Math Hombre

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

Gretchen Zeuch developed Fraction Reaction to be a simple to learn, easy to play, fast game that works on fraction magnitude and mixed number fraction equivalence. 

She writes:

The process of making this game had many stages. The first stage was deciding what kind of materials I wanted to use in my game. I decided to use a standard deck of cards because I really wanted to make a game that was accessible to every classroom. I then had to pick the mathematical content I wanted my game to be based on. I started by just laying out all the cards in a standard deck and brainstorming different mathematical content. I finally landed on fractions because I liked the students being able to physically see it. I then decided that making the connection between improper fractions and mixed fractions would be the most helpful. I then went through a lot of trial and error by playing the game with a variety of people. This helped me decide how points would work, specialty cards, and general playing rules.

This game is great to teach in a classroom when students are learning about improper and mixed fractions. It is very easy to teach to students as well as all students will be able to play at the same time because of the accessibility of the materials. This game will help students make the connection between an improper fraction and a mixed number. They will also be able to compare the sizes of mixed numbers and improper fractions so identify which is larger and which is smaller. Overall, this game is simple to understand and helps to solidify students' understanding of improper fractions and mixed numbers.

There are a few different uses for this game in a classroom. The first use is that, while students play, you can have them record all of their improper fractions turned into mixed numbers and then have them sort them on a number line. Another use is for students to record their answers during the game and then answer some comparison questions at the end. Lastly, another in class use for this game is to have students discuss the differences between fractions and mixed numbers and how they relate to each other.

Rules - https://bit.ly/FractionReactionRules

In addition, Gretchen made a video to promote the integer game, Zero Rummy. She  writes: This is a great game to use with young children to get them working on their addition and subtraction or to help introduce the concept of negative numbers. This game should be used as a fun exercise rather than to teach a skill. The great thing about this game is that it is stimulating for children so that they are doing math without knowing they are. It is very easy to use in the classroom with minimal materials and does not take up a large chunk of time. Children really enjoy this game and it is a very easy game to play for many ages with multiple variations.

Rules: https://bit.ly/ZeroRummy-rules

• CCSS.MATH.CONTENT.2.OA.B.2 Fluently add and subtract within 20 using mental strategies. 
• CCSS.MATH.CONTENT.2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Cf. Kent's original post on Integer Solitaire

Preamble
Wow - I missed a blogging month. And I had so much to say about it... we did Math in Your Feet, some excellent student projects, lots of new lessons, assessment thoughts... So I was thinking about resolving to blog less big, more frequently. Then Sue Van Hattum blogs her #edustory, and I think challenges me and a few others...

I've been thinking about doing more microblogging - and maybe I'll try it. I get stopped by "nobody wants to read that" which makes me forget that I'm writing out my own understanding, so that shouldn't matter. I'm not an author who's trying to please a fan base, I'm a teacher trying to work my way to understanding.

Actual Post
But what's really on my mind is embodied cognition. Last summer I got to try a session with Malke Rosenfeld and Christopher Danielson at TMC14 on Embodied Cognition.  (My account.) Outside of their session, Malke worked with Michael Pershan and Max Ray and others on doing a life size complex plane and number line. I wasn't even a part of that but it got me wondering. Malke and Max have continued to work on the idea, and there is an MTMS article in the works. They were willing to share their writing on that so I could try things in class.

The course is for preservice middle school teachers. I start off with negative numbers (and probably end with negative numbers too if you know what I mean. Where's my Dangerfield font?) because it is a good setting for talking about operations as story and action (exposing them to the CGI structures), and rolling in some content from our preservice elementary classes on fact families and operation strategies, models and landscapes of learning.


For the Cognitively Guided Instruction stories we watched the Kindergartener uses Direct Modeling video from the Heinemann site. Then sorted these stories.  Usually students sort them by operation needed to solve them, but the video was a great focus, because they really did a great job discerning actions. The idea is that young students encountering stories before direct operation instruction classify stories by what's happening in them. Are amounts increasing or decreasing? Are we comparing separate amounts or looking at static groups of different types? They then model and invent strategies that fit the contexts. For example, students who are taught addition first but then have to use it in a decreasing context often have difficulty solving the story problems. (James gave away 3 pencils but still had 5 left. With how many did he start?)

We followed that by brainstorming contexts  for negative numbers: money, debt, bills, weights/balloons, depth/sea level, golf, football yardage, temperature… the usual suspects but a good variety. When we tried writing stories for them, it was challenging to ask the questions in a way that the answer was negative. I nudged them towards the idea that one of the strongest contexts for negative numbers is when the numbers are describing change rather than direct quantities.

• where she was compared to where she started
• tell us how far away from someone they were, and in what direction
• a plan for how everyone could, in a coordinated way, get from their home position to their spot that was the same distance away but in the opposite direction
• identify if there was someone who was the same distance away from Shane, but in the opposite direction
• give students a target result and ask them to come up with a series of moves that resulted in the given displacement

• 1st group: stood at 8, 5 and -3. Class brought up person at 8 could be change: how big a change when walking from -3 to 5. Another said -3 could be how big a change from 8 to 5? Discussion: no way for 5 to be the change in that situation. 
• Next group of 2 stood at 14 and -7. Their story was: Samantha climbed a 14 foot hill and jumped in the water, sinking to 7 feet below the surface. How far did she dive? Someone brought up 14, and dove 21 feet, where is she? (“Dead”)

• http://math4teaching.com/2010/02/04/assessing-conceptual-understanding-of-operations-involving-integers/  
• https://alg1blog.wordpress.com/2012/07/29/some-approaches-to-negative-numbers/ 
• https://mathymcmatherson.wordpress.com/2012/04/13/lots-of-thoughts-on-integer-operations/

Day 2 we set the number line back up and started thinking about how to explicitly model addition and subtraction. The first group shared the idea to face positive: if addition of positive move forward, if subtraction of positive turn back, if negative turn... I raised my opposition to things that just feel like more rules and not connected to ideas. In discussion, the idea was raised to face neutral by default, turn positive to add and negative to sutract, walk forward for positive, walk backward for negative. I brought up how the class deciding this idea for themselves is probably the most valuable part.

Once we had decided on the representation, we got to some exciting stuff. We had students do a walk on the number line. (Start, turn, walk, stop) and then we wrote it down. 5 + -4 = 1. We discussed how non-threatening it was to walk for something like this, and it was a place where you were really free to experiment. When someone brought up different options, 7 - 13 or 7 + -13, we talked about how you can tell and was there really a difference. Then we hit on the idea that you could walk out equivalences. Are these two things equal? Let's try them! Someone had the idea to try commutativity. What would associativity look like? (Hard to walk.) There was an interesting side effect: some common student errors are impossible to walk. They just don't make sense in embodied cognition land


I brought a game idea, of course. Take a deck of cards, remove everything but A to 7, red is negative and black is positive. 

End of the Line (game)
Shuffle, and deal two decks, like for War. Both players start at zero, facing each other. Flip the top card of your deck. Player with the smaller magnitude number goes first. You can add your number to yourself or subtract it from your opponent. The first player's move decides whether they are going positive or negative, and the 2nd player is going the other way. The goal is to get off the number line. (Ours went to +15 and -15.)

Sample turn:

• player Positive is at 2 and flips red 6. Negative is at -3 and flips red 2. 
• Smaller magnitude goes first, so Negative adds -2 to their position. 
• Faces positive and walks back two squares to -5. Positive player doesn't want to add -6, so makes Negative subtract. Negative faces negative, then walks backward 6 squares to 1.

In the picture there are two different games going on. Other students tried the game or modeling stories with a chip model. People liked the game pretty well with the numberline, and got them a lot of practice thinking about adding and subtracting positive and negative, as well as use on the numberline.

Day 3 involved no embodied cognition. We discussed fact families and addition and subtraction strategies, and then discussed how to show a variety of strategies for integer addition and subtraction in symbolic records and in number lines. The number lines really showed the benefit of the previous classes' movement as they felt the directions really made sense.







All in all, great start to the year. I've only become more convinced of the need for more opportunities to embody mathematics, and the value of the intuition that this helps build through experience. And, of course, I'm interested in your stories about this, or ideas for what else you might have tried.

EDIT/postscript: several of the preservice teachers are blogging about this.

• Sam - clearer explanation of the game.
• Brittany - thinking about the human number line in terms of variety
• Dakoda - trying it out in her tutoring
• Kevin - finding some online number line games

Play the game!

I've been filling up the blog with my class notes from Learning Creative Learning, but an assignment from there really turned into a fun project.

In Week 6 we were supposed to remix Scratch projects from other users. I - of course - looked for math games. First I found 15 Seconds by jOHEEN_c which was an integer game. I wanted more leveling, so found Maze Levels 1-5 by Cats_Are_Awesome. I didn't wind up using their code, but I did learn a lot about Scratch by working through their programs.

I wanted the levels because increasing challenge is key to engagement. I have about 10 screens of increase, after which it levels out. The movement is a little challenging for me as an old guy: the cat follows the mouse instead of being controlled directly. The game forces you to add and subtract positive and negative, but gives you choices on how to get the target. I did add a restart button for if you got stuck that lets you proceed without having to play all the levels over. Is it a problem that the game never really forces an end by becoming impossible?

The game is on the Scratch website, where you can play it or download the code. (Scratch is free to download, of course.) All Scratch submissions are cc 3.0, which is very nice. (Direct link to video, made with Screenr.com.)
(You can embed the Scratch program directly in a webpage, but it starts immediately, which can be aggravating.)  I would be very interested in your feedback on the game, and delighted if you or your students would be interested in remixing it.

Music credit: Upbeat Ukelele Song by Akashic Records, via Jamendo.

Previously: planning and coaching on inequalities.

The origin of this game was trying to think about a game that helped give students enough experience to intuit the rules for how operations with signed numbers affect inequalities. I think this is doubly hard for students because they don't have much intuition for signed numbers, and learn both integer operation and inequality rules by memorization rather than understanding.

I love playing cards as a material for signed numbers because the red and black make such a nice positive and negative visual. (For more integer games, see this collection elsewhere on the blog.) I thought about students somehow constructing expressions, but I couldn't think of anything not clunky. I knew we wanted comparison, and I like War as a comparison structure. That made me think of the exciting moment of war, playing down extra cards on a tie.

When I played around with the cards, though, it didn't get at inequalities because each player was changing the value of only one side of the comparison. A big no-no in the context we want. So the played cards had to effect both players' values. It seemed confusing to have both players reveal at the same time, so I decided that - at least to start - players should take turns, even though that makes the optimal strategy pretty determinable. (If it's not a word it should be.)



After students get the strategy, go to the variation where both players flip at the same time or the dealer has one less card. If playing with middle schoolers for integer operations practice, try the flipping off the top variation.

I think this is a good educational game, but only close to a good strategy game. I can't quite figure out what's missing, so if you have a variation or adaptation to try, please let me know.

Evaluating this on my game design framework:

1. Goal(s). Gain experience with effect of integer operations on inequalities. Works well. Also good for gaining experience with integer computation.
2. Structure. The game generates a lot of these situations and the cards guarantee a mix of operations and values.
3. Strategy. Pretty simple. On variation becomes a pretty nice bluffing game, but not intensely strategic.
4. Interaction. What you do completely depends on opponent.
5. Surprise. Hidden information and opponent plus randomization of cards helps here.
6. Catch-Up. Victory is almost always possible.
   
7. Inertia. Might be too simple for loads of play, but good enough for the objective.
8. Rules. The idea of applying your card to both is non-intuitive, and remembering suits-operations connections is hard. You might want specialty cards
9. Context. No context, but all the variations generated engagement from the preservice teachers.

One thing I want from this experience is a deck with operations for suits. Not sure if it should have all 4 operations, or multiplication and addition with positive and negative numbers or some other variation. What do you think?

Photo credit: Abulic Monkey @ Flickr

Sorry I haven't been writing more. There are many things I want to get to here, but there are many things in my way.

I had a conversation with a commenter that resulted in his writing a post to share here. He is selling something, which makes me disinclined to share it; but he is a teacher sharing his connections and story, and attempts to innovate, which I like. I've received no compensation for this. Please let me know in the comments if it is an inappropriate use of the blog, to you.

Jumping Joey's Numberline


Math Facts for Kids or What’s More Important the Answer or the Process?

By Matthew G. Mandelbaum, MA, MSEd, PhD Candidate, Learning Specialist

Two middle elementary students sit together faced with the following problem: A girl has some jellybeans that she wants to share with her friends. With 3 other friends, she has 1 left over; with 5 other friends, she has 1 left over; and with 11 other friends, she has 1 left over. How many jellybeans does she have?


After reading this problem, the two students are left with choices for problem solving. Should they take a trial and error approach? Should they continue to re-read the problem over and over again, in hopes of some insight? Or should they use a tool to help them learn? Seeking to improve frustration tolerance, perseverance, and the value of process, I suggest they use a tool. They turn to JumpingJoey’s NumberLine® Multiplication and Division Book, which they have been using to learn both operations. “It’s not just for learning number facts,” one student says to the other. “Right! Let’s figure this problem out!” he replies. They now re-read the problem with purpose, hunting for clues. “4, 6, and 12 seem important,” one student says. “Because we have to add the girl and her friends. Let’s investigate them as factors.” The students turn to each of the factor’s string of multiples and analyze the collection of number facts. “Do they share a common multiple?” the first asks. “Let’s see,” flipping back and forth through the book’s pages. “Hey, each of them has 48. Look, on the four’s page, there’s 48 (4 x 12) on the six’s page there’s 48 (6 x 8) and on the 12’s page there’s 48 (12 x 4). The common number is 48!” “Cool!” says the other. “If they all share 48 and there’s 1 left over, that means there are 49 jellybeans in total!” “Let’s check: 49/12 = 4 remainder 1; 49/4 = 12 remainder 1, and 49/6 = 8 remainder 1. That is it!” “We did it; we solved the problem! That was awesome!”

They showed a great deal of pride. I asked them if they felt confident with their answer and if they liked the process of using a tool. They said they did, because it made them feel like mathematicians; they weren’t afraid and they did not give up. In this example, a challenging problem using math facts led to an enjoyment of the process, a sense of satisfaction, and a chance to have a mastery experience where the students could take on something difficult and with there own effort, and only a small amount of adult guidance, to develop increased self-efficacy for math.

This is a path towards a sound math foundation, where process leads to performance. However, I often see that in the quest for math achievement, parents can assume that getting the right answer is the most important thing about knowing math facts, when in fact a child’s decision process that leads to the answer is what should be looked at most carefully.

Memory can work a few ways. Either there is a weak association among information or a strong association. Both types can lead to the right answer when the situation is not so difficult. However, when the challenges begin to mount, having a strong association among information will yield to fewer errors than a weak association.

The key to math facts for kids is strong conceptual understanding. Parents looking to help their children grow should seek to have benefits over the long-term in addition to short-term grades on little assessments. In order to reach this goal, parents can supplement scholastic efforts with at-home tools that provide an organized number line framework to promote conceptual understanding, mathematical fluency, plus a strong foundation for learning math facts. We feel that by using a product like JumpingJoey’s NumberLine, students build a coherent mental number line, which they use flexibly to solve a range of problems. This ability is important because, as reported in the journal Psychological Science of the Association of Psychological Science, scientists found that the quality of mental number line in children and pre-adolescents strongly and positively correlates with arithmetic aptitude, math achievement-test scores, and overall math grades.

Math facts are the building blocks for arithmetic, which form the foundation for higher math throughout the grades. It is important for the child to interact with numbers and consider their meaning with deep contemplation, because new topics will be built upon these numbers to form new knowledge. The child will need to relate what he knows to what is being taught. This relationship needs to be processed on a deep level.

When looking to help support a child’s learning math facts, parents can ask questions like “How did you get that answer?” or say “Show me your thinking.” Within this process, parents need to reward their children’s effort instead of their ability so as to build mathematical competence. To support the last goal, parents can choose products like JumpingJoey’s NumberLine that help their children see themselves as capable mathematicians who are actively engaged in the learning process. Such products should be intrinsically rewarding and promote intellectual curiosity.

Arithmetic is a fascinating subject of study. Numbers possess a lot of power. The Pre-K through elementary years are formative in establishing a child’s sense of self as a learner. Throughout these years, students are met with challenges, like learning math facts. Of vital importance, is their ability to persevere despite obstacles so that they may approach tasks with a realistic sense of confidence and openness to what is new.

Einstein said, “Never regard your study as a duty, but as the enviable opportunity to learn to know the liberating influence of beauty in the realm of the spirit for your own personal joy and to the profit of the community to which your later work belongs.” Children deserve to have this orientation towards learning. As parents and educators, we owe it to them to create an environment in which they can think this way. After all, it worked for Einstein, right?

BIO:
A New York State certified educator in Childhood General and Special Education, Matthew has over 13 years of varied experience working with students of diverse ages in a range of settings from pre-kindergarten to college, in public, private, parochial, afterschool, and tutoring environments. Together with his wife Jamie Cohen, he founded PsySoEd Dynamics® LLC, a company dedicated to developing high quality educational products whose first line of JumpingJoey’s NumberLine products helps students learn math facts and concepts using a fun, multi-sensory approach. As parents of a young girl, they’re even more committed to making math for all and fostering academic success and achievement for children. You can read Matthew and Jamie's Statement of Philosophy for Teaching, Learning and Educational Product Development here.

Copyright 2011 Matthew G. Mandelbaum All rights reserved.

As we consider games for the classroom, there are several possible purposes:

• they’re fun.
• skill practice with engagement that worksheets can’t match.
• sometimes the game can support the underlying concept development.
• sometimes they can be the context to help conceptual understanding.
• provide an opportunity for problem solving in the context of game playing strategy.

I often launch a game by playing me vs. the entire class. It tends to communicate more of the rules than just explaining them. I’ll often have students play in two person teams to start, as their discussion helps work out understanding of the game and the mathematics. After the lesson, I’ll try to engage students in a conversation about what they noticed, what their strategy was, and if they would change anything about the game.

The games for today: you will rotate through the tables, spending a few minutes trying out the game at each stop. This may not be enough to finish a game, but will hopefully give you an opportunity to get a good taste.

Game - Presenter - Content

• Consecutive Capture - Emily Trybus - Integer representation on the numberline
• Tug of War - Anne Harkema - Integer small number addition, especially positive + negative.
• Close to Zero - Jill Beauchamp - Two digit integer addition, especially positive + negative
• Zero Rummy - Cassie Becker - Integer addition with more than two summands, especially zero pairs and sums.
• Gridfight - Kirsten Clemans - Integer multiplication
• Honeycomb - Nick Smith (game coauthor) - Integer multiplication and addition
• +/−24 - Emily Scothorn - Integer operations mixed plus order of operations

At the end we’ll try to come back together to discuss which games you liked the best for your classroom and why.


Files for the day.  Click on these links to see or download the games. (Finally updated to Google Drive to get rid of Scribd links. Some PDF, some Word.)

• Handout - including the Product Game and adapted Integer Product Game
• Treasure Hunt
• Integer Card Games - includes Consecutive Capture, Close to Zero, Zero Rummy and +/- 24
• Tug of War
• Gridfight
• Honeycomb

I tinkered a lot with the Product Game to adapt it for integers, and then found the exact same version I came to on the Connected Math Project website.  Sigh.

The most interesting games to me are Gridfight and Honeycomb.  My preservice middle school math teachers helped with the playtesting of these games, and gave a lot of valuable feedback.  Consecutive Capture is a nice variation of Fraction Catch... which I was about to link to, but I guess I haven't written about yet... and the preservice teachers were fond of it.  Most of the games had someone who really liked them, with the possible exception of Treasure Hunt, which is meant to be a quick and easy introduction game, and indeed the preservice teachers found it simple.

We are beginning our middle school math for teachers course thinking about integers.  What a miserable topic, on the face of it. Trying to think about why K-12 students might need or see negative numbers, these pre-service teachers came up with three spot on suggestions:

• Answers unanswerable problems: 5-7
• Gives relative position or quantity
• Alternative way to think about subtraction

Really sharp thinking, I thought.  And if you think about why mathematicians invented them (or worked out how they work), this is definitely it.  Completeness and closure, and then a more sophisticated understanding of operations.  My suspicion is that these make poor justifications for middle school students.

As I thought through contexts, I thought that the relative position or quantity situations were the strongest.  Something got me thinking about keeping track of change as the most interesting of these.  That and our heavy family diet of fantasy led to this story.  Apologies to Mr. Python and to you, the reader.

Task:  Archaeologists definitely did not unearth this journal from the middle ages.  But as you read through the account of this thrilling fictional adventure, please keep track of the number of knights.  Make two graphs:

1. A graph of days on the quest vs. the number of knights.
2. A graph of days on the quest vs. the change in the number of knights from the previous day.

A mathematician would probably mark Jan. 11th  as day zero, but what you do is up to you.  As the graphs are finished, label the important points with what was happening in the story.  Which graph do you think shows most clearly what happened?  Why?

Quest for the Holy Snail
Diary of Sir Vaysez

Jan. 11th, year of our Lord 1011.
We set out for Gudtonoya today. Our company numbered 100 good knights, in search of the Holy Snail. It is hoped that recovery of the Holy Snail will bring peace to our land.

Jan 12th. As we topped the hill outside Dentite-on-Wails we encountered a party of Orcs. 10 men lost. Ran away screaming at the sight of the beasts. The orcs waved us on, wanted to know if we had extra biscuits. Not for their ilk!

Jan 13th. Faced an enemy party of evil knights from Notusistan. The blocked our path and challenged us to battle.  We shouted as loud as possible, they were overwhelmed and joined our party. 25 converts to our cause!

Jan. 14th. Sad morning. Turns out Notusistannis were playing a trick. They left and convinced 50 gullible men to go with them. Silly knights, such tricks are for children!  King seems depressed.

Jan 15th. Wandering through the Moors of Lesthan. Doing nothing for morale. More knights returned home today. The company is down to 35. Still mostly mighty!

Jan 16th. Surely this day shall be remembered always. Camped at the edge of a pond, the King heard a voice calling to him. When he looked into the water, a maiden stared back at him! (Not a reflection, as he is quite, um, "rugged" might be polite.) This maid said that the heavens did find favor with him, and she reached out from the water holding the Spear of Justice! Hurrah!

That said, 10 more men left. Mumbling something about “watery women handing out mystical weapons is no basis for any kind of representational government.” Good riddance, say I! 

Jan. 17th. 12 new knights have joined. We marched today to the Castle of Awwshux. The nobles there, inspired by our majestic liege and Expointsalot, have joined our quest. (That’s what he’s calling the spear; doesn’t seem like good namesmanship. What’s wrong with ‘Spear of Justice’?) They also had information that …

Jan. 18th. Sorry I never finished yesterday. Dragon attack!  Lost a few brave warriors, but the dragons were repelled. Our company numbers 33 hale and lightly toasted fellows.

Jan 19th. Marching on Notusistan, as the Shuxters are sure these dark knights hold the Snail. We’re worried, as we hear they eat snails. Might as well be French, right? Thrilled by promise of action, many knights have joined our company. We are 51 knights strong, and should enter Notusistan on the morrow.

Jan 20th. Exhausted.  This was a major battle.  Fully armored knights clashed, bashing sword on shield, lances driven forward. So glorious! Surprisingly, we still have 51 knights. Don’t think they lost any warriors either. But it was glorious, I say!

Jan 21st. Turns out Notusistannis are good cooks.  Might as well be French, right? They invited us in for a meal. We recounted the glorious deeds of the battle from the previous day. 8 men left because the food was "too spicy."  Have a palate, man! But 5 brave Notusistannis have joined us. Turns out they didn’t even know the Snail was here. They said that if it’s anywhere, it must be in the castle of the Wizard King of Wartshog.  It is rumoured we shall face the undead.  Then 6 more men left.

Jan. 22nd. I write these words weary, but victorious. 10 more knights joined us for the final assault. But the legion of skeleton fighters claimed heavy tolls.  Relentless, they were.  Then the Wizard King smashed 5 brave knights as we rushed him together. But 22 knights, including our King, Herbert, made it through. The King smashed the evil warlock's dread terrarium and recovered the Holy Snail!

Rejoice all you lovers of Blessed Mollusks!  Once more our land shall know peace.

Chapel of the Holy Snail

Writing this made me think of Denise and her adventure math stories.  Check them out at Let's Play Math!

Photo Credits (Flickr): GraphicReality, Ton MJ, ElitePete, estherase, modowd, greyloch, rogersanderson

Where do you get good problems for your students?

One source is that problem-of-the-day widget at the bottom of the blog. A couple times a week, I'm copying those, put them into a Word document, and then save them for a good opportunity.

But my all time favrite source is from the English (or British?) parallel to the NCTM: Nrich. Problems are sorted by content, tagged, by grade band (stage) and challenge level (number of stars). Some are unsolved, but accessible. Almost all are clever and/or interesting. Soooo nice! Give them a try. Here's an account of a teacher and how they use Nrich.

Here's one that I gave on a math for middle school final this semester:
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Take, for example, four consecutive negative numbers, say
Now place + and/or − signs between them. e.g.
There are other possibilities. Try to list all of them. Now work out the solutions to the various calculations. e.g.
Choose a different set of four consecutive negative numbers and repeat the process. Take a look at both sets of solutions. Notice anything? Can you explain any similarities? Can you predict some of the solutions you will get when you start with a different set of four consecutive negative numbers? Test out any conjectures you may have. Try to explain and justify your findings.