Summarize the strategic hints for problemsolving, and give an example of the meaning of each one.

Which of the following best summarizes the strategic hint"There may be more than oneanswer" and provides an example of themeaning?

A.

Every problem in mathematics has more than one answer. Forexample, both x equals 2

x=2 andx equals negative 2

x=2 are solutions to x squared

x2equals

=4.

B.

If a problem has more than oneanswer, it is not valid. Every problem in mathematics will have one unique answer. Forexample, x equals 2

x=2 is the only solution to x plus 1 equals 3

x+1=3.

C.

There may be several ways to get an answer. Forexample, you can find a solution for one third x equals 4

13x=4 by multiplying each side by 3 or by dividing each side by one third

13.

D.

Some problems in mathematics have more than one answer. Without furtherinformation, there is no way to determine whether both solutions are valid for a particular problem. Forexample, both x equals 2

x=2 and x equals negative 2

x=2 are solutions to x squared equals 4

x2=4.

Which of the following best summarizes the strategic hint"There may be more than onestrategy" and provides an example of themeaning?

A.

There may be several strategies to solving a problem and all of the strategies are equally efficient. Forexample, there is more than one strategy to solve x squared plus 3 x plus 2 equals 0

x2+3x+2=0, all taking the same amount of time and work.

B.

If there is more than one strategy that can be used to find ananswer, the problem is not valid. Forexample, since there is more than one way to solve five fourths x plus 7 equals one eighth

54x+7=18, this problem is not valid.

C.

There may be several strategies to solving aproblem, however only one of them will find the correct answer. Forexample, there is more than one strategy to solve x squared plus 3 x plus 2 equals 0

x2+3x+2=0, however, only one of them will provide the correct answer.

D.

There may be several strategies to solving a problem.However, not all of the strategies are equally efficient. Forexample, there is more than one strategy to solve x squared plus 3 x plus 2 equals 0

x2+3x+2=0, however, using the quadratic formula involves more time and work.

Which of the following best summarizes the strategic hint"Use appropriatetools" and provides an example of themeaning?

A.

Choosing only the tools most suited to the job will make tasks much easier. Forexample, if a problem can be solved one way with calculus and another way withmultiplication, using multiplication will make the solving process easier.

B.

For anyproblem, the appropriate tools needed will always be stated in the problem statement. Forexample, if a problem can be solved one way with calculus and another withmultiplication, the problem statement will provide that information.

C.

For any giventask, there will be a choice of tools to use in any problem. Choose only the most basic tools so the problem does not get too difficult. Forexample, one fourth x equals 6

14x=6 can be solved with multiplication or division. Use only multiplication because it is easier.

D.

When given a choice of tools for aproblem, use as many tools as possible. This will make theproblem-solving process much simpler. Forexample, one fourth x equals 6

14x=6 can be solved with multiplication or division.Thus, both methods should be used.

Which of the following best summarizes the strategic hint"Consider simpler, similarproblems" and provides an example of themeaning?

A.

The exact strategy used to solvesimpler, similar problems is almost never used to solve the more difficult problems. Forexample, if a problem involves subtracting decimals to 3 places and simplier problem involves subtracting decimals to 1place, the exact strategy is not used to find a correct answer.

B.

Sometimes, it is easier to considersimpler, similar problems. If there are no simplerproblems, the question has no solution. Forexample, if a problem involves addingmonomials, there is nosimpler, similar method.So, there is no solution to that particular question.

C.

Sometimes, the insight gained from solving the easier problem may help to understand the original problem. Forexample, instead of picturing molecules mixingtogether, try to picture marbles mixing together.

D.

The exact strategy used to solvesimpler, similar problems is always used to solve the more difficult problems. Forexample, if a problem involves dividing polynomials and a simplier problem involves dividingintegers, model that exact strategy to find a correct answer.

Which of the following best summarizes the strategic hint"Consider equivalent problems with simplersolutions" and provides an example of themeaning?

A.

A useful approach to a difficult problem is to look for an equivalentproblem, which does not have the same numerical answer but may be easier to solve. Forexample, suppose you cannot remember the formula for surface area of a cube.Instead, compare the dimensions to another cube with a known surface area.

B.

Consider equivalent problems with simpler solutions in any case except for when looking for numerical answers. Forexample, instead of finding an equivalent problem to find surface area of acube, look for any simpler way to solve the problem.

C.

A useful approach to a difficult problem is to look for an equivalentproblem, which has the same numerical answer but may be easier to solve. Forexample, suppose you cannot remember the formula for surface area of a cube.Instead, find the surface area of each square and add them together.

D.

A useful approach to a difficult problem is to look for a similarproblem, although it may have a different answer. Forexample, suppose you cannot remember the formula for surface area of a cube.Instead, use dimensions of another cube with a known surface area and solve.

Which of the following best summarizes the strategic hint"Approximations can beuseful" and provides an example of themeaning?

A.

An approximation should always be within one tenth of the exact solution. If this is nottrue, something has gone wrong. Forexample, if an exact solution is 7.232 and the approximation is7, something is wrong in the solution process.

B.

Approximations can always be used instead of an"exact solution", unless otherwise noted. Forexample, if an exact solution is3.1, an approximation of 3 can always be used.

C.

Mostreal-world problems approximate numbers to beginwith, so an approximation is often good enough for a final answer. Approximations also provide a useful check. Forexample, if areal-world problem involves long lengths with severaldecimals, approximating the final answer is often good enough for a final answer.

D.

Approximations are only used to check solutions and are never good enough for a final answer. Forexample, if areal-world problem involves long lengths with severaldecimals, approximating the final answer is never appropriate.

Which of the following best summarizes the strategic hint"Try alternative patterns ofthought" and provides an example of themeaning?

A.

Avoid patterns of thought that tend to suggest the same ideas and methods.Instead, create new ways to solve eachproblem, avoiding the fastest way to solve a problem. Forexample, when solving a problem that requires morethought, always create your own ways to solve the problem.

B.

Have a list of basic solving methods to solve problems. Keeping a narrow list of options is more efficient. Forexample, when solving a problem that requires morethought, use basic solvingmethods, making the question easier to solve.

C.

Avoid using any common patterns to solve problems. Common patterns create common mistakes and may yield an incorrect answer. Forexample, when solving a problem that requires morethought, avoid the most common way to solve the problem to avoid an incorrect answer.

D.

Avoid patterns of thought that tend to suggest the same ideas and methods.Instead, approach problems with an openness that allows new ideas and solving methods. Forexample, when solving a problem that requires morethought, a moment of insight may be useful and make the problem easier to solve.

Which of the following best summarizes the strategic hint"Do not spin yourwheels" and provides an example of themeaning?

A.

Often the best strategy in problem solving is to put a problem aside for a few hours or days. Forexample, if you are struggling with a problem involving long division withpolynomials, set it aside and come back to it in a few hours.

B.

Often the best strategy in problem solving is to put a problem aside and immediately ask for help. Forexample, if you are struggling with a problem involving long division withpolynomials, set it aside and go find help.

C.

Do not try problems that you have not seen before. Forexample, if you have never seen long division withpolynomials, do not attempt it.

D.

Do not use different strategies to solve problems. The known strategy is always the most efficient. Forexample, if you are struggling with a problem involving long division withpolynomials, only use strategies you know to solve the problem.