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

I have no idea what I was going for here:

WODB? Cuisenaire Rods

At that time, I was creating Which One Doesn’t Belong? sets. Cuisenaire rods didn’t make the cut. Nor did hundreds/hundredths grids:

WODB? Hundreds 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:

Alike&Different.006

Another that elicits equivalent fractions and place value:

Alike&Different.007

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.

Alike&Different.009

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:

Alike&Different.013

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:

Alike&Different.015

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.

Alike&Different.017

Alike&Different.023

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.

 

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Survivor: 100 Chart Challenge

I don’t watch Survivor. Stopped watching after Richard Hatch, often competing naked, won the first season.

Channel surfing last week, this grabbed my attention:

Host Jeff Probst:

“Alright, let’s get to today’s duel. For today’s duel you’re gonna race across a balance beam, collecting bags of numbered tiles. You must then place the tiles in order, one to one hundred.”

(Aside: If there are three opponents, is it still called a duel?)

The reaction online was swift and harsh:

“It is seriously the most idiot-proof puzzle in the history of puzzles. You basically have to know how to count and that’s pretty much it.” (source)

But that’s not pretty much it. I mean, it is counting from one to one hundred (and that is how the contestants solved the “puzzle”), but it could be more than that. A better strategy involves comparing numbers, understanding place value, and identifying patterns found in tables.

At 1:49 and 2:02, we see two contestants, Laura and Brad, respectively, place 25 from the second bag (11 to 30).

26A literal translation of “You gotta put ’em in order”? Each competitor places 25 only after placing 24. Then, he/she tries to find 26 in his/her pile o’ tiles. Some tiles are facing down. Suppose a player turns over a tile and finds 28 rather than 26. He or she should take advantage of another pattern and place it under 18.

At 3:11, it’s down to Brad and John for the last spot. At 3:18, Brad places 87 after 86.

87

He could have caught John if he had an understanding of place value. Suppose Brad turns over 94 before finding 87. Should he drop 94 and continue looking for 87 or just place 94 in the 9th row (9 tens) and 4th column (4 ones)?

This challenge reminds me of an activity I’ve used in Grade 3 classrooms. Take some 100 charts. Cut each chart into “puzzle” pieces. Place in a Ziploc bag. In pairs, have students reassemble. Ask students to describe how they solved their puzzle. This activity is much more engaging (and puzzling) than it has a right to be.

100 Chart Puzzle

Don’t be surprised if you see some completed 100 charts that look like this:

100 Chart Puzzle 2

More Decimals and Ten-Frames

What number is this?

123

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.

blank 100

5 tenths 50 hundredths

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:

two quarters

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.

annotated 500 thousandthsIn 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:

37

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:

48 plus 3650 plus 34

  • 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:

Ten-Frames – Full
Ten-Frames – Less-Than-Ten
Ten-Frames – Place Value Mat