Using the chart paper,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.draw a squarewith an area of area 40 units^{2}

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)*

2x20

6.6 x 6.6

4x10

2x20

6.6 x 6.6

4x10

*7.5 x 7.5*

6.7

6 x 7 =42

6.7

6 x 7 =42

What’s 40 divided by…

Can we do point 5’s

HAS TO BE A SQUARE RIGHT?

Factors of 40

40 is NOT A PERFECT SQUARE!

6 is closest to 36.

__Class C__Guys, it’s 40 divided by 15

2.66

Wait, 40 isn’t a square.

20 x 20 that’s it...no wait.

20 x 20 is not equal to 40...it’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 units

^{2}. 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?

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

ReplyDeleteThere 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"...it 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!

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

DeleteThe 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.

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

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

ReplyDeleteThe 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.

DeleteIt 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!

ReplyDeleteIt 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

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

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