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?
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.
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:
- For students just beginning to learn about area, they might use graph paper and count the number of rectangles by hand.
- 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).
- 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?
- What if you have access to two different sized tiles? What if your tiles or board are not rectangular?
. . .
- 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.
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):
I kept a supply of Think Fun puzzles. These are solitaire brain teasers that the students enjoyed.
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
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.
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.
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.
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.
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
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.
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.