# Average speed for two part journeys

Examiner reports frequently show only the most able students achieve full marks on questions about average speed for two part journeys. This is strange when the same reports show the majority of students often achieve full marks for single stage journeys.

When attempting questions on speed, distance and time for two part journeys students often fail to, add different units of time correctly, use consistent units of speed or separate the journey into two or more parts clearly enough to identify the key information. Click here to watch the video.

## Average speed for two part journeys

I created this lesson to help students break down the information from a multiple stage journey into an easy to understand diagram.  This way, they will be able to focus on the mathematics rather than trying to comprehend the problem.

## Scaffolding the types of questions

At the start of the lesson the questions include the timeline as part of the information given to the students.  As the learning progresses the time line is removed and replaced with worded descriptions.  Later, questions require students to model the journey using algebraic notation to set up and solve linear equations.

## Using breakout rooms to identify misconceptions

I first taught this lesson using Microsoft Teams during lockdown with students and I working remotely from home.  I encouraged students to take part using either the chat facility or by having their microphones on.  We also used breakout rooms when multiple had a problem with the same question.

Here is a sample of three questions that are included in the lesson and the misconceptions that arose through the discussion in breakout rooms.

## Average speed for two part journeys with a time line

Misconceptions

• Some students failed to realise the significance of the different speeds and the need to calculate the time for the individual stages.
• When working out the time for the first stage of the journey some students correctly calculated 12 mi divided by 40 mph as 0.3.  However, the 0.3 was interpreted as 30 minutes as the journey time between Ashton to Lees.

When it worked well

Most of the students who had correctly calculated the times for each stage of the journey went on to solve the problem.

## Average speed for two part journeys with no timeline

Misconceptions

• A few students converted Jackie’s time of 3 hours 15 minutes as 3.15 hours rather than 3.25 hours.
• While most students drew an appropriate diagram, some failed to identify the 36 km point as an equal distance for both Jackie and Lee.
• A few of the students who correctly calculated Jackie’s average speed as 24 km/h did not use this to find the time it took her to travel 36 km.  Therefore, they were unable to work out how long it took Lee to travel 36 km.

When it worked well

Those students who worked out Jackie’s average speed and used this to find the time it took her to reach 36 km went on to solve the problem correctly.

## Modelling two part journeys algebraically

Misconceptions

• The most common mistake was to write Lauren’s time as T + 2 hours rather than Hannah’s.
• Only a few students realised to write the distance travelled as an algebraic product of time and speed involving T.

When it worked well

Those who had correctly written the distance Hannah and Lauren travelled, at the point Lauren caught up to Hannah, in terms of T were able to set up an equation to find the time.

## Teach this lesson

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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