The Strategies of Looking for a Pattern and Making an Organized List Study Guide (page 2)
Introduction to The Strategies of Looking for a Pattern and Making an Organized List
Mathematics is written for mathematicians.
—NICHOLAUS COPERNICUS (1473–1543)
Having a plan in mind is an important step in solving math word problems. The two strategies presented in this section will show you how to look for patterns and make organized lists.
Looking for a Pattern
Some math problems can appear very complicated when you are first reading the question. However, with many problems, a pattern or a trend may emerge to make solving the problem easier. Often, you can extend the pattern, so finding out what comes next in a series of numbers or the pattern may simplify what is actually happening in a problem. Looking for a pattern is one key strategy that will be discussed in this section. You will also find examples of problems that can be solved by this method.
One type in this category is number pattern problems. In this type of question, a series of numbers is given, and you may be asked to find the next number in the sequence. For example, take a look at the following pattern.
- 2, 4, 6, 8, 10, _____
What is the next number in the list?
This is a list of even numbers, or numbers beginning with the number 2, with 2 added each time to find the next number. The next number in the list would be the next even number, 12, which is also equal to 10 + 2.
The pattern may also be decreasing, as in the following list.
- 100, 80, 60, 40, _____
In this list, 20 is subtracted from the previous term to get the next term. The next number in the list would be 40 – 20 = 20.
If a pattern appears to be increasing, check first to see if addition or multiplication is the rule. If the pattern is decreasing, check first to see if subtraction or division is the rule.
Number patterns can also be found in a context or situation, which is the case in the following question.
The end of the regular school day is 2 P.M. at Hanford High School. There are 1,200 students at the school. During the first hour after school, half of the students in the building leave and the other half remain for extra help and various after school activities. During the second hour after the school day, half of the remaining students leave. If this pattern continues, how many students are left in the building four hours after the end of the school day?
To solve this problem, use the steps to solving word problems explained in Lesson 1.
Read and understand the question. The number of students in a school decreases each hour after the school day ends. The question asks for the number of students in the building four hours after the school day ended.
Make a plan. Start with the total number of students in the building and make a decreasing pattern. Continue the pattern until the number of students remaining four hours after school is reached.
Carry out the plan. The number of students in the building at the end of the school day is 1,200. Thus, after one hour only half remain, or 600 students. After the second hour, half of 600 remain, or 300 students. After the third hour, half of 300 remain, or 150 students. Finally, after the fourth hour, half of 150 remain, or 75 students. In summary, the pattern is: 1,200, 600, 300, 150, 75. There are 75 students left in the building four hours after the end of the regular school day.
Check your work. Start with the 75 students remaining after four hours and work back to the end of the school day: 75 × 2 = 150 after 3 hours; 150 × 2 = 300 after 2 hours; 300 × 2 = 600 after 1 hour; and 600 × 2 = 1,200 at the end of the school day.
Patterns can also be displayed in table format. The table may be horizontal or vertical, but both are solved in the same manner. Take the following table.
What is the missing value in the table?
Use the steps to solving word problems to work through the problem.
Read and understand the question. You are looking for the missing value in a table where most of the numbers are given.
Make a plan. Look for a pattern in the table to help you figure out the missing number. This pattern can be a horizontal (across) or vertical (up/down) pattern.
Carry out the plan. In this table, each of the values is filled in except for the location of the question mark, so look for a pattern with the other numbers. For each given x-value, the corresponding y-value appears to be three times the x-value. Be sure to check each row to make sure the pattern holds true for all numbers in the table. By using this pattern, the missing number is 3 × 5 = 15.
Check your answer. Another way to view this table is vertically. As the y-values increase, each number is three more than the previous one. By using this method, the missing number is 12 + 3 = 15, which is the same answer found by the other method.
Making an Organized List
Another strategy that can be used to solve many math word problems is making an organized list. This strategy is similar to looking for a pattern, but this time, you will make a list that will include all possibilities in a situation. Take a look at the problem below.
Richard has 3 different ties: blue, purple, and red, and four different dress shirts: green, yellow, white, and black. How many shirt-and-tie combinations can be made by selecting 1 shirt and 1 tie?
Use the problem-solving process to solve this question, along with the strategy of making an organized list.
Read and understand the question. The question is asking for the total number of possibilities when selecting 1 shirt and 1 tie. There are 3 ties and 4 shirts from which to choose.
Make up a plan. Make a list of all of the combinations of 1 shirt with 1 tie. Then count the total number of possibilities to find the answer.
Carry out the plan. Make an organized list pairing each tie with each shirt. An organized list could appear as follows:
blue—green purple—green red—green blue—yellow purple—yellow red—yellow blue—white purple—white red—white blue—black purple—black red—black
This is a total of 12 different pairs; therefore, there are 12 different possibilities.
Check your work. Since there are 3 different ties being paired with four different shirts, there should be 4 + 4 + 4 =12 possibilities. This is the same as the number of items in the organized list.
When making an organized list, you can also use abbreviations for the words in your list. For example, b—g would represent the blue tie with the green shirt. This may make it easier and faster to construct your list.
Find practice problems and solutions for these concepts at The Strategies of Looking for a Pattern and Making an Organized List Practice Questions.
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