Perimeter formula – Grade 4

[Duane]

4.MD.3 requires students to apply the area and perimeter formulas for rectangles. The Measurement Progressions (p. 21) seems to suggest that students should know and use P = 2(l + w), with simpler formulas of P = l + w + l + w or P = 2l + 2w as alternatives that lead towards it.

Is it expected that all students use P = 2(l + w) by the end of Grade 4 or can they use one of the other formulas? They all give the same answer of course. It’s just that P = 2(l + w) and even P = 2l + 2w require students to omit the multiplication symbol. This appears to be the first and only place where it occurs in all of the elementary grades. Even the volume formulas given in 5.MD.5b use the multiplication symbol instead of V = bh.

[Bill McCallum]

The measurement progression should include the multiplication sign in these formulas, thanks for pointing that out. In a case like this, the standards trump the progressions, and the standards are not insisting on leaving out the multiplication sign.

[Bill McCallum]

Phew! Thanks for the clarification on the multiplication symbol, Bill. So should students be using P = 2 ×(l + w) by the end of Grade 4 or can they use one of the other formulas I mentioned?

[Bill McCallum]

I think worrying about which formula to use is the wrong focus here. To quote from the progression:

“Apply the formula” does not mean write down a memorized formula and put in known values because at Grade 4 students do not evaluate expressions (they begin this type of work in Grade 6). In Grade 4, working with perimeter and area of rectangles is still grounded in specific visualizations and numbers.

We don’t want students in Grade 4 to think of any of these formulas as a way you have to pass through in order to find the perimeter (in fact, I’m not sure we ever want that). Rather, formulas are a way of describing the calculation you do in order to find the perimeter, and different versions of the formula correspond to different ways of doing that calculation. But the basic understanding of perimeter should not be mediated through a formula; perimeter should be understood as the distance around the sides, which you can find without knowing any formulas.

• This reply was modified 11 years, 5 months ago by Bill McCallum.

[Bill McCallum]

Thanks again Bill, I’d missed that bit in the Progressions (p. 22 for others who are interested). Without reading that part of the Progressions I’m sure many teachers will mistakenly read “apply the formula” in the Standards as exactly what you advise against. I’m a little puzzled how it relates to these statements on page 21, “… remembering the basic formula can help to prevent the usual error…” and later on that page “…discussion of formulas such as P = 2l + 2w…”.

I’m guessing that the formulas have value in Grade 4 and 5 as a “summary” of the thinking that students must use – you can discuss them but students aren’t expected to use them in the sense that they substitute numbers into a formula. Am I on the right track?

[Cathy Kessel]

Duane, I think that your earlier comments are partly about the meaning of “the” in 4.MD.3:
Apply THE area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
It’s fairly easy to see l x w and w x l as being two variants of the area formula, if you’re aware of the commutative property.

It may be less easy to see the various forms of the perimeter formula as the same because they involve the distributive property. That’s also complicated by the way in which the word “formula” is used in the text and discussion of “various formulas” on p. 21. (Sorry about this! My job is to try to catch these things, but remember the drafts of the progressions are called drafts for a reason.)

Note that the text on p. 21 of Geometric Measurement progression draft talks about “specific numerical instances” of the various different forms of the perimeter formula. That’s one reason to write the equation with the different forms as an adult would, without the multiplication symbol. Anyway, your comment suggests that the progression needs to give more detail, perhaps with an example in the sidenotes.

• This reply was modified 11 years, 5 months ago by Cathy Kessel.

[Bill McCallum]

Thanks for helping, Cathy. I guess my initial question was how to write the formulas and which ones are expected. It then moved on to wondering whether formulas written with letters should be  used at all in Grades 4 and 5. The reference to “specific numerical instances” gives some guidance as to how to discuss how terms can be rearranged but I’m mainly concerned about whether I should write P = 2 ×(l + w) (and other formulas) on the board or not.

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