## Desmos: An Introduction to Irrational Numbers

Recently, I co-taught a lesson with a colleague where students  investigated square roots.  The lesson revolved around the following  Desmos Activity Builder task. Afterwards, I blogged about the experience and sent the post to my teachers.  Peytra, a six grade teacher responded.  She inquired about running the lesson in her classroom. My heart sank. I felt I had misled her since the lesson was designed for 8th graders. The remedy – create  a lesson appropriate for 6th graders.

This was a fun lesson to share with teachers and students!!

Acknowledgements:

• Peytra:  For being my inspiration to create this lesson.
•  Michael Fenton (Lead Instructional Designer at Desmos).  Your activity builder, Square Builder sparked an idea.
• Allison, Carmen, Peytra and Shannon: For letting me refine the activity with your students.
• Tim:  For bringing the lesson to classes when I couldn’t!

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## Day 1 – The Background Work

Day 1:  The structure of the lesson comes from the following Desmos Activity Builder task.

## The Breakdown of Slides 1 – 4

In Slide 1: Students were to create a square that has an area of 9 square units.

In Slide 2:  Students were to explain how they arrived at their answer.

Discussion Time #1:   I scrolled through their slide 2 submissions and selected a few student responses to highlight – deliberately picking ones who mentioned an anticipated response.  Those students read their responses out loud as I recorded on the board.

• I counted the boxes inside
• I saw 3 groups of 3
• 3 x 3 = 9
• Length = 3, width = 3
• S = side.  3² = 9
• I moved the orange point until the area box equaled 9.
• I moved the slider until it equaled 3.

Bonus: The square root of 9 is 3.  I wasn’t looking for this but this was a student’s response.

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Slide 3:  Same as Slide 1 but with 16 square units.

Slide 4:  Explain how you arrived at your answer.

Stop – Discussion Time 🙂

Even though the official stopping point was after slide 4, I did call a whole group discussion after slide 2 (mentioned above).

Why the added whole group discussion?

1. To create flexibility.
1. Students who zipped through slides 1 & 2 could move on while the other students finished.  The group discussion highlighted various strategies which were recorded on the board.
2. To guide the conversation toward the learning goal.
1. After the whole group discussion, students were to complete slides 3 & 4 with the criteria to use a different reason/strategy when describing their process. Students who had already finished, now edited their initial slide 4 submission while their classmates finished.
2. When we reconvened, I called on students to apply the previous strategies to the new problem.

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## The Breakdown of Slides 6 – 7

Slide 6:  Same as 1 and 3 but with a decimal area value.

Again, we applied all the strategies toward an area value containing a decimal.

• length = 2.5 units  width = 2.5 units
• (2.5)(2.5) = 6.25 square units
• S = (side) = 2.5
• S² = 2.5² = 6.25
• Counting the number of squares, joining 1/2 squares and identifying the 1/4 square.
• Checking the area box
• Moving the orange point/slider to find the answer

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## The Breakdown of Slide 8

Slide 8:  Same as 1, 3 & 6 but with an non-perfect square area.

• Seconds after working on slide 8, I hear
• It won’t let me
• I can’t get 12 exactly
• It’s broken
• I can get close but not exact
• It’s impossible
• I can only get 11.9716
• I can get 12.25

Ah, music to my ears!

My two responses were…

• Are you sure you can’t get closer?
• Why not?  Why can’t you get 12 exactly?

By the time we reached this point, we didn’t have a lot of time before the bell rang.  The discussion was rushed.

Therefore, I’ve learned to stop at this point and leave the lesson as a cliffhanger.

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## Day 2 – Discussing Square Roots

1. Students logged back into the Activity Builder assignment.
2. Recap:
• (3)(3) = 9                            3² = 9
• (4)(4) = 16                         4² = 16
• (2.5)(2.5) = 6.25           2.5² = 6.25

3.  Students jumped to slide 8 and try to get as close to an area of 12 square units as possible. Here’s a memorable conversation.

• Student: There isn’t a whole number that when multiplied by itself equals 12.
• Me:  Can the side have a decimal value?
• Student:  N0.
• Me:  What was the side length for the square with an area of 6.25 sq units?
• Student:  2.5.  Oh yea  (thinking) The decimal’s gotta be really long

In a different class ….

• Me:  Why is finding the side length for a square with an area of 12 so difficult?
• Group 1:  12 is not a perfect square
• Group 2: 12 doesn’t have a square root
• Group 3:  There isn’t a number times itself that equals 12.

4.  Collect trials

• (3.5)(3.5)       =  12.25
• (3.4)(3.4)       = 11.56
• (3.47)(3.47)  =  12.0409
• (3.48)(3.48)  =  12.1104
• (3.45)(3.45)       = 11.9025
• (3.464101615)(3.464101615) =11.999999999 (Kai zoomed in to get this result)

4. Square root discussion

• Me:   Do you want to find a more exact side length?
• Students:  Yes!
• Me:  We are going to talk about a symbol known as the square root.

Students entered √12 into Desmos. They get:  3.46410161514.  I wrote in that number for  the square’s side value.

In Kai’s class, we compared the his answer with desmos’ answer for √12.  Students quickly noticed that the Desmos decimal was similar to Kai’s value  but longer.

5.  Addressing a misconception.

In the class, where students stated, “12 doesn’t have a square root.” I asked:

• Me:  Do you think 12 has a square root?  (and then circulated the room)
• Students:  yes!, no! yes? no?
• Karissa: No
• Mariah (towards Karissa) yes.  It’s right there. (pointing to Karissa’s screen)
• Me: (taking my cue from Mariah)  If you think the answer is yes, then prove it.  Point to the square of 12.

Slowly students pointed to:

After circulating, I address the class.

• Me:  Does  12 have a square root?
• Class:  Yes!
• Me:  What is the square root of 12?
• Class:  Not in unison 🙂:  3.46410161514

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6.  Web 2.0 calculator to find √12.  Once kids reached the site and entered √12, I hear:

• It’s the same number
• It’s a different number.
• Why is it different?

These comments guided my next move.  I started at the 3 and asked

• Is the 3 the same?  Student:  Yes
• Is the decimal the same?  Students:  Yes
• The 4?  Students:  Yes
• The 6?  Students:  Yes
• (Continued to the last 4)
• The 4?  Students:  No!
• What should I write here?  Students:  3!

I updated the square on the board.

I then showed them:

The web 2.0 calculator is deep enough to show students that even rounding to 16 spaces after the decimal, the product is not exactly 12.

We talked about why the number changed.

• Larry (summarizing):  The number grew because the calculator window grew.
• Me: What does that say about the √12?  Discuss in your groups.
• Groups:  It never ends.

This conversation set me up to introduce the concept of irrational numbers.  We briefly discussed irrational numbers before the bell rang. The second part of the lesson was taught on a minimum day, so the class time was reduced, therefore the irrational number discussion was informal and varied depending on the group.  Here are some student comments:

• The square root of 12 seems like Pi.
• Is it Pi’s cousin?
• What do mathematicians do if they want to use the whole number?  (Talking about the decimal version of √12.  What a great Question!  I did have time to answer.)

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## Closing Thoughts

On a personal note, I loved teaching this lesson. Teachers and

Allison

students enjoyed both the content and the interactive piece. During the slide 8 discussion, Allison (teacher) hopped onto the demo computer and started tinkering with the side value.  Her students were glued to the screen watching her test out numbers.  Soon they started to do the same.

Student excitement and learning showed through their reactions to the activity builder tasks, small group and whole class discussions.  Here are a few comments:

• Larry (after day 2):  That was intriguing.
• Kayley (running up to me on day 2):  I can’t wait to do math today
• Lexi (after day 1) That was fun
• Carmen (teacher) The students enjoyed the lesson.
• Rachel:  I liked your lesson

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Other Posts:

A link to my Desmos Page which contains other Activity Builder tasks, Polygraphs and blog posts written about lessons incorporating Desmos

Tap Tap Trigonometry