1. Making Sense of Graphs That Tell A Story
Grade 10 Mathematics Literacy

2. What is a graph?
A graph is a mathematical picture of the relationship between two variables such as temperature and the day of the week.
The benefit of graphs is that you can see and understand the whole picture at a glance.

3. Graphs going up and going down (increasing and decreasing)

4. Looking at a graph in a newspaper
Tessa saw the graph in a financial magazine – What information can Tessa extract from this graph?
It shows the inflation rate for each year for seven years.
Looking at the line of the graph, you can see that inflation goes up in general.
The graph is steepest in the last year. Therefore inflation increased the most in the 7th year.

5. What is Inflation? The increase in the price of goods in a country.
If a graph is increasing, the slope goes up from left to right.
If a graph is decreasing, the slope goes down from left to right.

6. How can you tell if one line is steeper than another line
How can you tell if one line is steeper than another line? You can see the difference by looking at the slope or gradient:
A steeper graph shows a quicker change.
A gradual slope shows a slower change.

7. Another example…

8. Looking at the graph of temperature over one day, answer the following questions:
1. At what time in the day is the temperature the lowest?
2. When is the temperature the highest?
3. Is there any time period during the day that the temperature stays the same?
4. Between what hours in the day do you see the biggest increase in temperature?
5. Is there a decrease in temperature? If so, what time in the day and explain.

9. 1. At 8h00, as the graph is at the lowest point.
2. At 14h00, as the graph is at the highest point.
3. Yes, between 18h00 and 20h00. We know this because the line of the graph remains constant during these 2 time periods.
4. The biggest increase in temperature takes place between 10h00 and 14h00.
5. There is a decrease in temperature from 14h00 to 18h00. We know this because the graph goes down the highest point.

10. Continuous and Discrete graphs
Some types of values can only be whole numbers, while others, like measurements, can have decimal fraction values.
Whole numbers must be shown by points on a graph, connected by dotted lines.
These kinds of graphs are called discrete.
Values which are continuous, such as length, should be connected by solid lines, to show that the values in between the points are included too.

11. Looking at the graph below, it shows the number of passengers on a bus for six different trips.

12. And this graph shows the distance that a bus travels for one trip.

13. Why does the first graph have dotted lines and the second graph have a solid line?
The first graph has whole numbers (discrete variables). The number of passengers and the number of trips can only be whole numbers, i.e. there can't be half a passenger on the bus.
The second graph shows continuous variables such as measurement values. For example, 50.5km. The solid line shows that all of the points along the graph are included in one relationship. Any measurement of time and distance would be valid, because the bus trip took place over a continuous number of minutes, and the bus drove all the way, along a continuous distance.

14. Interpreting Graphs…
Graph 1: Chantal and Sarah went on a training run and drew this graph to show their progress:

15. One part of the graph is steeper than the others. Identify this part.
What was the total distance of the training run and how many hours did it take?
Give the times when Chantal and Sarah were resting (where the distance stayed constant).
One part of the graph is steeper than the others. Identify this part.
GRAPH 1
1. Total distance is approximately 5.5km and total time is 8 hours, 30 minutes
2. Between 07h45 and 08h00
3. From 07h00 to 07h15

16. Graph 2: Daniel's car takes 45 litres of petrol
Graph 2: Daniel's car takes 45 litres of petrol. The graph below shows the amount of petrol in the tank over one week.

17. Is there any time when his petrol tank is completely empty
Is there any time when his petrol tank is completely empty? How do you know?
Daniel was sick for two days during the week and stayed at home. Which days was he sick? Explain your answer.
How many times does he fill up his car with petrol? Where do you see this on the graph?
GRAPH 2
1. No. At no point does the graph touch the horizontal axis
2. Tuesday and Wednesday - his petrol consumption did not change at all, this suggests he did not use his car, and was therefore at home.
3. Once. Tuesday shows a sudden spike in the amount of petrol in the tank.

18. Graph 3: This graph shows the temperature in Grahamstown, measured over one week in September.

19. Using the graph on the previous slide, answer the following questions:
Is this graph continuous or discrete? Explain.
What was the highest temperature recorded during the week? On what day was this?
What was the lowest temperature recorded during the week? On what day was this?
Write down the maximum and minimum temperatures on Wednesday. Calculate the difference between them.
1. Continuous - there are no gaps in the graph, temperature is measured all day, from Friday to Thursday.
2. 30°C, on Wednesday
3. Approximately −2°C, on Sunday.
4. Minimum temperature is approximately 7°C maximum is approximately 30°C. 30°C - 7°C = 23°C difference.

20. Dependant and Independent Variables:
Variable: The quantity you are measuring or calculating.
Independent Variable:
a variable that stands alone and isn't changed by the other variables you are trying to measure.
Dependant Variable:
depends on other factors.
INDEPENDENT VARIABLE
For example, someone's age might be an independent variable. Factors such as what they study, how they get to school, and how many books they have read are not going to change a person's age.
DEPENDENT VARIABLE
For example, your marathon time could be a dependent variable because it could change depending on several factors such as how much you trained, whether you ate the right nutrition, how much sleep you got the night before you ran the marathon, or whether you experienced any injuries.

21. An easy way to remember which is the dependent variable and which is the independent variable is to put the names of the two variables you are using in a sentence in a way that makes the most sense. Then you can see which is the independent variable and which is the dependent variable.
When we plot graphs of variables, we usually put the independent variable on the horizontal axis and the dependent variable on the vertical axis.

22. Touching the axes
What does it mean when a graph touches the horizontal axis or the vertical axis?
If the graph touches the vertical or "y  "-axis, it means that the quantity on the horizontal axis has reached 0.
If a graph touches the horizontal or "x  "-axis, it means that the quantity on the vertical axis has reached 0.
Another name for the horizontal axis is the x  -axis. We plot the independent variable in a relationship on this axis.
Another name for the vertical axis is the y  -axis. We plot the dependent variable in a relationship on this axis.

23. Interpreting a graph that touches the vertical axis
Jane buys chocolate as a present for her dad. Look at this graph of the price of chocolate per weight.

24. Explain which is the independent and which is the dependent variable.
What is the price when the weight of the chocolate is 0 kg? Explain your answer.
Explain which is the independent and which is the dependent variable.
1. The price of buying the chocolate at 0 kg is R 10, this is where the graph touches the vertical axis. This means that there is a starting cost of R 10, which is constant, no matter how much chocolate is bought.
2. The cost depends on the weight bought, so weight is independent and cost is dependent.

25. Interpreting a graph that touches the horizontal axis
Mark empties his 500 ml water bottle at a constant rate.
Describe what you see in the graph below.
Explain which is the independent and which is the dependent variable.

26. The volume of the water starts at 500 ml
The volume of the water starts at 500 ml. It decreases steadily as Mark empties it. At 5 minutes, it reaches 0 ml. This means it has taken 5 minutes to empty the bottle completely.
The volume changes with time, so time is the independent variable, and volume is dependent.

27. One last problem to work on…
Reading graphs: Helen has a long walk to school today and takes a one litre bottle of water with her. Look at this graph carefully and then answer the questions on the next slide.

28. 1. What are the two variables plotted on this graph?
2. Which variable is dependent and which is independent? Explain fully.
3. What happens to the amount of water in the bottle during the first two hours?
4. What happens at hour number 5? Explain.
5. Between which two hours does Helen drink her water the fastest?
6. Is there a point at which she finishes all the water in her bottle? How do you know this?

29. 1. Time is on the horizontal axis, and the volume of water in Helen’s bottle is on the vertical axis.
2. The volume of water is dependent on time, the independent variable.
3. It remains at the same level.
4. The amount of water in the bottle increases suddenly which suggests that Helen refilled her water bottle.
5. Between the hours of 8 and 10.
6. No. At no point does the graph touch the horizontal axis - i.e. at no point is the volume of water in the bottle 0 ml.