In my Pre-Algebra classes last week, we worked on developing an understanding of irrational numbers. Specifically, we focused on irrational numbers like √2 and -√73. Among the various elements to developing this concept, I had students work with finding square roots of perfect squares and estimating the closest integer for an irrational square root. Plotting approximate locations for irrational numbers was also a part of the scope of this concept, so I knew right away what I wanted to do with this topic when I was thinking about it at the start of August: Clothesline math.
I used the Clothesline activity last year for my 6th grade students when we learned about integers. The activity was the perfect way to have students see negatives as opposites of positives and introduce absolute value. It was awesome to see a student place -2 far away from 0, only to have another student protest saying, “-2 needs to be just as far from 0 as +2.” I knew using a Clothesline activity with irrational numbers would be just as valuable for my 8th grade students.
The set up:
- Cheap clothesline ($1.98) from a hardware store hanging from the front and back staple strips above the whiteboards in my room
- Clothespins from a dollar store
- Cards with integer values -7 to +7
- Cards with perfect square roots (-√25, +√16, etc.) and other rational numbers (1.5, -2.5, etc.)
- Cards with irrational square roots (-√22, √2, √3, √43, √48, etc.)
- Purposely vague directions
I started the lesson by asking students what they thought the clothesline was going to be used for that day. Not surprisingly, a student guessed a number line because we drew a couple in our notes the previous day. I told students the activity would take a while and require everyone to critique each other’s work, then I directed the people with -1, -2, 0, +1, and +2 to hang their cards with a clothespin.
The 4 students made quick work of this task and I noticed two of them using their hands as measuring tools to place the cards about a half foot apart from each other. I asked them to explain to the class what they were doing and a student responded, “The numbers need to be the same distance apart.” I moved the cards slightly so there would be plenty of room in between the numbers (and so the final number line would use the whole length of the rope), then I moved onto the interesting part.
I directed anyone with a card with an integer value to put it up on the clothesline. Immediately, the students with integer cards (-5, -4, +3, etc.) got up and started coordinating to place their cards at the appropriate locations. A moment later, a student said to me, “My card has √16. That’s just 4. Am I supposed to put my card up there, too?”
In keeping with my style, I answered with a question of my own, “Does your card have an integer value?”
Realizing the vagueness of my direction, the student got up and clipped his card over +4. A couple other students asked me similar questions and I gave similar responses. Within a minute or two, all of the students in the room with perfect square roots realized that they needed to put their cards up on the clothesline. It was pretty cool to see the realization dawn on student after student.
After a little discussion about what students did and asking about some cards (a couple students with perfect square roots clipped their cards next to an integer, but were able to tell me it should be in the exact same place as an integer), I directed any student with any remaining card to put hang it up on the clothesline.
I didn’t say much as students moved around and I watched intently as I saw students stopping, putting a card for an irrational square root between the two closest integers, then moving it closer to a particular integer. One student put √48 really close to +7, while another put √43 close to halfway between +6 and +7. A student with -√11 changed his mind about whether it should go between -2 and -3 or -3 and -4. One student still working on the square root thing put √5 on 5. It was really interesting to see two students repeatedly swapping the order of √2, 1.5, and √3 over and over again. They finally settled on placing √2 closer to +1, 1.5 halfway between +1 and +2, and placing √3 near where 2/3 would have been.
I tweeted a picture of part of the final product:
— Tom Hall (@trigoTOMetry) September 7, 2016
When it was all settled, I brought everyone to attention and began asking about specific cards. Everyone was in agreement about some cards. For example, everyone agreed √48 should be slightly before +7 because +7 is √49. Similar reasons were given for other cards, but some cards led to disagreement. For instance, I asked about the placement of √2, 1.5, and √3. One student said √3 should be on 1.5 because 1.5 was half of 3 (we’re still working through that misconception). There was some buzz in the room about the cards until one person raised a hand and calmly said, “2 is closer to 1 than 4, so √2 needs to be closer to √1 than √4. It’s the same idea for √3.”
There was at least one audible, “Oh, I get this now,” after that statement.
Discussing a couple more cards, I concluded by emphasizing to students the thinking process they did for plotting irrational numbers. They created a number line with equal intervals. They matched perfect square roots with integers, then they used those values to estimate where an irrational square root needed to be placed. It wasn’t random. Even though we’ll never be able to place the irrational square roots in their exact locations on the number line, we got close because of the thinking we did with the number line we made.
Some students were slipping into social conversations as the activity passed the 20 minute mark. Next time, I think I’ll either use more cards (some students only placed one card the whole activity) or set up multiple clotheslines in the room. Some of the discussion I had with the whole class might have happened naturally if I challenged groups of 4 to create their own clothesline. Also, I want to slide into approximating irrational square roots with mixed numbers on the same day. I needed to pick up with approximating the following day, which was fine; however, I think the process would have been easier if I was able to ground it in the concept of plotting on a number line.
Overall, the Clothesline activity was a good way to get kids thinking about irrational square roots using a number line. There were elements of concept building, practice, and revision built into the activity. I also liked the way I was able to see the product of kids thinking as they moved cards they originally placed elsewhere.