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I've begun a "Question of the Day" (QOTD) contest in my grade 8 class where I post a problem solving question on the wall each day and students are challenged to answer it in their own time. I need more ideas for good questions.

Question: Where (either online or in print) can I go to find a long list of short and simple problem solving questions for my students?

Here are my criteria for QOTD questions:

1) Non-routine questions for grade 7-9 students, most often based upon content in the syllabus. Questions can range widely from easy to difficult.

2) Very important: Questions should be short and easily understood so that they fit on an A4 paper with large font and I don't have to spend time explaining what the question means.

3) Questions should have clear, concise solutions.

Here are some examples of questions I've given in the past that have worked reasonably well:

  • What is the measure of an interior angle of a regular decagon?

  • How many pairs of parallel edges does a rectangular prism have?

  • If I flip a coin 3 times in a row, what is the probability of getting heads all 3 times?

  • Find the next number in the pattern: $1, 1, 2, 3, 5, 8, 13, 21, ...$

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    $\begingroup$ I'll think about it, but I must say that "what's the next number" problems are often really really annoying for students because they don't have any useful strategies to attack them. $\endgroup$ – DavidButlerUofA Aug 31 '14 at 23:54
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    $\begingroup$ @DavidButlerUofA I also must add that they're hardly well defined since what constitutes a pattern is pretty subjective (e.g. people who claim to see religious deities in their pancakes) and also because there exist an arbitrary number of sequences which interpolate any given sequence of integers, for example take the sequence $1,2,3,4...$ one would probably guess the next term is $5$ though it could actually be $9$ if we let the sequence be defined by:$$a_1=1$$$$a_n\stackrel{}{=}\begin{cases}2a_{n-2}+1&\text{if } a_{n-1}\text{ is even }\\a_{n-1}+1&\text{if } a_{n-1}\text{ is odd}\end{cases}$$ $\endgroup$ – Ethan Sep 2 '14 at 5:51
  • $\begingroup$ I think offering a few pattern problems in this POTD format is actually quite valuable for students. It introduces them to the aesthetic side of mathematics that is all too often brushed aside to focus on mechanics. Learning that some math problems are open-ended and have multiple interpretations is a good thing! $\endgroup$ – Michael Joyce Sep 5 '14 at 15:04
  • $\begingroup$ @Ethan I like your example but he next number in respect to your last sequence is 7 not 9 because $a_{5-2}=a_3=3$! But yet $7\neq 5$. $\endgroup$ – AmirHosein SadeghiManesh Sep 13 '14 at 12:47
  • $\begingroup$ @AmirHoseinSadeghiManesh Oops, yeah I see where I made that error. To late to edit now, thanks anyway though. $\endgroup$ – Ethan Sep 13 '14 at 22:56
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A decent source of questions is the AMC 8 math competitions that are held yearly. The competition consists of 25 questions in roughly increasing order of difficulty. There are also 10th and 12th grade versions of the competition (titled AMC 10 and AMC 12, respectively). Here is a link to aops which presents the questions and solutions by year.

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    $\begingroup$ This is exactly what I'm looking for. There are an abundance of short, interesting questions, all of which are appropriate for my grade 8 students. Thanks! $\endgroup$ – David Ebert Sep 8 '14 at 9:36
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    $\begingroup$ Wow. These are great! $\endgroup$ – DavidButlerUofA Sep 11 '14 at 23:34
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  • This website has a host of great problems of various levels of difficulty, and it's actually called "problem of the week". Note that many of them are higher than the level of your students, but many are ok. http://mathforum.org/wagon/

  • Henry Dudeney's books are very cute and are available on project Gutenberg. They were published in the 19-teens and have a certain charm about them. We have particularly enjoyed the Canterbury Puzzles at our puzzle club here at Adelaide Uni: http://www.gutenberg.org/files/16713/16713-h/16713-h.htm and http://www.gutenberg.org/files/27635/27635-h/27635-h.htm .

  • The Mathematics Task Centre has activities that are actually "tip of the iceberg" problems that have nice closed problems to begin but a lot of extension after. They can often be adapted to puzzles. http://mathematicscentre.com/taskcentre/

  • On top of these, many things that come up in the normal curriculum can turn easily into problems that are quite interesting. The advantage of these is that they can create several puzzles along the same theme. Some examples:

    • Draw the diagonals of some a regular hexagon or other shape -- how many triangles are formed? how many right-angled triangles are formed?
    • Construct a strange-looking shape out of bits of various shapes and ask them to find the area and perimeter. Good ones are those that require you to take away bits (eg a square with triangles removed).
    • Give them a property of numbers and ask them how many numbers from 1 to 100 (or 1 to 1000) have the propery. How many numbers are there such that they are both a square and triangular number? How many have digits that sum to 6? How many are more than the sum of their factors? How many are multiples of 7 but not 8? How many can be written as the sum of two consecutive numbers? This can lead into all sorts of interesting discussion about numbers.
    • Give them a long calculation that will simplify if you do some rearranging first.
  • One final comment: avoid questions that are simply annoying. There is nothing worse than seeing the answer and groaning because it's a bad pun, or thinking "I suppose that makes sense, but where the hell did that inspiration come from?" You don't want to give the impression that problems can only be done with divine inspiration or are about guessing the mind of the problem-poser. Examples of such problems are "what's the next number in the sequence?" problems.

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What is the most number of confined regions you can form by drawing an equilateral triangle on top of a square?


          SqTri
One can concoct many such problems, and some can be quite difficult, especially in 3D. "Confined region" might need some explication. You could change to count the number of triangular regions, the number of edge-edge points of intersection, etc.

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I don't care for their motto, but the problems at Five Triangles are great. Some are too hard, I think.

Also, you might like Mathematical Circle Diaries, Year 1: Complete Curriculum for Grades 5 to 7, by Anna Burago.

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  • $\begingroup$ +1 for the link, and agreed about the tagline $\endgroup$ – Benjamin Dickman Sep 11 '14 at 22:08
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You could teach your students a little about telescoping series':

First show them that: $$n^2=-0^2+n^2=(-0^2+1^2)+(-1^2+2^2)+(-2^2+3^2)+(-3^2+4^2)+\cdots (-(n-1)^2+n^2)$$ $$=(1)+(3)+(5)+(7)+(9)+(11)+\cdots (2n-1)$$ $$=(2-1)+(4-1)+(6-1)+(8-1)+(10-1)+(12-1)+\cdots (2n-1)$$ $$=(2+4+6+8+10+12+\cdots 2n)-(1+1+1+1+\cdots1)$$ $$=2(1+2+3+4+5+6+\cdots n)-n$$ $$\text{ Then adding } n \text{ to both sides gives: }$$ $$n^2+n=2(1+2+3+4+5+6+\cdots n)$$ $$\text{ And dividing both sides by } 2:$$ $$\frac{n^2+n}{2}=1+2+3+4+5+\cdots n$$

From here you could ask your class if they can find a simple formula for the sum: $$1^2+2^2+3^2+4^2+5^2+\cdots n^2$$

And as a hint tell them that: $$-(k-1)^3+k^3=3k^2-3k+1$$

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  • $\begingroup$ Telescoping series are a nice idea. I wouldn't use them with $\sum n$ or $\sum n^2$, since they seem a bit messy for 8th graders and there are nicer proofs for those sums. But showing them $\sum 1/(n^2-n)$, and having them figure out $\sum 1/(n^2+n)$ could work out nicely. $\endgroup$ – user173 Sep 3 '14 at 2:14
  • $\begingroup$ @MattF. Thanks. I had written out a proof similar to this before using sigma notation so I pictured the proof being pretty contained. Then I realized half way through they probably aren't familiar with that so I removed it and everything got messy. $\endgroup$ – Ethan Sep 3 '14 at 4:24

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