# The seven benchmark numbers?

I was presented with a power point slide by a friend about math education and one of his slides talked about "the seven benchmark numbers". He said that:

The seven benchmark numbers to develop a "complete" number sense are: $0, \frac{1}{10}, \frac{1}{2}, 1, 10, 12,$ and $100$. These numbers form the foundation of the mathematics curriculum in primary and secondary education.

Unfortunately, when pressed to do so, my friend was unable to explain why these numbers were "benchmarks." Does anyone know what he may be referring to or, better yet, does anyone know where he gets this information from?

• Why didn't you ask him the source? Strange, he's presenting material he can't explain. – JTP - Apologise to Monica Jun 5 '14 at 2:20
• To me (and others) a benchmark number is a useful one on which to base estimates. E.g., 1/2 is a good benchmark and helps us understand where 3/8 is on the number line relative to 1/2. I'm not sure what 12 is doing there, though. And this particular list seems arbitrary. – ncr Jun 5 '14 at 2:22
• Most of them are pretty straightforward to guess the motivation for, but surely the numbers alone are insufficient to develop any sort of "complete" number sense. @ncr The one seemingly arbitrary number, 12, is likely due to the non-metric system in which, e.g., one has a dozen (12) or - not long ago - a gross (144). Plus 12 inches in a foot, 12 hours in each half of the day, and many students in the United States learn the 12 by 12 multiplication table. I can't say anything else definitive about this list of "benchmark numbers," except that I have never seen the collection discussed formally. – Benjamin Dickman Jun 5 '14 at 2:27
• He was unable to provide me with the source (making me even more interested in this) – illysial Jun 5 '14 at 2:36
• This strikes me as very arbitrary. As a mathematician, I would not give any special significance to these numbers. Especially $12$ would not be important in many parts of the world where metric system is used. It is somewhat arbitrary to include $100$ but not, say, $1000$. Also, why include $1/2$ but not $2$? – Jakub Konieczny Jun 5 '14 at 17:24

A decent volume on elementary mathematics is Mathematics for elementary teachers (Beckmann, 2010). The book is intended to help strengthen teachers knowledge of the mathematics behind the ideas in elementary curricula (especially reform curricula, I think). As such, it's often a good place to check for things like this.

Benchmarks (also called "landmarks") are introduced in the context of comparing fractions. When students are trying to determine which fraction is larger, $\frac{4}{9}$ or $\frac{3}{5}$, one strategy suggested is for students to reason about their relationship to some other number, like the fraction $\frac{1}{2}$ :

When we compared $\frac{4}{9}$ and $\frac{3}{5}$ by comparing both fractions with $\frac{1}{2}$, we used $\frac{1}{2}$ as a benchmark (or landmark). The fractions $\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$, $\frac{1}{3}$, and $1$ are good to use as benchmarks. (p. 73)

It's clear from this text that the numbers are somewhat arbitrary; there's not meant to be a definitive list of benchmark numbers. Students would choose a fraction benchmark that helps them compare.

I can't say whether others use benchmarks the same way (a quick look at some other books I have within arms reach doesn't show up the term). However, the use here is clear: a benchmark number is a number useful in reasoning about a problem. In this case, the benchmark is used as a reference point for comparison of fractions.

The intent is to encourage reasoning rather than procedure. There are algorithms some students are taught to use for fraction comparison, which allow them to replace mathematical reasoning with a couple of memorized steps and some arithmetic. But reasoning allows them to practice conjecture, work through coming up with a justification for their answer, and eventually have a way to defend their answer other than "this is what the procedure produced."

I should think any useful number used in reasoning could be called a benchmark. For example, in my response to another question (seen here), I wrote about student reasoning that transforms a subtrahend into the number $2000$. In that case, $2000$ is useful.

Another type of mathematical reasoning that might benefit from a benchmark is estimation. Numbers can be replaced by nearby benchmarks that make for quicker calculation, if the aim is to just ballpark an answer (an often quite useful strategy for many real world applications).

In summary, I don't think there is support for a definitive list of benchmarks. The ones Dr. Beckmann provides are suggestions ("good to use") but the real test is whether they are useful to the thinker in the midst of their mathematical reasoning.

Works Cited:

Beckmann, S. (2010). Mathematics for elementary teachers. New York: Pearson Addison-Wesley.

• maybe it's just me being lazy, but as a child, I think I would just calculate the decimal expansion to compare two fractions. I've read some history of physics which echos this sentiment... that the decimal number system was extremely important to the approximation aspect of Newton's thinking... but, I'm no expert. – James S. Cook Jun 6 '14 at 3:01
• @JamesS.Cook It's not lazy to use the representation that best fits your skills and the application at hand. Classroom work has an additional learning goal, of course. In this case, turning to reasoning for the comparison (in that, it stands in contrast to some other "trick" methods). Out of curiosity, when you were comparing fractions with decimals as a child, what reasoning connected the fractional and decimal representations? In other words, how did you informally prove to yourself that the decimal representation was truly the same number? – JPBurke Jun 6 '14 at 11:48
• If I recall, and that is debateable, I believe it was the standard meaning. For example, $1/4 = 0.2+0.05$ so we build the decimals from adding integer multiples of $10,1,1/10, 1/100$... together. The need for series was only appreciated much later, approximations sufficed for my purposes as a child, I don't recall pondering convergence on the playground. – James S. Cook Jun 6 '14 at 16:29
• @JamesS.Cook So the sort of "atomic" knowledge here is that $\frac{1}{10} = 0.1$ (and so on for other fractions involving powers of ten). But also, you would have to justify that $\frac{2}{10} + \frac{5}{100} = \frac{1}{4}$. On the face of it this looks more sophisticated than comparing two fractions based on a benchmark (i.e. you'd be beyond needing that benchmark strategy at this point). Your power-of-ten-denominator fractions is obviously a vital part of understanding of how place value applies to fractional values. – JPBurke Jun 6 '14 at 18:31

I can't back this up, but here's a thought as a mathematician and father of school aged kids (in order that the benchmarks arise):

1: Represents the whole idea of what a number is. Once you get 1, you just have to memorize 2, 3,..., 9.

0: Represents understanding that nothing is a quantity/number, too.

10: At first "10" is just another symbol for a number like "7". But if you really get that it's a 1 and a 0, then the symbols 11,...,99 become immediately understandable.

100: Understanding "ten" is one thing. The next step is understanding that there must be a new name for ten 10s. Once you get "hundred", then "thousand", "ten thousand", "million", etc. become memorization.

1/2: Being able to truly understand 1/2 means you get what fractions are. I know students really struggle with fractions, but it all starts with 1/2.

1/10: Once you get fractions, the question of decimal representation is natural. So I'm guessing 1/10 should really mean understanding 0.1.

12: A bit of an odd ball on the list. My guess is one of two possibilities: It's important because most students memorize multiplication tables to 12x12, or because in English, "twelve" is the last number whose name tells you nothing about its decimal representation, e.g. Maybe it should've been called "seconteen".

• If you look closely, "twelve" does at least contain a form of "two." See also etymonline.com/index.php?term=twelve. – J W Jun 11 '14 at 4:32
• Twelve is the first abundant number, and also key in the clock model some teachers use for fractions. I don't know if that is why it's on the list, but certainly it makes some sense why it might be on a list of important numbers in 4th and 5th grade. – JPBurke Jun 12 '14 at 1:56
• The whole number "1" is the Universal Multiplicative Identity. Although "2" isn't needed as a basis for whole numbers, I would consider the fact that multiplying anything by the whole number two is the same as adding it to itself is pretty important. I would consider "4" important because it multiplying something by four is the same as adding a something to itself and adding the result to itself, while "3" is important because multiplying by three requires adding something to itself and then adding the result to the original thing. – supercat Jun 20 '14 at 15:34