Monday, October 15, 2012

What happens when you've never even heard of a square root?

On the board,
Using the chart paper, draw a square with an area of area 40 units2
I circulated around the class and recorded the conversations. I didn't answer any questions but I did repeat, 'it has to be a square' a few times.

I hadn't even asked a question yet and this is what I got in return.

What I heard and what I saw:

Class A
Wait, how big is that?
Can we use a calculator?
Does it have to be a square…can it be a rectangle?
I know what it is! ~ (…what does he mean by it?)
But 40 is not….
Is 40 a perfect square!
Points…we have to use points… point 5
6 x 6 = 36.
40 is not a perfect square.
It must be between 6 and 7
What's 40 divided by…
(and again)
What’s 40 divided by… (not sure how to complete this sentence)
Does it have to be a perfect square?
What can be divided by 40 so that it makes a perfect square?
What number by what number will get us 40…but it has to be the same number.

Class B
Does it have to be a square?
No rectangles?…is a rectangle a square?
It has to be times by the same number
(found written on a page)
6.6 x 6.6

7.5 x 7.5
6 x 7 =42

What’s 40 divided by…
Can we do point 5’s
Factors of 40
6 is closest to 36.

Class C
Guys, it’s 40 divided by 15
Wait, 40 isn’t a square.
20 x 20 that’s wait.
20 x 20 is not equal to’s like...400
Let's find something that = 40
Something that multiplies to give 40
What times itself = 40, let’s start with that....
Has to be lower than 6.5
Try multiplying 6.5, 6.8, 6.7
More than 6, because 6 x 6 is 36.
6.3 is too low.

Class D
A rectangle is a square
When you say a square what do you mean?
Area is 40 units (2)
Rectangle is a square or a square a rectanlge?
2 of the same...numbers
Something times something....= 40
It needs to be squared
Can we go to decimals
Definitely not 7, no wait, definitely not 6
6 x 6 = 36
6.5 x 6.5
What’s on these sides? (points to the sides of the square)
It’s a decimal point!
8 x 5 = 40
6.3 x 6.3

Some End Results

I also noticed a surprising amount of reluctance at not being exactly at 40 units2. So much so that some groups were paralyzed and did not show any numbers, like this:

And, to be honest, I even saw some of these (despite work being done somewhere on the side.)

And then, after all that, I mentioned square roots.
I feel like my return on investment is very high here.
Which of these student comments stands out for you?
Are there some comments worth noting more than others?


  1. I love the "what's 40 divided by..." and watching them zero in on rational approximations. This must have been fun for you.

    1. There was something about the pause that I really enjoyed. The struggle I guess. I appreciated the struggle to find the answer. The fact is, he had to reword the sentence. Instead of saying, "What's 40 ÷ N = N" he reworded it and said, "N x N = 40" came out in words like this, "What number by what number will get us 40…but it has to be the same number." That struggle was fun to watch. Someone called it, 'Taxing their ingenuity.' Thanks for the comment. I did enjoy, mucho!

    2. The struggle is the best part. I can imagine your follow-up lesson. A little consolidation as all the hard work has been done!

    3. The consolidation included me telling them that sqrt(40) which gives 6.3245553 on the calculator is actually a decimal that goes on forever and doesn't have a pattern. A crazy number. Completely irrational.

    4. Maybe a little U2 playing in the background. I think the song is obvious!

  2. I know it wasn't the point of your exercise, but the girl who got 6.325 gives me hope for humanity.

    1. The group of 4 students divided up the work trying 6.3, 6.4, 6.35, 6.32..and so on until they ran out of time. Stopping at 6.325. They wanted to go further. I repeat...they wanted to go further. Hope for humanity indeed.

  3. It seems like a lot of the students really DID know what a square root was. They just didn't have the language for it. Love it!

  4. It would be interesting if you tried this again with graph paper; then kids can get to the exact area (by slanting the square), but not the exact side length

    1. New on my list for ordering supplies: Large Chart Grid Paper.