I'll start you off with some methods that might help certain people in your group. I'd like you to think of I notice/I wonder (or questions in your own words) that will help students struggling along that path (or get a student who doesn't know where to start, started), for each method.
Writing "Noticings" As Algebraically As Possible
* As you go around the table, students can try to write as many noticings as they can as algebraic expressions * Note that the expressions don't have to be said algebraically, just written algebraically * The added bonus is that suddenly each student becomes responsible for the others' noticings * This may not be possible for everyone, or it may not lead them anywhere * For other students, once they have all the relationships written down, they may need a little help figuring out what to do next... what could you help them notice/wonder/do?
Drawing & Labeling a Diagram
* This is clearly a visual problem, so many students might want to draw a diagram * Good labeling is going to be key... * What can you help them notice/wonder/do to make sure that they have good, helpful labels?
Guess & Check
* Guess and check is in fact an awesome strategy for working your way towards an algebraic solution... * ... If you write out your guess, calculations, and check very clearly * Guess and check helps you get a concrete idea of what must be equal, and what the constraints are on that equal thing * What could you help a student notice/wonder/do to encourage them to begin guessing and checking for this problem? * What could you help a student notice/wonder/do to encourage them to move from guessing and checking to solving algebraically?
Any Other Strategy You've Witnessed these 5 Weeks?
Do you have any wonderings or noticings you can add?
Our notes for supplemental sessions:
Chicken Wings: http://mathforum.org/librarypow/puzzles/index2.ehtml?back_to=lpow&puzzle=4532
Week 5 - July 23 2008
712 - All Wet: http://mathforum.org/librarypow/puzzles/index2.ehtml?back_to=lpow&puzzle=712
Guess and Check:
Please add any other strategy that you can think of to solve the problem and any noticings and wondering you may have about the problem.
1182 - Supermarket Shelf: http://mathforum.org/librarypow/puzzles/index2.ehtml?back_to=lpow&puzzle=1182
One thing for you to decide as you plan ahead for the week is what variety of strategies might you see? Will most people move through the strategies in the same order (e.g. first act it out, then draw a picture, then assign variables, etc.)? How are you going to manage various groups working on various strategies?
Here's the Math Forum @ Drexe's list of famous problem-solving strategies, some of which your table may use. Below some of the ones I think are likely to be applied to Supermarket Shelf, I've started brainstorming key issues that might come up.
Check your understanding Look up new vocabulary:
Calculate as you go; don't always worry about how to solve the whole problem
Act it out
Systematic guessing and checking
Solve a simpler problem
Divide into cases (solve it for specific situations)
Consider extreme cases
Draw a picture/diagram
Make a list/table/chart
Label, assign variables, formulate expressions and equations
Look for a pattern
Use logical reasoning (e.g. show the opposite can't be true)
Make a model
Take advantage of symmetry
Set up an equation
Use odd/even to analyze the situation (parity)
Please add questions you could ask, suggestions you could make, the math behind the strategy, possible alternative approaches that might be harder to work with, errors you might have to diagnose, etc.
Week 3 - July 9 2008
554 - Car Rental Quandary: http://mathforum.org/librarypow/puzzles/index2.ehtml?back_to=lpow&puzzle=554
I notice: that this problem can be solved algebraically and by graphing. Algebraically: set-up an equation for both scenarios, generally: Y(cost) = [(cost/day) * (3 days to be traveled) + (rate/mile) * X(miles)]. Both equations will have 2 different line equations, more specifically, different slopes. These two lines will eventually cross because they are not parallel (different slopes); we equate each equation and solve for X. -PC
I notice: that no matter how many miles were driven each day or during the entire trip, the number of days in the trip is held constant at 3 days. -SB
I notice that I can read the first paragraph and make some notes before looking at the next paragraph. In fact, I might make a chart to compare the cost if I traveled 50 miles, 150 miles, 200 miles before even reading the rest of the problem, it might give me a sense of what is happening with the two scenarios. I could do that for 1 day and maybe 2 days, and 3 days, too. Then when I read the next part of the problem I'll have some "story" to use to continue thinking about the problem. -SA
I wonder: I wonder if another good question would be to figure out the cost at the given amount of miles where both companies would charge the same amount. We could hint at this as a way to check to see if the number of miles found actually produces the same cost for both companies. -SB
I wonder if students might be able to get a handle on the algebraic way to think about this problem if they first make a chart and look for a pattern.
I wonder whether the students would better understand if they were to graph the two equations?
Sarah's Warm-up Problem: 2899 - Odds vs. Evens: http://mathforum.org/librarypow/puzzles/index2.ehtml?back_to=lpow&puzzle=2899
I notice that the problem says that you can play it with dice instead of spinners. -SA
I wonder if leaving the question off, bringing dice for each table, and having the students play the game might help them have a context to talk about the math? -SA
628 -The Function Challenge: http://mathforum.org/librarypow/puzzles/index2.ehtml?back_to=lpow&puzzle=628
I notice: that this problem uses logical reasoning with background in algebra. Students can answer this problem by describing what the functionality are for each component in the 5 equations [A(x),...E(x)], how can they contribute in producing the largest output, and then, moreover, describe how each component can produce a large output for different inputs. -PC
I notice: Logic can be used to divide the equations into 1st and 2nd degree functions accompanied by plugging each function from each group into the other functions within the group. The second method may also be a good way to help check if an answer is correct. -SB
I notice that plugging one equation into another and then doing it backwards will tell you the order those two equations will go in but I wonder whether it will work for any number - SB
I wonder how would you decide which equation to plug into which -MS
I wonder: if there is a way to solve the problem using only pure logic
I wonder: if you can use only algebra to solve the problem. -MS
I wonder: if rate of change would help justify a solution to this problem. -LR
Response : I think that the impact of the rate of change in the linear functions has to be weighed against the impact of the constant term when deciding on the order of the functions. -MS
I wonder: if linear functions have constant rate of change, does it matter what order you use them in? -LR
Response: Yes, they may all be constant but they are different from each other. - SB