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I am a math teacher for sixth graders and I am trying to think of some strategies to keep the students who finish their work quickly productively occupied. I would like to have a selection of activities - including games, projects, etc. Any ideas of what I can include? Maybe games like Set? Or Q-bitz? What else can I include?

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    $\begingroup$ What’s wrong with letting them get a head start on your homework? and if they finish that, their other homework? For me, a gifted student, if the consequence of finishing the assigned work promptly was getting more work that other kids aren’t expected to do, I’d just start pretending to take longer—in fact I’ve done this. $\endgroup$ Sep 9 '20 at 2:24
  • $\begingroup$ Giving more practice questions may be a better option too, after all, "practice makes a man perfect". The ones who are quick to solve can try to solve more difficult ones and enhance their learning to a higher level. $\endgroup$ Sep 14 '20 at 19:47
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    $\begingroup$ @RutvaBabaria the problem with this is that the slower ones usually are the ones who actually need the extra practice and the cycle can just go on and on. $\endgroup$
    – Burt
    Sep 15 '20 at 21:07
  • $\begingroup$ @gen-ℤreadytoperish I agree with your idea that giving more work may not be the best solution. That's why I am looking for alternate solutions. Allowing them to work on homework can sometimes work, but homework is sometimes helpful in ensuring that the students practice after a pause - so that it is a second time going through the material. (If they start pretending to take longer, that does solve my problem - I need them to be productive and busy.) $\endgroup$
    – Burt
    Sep 15 '20 at 21:09
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    $\begingroup$ @Burt I agree with the point that slower one might need the extra practice. I think group activities will be better to conduct. Sometimes the best tutor can be friends. The ones completing their work quickly can help their classmates to improve through group activity. $\endgroup$ Sep 16 '20 at 13:14
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Stuff for fast finishers/gifted kids should be out of the stream --stuff that isn't what you're covering next week or next year.

Look at puzzles -- especially geometrical ones. Graph theory -- Bridges of Konigsburg, three houses, three wells. Chess board problems. There are lots of good puzzlebooks out there. Logic puzzles too.

Mechanical puzzles too. Rubik's cube, sliding block puzzles. Ring and string puzzles. You will have to figure out how to keep them from wandering away permanently.

Scientific American had a whole series of books, mostly extracts of the columns of Martin Gardener of puzzles and activities.


Paper and scissors activities.

  • Properties of a mobius strip.
  • hexaflexagon. (Use adding machine tape)

Number theory conjectures:

Pick a positive whole number. If odd, triple it and add 1. If even cut in half. Repeat.

E.g. 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

Any number that results in 4-2-1 is said to be wonderous. Are there any non-wonderous numbers? Write a program to test numbers. Is there any pattern to how many steps it takes to reach 4-2-1


  • Can any even natural number be expressed as the sum of two primes? (You need to consider 1 a prime for it to work with 2.)

  • Can any natural number be expressed as the difference of two cubes? Why is this one intrinsically harder?


For something bigger, get them started on geometry, follow that up with coordinate geometry.

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    $\begingroup$ Regarding geometry, I suggest the Euclidea app. It allows the user to do straightedge and compass constructions to solve puzzles: play.google.com/store/apps/… $\endgroup$ Sep 8 '20 at 13:53
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    $\begingroup$ I strongly disagree about the mechanical puzzles. I think gifted students will immediately ask themselves “What is the point of this? How does it relate to what the state expects me to be able to do for the stabdardized test or what my job down the road will expect of me?” and feel like their time is being wasted. Personally I would find it demeaning if a sixth grade math teacher put a sliding blocks puzzle in my hand instead of teaching me—I’m a student, not a child at daycare. $\endgroup$ Sep 9 '20 at 17:38
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    $\begingroup$ @gen-ℤreadytoperish It is sad to think of any students never taking a moment to play, and instead worrying about standardized tests and employability. What a horrible corruption of the human spirit. $\endgroup$ Sep 9 '20 at 21:39
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    $\begingroup$ @gen-ℤreadytoperish I strongly disagree with your assessment of gifted students: I was one, and always enjoyed opportunities to go above and beyond with fun assignments like that. One alternative is to make some sort of project required for all students, and permit the choice of a large variety of projects, from straightforward ones any student could complete to ones that a gifted student would find challenging and interesting. Then it would be up to them what level of engagement they put in to the project. $\endgroup$
    – Opal E
    Sep 14 '20 at 4:43
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    $\begingroup$ Having kids work ahead postpones the problem. Now they know next week's/month's/year's lessons too, and so they are bored then too. Side tracking them onto math that isn't part of the curriculum works better. I too was a gifted student who knew the entire math course by October. Rest of the year Math was boring, so I read physics books. $\endgroup$ Sep 14 '20 at 13:47
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One option is to assign tasks with a low floor, but high ceiling. That means that almost everyone has the background to start tinkering, and accomplish something, but there is enough complexity and variation that one could keep discovering new things for a very long time. This naturally differentiates instruction. Engaging tasks of this form are hard to craft, and find, however.

An example: When teaching about area, you could assign the task "How many 2 inch by 3 inch rectangles can fit in a 60 inch by 81 inch rectangle?". Progression of associated tasks:

  1. For students just beginning to learn about area, they might use graph paper and count the number of rectangles by hand.
  2. More advanced students might jump immediately to using division. For them, you can modify the problem to 61 inches by 84 inches. Now the small rectangles do not tile evenly: or do they? Try different arrangements to see how many you can fit. Still "easy" to answer if you allow "fractional tiles", but not if you restrict yourself to whole tiles (note: this will not be easy if knowledge of fractions still needs work, and hence is a good task in 6th grade).
  3. If a student solves the previous task to their satisfaction, you can ask them to change the size of the board and the size of the tile. Can they find a formula or procedure to find the maximum number of tiles which fit?
  4. What if you have access to two different sized tiles? What if your tiles or board are not rectangular?

. . .

  1. This train of thought can keep you occopied for a whole lifetime. https://www.ics.uci.edu/~eppstein/junkyard/polyomino.html

Some students might spend their entire class period stuck at 1, or progressing from 1 to 2. This would be time well spent. The fast finishers would have plenty to chew on, which could occupy them for the rest of the class period, possibly the rest of the semester, or (depending on how hard these problems end up being) the rest of their lives as professional mathematicians.

I just came up with this, so I do not know the answer to question 3 or 4. They have the flavor of something I should be able to solve with elementary number theory (linear diophantine equations), but I have never tackled these particular problems. In particular, the geometric considerations make this possibly much harder than I would anticipate (how do you decide when to lay a tile vertically or horizontally?). It could certainly be fun to spend some time thinking about them whether you were a 6th grader or a professional mathematician.

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I taught gifted students and had many fast finishers. Here are some of things we did (most require no preparation on your part but some require resources that you may not have):

  1. I kept a supply of Think Fun puzzles. These are solitaire brain teasers that the students enjoyed.

  2. Some of my classes had access to tablets/computers and the students were allowed to work om Khan Academy polishing their skills, by working on what I assigned. There were also games on NCTM Illuminations that I allowed them to play

  3. My classes had hand-held Flashmasters - the students practiced their facts on them. Those students who used them regularly developed amazing speed and accuracy. Note they now have an app instead of the hand-held devices.

  4. The students were given 5 dice and they had to use the 5 numbers that they spun to make a target number (usually 100). They had to use each number exactly once and were allowed to use: decimal points, the four operations, exponents, parentheses, and factorial. They were also allowed to combine two of the numbers to make a two digit number (e.g. if they spun a 3 and a 6, they could make 36 or 63). The students had to write out their expression using order of operations. This was a big hit with the students. Alternatively you can the game 24 and they can record their answers using order of operations. 24 is quieter than dice.

  5. The students worked on puzzles from MOEMs (math olympiads for elementary and middle schools) and from Math Counts. These are two different competitions for students of this age. You might consider entering a team in either competition or you can just give them the problems for extra challenge.

  6. I kept a collection of math books that students enjoyed in my classroom. I would particularly recommend anything by Brian Bolt (e.g. Mathematical Arcade), The Boy Who Loved Math, I Hate Math by Marilyn Burns (look for others by Marilyn Burns), The Cat in Numberland. There are many others, but Brian Bolt writes great books of activities and if you get one book, I suggest you get one of his and assign some of the activities to your students.

Some words of caution - some of these activities are a lot of fun (e.g. Think Fun games) and you don't want students to rush work so that they can do these. I would usually check their work before allowing them to enjoy these activities, but if you have a big class, that might not be possible. You will have to judge in that case if you should avoid the activities that are the most fun.

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    $\begingroup$ I like the idea of thinkfun games. I'm going to check into those books too! $\endgroup$
    – Burt
    Sep 9 '20 at 12:37
  • $\begingroup$ Along the lines of #5, the UKMT also has large archives of past questions at various levels available online. $\endgroup$ Jun 19 at 19:07
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As someone who was not too long ago in middle school, I have fond memories of working on AMC 8 or AMC 10 problems once I finished my school work. These are competitions for middle and beginning high school students that typically require much more thought than standard classes. These problems would also make for great discussions if you wanted to steer your class in that direction.

There is also a series of books by Art of Problem Solving (Volume 1 and the introductory series would be appropriate at this level).

Here are some links:

https://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions https://artofproblemsolving.com/store/item/aops-vol1

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    $\begingroup$ I love AoPS, but I wouldn't suggest this for everyone. Half of the early finishers might enjoy doing more advanced problems, other early finishers will see this as more of a punishment. "Why should I finish this work early, just to do more work?" It might actually discourage them to work to their full potential and complete assignments early if their "reward" for doing so is more problems. $\endgroup$
    – ruferd
    Sep 9 '20 at 12:34
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When I was in my high-school, I was a kid that solved all the questions quickly. This thing was good practice for me and I loved it when I was appreciated. But what I think personally is, healthy competition is always a good part of the learning process. I have a suggestion for the similar :

Conducting group activities

You can divide the class into groups and provide some points, say 10, if 3 students out of 5 in the group solves the question, their team scores 30 points. This may help students to learn from other students of the class, also fun activities increase love for the subjects.

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Why simply let students whatever they want. Make groups of students based on what they want: equations, fractions, geometry, etc. Some students need more patience and some students are fast. The last group of students could switch to other topic. Why we have to teach the same thing at the same time to everything?

The idea is picked from the book of Salman Khan.

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    $\begingroup$ True, but sometimes students need to learn something. For example - if the class is learning ratios, everyone needs to learn them. Perhaps some can learn it on a higher level than others, but I can't just skip it for some people. $\endgroup$
    – Burt
    Jun 11 at 12:54
  • $\begingroup$ "If the class is learning ratios". This is the point. The "class" don't learn nothing. The individual persons do. So, we need to provide something different to each students. Why we have to teach Monica ratios if she already knows? Perhaps she need to know something about equations. If she could use time of ratios learning equation, then she has 2 chances to learn.... Just an idea. $\endgroup$ Jun 12 at 20:01

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