. I do not need students to draw a detailed number line, nor does it have to be to perfect scale. However, I do want students to include 0 on the number lines and place the values in correct order. Thus, I will only need to see 5 points on a number line.

I still expect some students to think 8/9 is greater than 0.9 because the 8/9 has a "longer" decimal. I will remind students that 0.9 has repeating 0. Writing 0.9 as 0.90000000.... under 8/9 as 0.7272727272 will help more clearly understand.

Now we can address the essential question again. The answers should be similar to what was discussed in the previous section.

5 minutes

The exit ticket has 4 questions. A successful student will answer at least 3 of the 4 problems correctly. Questions 1-2 are straight forward conversion problems.

Question 3 ask students to identify the equivalent value of 3/16 in a multiple choice format. It is an even better question if their is time to have students write or discuss how they know the wrong choices are incorrect. This gets students estimating with fractions and applying what they already know. For examples 5.3333.... is too large because 3/16 is less than a whole. 0.316 is too large because 3/16 is less than 4/16 which is 1/4 or 0.25. That means that 0.0316 is way to small. So they correct answer must be D.

If there is no time to have students write such responses on the exit ticket, it is worth quickly discussing the next day.

The final question gets to the heart of the essential question. I have put a constraint on the question by asking students to focus on the remainder. This can avoid students fixating on particular divisor values as being prone to repeating or not. I think this point could be explored using a simple computer program or excel!

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