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).
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,
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
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.
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:
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
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.
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:
- The numerator is the top number in a fraction, whereas the denominator is the bottom number in a fraction. Remember that "numerator" and "top" go together because they begin with letters that are close to each other in the alphabet. Similarly, "denominator" and "bottom" also begin with letters that are close together in the alphabet, plus the letters "d" and "b" look very similar in form.
- A polynomial is a series of one or more terms that are added or subtracted, such as 3x + 2y - 4. To associate this word with its definition, try this visual association: Picture a prison inmate in a black and white striped outfit whose prison term involves adding and subtracting a bunch of parakets named Polly.
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.
Try drawing or visualizing math symbols as characters in order to remember their meaning. For example,
- A cursive M stands the for mean of a population. Draw or picture in your head a bunch of angry-looking M's to remember this symbol.
- In the equation I = Prt, the P stands for the principal (amount of money) invested. Draw or picture in your head a large P that will remind you of your school principal - a face in the loop of the P and arms holding a ruler or some other significant object. Have little dollar signs floating around the P to help you remember the symbol represents a sum of money.
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
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,
- This association based on fundamental moral principles helps one to remember the rules for multiplying signed numbers (REFERENCE). "Good" things in this association represent positive numbers and "bad" things represent negative numbers.
- A good thing happening to a good person is good.
[positive times positive equals a positive]
- A good thing happening to a bad person is bad.
[positive times negative equals a negative]
- A bad thing happening to a good person is bad.
[negative times positive equals a negative]
- A bad thing happening to a bad person is good.
[negative times negative equals a positive]
- A good thing happening to a good person is good.
- The rules for converting decimals to percents may be remembered using a variety of associations.
- Use common experiences in the association: Think of common percentages we see in our everyday lives, such as sales (50% off and 20% off) or runaway inflation rates (100% or 150%). These are big numbers. Decimals are small numbers (0.5, 0.2, 1.0 and 1.5). How do you make a large number smaller? By dividing. How do you make a small number larger? By multiplying. So to change from percents to decimals (large to small), you divide by 100. And to change from decimals to percents (small to large), you multiply by 100.
- Use alphabetic associations to remember the rules: To change from percent to decimal, you move the decimal point two places to the right. When you start with a percent you move to the right - p and r are close in the alphabet. To change from decimal to percent, you move the decimal point two places to the left. When you start with decimal you move to the left - decimal ends in l and left begins with l.
- Use a variety of associations to keep straight the equations for the perimeter (P = 2L + 2W) and area (A = L * W) of a rectangle.
- Associations based on real-life experiences can be used to remember the equations. When ordering fence to go around the perimeter of your yard, you would order so many feet or meters - the units are raised to the first power. How do you keep the units of something in the first power? By adding - so use the equation with the addition sign. Now, when ordering carpet to cover the area of your room, you would order so many square feet or square yards - the units are raised to the second power. How do you get units to the second power? By multiplying - so use the equation with the multiplication sign.
- A simple association based on the length of the equations might help you to keep them straight. The word perimeter is a long word and it corresponds to the longer of the two equations. The word area is a short word and it corresponds to the shorter of the two equations.
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 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.
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.
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.
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.
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.
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:
- This cue word stands for the steps in multiplying two binomials: multiply the First terms, then multiply the Outer terms, then multiply the Inner terms, and finally multiply the Last terms.
- Please Excuse My Dear Aunt Sally
- This cue phrase helps in remembering the order of operations: Parantheses, Exponents, Multiplication, Division, Addition, and Subtraction. Combine it with a mental image of your aunt doing something rude in an operating room to enhance your memory.
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,
- Problem: Find the equation of a line that passes through the points (8, -3) and -2, 1).
- Key Words: equation of line, through two points
- Steps in the Solution: find the slope, use the point-slope formula, solve for y
- Visual Association: Picture the slope equation at the top points of two mountain peaks [step 1], go down the mountain slope to the point-slope formula [step 2], and move to the Y of a clear mountain stream to find your equation [step 3].