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Reference: Alternate Angles.
Collecting data is the gathering together and recording of information, which in many cases is numerical, in order to analyze it, draw conclusions, and use it to help make decisions or predictions. Data are collected in response to a problem, and the method of collecting the data depends on the type of problem.
Example. The highway going past Washington High S chool is busy all day with cars and trucks. At the end of the school day, students leaving the campus on motorcycles, in cars, or as pedestrians have to j oin the traffic or cross the road. There is a possibility
that a student may be injured, despite being constantly reminded by teachers to be vigilant. The school’s Parent Teachers Association has decided that something must be done. But what?
Solution. First, how bad is the situation? The traffic numbers should be counted, at the appropriate times of day, to establish just how great the volume of traffic is. The problem may be no worse than in other areas of the city, except that this is outside a school.
Second, the opinion of parents and members of the public should be sought to find their reactions to the following suggestions:
♦ Pedestrian crossing
♦ Foot bridge over the road
♦ A tunnel under the road
♦ Speed restriction for vehicles
♦ Speed bumps at various intervals along the road to slow the traffic
♦ An entrance lane joining the school gate to the highway
♦ Reroute traffic away from the school
♦ Traffic lights at the end of the school drive
Third, ask the public and/or students for alternative views and suggestions. These data are collected in the form of a survey. This may take the form of
♦ An observation: traffic count at the side of the road outside the school gates
♦ A questionnaire: the public’s reaction to various options
♦ An interview: to gain alternative views
References: Data, Questionnaire, Survey.
A set of points is collinear if all the points lie on the same straight line. A straight line can always be drawn through two points, but for three or more points this is not necessarily so.
Example 1. Prove that the three points A(l, —2), J5(4,4), and C(6, 8) are collinear.
Solution. The points A, B, and C are plotted on axes, and squares are counted to find gradients (see figure a). Write
Gradient of BC =----------
_ 4 “ 2
Gradient of AC =----------
_ 10 - y
The line segments BC and AC are parallel, because they have equal gradients. Also, they have a common point C, which makes the points A, B, and C collinear.
Word similar to collinear is concurrent, which means that two or more straight lines or curves all pass through the same point. In other words, all the lines intersect at a point. Collinear and concurrent are two of the four options for a set of straight lines:
1. They are parallel.
2. They are concurrent.
3. They are skew, which means not parallel and not concurrent.
4. They are a combination of some of these three.
84 COLUMN GRAPH
Example 2. Figure b is a sketch of Jacob’s room. Find one example for each of the following types of lines:
1. Three parallel lines
2. Three concurrent lines
3. Two skew lines
1. AB, DC, and GH are parallel lines.
2. DC, HC, and BC are concurrent lines, and their point of concurrence is point C.
3. CD and FH are skew lines, because they are neither parallel nor concurrent.
References: Concurrent, Gradient, Parallel, Simultaneous Equations.
Reference: Four-color problem.
A column describes a vertical or upright line of numbers, or terms, in a table or vector. A row is at right angles to a column.
References: Table of Values, Vector.
This is the same as a bar graph, except that the bars are drawn vertically as columns, whereas the bars on a bar graph can be drawn either vertically or horizontally.
Reference: Bar graph.
Another name for combinations is selections, which is really a more appropriate word to describe the process. A selection is the number of ways a set of r elements can be chosen from a set of n elements when the order of the choice of the elements is not taken into account. This difficult definition is best explained with an example.
Example 1. Ann asked for volunteers for her debating team at Plato High School. There are 3 students needed to form a debating team, and she had 5 volunteers. In how many different ways can the team of 3 be selected from 5 students?
Solution. Suppose the 5 volunteers were A, B, C, D, and E. The selections, or combinations, of 3 students from the 5 volunteers can be written down in the following orderly way:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
This makes 10 combinations in total, so there are 10 ways of selecting 3 people from 5. This combination of selecting 3 people from 5 people is written as 5C3 = 10. This value of 5C3 can be worked out to give the answer 10, without writing down all the combinations, using a scientific calculator. An alternative way of writing the combination 5C3 is (3), both notations are in common use. Another term that is sometimes confused with combinations is permutations, which are often called arrangements. An arrangement is the number of ways a set of r elements can be chosen from a set of n elements when the order of the choice of the elements is taken into account. Suppose we extend the example of the debating team further with the following example.