# How would you solve this problem without using square roots?

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I found myself stuck on one of my daughter's math problems.

I mean-- I can solve it using square roots, but she hasn't learned those yet so I know there is another method.

This is from the 4a intensive practice Singapore book.

Bonus points if you can show me how to use the bar model for it...

"The length of a rectangle is four times its width. If the area of the rectangle is 196 cm^2, what is the perimeter of the rectangle?"

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Here's what I got... once you simplify to 49=x times x, a 4th grader should be able to recognize that 7 times itself will be 49 without discussing square roots.

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I'm not sure that a 4th grader would know about simplifying by dividing by 4... but that is a challenge problem, right?

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I drew a rectangle roughly four times as long as the height and then chopped it into four pieces (they visually looked like squares)

Next, I took the area of 196 and chopped it into 4 pieces (aka dividing it by four) and wrote the answer 49 in each box

I believe a 4th grader doing Singapore challenge problems should (or with some prompting) recognize that a square with an area of 49 has a side of seven. I wrote a 7 on the vertical height and a 7 on the base of the first square.  Since the squares making up the large rectangle are all equal, they should see that you add up (or multiply) all those 7s to get the perimeter.

unless we had done something recently, my older kids would likely have needed a quick prompt along the lines of “what’s the definition of perimeter?”

I didn’t start early with Singapore math and I struggled with those bar drawings and figuring ways to solve the problems without algebra.  Variables would have had some blank looks in fourth grade.

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I’d just try some numbers? It’s pretty obvious that if you increase the width, you increase the area so if you try 1 and 10, say, you’ll see it’s in between somewhere... then try a few more.

I don’t know if that’s what’s intended, but in my experience, the “just try it” problems are wonderful for a kid’s number sense and make kids think about interactions between operations more than calculational and algorithmic problems.

Edited by square_25
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Just throwing in that my 4th graders would have thought "what would be an easier number to quarter than 196?" (200), and correct the answer (50) by a fourth of the added 4 (1). Dh likes the approach of "solve an easier problem in your head instead."

The single digit products and the squares through 16^2 are easy to have memorized at that age without learning the terminology about square roots (dh liked to use the word "unsquare" instead). So 49 immediately yields 7.

I never really understood the Singapore bar approach, but this looks like a good problem for starting with drawing the rectangle.

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I showed the picture of the rectangle with 1 x on the side and 4 xs along the wide part to my third grader and she immediately caught that the big rectangle was 4 squares. She doesn't have enough multiplication to know 7x7=49 yet but she figured out 5x5 four times was too small and 10x10 four times was too big.  (PS I didn't ask her to figure out the perimeter. Just the lengths of the sides)

Edited by vonfirmath

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My fifth grader would just start plugging numbers, and I think that’s okay. She might start at 5x20, then 6x24, then 7x28, and then she’d know the outside measurements and figure it from there. It seems appropriate for a child to work the slow tedious way that makes sense given what they know. Later they can learn a quicker way that gives them the correct answer.

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Permiter of Rectangle = 70

Length = 4x = 28

Width = x = 7