The Strategies of Making a Table and Drawing a Picture in Word Problems Study Guide (page 2)
Introduction to The Strategies of Making a Table and Drawing a Picture in Word Problems
Most people spend more time and energy going around problems than in trying to solve them.
—HENRY FORD (1863–1947)
At this point, you have seen that by using certain strategies, even difficult word problems can become easier to solve. The strategies presented in this lesson will focus on making a table and drawing a picture. Since "a picture is worth a thousand words," this seems like a great place to start.
Do not want any longer. Use your time and energy wisely, and rely on the strategies presented in this lesson to solve the word problems that lie ahead.
Making a Table
Many times, the data in a problem needs to be organized in order for you to find a potential relationship. One way to efficiently arrange this information is in a table format. This way, if there are certain patterns in the numbers, you will be more likely to see them. This method also allows you, the problem solver, to better organize larger groups of information, as seen in the following example.
At a carnival, it costs $2 to play each game and $3 for each ride ticket. Sandy spent $17 at the carnival. What is the largest number of ride tickets she could have purchased?
Read and understand the question. The question is seeking the largest number of ride tickets that could be bought when $17 was spent on both games and rides. Each game costs $2 and each ride costs $3.
Make a plan. Make a table to organize the information and to solve this problem. Have one row in the table for ride tickets and one row for game tickets. Then, use the table to find the total number of tickets bought when the total cost is $17.
Carry out the plan. Make the table. An example of the table is shown next.
To complete the table, divide $17 by 3 to find the largest number of ride tickets that could have been purchased: 17 ÷ 3 = approximately 5.6666. Put a 5 in the column for the number of ride tickets. The total money spent on ride tickets: 5 × $3 = $15. Since Sandy now has $17 – $15 = $2 left over, she must have spent this amount on games. Now, fill in the row for games using this leftover money. A possible completed table is shown next.
Use the data in the table to check to see if the amounts add to $17: $15 + $2 = $17.
Check your work. To check for the largest number of rides, try a larger number such as 6 for the number of rides. If Sandy bought 6 ride tickets, this would be $3 × 6 = $18, which is over the amount of money she spent.
When using the strategy of making a table, put the labels to the left in the table and make a row for each category. Then, you can use the table to work across and down in columns, depending on the question.
The question that follows involves a series of numbers in a pattern. Read through the problem and answer explanation to see how using a table can make this pattern clearer and the solution easier to find.
Karl betters his time swimming in an event by 3 seconds in each of the 5 races of the season. If his time for the first race of the season is 58 seconds, what is his best time at the end of the season?
Read and understand the question. This question is looking for Karl's best time at the end of the season. Each time he betters his time, subtract 3 seconds from the previous time.
Make a plan. Start with his time at the start of the season, and then subtract 3 seconds for each of the 5 races. At the end of the 5 races, this is his time at the end of the season.
Carry out the plan. Make a table that includes his five races of the season, and his time for the first race of 58 seconds. Fill out the table, subtracting 3 seconds for every subsequent race. Continue this for all five races. The table could look like the following one.
The time for race 5, the final race of the season, is 46 seconds.
Check your work. Work backward to check this problem. Begin with his best time of 46 seconds and add 3 seconds until you reach his time for race 1: 46 + 3 + 3 + 3 + 3 = 58 seconds. This was his time at the start of the season, so this answer is reasonable.
Drawing a Picture
The strategy of drawing a picture to solve a word problem may also shed light on patterns that may occur. It also gives a bird's-eye view to what is going on in certain problems. Take, for example, the following question. The method of solution will give a picture, or diagram, about the locations and distances used in the question that will make finding the correct answer much easier.
Joe leaves school and travels 3 miles directly west to his home. Later that night, he leaves home and goes north 2 miles to the public library. When he has finished at the library, he travels 5 miles east to his friend's house and then 2 miles south to his grandmother's house. If his grandmother lives 2 miles from his school, how far away is his grandmother's house from his house?
Read and understand the question. This question asks for the distance between Joe's house and his grandmother's house.
Make a plan. Joe's route after school is detailed in the problem. Draw a picture of the route Joe took after school. Use these details, along with knowledge of east, west, north, and south to find the distance between points.
Carry out the plan. Draw a picture of Joe's route. Recall that on a map, west is to the left of north, north is to the left of east, east is to the right of north, and south is to the right of west. Use these facts as you retrace the path. A possible picture of Joe's path is shown in the following figure.
His path is 3 miles west, then 2 miles north, 5 miles east, and 2 miles south. At this point, he is 2 miles directly east of school. Because he lives 3 miles east of school, his home is 2 + 3 = 5 miles from his grandmother's house. The distance between Joe's house and his grandmother's house is 5 miles.
Check your work. Compare the distances to check this problem. Joe went 2 miles north and 2 miles south, so these two distances cancel each other out. He went 3 miles west, and 5 miles east. This is a difference of 2 miles. However, since his trek did not begin at his house, add 3 miles between his home and school: 2 + 3 = 5 miles. This solution is reasonable.
When you are drawing a picture or diagram to solve a problem, try to label in the figure each of the details given in the question. That way, important information will not be left out when you are trying to find a solution.
Find practice problems and solutions for these concepts at The Strategies of Making a Table and Drawing a Picture in Word Problems Practice Questions.
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