Topic 3: Graphs

By the end of this topic, you will be able to:
(a) tabulate values from the given relations.
(b) plot and draw lines through the given points.
(c) choose and use appropriate scales.
(d) draw, read and interpret the graph (e.g. distance-time and speed-time graphs to estimate distance, speed and time).

Introduction

Graphs provide an effective way to gain a real understanding of data. Graphing in mathematics helps learners develop deeper analytical skills towards problem solving, especially where a decision is to be made. Graphs are mathematical structures used to model pair wise relations between objects from a certain collection. Graphs are a means of summarising data so that the results may be easily understood on a fine grid graph paper.
This topic will enhance the knowledge of plotting, interpreting and using graphs to solve problems.

3.1 Tabulating values from relations

In this section, you will develop an understanding of patterns and relationships, which you will use to represent and explain real-world phenomena. A table of values can guide in finding the maximum and minimum points so as to decide on the scale to use in drawing the axes. Points on the graph can be plotted using the values in the table. Therefore, it is important that a table is drawn before the drawing of the graphs.

Activity 3.1 Drawing table of values
The table below shows a set of values based on a relation. Copy and complete the table below.

1. Explain how the values in the second column of the table were obtained.
2. Is it possible to obtain a set of values without a relation? Give reasons for your response.
3. What relation have you used to complete the table?
4. Explain how you used the relation to fill in each missing value
   in the table.

Exercise 3.1

1. Study the table below and derive the relationship that is used in each of the tables.
2. Find the answers when you multiply a two-digit number by 10, 100 or 1000.
   • (a) Develop a “rule” that you think will help you perform this type of multiplication.
   • (b) Test your rule on some new problems and check whether your rule works by multiplying the numbers on the calculator.
3. Okellowange had two job offers with the following salary scales. She needed to decide which is a better deal. One job offered a salary of UGX 1,250,000 per week or UGX 30,000 per hour. Create a table comparing the pay for different numbers of hours worked and decide at what point the hourly rate becomes a better deal.
4. An electrician charges a basic fee of UGX 250,000 plus UGX 180,000 for each hour of work. Create a table that shows the amount the electrician charges for 1, 2, 3 and 4 hours of work

3.2 Plotting and drawing lines through given points

In this section, you will be able to use a relationship to plot and draw a graph.
Activity 3.2 Investigating a relation between quantities
Identify at least any 3 shapes around you that is in square form. Use them to carry out the following activity.

1. Measure the length of one side of each square in cm.
2. Find the perimeter of square one, square two, square three, and so on.
3. Describe a rule which relates the perimeter to the lengths of squares.
4. Make a table of values.
5. Predict the perimeter of a square of length 10 units.
6. Explain how you were able to predict the perimeter of the square in (5).
7. Plot and draw a graph that relates the perimeter to the length of the side of squares.
8. Describe the kind of graph you have drawn. Give reasons for the type of graph you have drawn.

EXAMPLE

The table below shows the results obtained by a group of senior two learners when carrying out an experiment to determine the boiling point of water.

Using a suitable scale, plot a graph of temperature against time.

At what temperature was the water after heating for 23 minutes?

Using the graph, determine the time when the temperature was 90 degrees

Exercise 3.2
1. The cost of a kilogram of sugar is UGX 3,000.
(a) Draw a table to reflect the cost of sugar up to 8 kg.
(b) Plot the points and join them.
2. Make a table of values that shows each of the following relations. In each case, plot the tabulated values and join the points. What type of graph do you observe in each case?
(a) y=2x+1
(b) y= 2x + 3
(c) y= 61-1

3. (a) Draw a table to represent the motion of a car moving at a constant speed of 60 kmh for 4 hours. Use the table of values to plot and draw the graph.
(b) Use the graph to predict the speed when the time is
(i) 18 hours.
(ii) 24 hours.

3.3 Choosing an appropriate scale in graph drawing

This section focuses on the details of constructing good graphs, and how to make appropriate choices of scales when drawing them on a graph paper.

Activity 3.3 Choosing an appropriate scale in graph drawing
Look at the grid below. Study it and use it to answer the questions that follow.

1. What can you observe about the size of the different strips?
2. How many yellow strips are in
   (a) the blue strip?
   (b)he orange strip?
   (c) the brown strip?

Activity 3.4 Finding out more about choosing scales in graph drawing.
Three learners A, B and C were given the following set of data to plot on a graph paper.

Each of the learners A, B and C had a graph paper of the same size.

The learners came up with the following graphs.

1. Which of the learners has the best graph in terms of coverage of the graph paper?
2. Explain how you were able to select the graphs with a bigger coverage.
3. Why is the graph of learner C different and yet the same coordinates have been used by learners A and B?
4. What can you notice about the distances apart from one coordinate to another along the x-axis and y-axis?
5. Which one of them is not uniform and why?
6. How would you help learner C to have a graph that is drawn to scale?

Learning points
1. Scales should be chosen in such a way that data is easy to plot and easy to read.
2. Scales should be consistent.
Research corner
Study the environment around you. Select a situation where scale is
applied.
(a)
Write a report on why it is important to use scale in the area you identified.

(b) What would happen if scale is not used in the area you have
identified?

Exercise 3.3
1. Study the diagrams below and use them to answer the questions
that follow.

Which of the following statements is/are true or false?
(a) A is drawn on using an appropriate scale.
(b) B and C are not drawn using an appropriate scale because
they are large.
(c) A and B are drawn using an appropriate scale.
(d) C is not drawn using an appropriate scale.

2. Prossy had the following data which she wanted to plot on a graph:
4, 26, 45, 68, 145, 300, 800, 1300
How would she choose the right scale?
3. The data shown in the table below are the circumferences and diameters of various circular objects. All measurements are in centimetres.
Diameter (cm) 4.4,6.6, 8.6, 10.3,23.0
Circumference (cm) 14.1, 20.9, 27.0, 32.3, 72.8
(a) Construct a graph to show this data, with the diameter on the horizontal (x) axis and the circumference on the vertical (y) axis.
(b) What scale have you used to draw the graph? Explain why you chose to use that scale.
(c) Draw a straight line through the points.

DISTANCE TIME GRAPHS

1. The table below shows the distance covered and the time taken by a given bus moving from Kampala to Malaba.

Plot the points on a graph paper. Draw an appropriate line through the plotted points and from your graph, calculate

1. The speed of the bus between
2.  Mukono and Lugazi
3. Mukono and Jinja
4. Iganga and Malaba
5. The distance covered after exactly
6.  hours
7.  hours

Example:

At 8:00 am, a car is moving at 40km/h and a bus 30 km behind it moving in the same direction at 60km/h.

On the same graph paper draw, draw the travel graph of the car and the bus.

Using the graph,

1. find the distance and the time at which the bus overtakes the car.
2. How far apart the bus and the car are at 11:00am.

(

Excercise 3.4

Jamila is going for a bike ride. Below is a distance-time graph that describes her full journey.

a) How long was she stationary?
b) What was the total distance travelled during her journey?

c)What was her average speed in kilometres per hour between 17:15 and 17:45?

2. Use the graph below to answer the questions that follow.

(a) What is the speed of the object between 25 and 60 seconds in the graph below?
(b) Explain the movements of the object at A and C.
(c) In which section is the object travelling fastest?
(d) In which section has the object stopped?
(e) From the description of Nelson’s journey below, construct a distance-time graph.
Nelson left home at 12:00 and after an hour and a half of moving at a constant speed, he had travelled 44 km. After 3 hours of not moving, he drove towards home at a constant speed and it took him 2 hours in total to get home.

3. Below is a distance-time graph describing a 1500 m race run by Cheptegei. Work out the maximum speed he reached during this run.

Field task
Materials Required
Task
Digital stop clock, a surveyor’s tape measure, pegs, oil/paint and field marks.
Measure off 100 m distance of a flat ground.
Select 5 students to run the distance involved in the 100 m race. Collect the data on the time each of the learners A, B, C, D and E akes after every 20-metre interval.

1. Tabulate the data.
2. Write down a relationship relating the distance and time.
3. Represent the findings on a graph (you may use ICT skills to represent the graphs).
4. How would you relate the slope of each graph with speed of each of the learners A, B, C, D and E?
5. Prepare a presentation of your work either by PowerPoint or otherwise.

ICT Section
Visit the website http://phet.colorado.edu/en/simulation/ moving-man which provides a downloadable simulation activity showing a distance-time graph for a person standing still, moving away from, and moving towards an observer
Write a report on what you learn from the website.

3.5 Drawing, reading and interpreting the speed- time graph

Activity 3.6 Interpreting the speed-time graphs
You will be travelling a designated distance by running, walking and choosing a third way to travel such as skipping, walking and running. Your partner will time you for each trial. You will keep track of your times in a table so that you can create a speed-time graph to represent your three trials on one coordinate plane. You will then be able to use your graphs to understand how your speed differed during each trial. Maybe you can predict how your speed will compare for each of the trials already. Can you predict how the graphs of each of your trials will compare?

1. Use the same distance for all the activities: running, walking and skipping.
2. Choose one group member to do all the three activities as the rest of the members take records.

Task
Record the data for your three trials in the table, then create a speed-time graph for each trial.

You should graph one line for each trial using the origin (0,0) as one point and your speed-time data (seconds, metres/second) as your second point. The result will be one coordinate plane with three lines beginning at the origin. Be sure to label the axes and graphs with titles.

(a) Use your speed-time graphs to compare the slopes for each of your trials and describe your observation
(b) Describe what the slope represents in this activity.

(c) If you looked at all the speed-time graphs that your classmates created, what would you look for to find out who is a very fast runner?

Exercise 3.5
1. Study the graph below and use it to answer the following questions.

(a) What is the speed of the object between 10 and 20 seconds
in the graph?
(b) Explain the movements of the object at A and C.

(c) In which section is the object travelling fastest?

(d) In which section has the object stopped travelling?

(e) What is the distance covered from the starting point to the destination?
(f) Explain what the gradients of A, C and D mean.

2. A ball is placed at rest at the top of a hill. It travels with constant acceleration for the first 12 seconds and reaches a speed of 4 m/s. It then decelerates at a constant rate of 0.1 m/s2 for 20 seconds. It then travels at a constant speed for a further 18 seconds. Draw a speed-time graph for the ball over the course of this 50 seconds.

3. Below is a speed-time graph of a cyclist during a race. Work out the total distance travelled by the cyclist over the course of the race.