Math = Love: Algebra 2 Skill 1: Classifying Real Numbers

Tuesday, September 1, 2015

Algebra 2 Skill 1: Classifying Real Numbers

For my first year of teaching, I made this nifty set of nesting boxes to illustrate the real number system.

First-year teacher me was too terrified to write on them with a Sharpie for fear of messing them up.  So, I held them up to illustrate, but they didn't actually end up getting much use.

Second and third year teacher me decided that the real number subsets weren't actually on the end-of-instruction exam so she didn't teach them.

Fourth year teacher me decided that she wanted to make sure her Algebra 2 students could talk about numbers using a more precise vocabulary.  So, the boxes made a new appearance.

This time, though, I wrote on them.  Is my handwriting perfect?  Nope.  But, I'm okay with that.

Note to self for next year: Label all four sides of the boxes!  (And, maybe the inside bottom!)

I also took some Dollar Tree neon index cards and chopped them in half.  Each card was labeled with a number to sort.

Fun colors make everything better.

I had a graphic organizer from my first year of teaching, but I decided to change up the look just the tiniest bit.

We glued this in our notebooks.  Then, I had the students write the definitions for rational and irrational numbers.  I gave them a quick introduction to the graphing calculator so they could type various numbers in to see what kind of decimal they produced.  We typed in pi.  We typed in e.  We typed in 2/3.  We typed in Radical 2.

A few minutes before class started, I had decided to let each student pick a sticker as they walked in the room.  I used a box of random name badge stickers that have been in my cabinet for ages.  Not sure exactly how I acquired these...

I wrote out a set for each of my Algebra 2 classes.

I made it so half were rational and half were irrational, but I didn't tell them that. ;)

It turns out that the stickers weren't that sticky, and they didn't stay on that well.  One tweep suggested I use paper plates in the future.  I really like that idea!

When kids walked in the door, I fanned out the pile of stickers in front of them and let them choose one.  They just saw the blank backs.  One girl picked the sticker that said pi.  The first words out of her mouth? "Ms. Hagan, I picked your favorite number!"  YES!

I guess I did tell them this on the first day of school...

After our introduction to the scary looking graphing calculators, I had them type the number off their name badge into their calculator.  Then, I gave them the task of sorting themselves into rationals and irrationals.

In retrospect, I made a mistake here.  I was really just wanting my students to practice using their calculator.  But, now a bunch of my students think that you have to type 2/3 into your calculator to see if it is rational or irrational.  Oops...

I went around the room and had each student read out their number.  The class decided if each student was in the correct category.  If I were going to do this again, I would have everyone sort themselves.  Then, I would do a quick survey and tell the class how many people were in the wrong place.  And, I'd let them discuss to figure it out.  Hind sight really is 20/20.

After we were all sorted correctly, we went back to our seats to finish our notes.

Next, I brought out the nesting boxes.

In 2nd period, I laid them out on the desk like this (but nested).  And, I had students come up, grab a card, and place it in the correct box.  It didn't work so well.

In 5th period, I handed the cards out to the kids (either 1 or 2) and had them walk up to the boxes and put them in once they had used their notes to classify the number.  SO much better!

Sometimes I feel bad that my morning classes are my guinea pigs since my afternoon classes are all repeats.  Every morning, I learn what not to do when teaching a lesson.

I took the cards out of the box one at a time.  On a blank notebook page, the students listed the number and what subsets it belonged to.  If it was a natural number, I would prompt them with "What box does the natural number box always fit inside?"  Whole numbers.  "What box does the whole number box always fit inside?"  Rationals.  And so on.

Seeing a 3-d representation really helped my students in a way that the 2-d version in their notes didn't.

We went through all of the numbers.  It was a bit more time consuming than I'd hoped.  If I do this again, I'd want to streamline this section.  Somehow.

The next day, students came in and found a dice activity to complete to wrap up this skill.

I always hated Always, Sometimes, Never in high school geometry.  But, I decided to give it a go with my students.  So glad I did!

Here's the sheet I gave them:

On the board, I wrote the numbers 1-6.  I asked them to list different types of numbers.  They quickly came up with 5: natural, whole, integers, rational, and irrational.  It took a few more moments before someone remembered that those are all types of real numbers.

I handed out one jumbo foam die to each table group.  The other math teacher I work with gave me these last year, and my students and I have had so much fun with them!

This picture shows me using two foam dice, but I just did that to show how I got each box filled in for my #Teach180 tweet.  I had my students just roll one die two times.  This meant they had to write the number subsets in a certain order instead of being able to choose which subset went in which slot.

I did an example of how the activity worked on the board for students to see.  Then, I set them loose to do 5 as a group.  I went around and stamped them after I checked them.

Look at these cute stamps I picked up at Dollar General!  They were 5 for a dollar.  I'm not sure how long they'll last, but the kids were definitely amused with them for this activity!

It was great to just be able to walk around and eavesdrop on their conversations.  When I noticed students were struggling, I brought over the nesting boxes for them to use.  One group had rolled this sentence: A integer is a natural number.  Half the group thought it was always true.  The other half of the group thought it was sometimes true.  So, I held up the integer box for them.  You have a number in this box.  Is it always in the natural number box or is it sometimes in the natural number box?  Looking at it this way, the answer seemed obvious to them.  Soon, I had groups fighting over who got to keep the nesting boxes sitting at their desks.  They informed me that I needed to make a set of boxes for reach group because they really helped.

I tried explaining that this was the same as the graphic organizer I had given them, but they weren't buying it.  I think rolling the die to make the problems was the real selling point of this activity.  I've found that anytime I pull out the dice that my kids get excited.  There's just something special about when the dice decide your problem instead of the teacher!  I found this was the case last year when working with point slope form!

Want to download my files for this lesson?  Here you go!


  1. Love, love, love the nesting boxes! How did you make them?

    1. I took cardboard boxes, cut off the flaps, and covered them in colored paper. I've seen others use aluminum cans, as well.

  2. I love this idea as well! I know sometimes the graphic organizer doesn't always click with students. I didn't have enough boxes to recycle, so I went to the dollar store, and bought various sizes of the aluminum baking pans. I was playing around with different sizes in the store to make sure they would stack how I wanted, and I was able to make it work. I'm not going to cover them entirely, yet, but I will color coordinate them and write on them to use this week. Thanks!

  3. I just want you to know that I heard about you on NPR before I graduated College, and I looked you up, and I am so impressed with your ideas and your enthusiasm. I just started my first year teaching at a small school, and I teach calc, pre-calc, algebra II, and Applied math. I am so excited to find so many hands on activities and great ways of explaining math on your website. Thank you for being an awesome teacher!

    1. Thanks Alexis! Hope your first year is going well! Let me know if there is ever anything I can help you with.

  4. I used your "Always, Sometimes, Never" assignment today with my Algebra 1 students. I heard some really interesting conversation and some good arguing (gotta love a good math argument!). Thanks!!

    I took to heart what you said about the students wanting to use the Real Number diagram. So I went to my local Dollar Tree and bought a bunch of different sized bowls. Then, students used post-it notes to label them. That way I could just buy 6 sets of bowls (for each group) and save some money. Here's a link to some pictures:

    I also did a card sort where they put different numbers into different sections. I think the model really helped them!