AmpRetreat

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?

* Noticings/wonderings/hints....

**Interpretation**

- I notice that the seal is made out of two pieces of a circle.
- I notice that the cut does not go through the center of the circle.
- I notice that the straight edge for the smaller piece and that for the big piece are of the same length.
- I notice that one of the pieces is bigger than the other.
- I notice that the gold is covering the surface area.
- I notice that the silver is covering the perimeter of each piece, the curved part and the straight part.
- I notice that it costs twice as much to cover the perimeter of the big piece than to cover the perimeter of the smaller pieces
- I notice that this problem is very difficult and just doing a good and thorough understanding of what’s going on might be a great accomplishment for the 90 minutes.

**Strategy**

- I wonder if the ratio of the price of silver for the two pieces is related to their perimeter and how.
- I wonder if the ratio of the price of gold for the two pieces is related to their surface area and how.
- I wonder if some students will think that the ratio of the price of gold for the two pieces is equal to the ratio of the price of silver for the two pieces.
- I wonder how to calculate the perimeter of each piece - What is it the perimeter related to? – How can I express it in terms of the parameters that I know and those that I don’t know?
- What role does the radius play in affecting our answers? Do you want to leave it in all your equations? Is it obvious to you that it will cancel out from the ratio equation eventually? What is the relationship between the straight part of the perimeter (cord) and the angle that intercepts this cord?
- I wonder if we can we add a triangle to the picture and use trigonometric techniques like the Law of Cosines to help solve for certain variables?
- I wonder if there are formulas that we are not sure about such as the area of a sector of a circle then we can Google them? (the students can use one of our computers for that if needed.)
- I wonder how can I solve for x in an equation such as ax – bcosx+c=0 (graph it)
- I wonder if I were not able to solve the equation, is it possible to leave the quantity x in my following steps.
- I wonder if it is helpful to draw my big and small pieces as part of a circle when thinking about their areas?
- I wonder how are the areas of the small piece and the big piece are related? What angles play a role in finding the various lengths in our pieces.
- I wonder what is the ratio of the price of gold that covers the small piece to the price of gold that covers the big piece.

Do you have any wonderings or noticings you can add?

- I wonder if the students may think that since the perimeter is double that the area will be double (if so we could suggest having them draw a more basic case in which they can choose their own perimeter and find the area so they can figure out on their own that their idea does not work) -SBlock

- I noticed that a few different equations are needed to solve the problem. For example arc length, area, perimeter, and trig functions.

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 **

** Interpretation:**

- I notice that the rectangular lawn is bigger in area than the surrounding flower border.

- I notice that the flower border has a fixed width.

- I notice that the flower border surrounds the rectangular lawn from all sides.

** Strategy:**

** Graphing:**

- I wonder what to draw.

- I wonder how would the lawn look like and how would the flower bed look like.

**Guess and Check:**

- I wonder what do I need to guess (length of lawn or width of lawn)

- I wonder how can I check my guess (all relationships have to check out- the addition of the lawn area and the flower border have to equal 1215 m^2)

**Equation strategy:**

- I wonder what dimensions I know.

- I wonder what dimensions I don't know.

- I wonder what variables I can assign for the dimensions I don't know.

- I wonder if there are relationships that I can write between my knowns and unknowns.

- I wonder if I can substitute one of the relationships into another to get an equation in one variable that I can solve.

- I wonder how can I solve a quadratic equation (guess and check; factoring; graphing; quadratic formula.)

- I wonder what do the roots of the equation represent and why?

- I wonder what do I need to know to find the radius of the sprinkler.

- I wonder how can I use the dimensions I found to find the radius of the sprinkler.

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: **

- I wonder what 15 degrees below horizontal means?

- I wonder what information is being asked for, as the final answer?

- I wonder how to deal with the two different heights given

- I wonder whether feet or inches will be easier to work with?

**Calculate as you go; don't always worry about how to solve the whole problem**

**Estimate**

- I wonder, are the estimates given in the problem good estimates? What would I estimate?

**Act it out**

- I notice that the people in the problem stand 3 feet away from the shelf, and tilt their heads down a little bit…

**Systematic guessing and checking**

- I wonder

- if I guess a height for the shelf, is the height itself part of the relevant right triangle

- if Law of Sines or that tangent ratio is the easiest way to check my guess?

- what I should compare to what

**Solve a simpler problem**

**Divide into cases (solve it for specific situations)**

- I notice that there seem to be two cases given in the problem

**Consider extreme cases**

**Draw a picture/diagram**

- I notice that the people in the problem stand 3 feet away from the shelf, and tilt their heads down a little bit

- I notice that people are at two different heights.

- I wonder how to draw a diagram that shows men and women?

- I wonder how best to label the height of the shelf? Should that be my variable?

- I wonder, if the height of the shelf is my variable, how to label the side of the right triangle opposite the 15-degree angle?

**Make a list/table/chart**

**Label, assign variables, formulate expressions and equations**

- I wonder what sides do we know? Why? What sides are we looking for?

- I wonder what angles can be figured out?

- I wonder what's the most efficient trig function, given what we know?

- I wonder how to set up and solve a tangent ratio problem

- I wonder if I could do the problem looking only for sine?

** Look for a pattern**

**Work backwards**

**Use logical reasoning (e.g. show the opposite can't be true)**

**Make a model**

**Take advantage of symmetry**

**Set up an equation**

-see above

**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