Think about a 4-by-5 rectangle. The rectangle contains infinitely many points$—$you could never count them. But once you decide that a 1-by-1 square is going to be “one unit of area,” you are able to say that a 4-by-5 rectangle amounts to twenty of these units. A choice of unit makes the uncountable countable.
Or think about two intervals of time. One is the period of Earth’s rotation about its axis; the other, the period of Earth’s revolution around the sun. Both intervals are infinitely divisible$—$a continuum of moments. But once you decide that the first period is going to be a “unit of time,” you are able to say that the second period of time amounts to 365 of these units. A choice of unit makes the uncountable countable.
More abstractly, think of a number line. The line is an infinitely divisible continuum of points. Zero is one of them. Now make a mark to show 1. The mark shows 1 as the indicated point; it also defines 1 as a quantity whose size is the interval from 0 to 1. This interval is the unit that makes the uncountable countable. We mark off these units along the line as 2, 3, 4, 5 and so on. (Later, we go the other way from zero marking off units: $–$1, −2, −3, −4, −5 and so on.) It is not for nothing that mathematicians call 1 “the unit.”
To a physicist, measurement is an active idea about using one empirical quantity (such as Earth’s rotation period) to “measure” (divide into) another empirical quantity of the same general kind (such as Earth’s orbital period). Mathematically, one can see that measurement is linked to division: how many units “go into” the quantity of interest.
Another way to say it is that measurement is linked to multiplication, in particular to a mature picture of multiplication called “scaling,” (5.OA) in which we reason that one quantity is so many times as much as another quantity. The concept of “times as much” enters the Standards in Grade 4 (4.OA.1, 4.OA.2, 4.NF.4), limited to whole-number scale factors.
The number line picture of this is that we are progressing from concatenating lengths to stretching them too. Thus, 2 is simultaneously “one more than one-more-than-0″ and “twice as much as 1.” These two perspectives on 2 are linked by the distributive property, which defines the relationship between addition and multiplication. $1 + 1 = 1\times 1 + 1\times 1 = (1 + 1)\times 1 = 2\times 1$.
By Grade 5, scale factors and the quantities they scale may both be fractional; the flow of ideas extends into Grade 6, when students finally divide fractions in general. At that point, we may consider a deep measurement problem such as, “$\frac{2}{3}$ of a cup of flour is how many quarter-cups of flour?”
The roots of all this in K$–$2 are 1.MD.2 and 2.MD:1–7, and 2.G:2,3.
When we reflect back on the geometry in K$–$2 from this perspective, we see that some of what is going on is learning to “structure space” by, for example, seeing a rectangle as decomposable into squares and composable from squares. Researchers show interesting pictures of the warped grids that students make until they get sufficient practice.
Already by Grade 3 we are dissatisfied with measurement. We want to know what happens when the unit “doesn’t go evenly into” the quantity of interest. So we create finer units called thirds, fourths, fifths, and so on. This is the intuitive concept of a unit fraction, $\frac{1}{b}$: a quantity whose magnitude is equal to one part of a partition of a unit quantity into $b$ equal parts. (3.NF) We reason in applications by thinking of the unit quantity as a bucket of paint, or an hour of time. We reason about fractions as numbers by thinking of the unit quantity as that portion of the number line lying between 0 and 1. Then $\frac{1}{b}$ is the number located at the end of the rightmost point of the first partition.
Because you can count with unit fractions, you can also do arithmetic with them (4.NF:3,4). You can reason naturally that if Alice has $\frac{2}{3}$ cup of flour (two “thirds”) and Bob has $\frac{5}{3}$ cup of flour (five “thirds”), then together they have $\frac{7}{3}$ cup (seven “thirds,” because two things plus five more of those things is seven of those things). The meanings you have built up about addition and subtraction in K$–$2 morph easily to give you the “algorithm” for adding fractions with the same denominator: just add the numerators. (And don’t change the denominator $\dots$ after all, you would hardly change the unit when adding 3 pounds to 8 pounds.)
Likewise, multiplying a unit fraction by a whole number is a baby step from Grade 3 multiplication concepts. If there are seven Alices who each have $\frac{2}{3}$ cup of flour, it is a bit like when we reasoned out the product $7\times 20$ in third grade: seven times two tens is fourteen tens; likewise seven times two thirds is fourteen thirds. Again the meanings you have built up about multiplication in Grade 3 morph easily to give you the “algorithm” for multiplying a fraction by a whole number: $n\times \frac{a}{b} = \frac{n\times a}{b}$.
The associative property of multiplication $x\times (y\times z) = (x\times y)\times z$ is implicit in the reasoning for both $7\times 20$ and $7\times \frac{2}{3}$. So is unit thinking. In $7\times \frac{2}{3}$, the unit of thought is the unit fraction $\frac{1}{3}$. In $7\times 20$ and other problems in NBT, the units of thought are the growing sequence of tens, hundreds, thousands and ever larger units, as well as the shrinking sequence of tenths, hundredths, thousandths, and ever smaller unit fractions.
The conceptual shift involved in progressing from multiplying with whole numbers in Grade 3 to multiplying a fraction by a whole number in Grade 4 might be aided by the multiplication work in Grade 4 that extends the whole number multiplication concept a nudge beyond “equal groups” to a notion of “times as many” or “times as much” (4.OA:1,2). The reason this meshes with the problem of the seven Alices is that those seven Alices don’t exactly have among them seven “groups of things,” yet they do among them have seven “times as much” as one Alice.
The step in Grade 4 from “equal groups” to “times as much,” along with the coordinated step of multiplying a fraction by a whole number, represents the first major step toward viewing multiplication as a scaling operation that magnifies or shrinks. Multiplying by 7 has the effect of “magnifying” the amount of flour that a single Alice has. In Grade 5, we will “magnify” by non-whole numbers, for example by asking how many tons $4\,\frac{1}{2}$ pallets weigh, if one pallet weighs $\frac{3}{4}$ ton. We will find, during the course of that study, that a product can sometimes be smaller than either factor.
This kind of thinking about scaling is connected to proportionality, as when we use a “scaling factor” to get answers to compound multiplicative problems quickly. For example, 1500 screws in 6 identical boxes $\dots$ how many in 2 of the boxes? What would be a multi-step multiplication and division problem to a fifth grader becomes, for a more mature student, a proportional relationships problem: a third as many boxes, so scale the number screws of by a third. We quickly have the answer 500.
A year isn’t exactly 365 days$—$nor is it exactly 365.25 days. Could any rational number express the number of days in a year? Students of mathematics run up against a similar problem when they ask how many times the side of a square “goes into” its diagonal. By Grade 8, we learn without proof that the diagonal cannot be written as any rational multiple of the side. In this way, irrational numbers such as $\sqrt{2}$ enter the discussion, and likewise $\pi$ for the quotient of the circumference of a circle by its diameter. The Greeks called the diagonal and the side, or the circumference and the diameter, incommensurable quantities. This ancient idiom, meaning not measurable by a common unit, underscores the importance of measurement thinking to arithmetic.
This is an excerpt from a larger document (almost two years old now). It’s revised here with input from Phil Daro and William McCallum. Also see the Progression document on measurement in grades K$–$5. For those interested in the scholarly literature about these questions, I’m sure it is vast$\dots$but I’ll pass along one article I came across just the other day (Thompson and Saldhana, 2003). $–$J.
