Reading Inequality Symbols

[lhwalker]

First graders learn to use inequality symbols, per the NBT progressions this way: “putting the wide part of the symbol next to the larger number.” I don’t see when, if ever, we tell students to read a symbol “greater than” or “less than.” Would it be in 6th grade 6.NS.7d when they have to …”recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.” Or is it expected that the students will always refer back to the idea of wide part next to larger number and think x > 6 is like 10 > 6 so read > as “greater.”

[Anonymous]

I kind of cringed a little when I read your post and then found that statement in the progression document. The whole idea of putting the wide part of the symbol next to the larger number doesn’t seem very mathematical. It seems more reminiscent of making the inequality symbol an alligator that eats the bigger number because it’s hungry. I don’t see how that helps students attach meaning to the symbols – it just becomes an exercise in identifying the larger number – which is fine if that’s the goal.

But since the goal in kindergarten is to identify sets as greater than or less than, I’m thinking the first grade goal should be to attach meaning to the symbols themselves. I would want the kids to see that we don’t have to write the words “greater than” or “less than” all the time because we have some notation that makes the job easier.

Can we get there with the wide mouth idea? Probably, but maybe a more direct route is helping the kids know the symbols and what they mean from memory. If the task is comparing 4 and 7, one thought process is “which way do I aim the hungry mouth” and the other thought process is “< means less than, and 4 is less than 7, so 4 < 7” or “> means greater than, and 7 is greater than 4, so 7 > 4”.

[lhwalker]

You raise a very important issue. We definitely need to get rid of mouths because x<6 is interpreted as x eats 6. Wide-part-as-large, in my opinion, seems mathematically sound and works very well with my Algebra students, particularly when analyzing word problems: “She needs to make at least $50, so what needs to be large?” After seeing <> for many years, many of my 14-year-olds still get them mixed up. I write <ess frequently, but still… So after my initial surprise to see wide-is-large in the progressions, I felt affirmed in my practice. It makes sense to me to have mental images like 8<10, 0.05 < 0.5, 1/2 > 1/3, driving the recall. We all have “hooks” that are necessary for retention. So if they can recall an example like 8<10, then they can recall “less than” if they pause a second to think 😉

[johnrmead]

It might be nitpicking, but another issue that pops up in some high school courses is the 0<x<1 notation. One could argue that it’s an abuse of notation, but it is common enough that at least some students will come across it sooner or later. In this setting, the preferred reading might be that x is between 0 and 1, a statement which completely omits any mention of “greater than” or “less than”. It has always struck me as imprecise when I’ve told students that “<” means “less than”. I would much prefer to call both signs order symbols and instruct that “4<6” could be read “four is less than six” or “six is greater than four”.

[lhwalker]

Ha! I just graded a unit test and I saw a lot of: sq root 11 < 3 2/3 > 3.51. Because I never know exactly what vocabulary my students will encounter, I read 0 < x < 10 as “x is between” and sometimes say, “Zero is less than x which is less than 10, but isn’t that a mouthful?” Between notation is succinct for describing domains and ranges. Is there a way to connect the terminology “order symbols” to a solution set like {x | 0 < x < 10}?

[Bill McCallum]

I guess in Kindergarten students might just be saying “bigger” and “smaller,” but I don’t think they need to wait until Grade 6 to see “greater than” and “less than.” In fact, you want bigger and smaller to go away sooner than that, because of the confusion this could cause with comparisons of negative numbers.

On “wide part,” I would say that’s a little different from alligators eating something, because it relates quantities visible in the symbol (the width of each end of the symbol) to the quantities in the comparison; there is no eating action here, which I agree is extraneous!

[Carole]

I realize that this thread is a few years old. As I kindergarten teacher, I am familiar with the “alligator” eating the larger number, and I’ve never thought that was a good way to teach the “less than” and “greater than” signs. The students don’t know what the sign really means. I remember how I learned the signs, way back in the 60’s (during the “New Math” era!), and that’s what I teach my students: Picture each sign as an arrow. Now, picture a number line or number path. The numbers to the right on a number line become bigger or “greater”, and that’s what “>” means. Back to the number line: the numbers to the left are smaller, or become less, and that’s what “<” means. The “Alligator” lesson only works when you are comparing two numbers. In fact, I just taught the arrow pointing down a number line to a substitute teacher the other day who said she can never remember which is which. She must have been taught the “alligator” way. Ha!

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