Opening the Mathematics
These questions are designed to promote decision, discussion, and debate. There are no wrong answers – only opportunities to explore and share.
Each question is designed to be prominently displayed as a focal point (through use of data projector or printed on a laminated A3 sheet).
For many of the questions students work in pairs or threes and have one plain sheet of A3 paper. They are encouraged to approach the question in any way they see fit. Diagrams, jottings and working out are requested.
The probing questions are to be used at the discretion of the teacher, either with individual groups or the whole class in order to facilitate progress.
After an agreed time groups are selected to feed back to the class. The use of the A3 sheet will help this feedback. Students are encouraged to ask questions and to provide their own comparisons.
The thinking behind the question is more important than any eventual conclusion. There are no wrong answers!!
Some questions may be more suitable for homework tasks with feedback in class.
Some questions may be more suitable for whole class discussion.
Some questions may only take up the equivalent of a starter whilst others may take the majority of a lesson.
Many questions need decisions to be made before they can be addressed. These decisions are to be made by the students and will not affect the mathematical processes.
Many questions will lead to unexpected opportunities for mathematical thinking.
Each question has an accompanying table of probing questions, opportunities for extension and other notes. These are not exhaustive and students will discover their own avenues for exploration.
If students can make decisions in groups and can confidently express their mathematical thinking and can appreciate others’ thinking processes through these questions then the mathematics will have been opened.
Designed by Peter Needham (Secondary Mathematics Consultant, Derbyshire)
The Questions
1. How many great-great grandparents did you have?
2. How many cigarettes does a smoker smoke in a lifetime?
3. How big is our school?
4. Is your age ever a factor of your parent’s/guardian’s age?
5. How many cats would weigh the same as a person?
6. How many words do you think you will need to write your autobiography when you are 80yrs old?
7. How steep is a flight of stairs?
8. What is the best size for a jigsaw puzzle?
9. The ages of the members of your family. Could they all be prime numbers one day?
10. How many people in the UK do not live in Chesterfield?
11. Will circles fit into squares?
12. Is it quicker to mow the grass on a football pitch or to paint the white lines?
13. How many people do you see in an average day?
14. How many times does your heart beat in a lifetime?
15. You have a secret and you decide to tell only 2 friends. But these 2 friends decided to tell 2 other friends each and these tell another 2 friends…. and so on. How many people get to know your secret in a day?
16. In a year how many doors do you go through?
17. How many steps do you take in a lifetime?
18. Are there more dead people than live people?
19. Is a fraction easier than a decimal?
20. Would there be room for all the people in the UK to stand on the M1?
21. Is Stratford – upon – Avon the centre of England?
22. There are 96 chairs to arrange for the audience. How could they be set out?
23. How many slices of bread would you need in order to give you enough calories for the day?
24. How many people will you make eye contact with in your life?
25. How many words will you read this week?
26. What proportion of your life will you spend outside?
27. How big is the earth?
28. Is it best to have 70% rather than 50%?
29. How many corners can you cut?
30. Is there enough petrol in a petrol tanker?
31. How many x’s and y’s make up 36?
32. Can a cube be the same size as a cuboid?
33. Would you prefer 5000 free lottery tickets per week or a gift of £50 per week?
34. If you saved all your bath water could you fill a swimming pool?
35. “Here’s your penny change” – is it worth taking it?
36. How far can you see?
1How many great-great grandparents did you have?
Probing Questions / Will a diagram help?
What about step-parents?
Is there a pattern as you build up from generation to generation?
Extension / Could you work out the number for 10 generations back without having to continue the sequence?
n generations?
How many great-great grandchildren will you have?
Notes / Tree diagrams will help students visualise the pattern.
The extension question about grandchildren has many hypothetical possibilities.
2
How many cigarettes does a smoker smoke in a lifetime?
Probing Questions / How long is a life time?
How many cigarettes does a smoker smoke in a day?
Are we looking for an exact answer?
How much will it affect your answer if you round off answers to calculations as you work towards a total number of cigarettes?
Will someone who smokes twice as many as someone else live half the time?
Extension / Total number of cigarettes smoked is 400,000. How many per day and for how many years could this represent?
How much will this all cost? What could you do with the money instead?
How much room would be taken up by all of these cigarettes?
Notes / Obvious links to health education, budgets, taxes etc.
3
How big is our school?
Probing Questions / What do we mean by ‘how big?’ What do we need to know to help decide?
Are there enough classrooms? Are the classrooms big enough
Could we fit in more pupils?
Are there enough teachers?
Extension / Could we rebuild the school such that there are no stairs?
What is the ideal teacher/pupil ratio?
Notes / Prepare statistics for total number of pupils, teachers and rooms and possible ground area covered.
This could develop into a project over a period of time.
Some students might interpret ‘how big?’ in quite a different way egs. Height of the building. Size in comparison to other buildings.
4
Is your age ever a factor of your parent’s/ guardian’s age?
Probing Questions / Are we talking about one parent’s age or both? (It might be best to work with just one parent)
Is there a way of recording your results?
Will it ever happen? If so, how many times?
When might it not happen?
Extension / What if we look at the situation where both parents are of a different age? Will your age ever be a factor of both at the same time?
What if we include grandparents?
Could the ages in a family all share a common factor?
Notes / Some prior work on factors might be needed.
5
How many cats would weigh the same as a person?
Probing Questions / How can we estimate the weight of a cat?
How many cats do you think will weigh the same as a student before we do any calculations?
Why do we know our own weight in imperial and not in metric?
What if we used dogs instead of cats? Is there a quick adjustment on the agreed ‘cats’ answer without having to redo all the calculation steps.
Extension / How many people would weigh the same as an elephant?
Notes / A bag of sugar or standard weights may be needed.
Weight of at least one student needs to be known.
Conversion rates between imperial and metric weights.
This task may be best done as a whole class exercise led by the teacher.
Ask a student (cat owner!) to hold a bag of sugar (or more) and to estimate how it compares to a cat.
Use of fractions and/or proportion might evolve when comparing dogs to cats.
6
How many words do you think you will need to write your autobiography when you are 80 years old?
Probing Questions / How many words on a page?…. in a book?
How many pages will you need if you decide to write your autobiography now? How many words will this be?
Will your autobiography at the age of 80 years be 4 times longer than one written at the age of 20 years?
Will ‘rounding off’ your answers to calculations have a great affect on the final answer?
How far back can you remember?
How far back can your parents remember?
What fraction/ percentage of your life have you not had to go to school?
What fraction/percentage of an adult’s life was spent at school?
Extension / If you wrote the biography of your friend would it be the same length as your autobiography?
What type of person would expect to write a lengthy autobiography?
How many different words do you know?
Notes / Bring some autobiographies and estimate number of words prior to this exercise.
Discussion about ‘how to spend one’s life’ might emerge.
The relative sizes of students’ answers are more important than the actual answers as these will be speculative anyway.
Share the number of words of some published autobiographies towards the end of this exercise.
7
How steep is a flight of stairs?
Probing Questions / What do we mean by ‘steep’?
How many steps would you need to ascend a height equivalent to the classroom?
What is an ideal height for one step?
What is an ideal angle of inclination?
Extension / How steep could a path be which does not have any steps? Does it matter if you’re walking up or down the path?
Notes
8
What is the best size for a jigsaw puzzle?
Probing Questions / Area of jigsaw or number of pieces?
Is a rectangle the best shape?
Is there an ideal number of pieces?
Could a jigsaw puzzle contain 1530 pieces?
Does the size of each piece affect the difficulty of the puzzle?
Extension / Is it possible to have a jigsaw where each piece is a hexagon?
Notes / Bring a variety of jigsaws.
9
The ages of all the members of your family. Could they all be prime numbers one day?
Probing Questions / How many prime numbers are there below 100?
What could the present ages be for this to never happen?
Could it happen more than once in any one family?
What if 2 families are combined.
Extension / Try this with square and/or triangular numbers.
Notes / Prior work on prime factors.
Students will need to know the ages of everyone in their families...including grandparents where possible.
Students might wish to start with just 2 members of the family before trying this with all the family.
10
How many people in the UK do not live in Chesterfield?
Probing Questions / What data is needed in order to work this out?
Is Chesterfield a large town? Where
How many people do you think this will be as a fraction and/or a percentage?
What if we included Derbyshire? Will it make much difference to your answer?
Extension / Choose other towns and cities. Estimate an answer for London.
Which county has the highest population? Smallest population?
How many people on Earth do not live in China? How does this percentage compare to the original question?
Can you find similar percentages for town/country and country/world?
Notes / http://www.statistics.gov.uk/census2001/pyramids/pages/17ud.asp
Data needed prior to the lesson
Obviously this does not have to be Chesterfield or Derbyshire.
Students may be asked to research the data prior to the lesson as a homework exercise.
11
Will circles fit into squares?
Probing Questions / What do we mean by ‘fitting in’?
Does each circle have to be the same size?
How much space is wasted if you fit one circle into a square?
Using just ‘one size’ circle… will you be able to fit in twice as many into a square that is twice as big?
Extension / Try other shapes. Circles into circles? Squares into triangles?
Notes / Possible use of formula for the area of a circle.
This can be interpreted in many ways. Some students may wish to cut up circles and fit them in. Others may want to investigate the problem similar to packing tins in a box.
Some students will tackle this by constructions/drawings only.
12
Is it quicker to mow the grass on a football pitch or to paint the white lines?
Probing Questions / Before doing any calculations, which do you think is going to be quickest?
Are straight lines quicker to paint than curves?
Extension / What about a rugby or hockey pitch?
Notes / Some data needed prior to the lesson on area of a football pitch and arrangement and lengths of lines.
Decisions to be made on the speeds of mowing/painting and width of mower.
The decisions made on speeds of mowing/painting do not have to be too ‘believable’…... they are for comparative purposes only.
13
How many people do you see in an average day?
Probing Questions / What do we mean by an average day?
Are there different times of the day when you expect to see more people?
What about a weekday compared to a weekend?
Will this result change much as you get older?
What type of person would constantly see a large number of people per day.
Extension / What fraction/percentage do you actually talk to?
How many of those you see actually see you?
Calculate the mean of all students answers?
Notes
14
How many times does your heart beat in a lifetime?
Probing Questions / How long is a lifetime?
What do you think might be a realistic estimate before you start?
Is it OK to round off your calculations as you work towards an answer?
Does a pulse stay constant as you get older?
Does everyone have a similar heart rate?
Extension / Number of breaths in a lifetime?
Notes / Pulse rates need to be known.
15
You have a secret and you decide to tell only 2 friends. But these 2 friends decided to tell 2 other friends each and these tell another 2 friends…. and so on. How many people get to know your secret in a day?