Year 8 Maths Time and Travel

Sketching Informal Graphs

20 practice questions 1 video lesson Theory + worked examples

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Sketching Informal Graphs - Video - Distance Time Graphs Watch

Theory

An informal graph tells the story of how something changes. Up means increasing, flat means not changing, down means decreasing, and steeper means changing faster. A drone that drops $40$ m in $10$ s is coming down at $40 \div 10 = 4$ m/s. This Year 8 maths page shows how to read and sketch these graphs.

Read the axes first: what is measured up the side, what goes across (usually time), and the units.

Up means the amount is getting bigger, flat means it is not changing, down means it is getting smaller. Steep means changing quickly, gentle slowly. Compare sections by how steep they are, not by how high up they sit.

A straight section is a steady rate: rate $=$ change $\div$ time, in units such as cm/min or $^\circ$C per hour.

Sketching from a story. Split the story into sections; for each decide up, flat or down, then steep or gentle. Draw them in order, joined up, with both axes labelled. A sketch needs the right shape, not exact numbers.

Curves. A curve getting steeper means changing faster and faster; one getting flatter means changing more and more slowly. A vertical jump means a sudden change; repeating humps mean something going up and down over and over.

Filling a container at a steady rate: a wide part rises slowly, a narrow part quickly. One width all the way up gives a straight line. Three times as wide across means nine times the area, so the depth rises nine times as slowly.

Six shapes to know, with time across. The first three are straight (steady rates), the next two are curved (the rate is changing), and the last has a jump.

Up, flat, down: the drone climbs, hovers, then comes down. The last section is the steepest.

Reading a section:

Section	Means
up, steep	increasing quickly
up, gentle	increasing slowly
flat	no change (stopped, waiting, hovering)
down	decreasing
curve flattening	change slowing down
vertical jump	sudden change

          Flat does not mean back at the start. On a distance-from-home graph, flat means stopped. Only a line down to zero means back home. On a speed graph, flat means a steady speed.
        

          Steeper means faster, not bigger. A high line means a large amount; how fast it changes is how steep it is.
        

Reading an informal graph

Read both axes and their units.
Split the graph into sections and describe each: up, flat or down; steep or gentle.
Calculate a rate for a straight section: change $\div$ time.

Sketching from a story

List the parts of the story in order.
Decide the shape of each part.
Draw them joined up and label the axes.

Example 1 — A return trip

The graph shows Ben’s bike ride to a friend’s house and back.
(i) When did he start?
(ii) How far is the friend’s house?
(iii) What was the total distance?
(iv) Did he stop anywhere?
(v) How long was the whole trip?

Solution

$8 + 8$	$=$	$16$
$15 - 11$	$=$	$4$

11008 km 16 km 4 hours

(i) At 11:00
(ii) $8$ km (reached at 12:00 noon)
(iii) $16$ km there and back
(iv) Not at the friend’s house, but he stopped for an hour on the way back, $5$ km from home, from 13:00 to 14:00.
(v) $4$ hours (11:00 to 15:00)

Example 2 — Reading the sections

The graph shows a drone’s height.
(i) What is the drone doing in the flat section?
(ii) Which section is steepest?
(iii) How fast does it come down, in m/s?

Solution

$40 \div 10$

$=$

$4$

hovering the last 4 m / s

(i) Hovering at $40$ m from $20$ s to $50$ s
(ii) The last: $40$ m in $10$ s (the climb took $20$ s)
(iii) $4$ m/s

Example 3 — Sketching from a story

A frozen pie sits on the bench, then goes into a hot oven, then is left to cool. Sketch its temperature against time.

Solution

On the bench: up, gentle. In the oven: up, steep. Cooling: down, fast at first then slower, so a curve that flattens.

A gentle rise, then a steep rise, then a fall that flattens out.

Example 4 — Filling a container

Water flows at a steady rate into a vase made of two cylinders of the same height. The top cylinder is three times as wide as the bottom one. The bottom fills in $1$ minute. Describe the graph of depth against time.

Solution

Three times as wide means $3 \times 3 = 9$ times the area, so the top takes $1 \times 9 = 9$ minutes and rises a ninth as fast.

A steep straight line for $1$ minute, then a gentler straight line for $9$ minutes.

Example 5 — Rates and a prediction

The graph shows Zara’s savings.
(i) What happened from month $4$ to month $8$?
(ii) Find her saving rate in the first $4$ months.
(iii) She keeps up the month $10$ to $14$ rate. When does she first have $\$600$?

Solution

$400 \div 4$	$=$	$100$
$(400 - 200) \div (14 - 10)$	$=$	$50$
$200 \div 50$	$=$	$4$

no change $ 100 per month month 18

(i) Flat: no change
(ii) $\$100$ per month
(iii) $\$200$ more at $\$50$ a month takes $4$ months: month $18$

Common pitfalls

Reading height as speed. How high a line is shows the amount; how steep it is shows how fast it changes.

Thinking flat means back home. On a distance-from-home graph, flat means stopped.

Getting containers backwards. A narrow part fills quickly, so its graph is steep.

Forgetting to label the axes on a sketch. Every sketch needs both axes labelled.

Frequently asked questions

What does a flat section on a graph mean?

The amount is not changing during that time, such as a car that is stopped.

What does a steeper line mean?

The amount is changing faster.

How do I find a rate from a straight section?

Divide the change in the amount by the time taken.

How do I sketch a graph from a story?

Break the story into parts, decide if each goes up, stays flat or goes down, and how steeply, then draw them in order.

Why does a narrow container fill faster?

The same volume of water spreads over a smaller area, so the depth rises more quickly.

\(8 + 8\)	\(=\)	\(16\)
\(15 - 11\)	\(=\)	\(4\)

\(400 \div 4\)	\(=\)	\(100\)
\((400 - 200) \div (14 - 10)\)	\(=\)	\(50\)
\(200 \div 50\)	\(=\)	\(4\)

Sketching Informal Graphs

📖 Theory

Reading an informal graph

Sketching from a story

Common pitfalls

Frequently asked questions

Theory