Data is a collection of facts, such as observations, measurements and other values that can be graphed and interpreted on a coordinate plane. For example, the temperature values recorded for a week in New York forms a data set for analysis.
A coordinate plane is a system for plotting points that is represented by two axes that meet and form a right angle. The vertical line is called the y-axis, and the horizontal line is called the x-axis. The point of intersection is called the origin.
The data points on a coordinate plane are written as an ordered pair and are written as(x, y). The x-value of a coordinate pair is shown on the x-axis, while the y-value of the coordinate pair is shown on the y-axis. For example, (36, 12) are data points on a coordinate plane, where the point 36 will be plotted on the x-axis and the point 12 will be plotted on the y-axis.
Line graphs are those graphs that exhibit line segments to display how the values of the data change over time. Here is an example of a line segment.
Numerical patterns can be defined as a sequence of numbers in a proper series, in which the numbers have a common relationship between them. For example, 0, 5, 10, 15… is a numerical pattern where the pattern is identified by skip counting by 5.
In order to identify the pattern of any sequence of numbers, we have to find the rule for the pattern. This is because number patterns are based on rules. Pattern rules establish the relationship between the numbers that are in sequence by using mathematical operations.
Let us take an example of 5, 8, 11, 14… where this pattern has consistently increased by 3. So, if we find the difference between any of the two consecutive numbers, it will be 3. As a result, we can get the next term by adding 3 to the previous term. Let us take another example where we have a table of data of days and hours worked and the amount earned and from that we have to derive the rule as well as the pattern.
In order to complete the table, we have to find the rule for this number pattern of hours and the amount earned. First, we will use the rule to find the sequence in the number of hours. As we can see, there is a growth in the number of hours by 4. We can get the next term by adding 4 to the previous term.
So, 4, 8, 12, 16, 20, 24 is the number pattern and the rule is to add 4.
Now, getting to the rule for the amount earned. In this sequence, the growth in the amount earned is by 20. We can get the next term by adding 20 to the previous term. So, 20, 40, 60, 80, 100, 120 is the pattern and the rule is to add 20.
From this example, we see that a rule tells us how numbers in a pattern are related.
Numerical patterns can be graphed on a coordinate plane when we have the ordered pairs of coordinates. Let us take the example from the previous question where we have a table of data.
Here, we have three quantities. So first, you have to decide which are the two quantities you want to compare or use for analysis.
Let’s plot a graph between the hours worked and the amount earned. The x-axis can denote the hours and the y-axis the amount earned.
We can get the ordered pairs from the above table and plot them on the coordinate plane. So for the first point, it will be (4, 20), for the second point it will be (8, 40), and so on.
The ordered pairs will be (4, 20), (8, 40), (12, 60), (16, 80), (20, 100), and (24, 120). When we graph these points, it will look like this:
Here, the x-axis shows the number of hours worked, and the y-axis shows the amount earned. The ordered pairs form a line segment together.
Example 1. The following table shows the quantity of medicines a chemist sells each day of a week. Draw a graph of the data.
What does the point (3, 5) represent?
Solution: From the above table, we will first write down the ordered pairs. (1, 6), (2, 10), (3, 5), (4, 10), (5, 8)
Now, we have to plot these ordered pairs on a graph. Here we will label the axes by putting the number of days on the x-axis and medicines sold on the y-axis. We will provide a title for the graph as well.
We have plotted a point for each ordered pair, and there are five points on the grid lines.
From the graph, we can say that the point (3, 5) represents the number of medicines sold on the 3rd day. It also tells us that in these 5 days on the 3rd day the least number of medicines were sold which is 5.
Example 2. Martha measured the weight of her pet for 5 months and maintained a table for it. Prepare a line graph based on the weight of her pet, as mentioned in the table. Can you find the two months in which Martha’s pet gained the most weight?
Solution: We will first write down the ordered pairs from the table. (1, 10), (2, 15), (3, 25), (4, 30), (5, 40)
Now, we have to graph the data. Here we will label the axes by writing the age in months on the x-axis, and the weight on the y-axis. We will also write a title for the graph as well. We are going to plot the points for each ordered pair and then connect the points with a line segment.
The most weight gain for Martha’s pet happens between the points (2, 15)and (3, 25) and (4, 30) and (5, 40), where the pet has gained 10 pounds compared to the previous month. So, the pet gains the most weight in the third and fifth months.
Example 3. The following table shows the number of sums solved by Emily each day, and the reward she gets each day for solving them. Use the rule to complete the table.
Solution: We have to create a pattern for the data and complete the table. For the first pattern for sums solved, we will use the rule of adding 5. As you can see, the growth in the numbers is by 5. So, the table will be 5, 10, 15, 20, and 25.
In order to find the number of rewards, we will use the rule of multiplication of 2. In this pattern, the growth is by multiplying by 2. So, the table will be 2, 4, 8, 16, and 32.
The number patterns will look like this:
If there is a discontinuity between the values on either the x-axis or y-axis, then the number line will have a break.
The scale tells us what 1 unit of the x-axis and y-axis represents. Hence, it is very important to decide on a suitable scale before plotting the points.
For example, say you have data values of 2, 4, 6, 8, 10… and each division on the x-axis is marked as 3, 6, 9… here the scale is 1 unit of x-axis represents 3 units of the data. Now this is not a suitable scale to plot the data values we have. We need a scale on the x-axis that the progression from one unit to the next represents an increase by 2. Hence, selecting an appropriate scale of the graph is vital.