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February 18th, 2018, 01:26 PM   #1
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"sorta hard" problems

I'm a private tutor. I have a couple students, algebra and prealgebra, who tell me that the teachers make the quizzes and homework easy, and then throw curveballs on the tests. My students wish they had some "curveball" problems to practice with before the tests.

I would like recommendations for prealgebra and algebra textbooks that are know for presenting creative problems. NOT "really hard problems", that would be too much for them. Just something beyond the basic, repetitive problems that you see everywhere.

Any ideas?

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February 18th, 2018, 06:39 PM   #2
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"Really hard" is in the eye of the beholder. One of the great things about tutoring is that you can make up problems that are suitable for your specific students. What one student finds hard may be simple to another.
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February 18th, 2018, 07:42 PM   #3
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Hi JeffM1,

I don't have a lot of experience as a tutor, especially at the lower levels like prealgebra or 7th/8th grade integrated math. Maybe it's just me, but I find it difficult to improvise problems that have a twist. By definition these are the kinds of problems that requires some thought and cleverness on the part of the test-writer.

Perhaps after a few more years of tutoring I will have seen it all, and can recall or make up such problems.

Here's an example. This is not much of a "twist" -- it's the best I can do from memory. (This is another reason I want to find some good textbooks, so I can study them and bookmark the most interesting problems).

Let's say we're talking about the area of shapes. If the student knows for a circle $ A = \pi r^2 $, then they may practice that a few times on straightforward problems. They see $r$ and they compute $A$.

Maybe the 7th graders are also learning to calculate the area of compound shapes. Perhaps a particular compound shape has a half-circle and a rectangle of some sort, placed together. Let's say that the students are supposed to infer the diameter of the circle by seeing it's part of one side of the rectangle, and they are given some of the lengths of the other segments.

So they infer the diameter $d$. This is where they might make a mistake by forgetting to use $r$ instead of $d$ in the area formula. Or maybe, because we're dealing with a half-circle, they forget to divide the area by 2.

In my example there are a couple ways to make a mistake. My 7th grade student says that the test problems are like this. The opposite of formulaic, instead presenting a new configuration of information on every problem.

Which is of course very cool. But it's almost too hard for him. And I don't like that the teachers give him plain problems on the quizzes so there's no preparation.

A couple years ago I tutored college algebra classes, and they always put the twisty-type problems on the quizzes, while making the tests straight-ahead. That seems more fair than the reverse.

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February 18th, 2018, 08:45 PM   #4
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For algebra, you could try this book.

By the way, Pick's theorem is a really cool way of finding the area of certain polygons.

Given its dimensions, as shown, what is the length of the perimeter of the line diagram below?
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February 18th, 2018, 10:12 PM   #5
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Thanks, Skipjack. Great that there's such a nice book available for free.

Pick's Theorem is very interesting.

That perimeter problem is a nice chance to reason creativity. It stretches a student who has only seen the simplest perimeter problems. I can imagine some ways of prompting them to see the method if they don't figure it out on their own. Like first showing a rectangle, then a rectangle with a smaller "rectangle bite" out of the corner. Can they see the "bite" doesn't change the perimeter?
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February 19th, 2018, 10:12 AM   #6
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I had the practice of putting, on my tests, about one third problems that had been on the student's homework, about one third slight variations of homework problems, and one third new problems (solvable using the ideas in the chapter).
I had a student who complained, bitterly, that I had included a problem of a type they had never seen before and that had nothing to with the chapter.

In fact, it had been one of the homework problems, one that I had gone over in class, and I opened the student's notebook to show exactly where that problem was in his notes. (He did take very good notes!)
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problems, sorta hard

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