Substitution
The substitution strategy is used for solving math problems, especially when the student is unclear about some component of a math equation or cannot set up the appropriate math equation to solve a word problem. With substitution, one simply replaces the unknown part of a math equation or problem with something known. Applications and examples of the substitution strategy are given below (D. Applegate, CAL).
Fraction
Math students are often confused when trying to solve math problems with fractions. Try substituting the decimal equivalent of the fraction whenever possible (as long as the decimal is not repeating). Simply divide the numerator by the denominator to get the decimal equivalent of the fraction. For instance,
1 2 (x + 4) = 14 0.5 (x + 4) = 14 0.5 (x) + 0.5 (4) = 14 0.5x + 2 = 14 0.5x = 12 x = 24 |
Variables
Sometimes the meaning or function of variables in an equation is unclear.
In this case, substitute an actual number for the variable(s) and work out
the problem. The numbers don't necessarily have to "make sense" mathematically
- they are just used to help you logically figure out the steps of the problem.
Then follow those steps to solve the actual problem with the variable(s).
For example,
Given I = Prt Find t in terms of the other variables.
Substitute numbers for the variables except t.
How would you get the numbers on one side?
What steps did you follow to get t by itself?
Use those steps to solve the real equation. |
Word problems
Students commonly experience difficulty with word problems, especially how to set up the equation using the informaton given in the question. Try substituting the unknowns or variables with actual numbers to help set up the equation. For instance,
Question: Two numbers add up to 15. If the larger number is twice the
smaller number, what are the two numbers?
Answer: First we need to assign variables. From the problem we know the relationship between the two numbers: the larger number is twice as big as the smaller number. If the smaller number is x, then the larger number is 2x. Now we need to write an equation using the variables plus the other information provided in the question. But how? Try substitution. Pretend one of the numbers is 2. If the two numbers add up to 15, as the problem states, the other number must be what? 13. How did you get this? This was determined by subtracting the pretend number from 15: 15 - 2 = 13. Now generalize. One number is equal to the total minus the other number. In other words, one number equals 15 minus the other number. This is your equation in English! Now you just have to put it into an algebraic expression.
Our two numbers are x and 2x. We replace these into our English equation
to get the math equation we need to solve the problem: |
Memory strategies
Math courses often require that four types of information be remembered by students on quizzes and exams. Strategies for encoding and retrieving terms and definitions, symbols, math equations, and problem solutions are described here (D. Applegate, CAL).
Terms and definitions
Key words
Highlight and focus on key words in the definitions. This reduces the amount of information to be remembered and helps one to identify words that may be omitted in fill-in test questions.
Association
Once the key words have been identified, try to associate the term with the key words. You can use phonetic associations, vivid visual associations, associations with prior knowledge, or other associations. Some examples are:
Flash cards
Flash cards are useful for registering definitions of terms into memory. Write the term on one side of the card and the definition on the other. Use the flash cards to test your recall. Practice recalling the definition when given the term and visa versa.
Running concept lists
Make a running concept list by writing all terms and definitions on notebook paper divided into two columns. The terms go in the left-hand column and the definitions with highlighted key words are written in the right-hand column. Fold the paper or cover one column to test your recall of the terms and their definitions.
Symbols
Characterization
Try drawing or visualizing math symbols as characters in order to remember their meaning. For example,
Flash cards
Symbols and their meanings may be summarized on flash cards and reviewed periodically to store them in memory.
Running concept lists
Make a running concept list by writing all symbols and their meanings on notebook paper divided into two columns. The symbols go in the left-hand column and the meanings are written in the right-hand column. Fold the paper or cover one column to test your recall of the symbols and their meanings.
Math equations and rules
Association
Try phonetic, visual, and other associations to remember math equations and rules. The goal is to associate the math equation or rule with something you already know or something with which you are familiar. For instance,
Flash cards
Math equations and rules may be summarized on flash cards and reviewed frequently to store them in memory.
Running concept lists
Make running concept lists of math equations and rules using notebook paper divided into two columns. The names of the equations or rules go in the left-hand column and the mathematical expressions are written in the right-hand column. Fold the paper or cover one column to test your recall of math equations and rules.
Problem solutions
Problem solutions refer to the correct order of steps required to successfully solve math problems. Herrman, Raybeck, and Gutman (1993, p. 192) offer the following suggestions for registering and remembering solutions to math problems. Associations (D. Applegate, CAL) may also be used.
Rehearsal
Repetitious review of the steps for solving a problem aids in registration in long-term memory. The effectiveness of this strategy is enhanced when rehearsals are done frequently and when rehearsals are made active by vocalizing, listening to recordings, or writing.
Practice
Working several practice problems for each solution set aids in registration. Try working sample problems from the book or problems for which answers are indicated in the book. Check answers to insure accuracy.
Solve forwards and backwards
Registration in long-term memory is enhanced when problems are solved forwards and backwards. Work the problem to find the answer, and then take your answer and work back to the original problem.
Procedure cards
Try using procedure flash cards to register problem solutions in long-term memory. On one side of the card write the type of problem and/or give an example. On the other side write the steps in English for solving the problem and actually show the steps for solving the example.
Explain problem to someone else
Remembering is enhanced when one explains or "teaches" the problem solution to another person. Try working with another student in the class, with a tutor, or with a friend or family member. Carefully and thoughtfully go through the solution process, step by step. Find an empty classroom and "teach" by writing the steps on the chalk board.
Frequent review
Review the solution often. Take flash cards with you to review while waiting in line or between classes. Explain the problem solution to a friend while walking to class. Frequent reviewing aids registration of information in your memory.
Mnemonics
Problem solutions may be registered in memory using mnemonics. Take the first letter of each step and form it into a cue word or cue phrase. The classic math mnemonics are:
Past experience
To remember the problem solution during a testing situation, think of specific practice problems that were similar to the test problems.
Key words and associations
Use visual associations or associations with real-life experiences to remember the key words in the steps for solving a particular problem. For instance,
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