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Where Can We Visit?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?


Can you find the values at the vertices when you know the values on the edges?

Babylon Numbers

Can you make a hypothesis to explain these ancient numbers?

Reaction Timer

Age 11 to 14
Challenge Level


Why do this problem?

The skills of making and testing hypotheses and analysing data are important both in mathematics and in scientific enquiry. This problem is an ideal starting point for developing these skills.

Learners need to make decisions about the information that is required to answer the questions posed, analyse the data that is collected, and decide whether the analysis supports the hypothesis.

Possible approach

If you have access to a computer and data projector, demonstrate the interactivity showing a few of the variations (e.g. varying the size makes little difference to the task, whereas varying the shape or varying location + click on shape make it harder).

To introduce the second experiment, ask a volunteer to come out to the front of the class and demonstrate dropping a ruler to test the speed of their reactions.

Once learners have seen both experiments, give them some time to discuss in pairs some hypotheses they could test, and then share these ideas with the whole class. There are some suggested lines of enquiry in the problem which could be shared with learners if they struggle to come up with good ideas of their own.

Ask students, perhaps working in pairs, to select a hypothesis and discuss:
  • whether they think it is true or false
  • how they could use the experiment(s) to test their views
  • what data (and how much data) they would need to collect

Give them time to collect, analyse and interpret their data and then prepare a poster for presenting their findings to the class. The task may take more than one lesson, so data collection could be done as a homework task.

One way of presenting their findings to the class is for learners to display their posters around the room and then take time to look at everyone else's work, perhaps annotating each other's work with post-it notes. Then the class could discuss which methods of collection, analysis and representation were most appropriate and effective in testing their hypotheses.

Another similar activity where students can make hypotheses and test them is Estimating Time.

Key questions

How many times do you think it would be useful to carry out the experiment(s)?
How will you represent and analyse your data to test your hypothesis?
Can you justify that your experiment is a valid way of testing your hypothesis?
Are your results reliable - could someone else replicate your results with their own experiment?

Possible support

Encourage learners to work in pairs or small groups and to support each other in constructing clear hypotheses which are straightforward to test. Each group could present their plan to the rest of the class before they start any data gathering, and the class could give feedback on what is good and what might need improving. This could be done using post-it notes as suggested above.

Possible extension

All of the hypotheses suggested in the problem could lend themselves to fairly detailed statistical analysis - there is the opportunity for learners to explore the idea of distributions, averages and measures of spread in order to compare data gathered from each of the two experiments and any experiments they devise for themselves.

A Stage 4 follow-up problem that investigates how to turn the results from the second experiment into reaction times can be found at How Do You React?