1.16: Trends in Data
Suppose that everyday you take the same number of vitamins so that the number of vitamins remaining in the bottle always decreases by the same amount. If you know how many vitamins were in the bottle to begin with, do you think you can determine the number of vitamins in the bottle after a certain number of days? Would it be easier if you looked at the data in a table, with the number of days in one column and the number of vitamins remaining in the bottle in another column? In this Concept, you'll look for patterns in data so that you can solve problems such as this.
Guidance
Problem-Solving Strategies: Make a Table; Look for a Pattern
This lesson focuses on two of the strategies introduced previously: making a table and looking for a pattern. These are the most common strategies you have used before algebra. Let’s review the four-step problem-solving plan from a previous Concept.
Step 1: Understand the problem.
Step 2: Devise a plan – Translate. Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart, or construct a table as a start to begin your problem-solving plan.
Step 3: Carry out the plan – Solve.
Step 4: Check and Interpret: Check to see if you used all your information. Then look to see if the answer makes sense.
Using a Table to Solve a Problem
When a problem has data that needs to be organized, a table is a highly effective problem-solving strategy. A table is also helpful when the problem asks you to record a large amount of information. Patterns and numerical relationships are easier to see when data are organized in a table.
Example A
Josie takes up jogging. In the first week she jogs for 10 minutes per day, and in the second week she jogs for 12 minutes per day. Each week, she wants to increase her jogging time by 2 minutes per day. If she jogs six days per week each week, what will be her total jogging time in the sixth week?
Solution: Organize the information in a table
Week 1 | Week 2 | Week 3 | Week 4 |
---|---|---|---|
10 minutes | 12 minutes | 14 minutes | 16 minutes |
60 min/week | 72 min/week | 84 min/week | 96 min/week |
We can see the pattern that the number of minutes is increasing by 12 each week. Continuing this pattern, Josie will run 120 minutes in the sixth week.
Don’t forget to check the solution! The pattern starts at 60 and adds 12 each week after the first week. The equation to represent this situation is \begin{align*}t = 60 + 12(w - 1)\end{align*}. By substituting 6 for the variable of \begin{align*}w\end{align*}, the equation becomes \begin{align*}t = 60 + 12(6 - 1) = 60 + 60 = 120\end{align*}
Solve a Problem by Looking for a Pattern
Some situations have a readily apparent pattern, which means that the pattern is easy to see. In this case, you may not need to organize the information into a table. Instead, you can use the pattern to arrive at your solution.
Example B
You arrange tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 8 layers?
One layer: It is simple to see that a triangle with one layer has only one ball.
Two layers: For a triangle with two layers we add the balls from the top layer to the balls of the bottom layer. It is useful to make a sketch of the different layers in the triangle.
Three layers: We add the balls from the top triangle to the balls from the bottom layer.
We can fill the first three rows of the table.
\begin{align*}&1 && 2 && 3 && \quad \ \ 4\\ &1 && 3 && 6 && 6 + 4 = 10\end{align*}
To find the number of tennis balls in 8 layers, continue the pattern.
\begin{align*}& \qquad \ 5 && \qquad \ 6 && \qquad \ 7 && \qquad \ 8\\ &10+5=15 && 15+6=21 && 21+7=28 && 28+8=36\end{align*}
There will be 36 tennis balls in the 8 layers.
Check: Each layer of the triangle has one more ball than the previous one. In a triangle with 8 layers, each layer has the same number of balls as its position. When we add these we get:
\begin{align*}1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36\end{align*} balls
The answer checks out.
Comparing Alternative Approaches to Solving Problems
In this section, we will compare the methods of “Making a Table” and “Looking for a Pattern” by using each method in turn to solve a problem.
Example C
Andrew cashes a $180 check and wants the money in $10 and $20 bills. The bank teller gives him 12 bills. How many of each kind of bill does he receive?
Solution: Method 1: Making a Table
\begin{align*}&\text{Tens} && 0 && 2 && 4 && 6 && 8 && 10 && 12 && 14 && 16 && 18\\ &\text{Twenties} && 9 && 8 && 7 && 6 && 5 && 4 && 3 && 2 && 1 && 0\end{align*}
The combination that has a sum of 12 is six $10 bills and six $20 bills.
Method 2: Using a Pattern
The pattern is that for every pair of $10 bills, the number of $20 bills is reduced by one. Begin with the most number of $20 bills. For every $20 bill lost, add two $10 bills.
\begin{align*}6(\$10) + 6(\$20) = \$180\end{align*}
Check: Six $10 bills and six $20 bills \begin{align*}=6(\$10) + 6(\$20) = \$60 + \$120 = \$180\end{align*}.
Video Review
Guided Practice
Students are going to march in a homecoming parade. There will be one kindergartener, two first-graders, three second-graders, and so on through \begin{align*}12^{th}\end{align*} grade. How many students will be walking in the homecoming parade?
Could you make a table? Absolutely. Could you look for a pattern? Absolutely.
Solution 1: Make a table:
\begin{align*}&& K && 1 && 2 && 3 && 4 && 5 && 6 && 7 && 8 && 9 && 10 && 11 && 12\\ &&1 && 2 && 3 && 4 && 5 && 6 && 7 && 8 && 9 && 10 && 11 && 12 && 13\end{align*}
The solution is the sum of all the numbers, 91. There will be 91 students walking in the homecoming parade.
Solution 2: Look for a pattern.
The pattern is that the number of students is one more than their grade level. Therefore, the solution is the sum of the numbers from 1 (kindergarten) through 13 (\begin{align*}12^{th}\end{align*} grade). The solution is 91.
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Word Problem-Solving Strategies (12:51)
- In Example A, what will be Josie's total jogging time in the eighth week?
- Britt has $2.25 in nickels and dimes. If she has 40 coins in total how many of each coin does she have?
- A pattern of squares is placed together as shown. How many squares are in the \begin{align*}12^{th}\end{align*} diagram?
- Oswald is trying to cut down on drinking coffee. His goal is to cut down to 6 cups per week. If he starts with 24 cups the first week, cuts down to 21 cups the second week, and drops to 18 cups the third week, how many weeks will it take him to reach his goal?
- Taylor checked out a book from the library and it is now 5 days late. The late fee is 10 cents per day. How much is the fine?
- How many hours will a car traveling at 75 miles per hour take to catch up to a car traveling at 55 miles per hour if the slower car starts two hours before the faster car?
- Grace starts biking at 12 miles per hour. One hour later, Dan starts biking at 15 miles per hour, following the same route. How long would it take him to catch up with Grace?
- Lemuel wants to enclose a rectangular plot of land with a fence. He has 24 feet of fencing. What is the largest possible area that he could enclose with the fence?
Mixed Review
- Determine if the relation is a function: \begin{align*}\left \{(2,6),(-9,0),(7,7),(3,5),(5,3) \right \}.\end{align*}
- Roy works construction during the summer and earns $78 per job. Create a table relating the number of jobs he could work, \begin{align*}j\end{align*}, and the total amount of money he can earn, \begin{align*}m\end{align*}.
- Graph the following order pairs: (4,4); (–5,6), (–1,–1), (–7,–9), (2,–5).
- Evaluate the following expression: \begin{align*}-4(4z - x + 5)\end{align*}; use \begin{align*}x = -10\end{align*} and \begin{align*}z = -8.\end{align*}
- The area of a circle is given by the formula \begin{align*}A = \pi r^2\end{align*}. Determine the area of a circle with radius 6 mm.
- Louie bought 9 packs of gum at $1.19 each. How much money did he spend?
- Write the following without the multiplication symbol: \begin{align*}16 \times \frac{1}{8} c.\end{align*}
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Here you'll learn how to answer questions about data by looking for trends in the data and by viewing the data in tabular form.