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forty-two

How do you teach "not enough information" word problems (without algebra)?

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By this I mean word problems that *seem* like they don't have enough information to find the answer, but when you work them out the unknowns all fall out of the problem - you didn't actually need to know them to get a definite answer.  An example we had in CWP yesterday (that had dd9.5 yelling about "not enough context"!!!!!):

 

 

Tank P and Tank Q had the same amount of water.  21 liters of water from Tank P were poured into Tank Q.  How much less water did Tank P have than Tank Q in the end?

 

 

 

I tried bar diagrams (but dd9.5 hates them and resists them), and idk that I showed it very elegantly anyway.  I tried working out the problem with manipulatives using several different starting amounts - showing there was the same answer each time.  And I finally resorted to algebra (which is the only way I truly know how to illustrate how the unknown falls out).  Eventually she sorta-kinda seemed to get the point, but idk what, if anything, she actually learned.  And this is not the first time she's had a screaming meltdown over a "missing information" problem (although after the last one, she's successfully done a few of them without problems, until yesterday).

 

Any ideas how to teach this?  Or how to model it with bar diagrams?  (I was using c-rods to do it, instead of on paper, because dd9.5 does better being able to physically set up (and re-set-up) the diagram instead of drawing it, but I think that might have contributed to the clunkiness of it.)

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Well, one way to solve it is by substitution, I.e. trial and error.

 

Say both tanks had 21 l. After pouring, one tank would have 0 l and the other tank would have 42 l.

 

Or try it with simpler numbers, say, substituting 2 for 21. Try the 2 l version with manipulatives. You could also use a dry erase sleeve with and insert of a picture of two measuring cups or tanks, with markings. Thane away x liters by drawing a colored line under the level on the first tank. Then add liters to second tank by drawing a colored line above the original level.

 

I realize this may sound juvenile, but it is one of the techniques in Kolya's Classic, How to Solve It.

 

ETA

 

My greatest help is a dry erase board or a dry erase insertable sleeve and colored markers. Much, much easier for kid to follow compared to a bunch of pencil marks on a piece of papaer. Also, I can follow along as problem progresses versus waiting till the end when there could be a cascade of mistakes.

Edited by Alessandra
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I would suggest doing it with some concrete examples then seeing if a pattern emerges. Try seeing what happens if the tanks started with 50 liters.  Then with 30. Then with 92. Obviously she will get the same answer and that may suggest to her what the full answer is. But more than that, she will probably begin to see why she doesn't need to know the starting volume. Seeing it several times will likely make the pattern very clear -- one tank goes down by 21, the other goes up by 21, and voila!

 

ETA: Alessandra beat me to it!

Edited by Cosmos
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Well, one way to solve it is by substitution, I.e. trial and error.

 

Say both tanks had 21 l. After pouring, one tank would have 0 l and the other tank would have 42 l.

 

Or try it with simpler numbers, say, substituting 2 for 21. Try the 2 l version with manipulatives.

 

I realize this may sound juvenile, but it is one of the techniques in Kolya's Classic, How to Solve It.

 

No, my math major sister suggested the same thing :).  And I did that with manipulatives - made the problem smaller and did a few iterations with different starting amounts.  The only problem was that, as soon as we moved to using manipulatives with the actual numbers in the problem, she latched onto my made-up starting number as *the* number - because she was so convinced we *had* to know it in order to solve the problem.  It seemed to be teaching her it was ok to invent numbers as needed to solve problems, instead of illustrating that the missing numbers in *this* problem were irrelevant to the solution.  Maybe I need to think of a way to illustrate this with manipulatives that solves it while "hiding" the starting amount the whole time - better mimic the math involved.

Edited by forty-two
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Cosmos

We are thinking the same way! I simplify difficult word problems by eliminating big numbers. Little brains may not have enough room to deal with concepts and operations at the same time.

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It seems like a pretty straightforward problem to solve via a bar model:

 

"Tank P and Tank Q had the same amount of water."

[             P             ]

[             Q             ] (obviously, the brackets should line up nicely to show equality)

 

"21 liters of water from Tank P were poured into Tank Q."

Tank P gets 21 "shorter" and Tank Q gets 21 "longer"

 

[            old P         ]

[        new P     ][21] (draw arrow to show movement of 21 liters from P to Q)
[            old Q         ][21]

[             new Q            ]

 

"How much less water did Tank P have than Tank Q in the end?"

From the bars it is clear that Q now has 42 liters more water than P.

 

Wendy

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The bar diagram difficulty may have been her general resistance to bar diagrams, and how, with the c-rods, she kept trying to see relationships between the "unknown starting amount" rod and the "declared to be 21" rod that would allow her to figure out the starting amount.  I illustrated it twice, with different rods for the "unknown starting amount", and she *did* get it in the end, more or less, but she was really invested in trying to figure out that starting amount, and kept bringing in outside relationships between the manipulatives to inform her understanding of the illustration - thus subverting what I was trying to show with the manipulatives :doh.

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No, my math major sister suggested the same thing :). And I did that with manipulatives - made the problem smaller and did a few iterations with different starting amounts. The only problem was that, as soon as we moved to using manipulatives with the actual numbers in the problem, she latched onto my made-up starting number as *the* number - because she was so convinced we *had* to know it in order to solve the problem. It seemed to be teaching her it was ok to invent numbers as needed to solve problems, instead of illustrating that the missing numbers in *this* problem were irrelevant to the solution. Maybe I need to think of a way to illustrate this with manipulatives that solves it while "hiding" the starting amount the whole time - better mimic the math involved.

The missing/irrelevant number is a fairly abstract concept. I give things a rest if there is a real block, but then I return as often as necessary to get the concept. I might use actual cups and water. Or, for us, anything with cookies often works. If you gave your brother two cookies, how many more would he have? Attention grabbing, lol. I would let dd work it out with various numbers of cookies. Slow and painful for me, but enlightening for her.

Edited by Alessandra
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Another option would be to rewrite the problem into a context that she could better visualize...or even better imagine herself physically moving the water.

 

Two mixing bowls have the same amount of water.  If you scoop one cup of water from bowl A and move it to bowl B, how much more water will bowl B now contain than bowl A?

 

Wendy

 

 

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I might have to do that next time - get out the bowls and water and let her have at it :).  (I remember doing that with some other tricksy capacity problem last year - the joys of pouring water overcame her "but it's *impossible*!!!!!  so I won't try" screaming meltdown resistance ;).)

 

Eta: It took a surprising amount of flexible math thinking on *my* part to figure out which measures we had that would scale up or down properly to match the problem, that was straightforward enough that *she* didn't have to already understand the problem in order to solve it with the manipulatives ;).  (It took a few iterations before I hit on the right set of containers.)

Edited by forty-two

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The bar diagram difficulty may have been her general resistance to bar diagrams, and how, with the c-rods, she kept trying to see relationships between the "unknown starting amount" rod and the "declared to be 21" rod that would allow her to figure out the starting amount.  I illustrated it twice, with different rods for the "unknown starting amount", and she *did* get it in the end, more or less, but she was really invested in trying to figure out that starting amount, and kept bringing in outside relationships between the manipulatives to inform her understanding of the illustration - thus subverting what I was trying to show with the manipulatives :doh.

 

Sometimes, I find my kids better understand a topic when I show an extreme example.  If she fixated on the 21, then write a similar problem that removes all numbers.

 

"Two strips of paper are the same length.  I cut a piece off the end of Paper A that is the width of my pointer finger.  I tape that piece onto the end of Paper B.  How much longer is Paper B now than Paper A, using the width of my pointer finger as the unit of measurement?" 

 

The advantage to that problem is that it uses "manipulatives" that mimic a bar model.

 

Wendy

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Whenever they start with the same amount, any movement from one to the other will result in the double of that amount being the total difference in the end.

 

You can prove this many times over using blocks, beads, volume, length of string. It always works.

 

I would agree with the poster above who suggested that more hands-on number sense needs to be developed here. I'd spend a full day of a math lesson with problems just like this and let her go for it. Let her write the word  problems for you to solve as well, to get some sense around this type of problem.

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Well maybe my suggestion isn't the greatest, but sometimes I just skipped some word problems.  If I felt like the non algebra explanation was too convoluted then I didn't bother. 

 

I severely disliked CWP.  We must have used 20 different word problem books over the years and that was the one series I found confusing more times than not.  I'm not sure why that is though.  I sometimes felt like the wording was unnecessarily confusing. 

 

 

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Hands on equations is great for these kinds of problems.

 

The student would start off with a pawn for Tank A and a pawn for Tank B.

 

Then the student would say that the amt in Tank A was a pawn -21 and that tank B was a pawn + 21.

 

They would realize that they needed to add 21 to both sides to see how much extra the other tank actually had.

Edited by 8FillTheHeart

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I made a little paper with strategies listed for figuring out word problems. Now that we do Beast this is extremely helpful. In the heat of the moment one of my girls freezes up and her first reaction is "I don't know" what I keep explaining is that she doesn't need to know. She needs to try. So number one is draw a picture. Number two is use manipulatives. Number three is use baby numbers. number four is move on and try it again later. Surprisingly coming back to the problem later really helps a lot of the time. When the kid's too upset I have them move directly to that step.

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