Sharing and Analyzing Rich Groupworthy Problems
Asilomar 2007 Focus Group

• What is a Rich Problem?
  • (1) drives/reveals deep student thinking and
  • (2) give a facilitator flexibility in dealing with a diverse student body.
• Is This Problem Rich or Not ? #1
  • How many 1 foot by 1 foot square tiles are needed to form a border inside a square space measuring 10 feet by 10 feet? Find a way to solve this without counting every tile.
  • The Activity
    • Do Problem 1 by yourself for two minutes. Compare your method to your neighbor.
    • Prompts: Can you find another way to do it? Try doing it for an S' x S' square.
    • Discussion involved the sharing of several different graphical ways to do the problem.
  • Discuss: What is Rich about it?
    • Various levels of approaches. Can "just draw it and count" or have more elaborate approaches.
    • Encourages seeing alternative solutions. (You are almost internally compelled to share out)
    • Leads into algebraic representation. Connect algebra symbols to diagram.
    • Seeing different expressions simplify to same thing. Concreteness.
    • Some more math lurking. Difference of squares.
    • Some possibly tricky algebra.
    • OK for ESL students because it's visual.
    • There are lots of avenues into it.
    • Lots of different ways to do.
  • Usage Notes
    • Be sure to ask for more than one way.
    • Should you give it without diagram...?
      • Make sure they know what a "border" is.
        • Point to physical objects. Use the tiling on the floor.
        • Use actual tiles. wouldn't even need the inner tiles.
      • Visual setup is accessible.
    • HOW do you get people to work together?
      • If it's too easy: make harder. Add some Qs? 10x10 , 20x20, generalize.
      • What if students just wait for someone else?
      • Tell them we want 3 solutions. Each person presents a different solution.
      • Set up roles.
        • Presenter - only Q asker.
        • Materials
        • Scribe
        • Checker - make sure explainer is ready.
      • Forcing collaboration
        • Ask random person to explain to you.
        • Don't take individual Qs only group questions.
        • Give problems too hard for one person.
      • Choosing Groups
        • "ability"
      • Give students time by self ???
  • Let's Ruin the Problem
    • Draw the diagram for them with squares. Ask how many squares there are.
    • In discussion, focus on the answer. The answer is 36, period.
    • Modeling too much before. Show them a 6 x 6 first.
      • Limit the methods in your presentation.
      • Showing any methods stops creativity.
      • Takes from their success. Don't steal their thunder.
      • Maybe do this as a hint later!
        • Helps them to generalize. Compare 6 and 10.
      • The size 10 matters. 10 discourages counting versus 2 or 3.
        • For extensions, maybe do 10, go to 100, then S.
      • (BTW. Does 4S-4 mess up at S=1?)
    • Don't allow discussion.
      • Get students to call our the answer fast.
      • Don't let them share solution in own way. Themselves.
      • Present answer as teacher.
    • Don't allow them to collaborate.
      • Collaboration make different students feel important, smart.
      • Collaboration builds community, caring.
    • Managing amount of TIME.
      • If too much time. Bored.
        • Suggest extensions.
          • Use formulas. Going more general.
          • Split into 2 10 x 5.
          • Cube border.
          • Have sudents WRITE explanation.
      • Too little time. Frustration, giving up.
        • Enough time for everyone to get SOMETHING.
• Rich or Not ? #4
  • Find the secret word using the following clues. Solve each equation for a value for x which corresponds to a letter using the code A=1, B=2, etc. Put the letters in the proper order to spell the secret word! (i) 2x = 6; (ii) 16 – x = 8; (iii) x/3 = 3; (iv) 7 + (x/6) = 10.
  • Activity: Do on own, discuss with partner, discuss whole group.
  • This is less rich.
    • Won't encourage "deep" student thinking.
    • Rewards linear thinking. Do equation, find letter. One way to do it.
    • What about students with English issues? UnJumbles are hard.
    • Hard to come up with mathematical extensions.
      • Make one on your own. Write a two-step equation.
      • Or write a word problem.
      • Or systems of equations.
    • Rewards shortcuts: Get halfway through, solve word puzzle, then don't finish.
    • As Practice for equation solving, not rich. Maybe could use as intro to coding.
      • But maybe there are richer ways into coding.
    • #1 had lots of connections. #4 not.
• Rich or Not ? #5
  • The price of a color printer is reduced by 30% of its original price. When it still does not sell, its price is reduced by 20% of the reduced price. The sales person informs you that there has been a total reduction of 50%. Is the sales person using percentages properly? If not, what is the actual percent reduction from the original price?
  • Activity: Do on own, discuss with partner, discuss whole group.
  • This has some rich aspects.
    • Easy to get wrong answers! Makes conversation.
    • Different representations? Do it numerically, or can look at a picture.
    • Interesting extension:
      • 50% is wrong. In fact .7 * .8 is correct. Is it a coincidence that the mistake is off by the product of the discounts .3 * .2?
      • (Not coincidence. Proof Left to Reader.)
    • Discount situation is real-life and can be understood by most people.
  • Eric's Comment
    • I came into the session thinking this was a problem of limited richness. The discussion helped me to value its rich aspects more. However, I am still concerned by two aspects. First, there is basically one numerical way to solve the problem, though one could certainly use percents, decimals or fractions and for some levels, this would count as different approaches. The graphical approach I am guessing would be very uncommon and would have to be prompted. Second, the problem rewards knowing the standard approaches and punishes not following those. That doesn't mean it is useless; there are times you want something like that. But for rich uses in a heterogenous classroom, these are disadvantages.
• Some Sources of Rich Problems
  • Integrated Mathematics Program (Key)
  • College Preparatory Mathematics (CPM)
  • Algebra/Geometry/Calculus Connections (for MS Teachers) (Pearson)
  • Jacobs, Mathematics: A Human Endeavor
  • Averbach and Chein, Problem Solving Through Recreational Mathematics
  • Stevenson, Exploratory Problems in Mathematics
  • Mathematics Teacher, Delving Deeper column
  • Math Forum, mathforum.com
  • Mason, Burton and Stacey, Thinking Mathematically
  • Stein, Mathematics: The Man-Made Universe
  • Beck, et al, Excursions Into Mathematics
  • Cuoco, Mathematical Connections
  • Ask Marilyn vos Savant. In Parade.
  • Gardner, My Best Math and Logic Puzzles
  • Cut-the-Knot.org
• Further Reading
  • These problems and some other interesting ones are discussed in a paper I co-authored: Using Rich Problems for Differentiated Instruction by Hsu, Kysh and Resek. From New England Mathematics Journal, 2007.