Teaching Real Numbers

I vividly remember teaching real numbers in my first year of teaching. It started off relatively simple with square roots and then slowly everyone (including me!) started to get confused when we got to the real number system and its subsets.

Thankfully for my students, I have grown more comfortable with the material and have learned how to teach it in a way that makes sense. Though I will say, it isn’t easy to make something like the real number system feel meaningful.

After students are good and comfortable with square roots and comparing and ordering real numbers, we move on to the Real Number System!

Irrational vs. Rational

Me: “Every number you see this school year will be a real number. None of them are imaginary.” Student: “Unless you erase it.”

Ask students to name types of numbers they’ve seen before, such as fractions, decimals, integers, etc. This is a great time to refer to the number of the day.

After a few years, I realized that students had a hard time seeing the difference between repeating decimals and terminating decimals. So, I decided to start with just that and created some fraction cards.

There was a fraction on both sides, one fraction had a repeating decimal and one side was terminating. I gave a card to each pair of students and had them convert them to decimals.

After about one minute, we used a T-chart to organize our fractions as either repeating or terminating. I explained that all of the fractions were considered rational and we learned the definition of rational numbers.

At this point, it felt like a good time to introduce them to irrational numbers.

I introduced it with an Irrational Numbers video from Generation Genius! (thankful for their 30 day free trial!) 

It provided a great explanation of irrational numbers and gave them a preview of the Pythagorean Theorem! I also had students complete the worksheets that went along with the video.

Some dialogue from class:

Teacher: What is the most popular irrational number?

Students: Pi

Teacher: What does pi look like?

Students: It never ends, and never repeats.

Teacher: Okay, what about the √20? What does that look like?

Note: Explain that it looks like it terminates on the calculator, but it does not! (Next year, I think I will have the kids do long division to see this…)

Teacher: So what does that mean for all other irrational numbers?

Students: They don’t terminate or repeat.

At the end of class, I gave them an exit ticket to identify various numbers as either rational or irrational.

Classifying Real Numbers

I always always always introduce this with the Real Numbers Song from Colin Dodds (search it up on youtube!). Play the song until your ears bleed, that’s how you get it to stick.

I started by giving each student a number before we watched the video. While they watched their task was to determine what type of number they had.

After watching the video, I took out the nesting bowls (you could do boxes) and went through each subset starting with natural, “If you have a natural number, stand up.” Then, they brought them to my bowl. After each subset, I went through the numbers with the entire class and we decided whether the numbers were classified correctly or not.

No nesting boxes or bowls? Try post-its and project a Venn diagram on the board.

We then took notes and students worked independently while I circulated the room to assist!

Review idea: Take your kids outside and have them do quiz quiz trade to practice classifying the real numbers!

Real Numbers Review

Our tests are standards-based, meaning all of the learning targets are tested separately. However, I still opted to do a mixed review of comparing and ordering real numbers and classifying. I used task cards and did a modified version of my prize bag activity. I did not have time to set up the prize bags so here’s what I did: students would pause after every 10 problems and get their papers checked by me.

If they had 10 checkmarks, I would write their name on paper and drop it in a basket. If they did not have 10 correct answers, I would give them a hint and they would go back to fix it.

At the end of class, I selected 4 winners from the basket. They could choose their prize, either a ticket, Jolly Rancher, or pick their seat for the day.

Now we are gearing up for the next unit: Consumer Math!