Solving problems {problem solving, algebra} requires understanding problem or question, realizing what you know already, and knowing answer type.
hypothesis
Solving formal problem requires testing hypothetical solution {hypothesis}.
assumptions
Problem has problem context. Solving problem requires using correct assumptions about context.
principles
Believe solution is possible. Do not feel pressured, confused, or anxious. Do not think about problem difficulty or time.
Estimate and approximate, before doing details. Always try something, do not just think. Talk while doing problems to aid thinking.
If solution fails, repeat procedure to check for errors and do not become frustrated or bored.
skill
Problem-solving skill involves ability to find rules, structures, or patterns that link known with unknown.
methods
Problem solving methods are similar in all cultures, though problem types differ across cultures. People try all known methods to see if one works.
Problem-solving methods include modeling, dimensional analysis, symmetries, physical-quantity analytic properties such as differentiable or power series, parametric methods such as perturbation theory, Scene Analysis, image filtering, contour smoothing, skeletonization, polar mapping, and structural descriptions.
example
Starting at 6 PM, car 1 goes east at 100 km/hr from X toward Y. Starting at 10 PM, car 2 goes west at 80 km/hr from Y toward X. X is 800 km from Y. X is 1000 km from Z. When will the cars meet? First, read problem and make sketch with X on left and Y on right, 800 km apart, with no Z. Then realize that times, t1 = 6 and t2 = 10, speeds, v1 = 100 and v2 = 80, and distance, s = 800, have values. Remember relation between time, speed, and distance, s = v*t, where time is interval, so s1 = v1*(t12 - t1) and s2 = v2*(t12 - t2). Then use the rule that whole equals sum of its parts, to realize that s1 + s2 = 800. Solve by substitution and algebra. Realize that it needs clock time, t12 = ?, not time interval. Check dimensions, logic, and size. Reflect on method.
steps
Solving problems requires steps, from known to unknown, with reasons or examples. Verify and correct step before going to next step.
steps: 1
Specify goal and answer-type output. Classify problem. Understand problem. Read whole problem. Visualize situation, draw picture or graph, or make concrete example. Write known information. Write variable for unknown information and note which variable type it is, such as measurement, number, word, or sentence. Work on only part of large problems.
steps: 2
Gather information and connect data. Specify assumptions. Gather information and connect data. Categorize problem. Remember previous or alternative solutions. Remember equation, relation, or definition between stated variables. Look for symmetries, analogies, and simplifiers. Use thought rules and logical relations. Remember related definitions, assumptions, concepts, data, history, and causes. Look for redundant data and for insufficient data.
One rule is whole equals sum of its parts.
steps: 3
Use input and output properties to find operations or transformations necessary to derive output from input. Perform analysis to get answer. Solve problem using solution type, relation, or rule found in step two. Use reasoning, insight, or trial and error. Use conclusion drawn from data. Think "if A then B" and "B", then A is probably true {heuristic reasoning, problem solving}. Make hypothesis and try it. Do overall and most important problem part first. Do problem step-by-step, properly and neatly. Check steps immediately. Put in numbers or details after feeling solution will work. Master manipulating, rearranging, substituting, using logic, and recalling facts, to solve quickly and accurately.
steps: 4
Evaluate solution and check result. Check answer against expected answer type. Check physical dimensions. Check answer magnitude. Check against real-world knowledge. Check details for accuracy. Check logic for accuracy. Test solution in problem.
steps: 5
Think about work. Try to find shorter solution path. Remember similar problems. Remember method steps. Classify problem. Think about what to do with knowledge gained. Note other solution effects.
Measurement units can transform into equivalent measurement units with more meaning {dimensional analysis}|, to give insight into problem. Constraints on units indicate answer type needed. Solution must have correct units.
Problem-solving methods {generate and test method}| {trial and error} can generate answers from a possible-answer space and test whether answer solves problem.
After studying and understanding problems, solvers can do something non-intellectual {incubation}. If solutions come to mind, they must be ready to recognize solution.
Algebra problems {word problem} can find unknown value from conditions, using definition, theorem, or fact.
a1 = f(a2) and a2 = g(a1) {age word problem}, where a1 and a2 are ages and f and g are functions.
d1 = f(d2) and d2 = g(d1) {digit word problem}, where d1 and d2 are digits and f and g are functions.
a = k(b) and f(a or b) / g(a or b) = h(a or b) / j(a or b) {proportion word problem}, where a and b are integers and f, g, h, j, and k are functions.
h2 = f(h1) and 1/h1 + 1/h2 = 1 {fraction word problem}, where h1 and h2 are integers, f is a function, and 1 is equivalent to 100%.
p1 = f(a or b), p2 = g(a or b), p1 + p2 = 1 {percent word problem}, where p1 and p2 are percentages, a and b are numbers, f and g are functions, and 1 is equivalent to 100%.
h1 = f(a or b), h2 = g(a or b), and 1/h1 + 1/h2 = 1 {percent mixture word problem}, where h1 and h2 are concentrations, a and b are numbers, f and g are functions, and 1 is equivalent to 100%.
h1 = f(a or b), h2 = g(a or b), and 1/h1 + 1/h2 = 1 {rate of work word problem}, where h1 and h2 are time, a and b are numbers, f and g are functions, and 1 is equivalent to 100%.
For triangle, A = 0.5 * l * h {area word problem}, where A is area, h is height, and l is side length. For rectangle, A = l*h, where A is area, and l h are side lengths. For circle, A = pi * r^2, where A is area, and r is radius.
P + P * r^t = T {interest word problem} {savings word problem} {investment word problem}, where P is principal, t is time, r is rate, and T is total.
For triangle, p = a + b + c {perimeter word problem}, where p is perimeter, and a b c are side lengths. For rectangle, p = a + b + c + d, where p is perimeter, and a b c d are side lengths. For circle, p = 2 * pi * r, where p is perimeter, and r is radius.
d = v*t {uniform motion word problem} {rate word problem} {time word problem} {distance word problem}, where d is distance, v is speed, and t is time. vf^2 = vi^2 + 2 * a * ds, where ds is distance, vf is final speed, vi is initial speed, and a is acceleration.
For box, V = l*h*w {volume word problem}, where V is volume, and l h w are side lengths. For sphere, V = (4/3) * pi * r^3, where V is volume, and r is radius.
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Date Modified: 2022.0225