## Alike & Different: Which One Doesn’t Belong? & More

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:

For more, see Brian Bushart’s Same or Different?, another single-serving #MTBoS (“Math-Twitter-Blog-o-Sphere”) site.

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.

The slides:

.pdf

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.

## Dividing by Decimals & Fractions: Ham & Ribs

I bought a ham. It was touch-and-go there for awhile. As I was picking up and putting down hams of various sizes, I was calculating baking times. My essential question was, can I have this on the table by six? Simultaneously, I was trying to remember if this was partitive or quotative division.

In partitive division problems, a.k.a division as (fair) sharing, the number of groups is known. This type of problem asks how many are in each group. In quotative division problems, a.k.a. division as measurement, the number in each group is known. This type of problem asks how many groups. For example: 6 ÷ 3 = 2 (partitive) means ♦♦  ♦♦  ♦♦; 6 ÷ 3 = 2 (quotative) means ♦♦♦  ♦♦♦. This distinction isn’t limited to collections of objects. Consider 6 ÷ 3 as cutting a 6 m rope into 3 parts (sharing) vs. cutting lengths of 3 m (measurement). Nor are these meanings limited to whole numbers. Which brings me back to my ham…

The directions read “bake approximately 15 minutes per pound (0.454 kg) or until internal temperature reaches whatever.” But here’s the thing:

Kilograms, not pounds. I could have converted from kilograms to pounds by doubling then adding ten percent of that. Instead, I divided 1.214 by 0.454. I know, I know, this still gives me the weight of my ham in pounds. But at the time, I interpreted 2.67 as the number of repeated additions of 15 minutes in my baking time. Either way, I determined how many 0.454s there are in 1.214. Quotative division. By a decimal.

As a math task, this is clunky. The picture book How Much Does a Ladybug Weigh? by Alison Limentani is a more promising jumping off point for quotative division in the classroom. On each page, the weight of one animal is expressed in terms of a smaller animal.

Using the data at the back of the book, we have 3.2 ÷ 0.53 = 6. We could ask children to make other comparisons (e.g., how many grasshoppers weigh the same as one garden snail?).

In the past, I have struggled with partitive division by decimals (or fractions). But I found the following example at The Fair this summer:

It’s not intuitive–at least to me–to think of 1/3 in 12 ÷ 1/3 as the number of groups. Take a step back and think about 26 ÷ 1 = 26. The cost, \$26, is shared between 1 rack of ribs; the quotient represents the unit price, \$26/rack, if the unit is a rack. This result should be… underwhelming.

Before we think about dividing by a fraction here, let’s imagine dividing by a whole number (not equal to one). What if I paid \$72 for 3 racks? (Don’t look for these numbers in the photo above–I’m making them up.) In 72 ÷ 3 = 24, the cost, \$72, is shared between the number of racks, 3; again, the quotient represents the unit price, \$24/rack. Partitive division.

So what about 12 ÷ 1/3? The cost is still distributed across the number of racks; once again, the quotient represents the unit price, \$36/(full) rack. The underlying relationship between dividend, divisor, and quotient hasn’t changed because of a fraction; the fundamental meaning (partitive division) remains the same.

We could have solved this problem by asking a parallel question, how many 1/3s in 12? And this quotative interpretation makes sense with naked numbers. But it falls apart in this context–how many 1/3 racks in 12 dollars? Units, man! If dollars were racks, a quotative interpretation would make sense–how many 1/3 racks in 12 full racks?

As a math task, this, too, is clunky. My favourite math tasks for partitive division by fractions are still Andrew Stadel’s estimation jams.

(Looking for a quotative division problem that involves whole numbers? See Graham Fletcher’s Seesaw three-act math task. For partitive, there’s Bean Thirteen.)

## More Decimals and Ten-Frames

What number is this?

123? 12.3? 1.23? One has to ask oneself one question: Which one is one?

Earlier this year, I was invited into a classroom to introduce decimals. We had been representing and describing tenths concretely, pictorially, and symbolically. We finished five minutes short, so I gave the students a blank hundred-frame and asked them to show me one half and express this in as many ways as they could.

As expected, some expressed this as 5/10 and 0.5. They used five of the ten full ten-frames it takes to cover an entire hundred-frame. Others expressed this as 50/100 and 0.50. They covered the blank hundred-frame with fifty dots. I was listening for these answers.

One student expressed this as 2/4. I assumed he just multiplied both the numerator and denominator of 1/2 by 2. And then he showed me this:

One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor.

In my previous post, I discussed fractions, decimals, place value, and language. To come full circle, what if we took a closer look at 0.5, 0.50, and 0.500? These are equivalent decimals. That is, they represent equivalent fractions: “five tenths,” “fifty hundreds,” “five hundred thousandths,” respectively. From a place-value-on-the-left-of-the-decimal-point point of view, 0.5 is five tenths; 0.50 is five tenths and zero hundredths; 0.500 is five tenths, zero hundredths, zero thousandths. Equal, right?

Hat Tip: Max Ray‘s inductive proof of Why 2 > 4

## Teaching Improper Decimals Using Ten-Frames

Professor Triangleman posed an interesting question a few weeks back:

If 15/10 is an improper fraction, then shouldn’t 1.5 be an improper decimal? Or is 1.5 a mixed decimal, having more in common with the mixed fraction 1 5/10? Both? Neither?

One definition of decimal:

A fraction whose denominator is a power of ten and whose numerator is expressed by figures placed to the right of a decimal point.

Thus, in 1.5, the implied denominator is 10 and the implied numerator is 5, the figure to the right of the decimal point. We read 1.5 as “one and five tenths,” a mixed decimal. The whole number part is treated separately, making an improper decimal an impossibility.

But what if we didn’t just look at the figures to the right? Nested tenths don’t stop/start at the decimal point. What if we looked at all the figures? We’d read 1.5 as “fifteen tenths,” an improper decimal.

Maybe the improper vs. mixed comparison is throwing me off track. Fractions can be classified as either proper or improper. Why not decimals? Decimals less than one, such as 0.5, would be proper; decimals greater than or equal to one, such as 1.5, would be improper (or, in Britain, top-heavy).

Christopher Danielson wasn’t trying to introduce new vocabulary to the world of math(s). Probably. Rather, he was making a point about place-value.

When we teach decimals using ten-frames we do.

If the whole is one full ten-frame, students may build 3.7 like this:

Students will describe 3.7 as “3 ones and 7 tenths,” “37 tenths,” or even “2 wholes and 17 tenths.” This mirrors what students know about place value and whole numbers: 37 can be described as “3 tens and 7 ones,” “37 ones,” or even “2 tens and 17 ones.”

Just like with whole numbers, thinking about place value makes calculations with decimals easier. For example, consider 4.8 + 3.6:

• 4 and 3 make 7
• 0.8 (“8 tenths”) and 0.6 (“6 tenths”) make 1.4 (“14 tenths”)
• 7 and 1.4 (“1 and 4 tenths”) make 8.4 (“8 and 4 tenths”)

Note the shift in thinking, not notation, from 1.4 as “14 tenths” to 1.4 as “1 and 4 tenths.” With fractions, it’s a shift in thinking and notation. Probably why we know about improper fractions but not improper decimals.

Blackline Masters: