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:

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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.

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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.

 

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“They’ll Need It for High School” (Part 3)

I’m picking “TNIFHS” back up. At the end of Part 1, I promised Part 2 would answer “What are the big ideas in elementary school mathematics that students will need for high school?” Instead, I talked times tables.

In this third half, I’ll refocus on these big ideas. Or one of them — one that came up in that initial “they’ll need long division for high school” conversation.

More than the standard algorithm, what students will need is an understanding of division as sharing (finding the number in each group) and measuring (finding the number of groups). More generally, what students will need is an understanding of the fundamental meanings of all four operations.

Here’s part of a task I presented to my secondary math colleagues:

Evaluate, or simplify, each set of expressions. Make as many connections as you can conceptually & procedurally, pictorially & symbolically.

fundamental meanings of all four operations

Sticking with division, in this task (−6) ÷ (+3) was chosen to bring to mind sharing (3 groups, −2 in each group) whereas 6/5 ÷ 3/5 was chosen to evoke measuring (3/5 in each group, 2 groups). (This often leads teachers themselves to revisit 6 ÷ 3.)

division

(−6) ÷ (+3) as sharing (top) and 6/5 ÷ 3/5 as measuring (bottom)

Flexibility is key. Consider (−6) ÷ (−3), 6/5 ÷ 3, 6 ÷ 0.3, 0.6 ÷ 3, 6x ÷ 3, 6x ÷ 3x, etc. (Note: division of fractions & integers are high school topics in Western Canada.)

(I’m not saying that dividing by a fraction — or decimal fraction — always means measuring. You can think sharing, which can be challenging. Andrew Stadel’s estimation jams are my favourite examples of this. How long is “All Along the Watchtower”?

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Did you see 2/3 in the picture? Did you divide by two, then multiply by three? In other words, did you invert and multiply? What’s the meaning of 2:40 ÷ 2/3 in this context?)

The subtraction set above is interesting. Teachers pick up that the expressions are variations on a theme: five “take away” two. Their pictorial representations tend to show subtraction as removal: “if you have five apples/quarters/x‘s/square root of two’s and I take away two…”

282px-Subtraction01.svg

Pictorial representations that show subtraction as comparison (the “difference”!) are less frequent, but maybe more helpful.

subtraction

5/4 − 2/4 as removal (top number line) and comparison (bottom number line)

Consider (+5) − (−2). To “take away” negative two from positive five means introducing zero pairs whereas the “difference” between positive five and negative two means understanding that positive five is seven greater than negative two.

This second meaning is probably more meaningful in high school. For example, subtraction as removal reduces (1.89t + 15) − (1.49t + 12) to an exercise in collecting like terms whereas subtraction as comparison has students contrasting rates of change (e.g., cost per additional pizza topping) and initial amounts (e.g., the price of a plain cheese pizza). Similarly, if F_1(C) = 9/5C + 32 and F_2(C) = 2C + 30, then (F_2 − F_1)(C) compares conversions given by an estimate and the formula. When solving |x − 5| = 2, it’s more helpful to ask “What numbers differ from five by two?” than think missing minuends in take away problems.

Addition and multiplication — as well as other big ideas needed for high school such as proportional reasoning (or “multiplicative thinking”?) — will be addressed in future posts.

More often than not, more is less

In the summer, Costco peddles a buttload of educational workbooks. You know the ones: collections of every worksheet necessary for your child to complete <insert grade here> Math. Can’t find them? Look over by the Christmas trees.

I picked up the Grade 3 book. Just browsing. Killing time. I opened to this page:

add:subtract words

I’m not a big fan of this approach. Forget about comprehension, just scan for the add or subtract words. See more, think add. But it’s not that easy. More shows up in five of the practice exercises. Try them.

  • In the picture, how many more 4-legged animals are there than 2-legged ones?
  • Peter has 39 goats.  He wants to have 64 goats.  How many more goats should he buy?
  • Peter has 68 animals on his farm.  He buys 23 more.  How many animals does he have now?
  • 413 gulls are joined by 311 more.  Then 136 more gulls come.  How many gulls are there altogether?
  • There are 576 gulls, but 153 fly away.  Then 283 more leave.  How many gulls remain?

A mountie (really?!) tells kids (Canadian, no doot) to decide on the operation.

mountie

From the answer key:

  • In the picture, how many more 4-legged animals are there than 2-legged ones? 15 − 12 = 3
  • Peter has 39 goats.  He wants to have 64 goats.  How many more goats should he buy? 64 − 39 = 25
  • Peter has 68 animals on his farm.  He buys 23 more.  How many animals does he have now? 68 + 23 = 91
  • 413 gulls are joined by 311 more.  Then 136 more gulls come.  How many gulls are there altogether? 413 + 311 + 136 = 860
  • There are 576 gulls, but 153 fly away.  Then 283 more leave.  How many gulls remain? 576 − 153 − 283 = 140

Subtraction is used to answer three of five questions with this ‘add’ word. Actually, kids will think addition for the first two questions (12 + 3 = 15 and 39 + 25 = 64) but that’s another post.