Comparing and Ordering Real Numbers

Before diving into comparing and ordering real numbers, we start with reviewing the square roots of perfect squares and estimating square roots.
As you likely know, number sense is crucial for being successful in a math classroom. This unit sets the foundation for the REST of the year. As a result, I have learned to take it slow and to do formative assessments with every single skill (exit tickets are my BFF).
After reading this post, you will have some lesson ideas along with teaching tips that will set your students up for success! Plus, there’s a freebie inside that you can use all year long!

Square Roots of Perfect Squares and Non-Perfect Squares

I love introducing this skill with color tiles. Start by giving students a bucket of color tiles. I like to provide them with 3 minutes to play with them before we start the activity. You can put them in partners if you are limited on color tiles!

After they finish playing, tell them to try to create a square with as few tiles as they can. This should be 1 tile. Then, ask them how many square tiles they need to make another square. This should be 4 tiles. Review how to find the area of a square.

Most will say A = l x w. Suggest they use s2 since we are working with squares. Then work your way up to at least 36 if you can. At this point, students will likely need to start sharing tiles.

Something new I added this year was an idea from my math coach and it is a KEEPER! We worked through this as we worked with the tiles.

Using a black pen, students could draw the radical to show the side length is the square root. This was actually kinda fun! In the future, I may have them make posters…somehow!
After this, we took guided notes on perfect squares of square roots so students would have a resource to refer back to.

We ended by playing square roots bingo, this was our first time for the year and they’ve been asking to play more bingo!

Estimating Square Roots

So what about non-perfect squares? We started with slides 11-20 of this free Desmos activity: Square Dance Desmos activity. It’s basically like using color tiles digitally! I had students work through it at their own pace for about 15 minutes and then we took guided notes (of course)!

One thing I was sure to reinforce this year was using a number line. So we started with a clothesline activity to see if students could put the non-perfect squares in order. I gave each kid a card and then put them in groups of 7. Their task was to organize themselves in order!

An idea from my math coach that I will try next year: Project a hundreds chart and have students color in the perfect squares. And then ask, “What about all of the other numbers? Do they have square roots?” I was able to try this for struggling learners after the lesson and it was super helpful! They were able to quickly see where the non-perfect squares were.

I then had students complete a scavenger hunt with a partner around the room where they had to place non-perfect and perfect squares on a number line. The number line was great because they received immediate feedback, if it spelled a word they did it correctly!

Independently, they completed an estimating square roots drag and drop activity to reinforce the idea!

Comparing and Ordering Real Numbers

Even though my students can use Desmos calculators (thank goodness), I noticed that they really struggled to understand that 41.2% was a percent and not a decimal in standard notation.
One downside of Desmos is they attempt to type the “41.2%” into the calculator and that does not work unless they type “41.2% of 1”, and they quite simply don’t remember to do that!
In addition, I had students looking at a number like “3” and saying, “there’s no decimal point…so it isn’t a decimal.” At this point, I realized I needed to do two things:

1. Change my language. instead of only saying decimal, I started saying “simplified into the standard format”. I also started doing a number of the day so they could visually see the different formats for writing rational numbers.
2. Introduce them to invisible math! I had students turn to the invisible math page of their binder and pointed out that any number without a visible decimal point has an invisible one and a zero. Well, the next day, this same student came back and said, “Didn’t you say 0.3 is the same as 3 because of the invisible decimal point?” YIKES. Language change again…”Any number without a visible decimal point has an invisible one and a zero BEHIND IT!”

Another super tip?
Use the same grid paper from square roots of perfect squares to help students them with comparing and ordering real numbers. Have them line up the decimal points.

Next, we will classify real numbers! Stay tuned for the next blog post all about that!