070. Line Graphs and Travel Graphs

Learning Intentions

• To understand that a line graph can be used to display continuous numerical data
• draw a line graph
• use a line graph to estimate values
• interpret a travel graph

Pre-requisite Summary

• Understand that numerical data can be discrete or continuous
• Know that continuous data can take any value within an interval
• Be able to read scales on axes accurately
• Know that graphs need a clear title and labelled axes
• Understand that points on a graph represent pairs of related values
• Be able to join plotted points in order when showing continuous change
• Understand that estimating means reading an approximate value between marked points
• Know that a travel graph usually shows distance against time

Worked Examples

Worked Example 1

a) Explain why a line graph is suitable for continuous numerical data.
b) State one example of continuous data.
c) State one example of data that would not usually be shown on a line graph.

Worked Example 2

The temperature at different times is recorded:

a) Draw a line graph for the data.
b) Label the axes and give the graph a title.
c) Join the points correctly.

Worked Example 3

Use the line graph from Worked Example 2.
a) Estimate the temperature at $3$ h.
b) Estimate the temperature at $7$ h.
c) Explain how the graph is used to make each estimate.

Worked Example 4

A travel graph shows distance from home against time.
a) Explain what the horizontal axis represents.
b) Explain what the vertical axis represents.
c) Explain what it means when the graph goes upward.

Worked Example 5

A travel graph shows:

• from $0$ to $1$ h, distance increases from $0$ km to $40$ km
• from $1$ to $2$ h, distance stays at $40$ km
• from $2$ to $3$ h, distance increases from $40$ km to $70$ km
  a) Describe the journey in words.
  b) State when the traveller stopped.
  c) State the total distance from home after $3$ h.

Worked Example 6

Use a travel graph to interpret:
a) when the traveller was moving away from the start point
b) when the traveller was stationary
c) estimate the distance from home at a given time between plotted points

Problems

Problem 1

a) Explain why a line graph is suitable for continuous numerical data.
b) State one example of continuous data.
c) State one example of data that would not usually be shown on a line graph.

Problem 2

The height of a plant at different times is recorded:

a) Draw a line graph for the data.
b) Label the axes and give the graph a title.
c) Join the points correctly.

Problem 3

Use the line graph from Problem 2.
a) Estimate the height at $1.5$ weeks.
b) Estimate the height at $2.5$ weeks.
c) Explain how the graph is used to make each estimate.

Problem 4

A travel graph shows distance from school against time.
a) Explain what the horizontal axis represents.
b) Explain what the vertical axis represents.
c) Explain what it means when the graph is horizontal.

Problem 5

A travel graph shows:

• from $0$ to $2$ h, distance increases from $0$ km to $60$ km
• from $2$ to $3$ h, distance stays at $60$ km
• from $3$ to $5$ h, distance increases from $60$ km to $100$ km
  a) Describe the journey in words.
  b) State when the traveller stopped.
  c) State the total distance from the start point after $5$ h.

Problem 6

Use a travel graph to interpret:
a) when the traveller was moving away from the start point
b) when the traveller was stationary
c) estimate the distance from the start point at a time between plotted points

Exercises

Understanding and Fluency

1. State whether each type of data is suitable for a line graph:
   a) daily temperature
   b) favourite fruit
   c) distance travelled over time

2. State whether each type of data is continuous or not:
   a) height of a plant
   b) number of students in a class
   c) water level in a tank

3. Draw a line graph for the data:

   $$\begin{array}{cc}\text{Hour}& \text{Temperature }{(}^{\circ }\text{C})\\ 0& 16\\ 1& 18\\ 2& 21\\ 3& 20\\ 4& 19\end{array}$$

4. Draw a line graph for the data:

   $$\begin{array}{cc}\text{Day}& \text{Rainfall (mm)}\\ 1& 2\\ 2& 5\\ 3& 4\\ 4& 7\\ 5& 6\end{array}$$

5. Use a line graph to estimate:
   a) a value halfway between two plotted points
   b) a value at a time between two marked times
   c) the highest value shown on the graph

6. Use a line graph to interpret:
   a) when the data is increasing
   b) when the data is decreasing
   c) when the value stays the same

7. A travel graph shows distance from home against time. State what each means:
   a) an upward sloping line
   b) a horizontal line
   c) a steeper upward sloping line

8. A travel graph shows:

   • at $0$ h: $0$ km
   • at $1$ h: $30$ km
   • at $2$ h: $30$ km
   • at $3$ h: $50$ km
     a) Draw the graph.
     b) State when the traveller was stationary.
     c) State the distance from home at $3$ h.

Reasoning

9. Explain why favourite colour is not usually displayed on a line graph.

10. A student plots the points correctly on a line graph but does not label the axes. Explain why this is a problem.

11. Explain why values between plotted points on a line graph can often be estimated.

12. A student says that a horizontal section on a travel graph means the traveller is moving slowly. Explain why this is incorrect.

Problem-solving

13. The water level in a tank is recorded every hour as $12,15,18,17,14$ litres.
    a) Draw a line graph.
    b) State when the level was highest.
    c) Estimate the level halfway between the second and third hour.

14. The temperature in a room is measured every $2$ hours as $20,22,25,23,{21}^{\circ }\text{C}$.
    a) Draw a line graph.
    b) Estimate the temperature at the time halfway between $2$ h and $4$ h.
    c) State when the temperature began to fall.

15. A travel graph shows a cyclist $20$ km from home after $1$ h, still $20$ km from home after $2$ h, and $50$ km from home after $4$ h.
    a) Describe the journey.
    b) State when the cyclist stopped.
    c) Estimate the distance from home after $3$ h.

16. A bus journey is shown on a travel graph. The bus is at $0$ km at $0$ h, $40$ km at $1$ h, $40$ km at $1.5$ h, and $70$ km at $2.5$ h.
    a) Draw the travel graph.
    b) Describe the bus journey.
    c) Estimate the distance from the start after $2$ h.

17. The height of a candle is measured as it burns: $18$ cm, $16$ cm, $14$ cm, $11$ cm, $9$ cm.
    a) Draw a line graph.
    b) Explain what the downward trend shows.
    c) Estimate the candle height halfway between the second and third measurements.

18. A runner’s distance from the starting line is shown over time. At $0$ min the distance is $0$ m, at $2$ min it is $300$ m, at $4$ min it is $300$ m, and at $6$ min it is $700$ m.
    a) Draw the travel graph.
    b) State when the runner was stationary.
    c) Estimate the distance from the starting line at $5$ min.

Potential Misunderstandings

• Students may use a line graph for categorical data instead of continuous numerical data
• Students may forget to include a title or axis labels
• Students may use an uneven or incorrect scale on one or both axes
• Students may plot points correctly but join them in the wrong order
• Students may not realise that estimates from a line graph are approximate rather than exact
• Students may think any joined graph must represent a journey, even when the graph is showing another type of continuous data
• Students may interpret the steepness of a travel graph incorrectly without referring to what the axes represent
• Students may think a horizontal section on a travel graph means slow movement, when it actually means no change in distance from the start

Next: 071. Stem-and-Leaf Plots