I have no idea what I was going for here:
At that time, I was creating Which One Doesn’t Belong? sets. Cuisenaire rods didn’t make the cut. Nor did hundreds/hundredths grids:
I probably painted myself into a corner. Adding a fourth shape/graph/number/etc. to a set often knocks down the reason why one of the other three doesn’t belong. Not all two-by-two arrays make good WODB? sets (i.e., a mathematical property that sets each element apart).
Still, there are similarities and differences among the four numbers above that are worth talking about. For example, the top right and bottom right are close to 100 (or 1); the top left and bottom right are greater than 100 (or 1); top left and top right have seven parts, or rods, of tens (or tenths); all involve seven parts in some way. There is an assumed answer to the question, “Which one is 1?,” in these noticings — a flat is 100 if we’re talking whole numbers and 1 if we’re talking decimals. But what if 1 is a flat in the top left and a rod in the bottom left? Now both represent 1.7. (This flexibility was front and centre in my mind when I created this set. The ten-frame sets, too.)
Last spring, Marc and I offered a series of workshops on instructional routines. “Alike and Different: Which One Doesn’t Belong? and More” was one of them. WODB? was a big part of this but the bigger theme was same and different (and justifying, communicating, arguing, etc.).
So rather than scrap the hundreds/hundredths grids, I can simplify them:
Another that elicits equivalent fractions and place value:
Another question that I like — from Marian Small — is “Which two __________ are most alike?” I like it because the focus is on sameness and, like WODB?, students must make and defend a decision. Also, this “solves” my painted-into-a-corner problem; there are three, not six, relationships between elements to consider.
The numbers in the left and right images are less than 100 (if a dot is 1); the numbers in the centre and right can be expressed with 3 in the tens place; the left and centre image can both represent 43, depending on how we define 1.
At the 2017 Northwest Mathematics Conference in Portland, my session was on operations across the grades. The big idea that ran through the workshop:
“The operations of addition, subtraction, multiplication, and division hold the same fundamental meanings no matter the domain in which they are applied.”
– Marian Small
That big idea underlies the following slide:
At first glance, the second and third are most alike: because decimals. But the quotient in both the first and second is 20; in fact, if we multiply both 6 and 0.3 by 10 in the second, we get the first. The first and third involve a partitive (or sharing) interpretation of division¹: 3 groups, not groups of 3.
Similar connections can be made here:
This time, the first and second involve a quotative (or measurement) interpretation of division: groups of (−3) or 3x, not (−3) or 3x groups. (What’s the reason for the second and third? Maybe this isn’t a good “Which two are most alike?”?)
I created a few more of these in the style of Brian’s Same or Different?, including several variations on 5 − 2.
Note: this doesn’t work in classrooms where the focus is on “just invert and multiply” (or butterflies or “keep-change-change” or…).
And I still have no idea what I was going for with the Cuisenaire rods.
¹Likely. Context can determine meaning. My claim here is that for each of these two purposefully crafted combinations of naked numbers, division as sharing is the more intuitive meaning.
Update: An edited version of this post appeared in Vector.