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  1. Strategies For Problem-solving In Counselling
  2. Strategies Problem Solving Inequalities
  3. Strategies Problem Solving Skills
  4. Strategies Problem Solving Math
Math Portal Home > Secondary Mathematics Resource Booklet > Problem Solving Strategies

It is a learned process intended to avoid bias, distortion, prejudice, and inconsistency, and is essential to effective problem solving. Critical thinking is required to evaluate potential solutions to a problem.

  • Look for a pattern
    Find the sum of the first 100 even positive numbers.
    The sum of the first 1 even positive numbers is 2 or 1(1+1) = 1(2).
    The sum of the first 2 even positive numbers is 2 + 4 = 6 or 2(2+1) = 2(3).
    The sum of the first 3 even positive numbers is 2 + 4 + 6 = 12 or 3(3+1) = 3(4).
    The sum of the first 4 even positive numbers is 2 + 4 + 6 + 8 = 20 or 4(4+1) = 4(5).

    Look for a pattern:
    The sum of the first 100 even positive numbers is 2 + 4 + 6 + .. = ? or 100(100+1) = 100(101) or 10,100.


  • Make an organized list
    Example:
    Find the median of the following test scores: 73, 65, 82, 78, and 93.
    Make a list from smallest to largest:
    65
    73
    78 Since 78 is the middle number, the median is 78.
    82
    93

  • Guess and check
    Which of the numbers 4, 5, or 6 is a solution to (n + 3)(n - 2) = 36?
    Solution:
    		Substitute each number for “n” in the equation. Six is the solution since (6 + 3)(6 - 2) = 36.

  • Make a table
    How many diagonals does a 13-gon have?
    Solution:
    Number of sides
    3
    4
    5
    6
    7
    8
    Look for a pattern. Hint: If n is the number of sides, then
    n(n-3)/2 is the number of diagonals. Explain in words why this works. A 13-gon would have 13(13-3)/2 = 65 diagonals.

  • Work backwards
    Example:
    		Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; 1/7 of what was left went to his son; 2/3 of what was left went to his butler; the man’s pet pig got the remaining $2000. How much money did the man leave behind altogether?
    The pig received $2000.
    1/3 of ? = $2000
    ? = $6000
    6/7 of ? = $6000
    ? = $7000
    1/2 of ? = $7000
    ? = $14,000

  • Use logical reasoning
    At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether?
    Solution:
    Hint: Draw a Venn Diagram with 3 intersecting circles.

  • Draw a diagram
    Example:
    Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; 1/7 of what was left went to his son; 2/3 of what was left went to his butler; the man’s pet pig got the remaining $2000. How much money did the man leave behind altogether?

  • Solve a simpler problem
    Example:
    		In a delicatessen, it costs $2.49 for a half pound of sliced roast beef. The person behind the counter slices 0.53 pound. What should it cost?
    Try a simpler problem. How much would you pay if a half pound of sliced roast beef costs $2 and the person slices 3 pounds? If a half pound costs $2, then one pound would cost 2 x $2 or $4. Multiply by the number of pounds needed to get the total: 3 x $4 = 12.
    Now try the original problem: If a half pound costs $2.49, then one pound would cost 2 x $2.49 or $4.98. Multiply by the number of pounds needed to get the total: .53 x $4.98 = $2.6394 or $2.64.

  • Read the problem carefully
    Know the meaning of all words and symbols in the problem.
    Example:
    		List the ten smallest positive composite numbers.
    Since positive means greater than 0 and a composite number is a number with more than two whole number factors, the solution is 4, 6, 8, 9, 10, 12, 14, 15, 16, 18. For example, 4 has three factors, 1, 2, and 4.

    Sort out information that is not needed.

    Last year the Williams family joined a reading club. Mrs. Williams read 20 books. Their son Jed read 12 books. Their daughter Josie read 14 books and their daughter Julie read 7 books. How many books did the children of Mr. and Mrs. Williams read altogether?
    Solution:
    		You do not need to know how many books Mrs. Williams has read since the question is focusing on the children.

    Determine if there is enough information to solve the problem.

    How many children do the Williams have?
    Solution:
    		There is not enough information to solve the problem. You do not know if Josie, Julie, and Jed are the only children.

  • Create problem solving journals
    Students record written responses to open-ended items such as those tested on FCAT in mathematics. Student identifies problem solving strategies.

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Thus, problem solving generally requires a commitment of staff time (and perhaps overtime). In larger organizations, the problem-solving team may be assembled and led by an outside consultant. One of the best problem solving strategies is to create a safe space for a group to share and discover practices, behaviors, and problem solving techniques that can help them find more effective solutions. Word problems require problem solving strategies. And more than anything, word problems require decoding, eliminating extra information, and opportunities for students to solve for something that the question is not asking for. There are so many places for students to make errors! Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with.

Math Portal Home > Secondary Mathematics Resource Booklet > Problem Solving Strategies

Strategies For Problem-solving In Counselling

A whole class of problem-solving techniques focuses on altering the given problem into a simpler form that is easier to analyze. Once a solution is found to the simplified problem, fresh ideas for attacking the original problem often become clear. This section will highlight multiple techniques for problem-simplification, as well as other useful techniques for problem-solving.

The first step, however, to problem simplification is to “give yourself the right” to alter the problem into a different form! Many students tend to avoid this, for fear of “getting off track” and losing sight of the original problem. What is needed is a spirit of adventure: a willingness and a confidence to explore the possibilities. Do not think you must solve exactly the problem that is given to you at first. Modify the problem, solve the simpler version of that problem, then apply the lessons and patterns obtained from that solution to the original (more complex) problem!

Limiting cases

Strategies Problem Solving Inequalities

A powerful method for analyzing the effects of a change in some system is to consider the effects of “extreme” changes, which are often easier to visualize than subtle changes. Such “extreme” changes are examples of what is generally known in science as a limiting case: a special case of a more general rule or trend, possessing fewer possibilities. By virtue of the fact that limiting cases have fewer possibilities, applying a limiting case to a given problem generally simplifies that problem.

Consider, for example, this Wheatstone bridge circuit, where changes in the thermistor’s resistance (with temperature) affect the output voltage of the bridge:

A realistic question to ask of this circuit is, “what will happen to (V_{out}) when the thermistor’s resistance increases?” If our only goal is to arrive at a qualitative answer (e.g. increase/decrease, positive/negative), we may simplify the problem by considering the effects of the thermistor failing completely open, because an “open” fault is nothing more than an extreme example (a limiting case Download flight simulator 2004my traffic 2004 rubackup. ) of a resistance increase.

If we perform this “thought experiment” on the bridge circuit, the circuit becomes simpler because we have eliminated one resistor (the thermistor):

With the thermistor eliminated from the circuit, we see that test point B has lost its connection to the negative terminal of the voltage source. This can only mean one thing for the potential at test point B: it will become more positive (less negative). If the bridge circuit happened to be balanced prior to the thermistor fault, (V_{out}) will now be such that B is positive and A is negative by comparison.

Analyzing the results of this limiting case even further, we can see that resistor (R_2) now carries zero current (thanks to the thermistor now being failed open), which means (R_2) will now drop zero voltage. If (R_2) drops no voltage at all, test point B must now be at the exact same potential as the positive terminal of the voltage source. This being the case, measuring (V_{out}) between test points A and B will be equivalent to measuring voltage across (R_1). Thus, the limiting case of (V_{out}) for an increase in thermistor resistance is (V_{R1}), with B positive and A negative.

Strategies Problem Solving Skills

Another realistic question to ask of this circuit is, “what will happen to (V_{out}) when the thermistor’s resistance decreases?” Once again, the problem-solving technique of limiting cases helps us by transforming the four-resistor bridge circuit into a three-resistor bridge circuit. The limiting case of a resistance decrease would be a condition of no resistance: a shorted thermistor:

With the thermistor shorted in this “thought experiment,” we see that test point B now becomes electrically common with the negative terminal of the voltage source. This, of course, has the effect of making test point B as negative as it can possibly be. More specifically, by making test point B electrically common with the bottom node of the bridge, it makes (V_{out}) equal to the voltage drop across (R_3). Thus, the limiting case of (V_{out}) for a decrease in thermistor resistance is (V_{R3}), with A positive and B negative.

Let us consider another application of this problem-solving technique, this time to the analysis of a passive filter circuit:

If the type of filter circuit shown here were unknown (i.e. the student could not identify it as a low-pass, high-pass, band-pass, or band-stop filter circuit at first sight), the technique of limiting cases could be applied to determine its behavior. In this case, the limit to apply is one of frequency: we may perform “thought experiments” whereby we imagine the input frequency being extremely low, versus being extremely high.

7 step problem solving process

We know that the reactance of an inductor is directly proportional to frequency ((X_L = 2 pi f L)) and that the reactance of a capacitor is inversely proportional to frequency ((X_C = {1 over {2 pi f C}})). Therefore, at an extremely low frequency ((f =) 0 Hz), the inductor will act like a short while the capacitor acts like an open:

Likewise, at extremely high frequencies ((f = infty)), the capacitor will act like a short while the inductor acts like an open:

From these two limiting-case “thought experiments” we may conclude that the filter circuit is neither a low-pass nor a high-pass, because it neither passes low-frequency signals nor high-frequency signals. We may also conclude that it is not a band-stop filter, because that would pass both low-frequency and high-frequency signals. This means it must be a band-pass filter, by eliminating the other three alternatives.

If we would wish to confirm the band-pass nature of this filter by a positive experimental result rather than merely by eliminating what it is not, we could perform one more limiting-case “thought experiment:” a condition where the signal frequency exactly equals the resonant frequency of the LC network ((f = {1 over 2 pi sqrt{LC}})). Here, we must recall the principle that a parallel LC network has infinite impedance at its resonant frequency:

In this “thought experiment” we see that the LC network will be completely “open” and allow 100% of the input signal to appear at the output terminals. Thus, it becomes clear that this passive circuit functions as a band-pass filter.

Strategies Problem Solving Math

As with the Wheatstone bridge circuit, the value of limiting-case analysis is that it acts to simplify the system by effectively eliminating components (replacing them with “shorts” or “opens”). Even in non-electrical problems, limiting cases works the same by simplifying a system’s behavior so that it becomes easier to apprehend, and from these simplified cases we may usually determine behavioral trends of the system (e.g. which way it tends to respond as some variable increases or decreases).