## Long Division and Decimals

12 Jan

Fraction-decimal-percent conversions are a fundamental building block for seventh-grade math, and after I browsed the Common Core standards and noticed this in 7.NS.2.d, I decided to do things a little differently this year.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

Don’t judge, but I didn’t fully understand why long division worked until well into my adult years.  I suspected my students didn’t either.  So, I created this lesson with these targets in mind – students will be able to:
– Explain why long division results in a decimal equivalent of a fraction.
– Predict when a fraction will result in a repeating decimal or terminating decimal.

The lesson is a workshop model, with a mini-lesson, cooperative investigation, catch-and-release, and summary.

I have a small set of base-ten blocks, but for this lesson, I found some terrific virtual base-ten blocks here. I liked them because with a click on the corner, the bigger blocks can be broken into ten smaller blocks.
http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html

I asked the students if they liked to understand why things work, or if they just like to know how to do something and don’t care how it works.  The poll results were interestingly split about fifty-fifty (why??) – but I said “for those of you that like to know why things work, this lesson is for you.”

I put a base-ten block on my head and told them “You need to know something about me.  I have a super power.  My secret identity is Decimal Woman.  My super power is that I can break things into ten equal pieces.  LOOK! It’s our special guest today, PEYTON MANNING!”  And I switched the bigger block out for ten flats. “Oh, Sorry you missed it. See how that works?  Ten equal pieces, each a tenth of the original.  Unfortunately, I’m a one-trick pony.  I can break anything, but I can ONLY break things into ten equal pieces.  It’s all I do.  Cakes, pizzas, enemies… ten equal pieces only.”

I put a place-value chart on the board and reviewed decimal place values.  Thousands, hundreds, tens, ones.  Each was explained in words and exponential form.  I moved to the tenths (10 to the negative first power), hundredths (10 to the negative second), thousandths (10 to the negative third), and so on down.  I told them my super power is useful in a base-ten world, where every place value is ten times as small as the one next to it.  Every time I break something into ten pieces, I can think of it one place value down.  I did some quick random questioning:  what’s a tenth of one cake?  A tenth of a tenth of a brownie?  A tenth of a hundredth of my enemy?

I explained that long division is a manifestation of my power, and I used the virtual manipulatives to put a cake on the board.  I picked four students.. let’s call them, Emma, Jackson, Caleb, and Grace… and offered to split the cake evenly between them.  How much cake does each person get?

One cake – would love to cut it into fourths! However, I can only cut it into tenths.

The students were excited. “One fourth!  One fourth!”  I stopped them. “I would love to give them one fourth of the cake, but I can’t!  I can only break things into ten equal pieces.  So I’m going to cut the cake, but cut it into tenths.  How many tenths does each person get?” The students thought about it and told me each person gets two tenths, but there would be two left over.  I divided up the tenths, put the results in the division problem above, and had the two tenths left over.

Everyone has two tenths, for a total of 8 that have been given away. But there are two tenths left over!

The students were excited again.  “Cut them in half!”
I stopped them again.  “I can’t cut them in half.  Remember my super power?   I can only cut them into…”
“Ten pieces!  Then give everyone five!”  They started yelling out the answer.  So I broke the tenths into twenty hundredths and gave each student five of them, adding the results to the long division problem.

The cake is successfully divided into fourths, simply by breaking it into tenths and then hundredths and dividing them up.

I pointed out that even though I can only break the cake into ten pieces each time, I could successfully divide up the cake evenly with my method.  Everyone gets two tenths, and I broke the leftovers into hundredths, which everyone got five of.  This is the heart of how long division works.  You divide up the pieces and then break the leftovers into smaller pieces.. ten times smaller each time.

We did the same experiment with a cake and only three students – poor Grace would not get any cake this time.  I had to use my super power to break a cake into thirds among Emma, Jackson, and Caleb, using the same method.  I could only break it into ten equal pieces every time.  We started with the tenths of a cake.

One cake.. attempting to divide into thirds using Decimal Woman’s power of breaking pieces into tenths.

We went through the same procedure as before. Everyone got three tenths, and there was one tenth left over.  I broke the tenth into tenths, which the students identified as hundredths.  Everyone got three of those, and there was one left over.  Even though everyone could see where the pattern was going and some students were yelling it out, I divided the leftover hundredth into thousandths and divided them up… each student got three of them, with one left over.  At this point, the base-ten blocks can’t be broken anymore, but the students saw where the pattern was going.

Three tenths and three hundredths and three thousandths… one left over every time!

This left a question hanging in the air, so Decimal Woman asked it.  Can you predict when a decimal will repeat and when it will terminate?  What kinds of numbers produce each kind of decimal?  I introduced the word “Conjecture” as something between an educated guess and proven statement.  It’s a well-reasoned statement that has patterns and evidence to back it up.  Their group’s product would be a conjecture on what makes a repeating decimal vs. a terminating decimal.

I passed out the group work, which I’ll attach.  The students were asked to use long division to find decimal equivalents of a lot of fractions, from halves all the way through twelfths.  I told them to divide up the work and look for interesting patterns, which would help them create their conjecture.

Worksheet:patterns_of_decimals

We had been practicing long division for a day or two, but many students were rusty on converting remainders into decimal equivalents.  Most students picked this up fairly quickly and the teaching was minimal.  The classroom management was not as big a challenge as I expected, because most students found the task really engaging, to my surprise!  They got excited when they found an interesting pattern, and when they discovered that a decimal repeated or ended.  A few students did not pull their fair share of work, but it was a better cooperative activity than usual, all things considered.  After I let them churn through decimals for 15-20 minutes, I did a “catch” and had them report out some of the things they found.  They reported:

– Eighths are interesting because you always end up with forty of something left over, which divides evenly into five – so all the eighths terminate with a 5.
– Tenths and fifths simply have one terminating decimal.
– Sixths all repeat except for three-sixths.
– Most twelfths start with a sequence and then repeat a single digit.
– Elevenths all have a two-digit repeating sequence, and that sequence is a multiple of 9.
– Ninths are cool because the numerator is the repeating decimal.  This led to a great discussion on whether nine-ninths was 0.999999… or simply 1.
– Sevenths have a six-digit repeating sequence, and the sequence starts from a different number each time.
– If you can reduce a fraction to one of the terminating decimals, then it terminates.

Students determined that halves, fourths, fifths, eighths, and tenths all terminated.  Thirds, sixths, sevenths, ninths, elevenths, and twelfths mostly repeated.  The students were very confused about the general trend that would make a decimal repeat vs. terminate.  It didn’t seem to be related to odds vs. evens, or primes.  I asked them for factors of ten, and the told me the factors were 1, 2, 5, and 10.  I pointed out to them that it was interesting that if I broke something into tenths, I could only divide it up evenly if I could then split it up into groups of 10, 2, or 5 – nothing else – and I prodded them to use that idea in coming up with their conjecture.

I gave them a few minutes more to finalize their conjectures and then we finished with a summary and a round-robin of what they learned.

I know not all students achieved the second objective, and could not immediately tell me without calculating if they had a terminating or repeating decimal.  However, I do think they are well on their way to being able to compute decimal equivalents easily and explaining why the long division method works – and they were engaged and excited with what they did.  I was really pleased with the lesson!