Running rates

Achievement objectives

NA4-4: Apply simple linear proportions, including ordering fractions.

NA4-5: Know the equivalent decimal and percentage forms for everyday fractions.

Required materials

See Materials that come with this resource to download:

• Running rates activity (.pdf)

Activity

In the 2014 Auckland marathon, the average time was 4:27:15 for the full and 2:22:53 for the half.

A full marathon is 42.195 km in length.

• How much faster (in minutes or seconds per km) was the average full marathon pace than the half?

The following prompts illustrate how this activity can be structured around the phases of the mathematics investigation cycle.

Make sense

Introduce the problem. Allow students time to read it and discuss in pairs or small groups.

• Do I understand the situation and the words? (Students may need support to understand the meaning of marathon and half marathon if they have no experience of these events.)
• What are the important numbers and what do they mean? (4:27:15 is interpreted as 4 hours, 27 minutes, and 15 seconds. 42.195 kilometres is interpreted as 42 kilometres and 195 metres.)
• Where else in my life/the world can I see this happen? (Speed of cars may be more familiar to students.)
• What will my solution look like? (The solution will be a difference in rates between the marathon and half-marathon averages. It will be expressed as minutes per kilometre or kilometres per minute, accompanied by clear working and justification.

Plan approach

Discuss ideas about how to solve the problem. Emphasise that, in the planning phase, you want students to say how they would solve the problem, not to actually solve it.

• What are the maths skills I need to work this out? (Strategies with rates involve multiplicative thinking.)
• What might the solution be? What is a reasonable estimate? What is not reasonable?
• How could I show this problem using numbers, pictures, graphs, tables, or materials?
• What strategies can I use to get started? (Ratio tables are useful in organising the conversion among quantities.)
• What tools (digital or physical) could help my investigation? (Calculators or spreadsheets will ease the burden on working memory and provide the calculations required.

Take action

Allow students time to work through their strategy and find a solution to the problem.

• Have I shown my workings in a systematic way?
• Have I included all the information?
• Are all my calculations carefully checked and match the conditions of the problem?
• Does my answer seem correct? Is it close to my estimation?
• How could I make sure that I haven’t missed anything?
• Have I recorded my ideas in a way that helps me to see patterns?
• Are there any patterns?
• How might I describe the pattern?
• Is there another possible answer or way to solve it? Which strategy will be most efficient?

Convince yourself and others

Allow students time to check their answers and then either have them pair share with other groups or ask for volunteers to share their solution with the class.

• What is the solution? Does it meet the requirements for the solution?
• Is my working clear for someone else to follow?
• Could I have solved the problem in a more efficient way?
• Which ideas would convince others that my findings answer the investigation question?
• Would my strategy work in a different situation? What kinds of situations would it work for?
• Which ideas or tools worked well in my investigation?
• What could I try differently next time?