Rational Numbers

This is for my 8th grade math teachers (or honestly, any person who can read a standard).

8.NS.1:  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

That highlighted portion stopped me in my tracks the last time I "unpacked" this standard.  I have read this standard so many times over the past 9 years.  When I teach this, however, I tell my students that rational numbers are either terminating or repeating decimals.  That is not what this standard says.  The standard says that rational numbers are repeating decimals.  

And it's true!

For those people who need some background knowledge.  Rational numbers are numbers that can be written as fractions where the numerator and denominator are both integers.  Irrational numbers cannot be written as fractions.  A common example of an irrational number is pi.  Pi's decimal expansion neither repeats nor terminates (ends).  

1/3 is a rational number because it can be written as a fraction.  Its decimal expansion is 0.3333333...

2/10 is a rational number because it can also be written as a fraction.  Its decimal expansion is 0.2.

The standard says that 2/10 is rational and that the decimal expansion repeats.  If students divide this by hand (hello, long division), they will see that there is a 0 that repeats.  

I don't know why, but this blew my mind.  I no longer have to say "rational numbers are terminating or repeating decimals."  All rational numbers are repeating decimals!

What do y'all tell students?  I mean, I still want them to have knowledge of terminating decimals, but I also want to have this discussion.  What do y'all think?


Until next time,

Cononiah W

Comments

Post a Comment

Popular posts from this blog

Grouping Cards

 For the month of August, I am going to try to make a quick blog post each day.  Every.  Single.  Day!  I initially saw the idea on Twitter by @druinok.  Here's the post that sparked my inspiration.   As many of you know, I am currently in school to receive my Ed.D.  Part of my dissertation will be looking at groupings.  I've seen where teachers do random groupings with playing cards, a wheel, an online platform, or just group numbers/letters written on a piece of paper.  I have done the latter in the past.  The only downfall was that students would swap their cards so that they were in a group with their friends.  Then, I saw grouping cards.  On a single card, there are multiple objects/words/pictures.  Students get a card without knowing how they'll be grouped.  Then, the teacher can group students however she wants.  This is how I will use them in my class.  Please note that I use VNPS (vertical, nonpermanent surfaces) in my classroom. Students will come into class and
Read more

Mild, Medium, & Spicy

Hey, y'all! It's been a little minute, but that's ok!  This will be quick. In math, students have to get some solid practice.  They just need exposure to the concept.  In the past, I've done levels 1, 2, and 3.  Students could do practice problems from each level.  They knew that level 2 was the goal and level 3 was the challenge.   Last year, another math teacher at my school explained how he did mild, medium, and spicy. I watched a Zoom call with Peter Liljedahl (maybe 2 weeks ago), and he explained how he also used mild, medium, and spicy. I tried it in class with my 8th grade algebra 1 students.  They LOVED it.  They were in groups of 3 and each group was working on the same problem at the same time.  I provided the answers for them so they immediately knew if they were correct or not.  Some of them immediately tried the spicy problems while others worked their way through each level.  They really enjoyed it and worked really hard. I'm trying it with my 8th grad
Read more

Be Flexible

Image
Something really weird happened today at school.  Projectors stopped working.  The wifi went out.  Some places were without lights.  The hallway smelled like death.  It was weird. It was at this moment that I realized I have become a new-age teacher.  I had to figure out how to teach my 4th block class with absolutely no technology.  I was definitely out of my element!  I mean, y'all.  I barely even write on my whiteboards.   Tomorrow, I have to teach Algebra 1 without an iPad.  Don't get me wrong.  I taught for 6 years without an iPad.  Last year, however, I'd connect the iPad to the Apple TV, and project solely from it.  I'd walk the room, let students write on the iPad, whatever I needed to do. I really had to ask myself "How did you teach without an iPad?  What did you do?"  It's funny and sad at the same time.  Before the fancy Prometheans, there were document cameras.  Before that, basic Epson Projectors.  Before that, overhead projectors (I still us
Read more