The purpose of this activity is to engage students in solving a problem involving investigating with whole numbers using graphical means and/or generalising.
The background knowledge and skills that need to be established before and/or during this activity are outlined in the diagram below:
- Using diagrams to represent a pattern
Show, in a diagram, the arrangement of bricks used to build a triangular stack with a base of 3, 4, 5...
- Using diagrams to find an unknown.
How many bricks are needed to build a triangular stack with a base of 12?
- Generalising a pattern.
How many bricks are needed to build a triangular stack with a base of n bricks?
- Making sensible assumptions when solving contextual problems.
If retirement age is 65 and a typical person expects to be employed for at least 80% of their working lives, how many weeks will they work after the age of 20?
- Solving context based problems with a diagram and/or by finding a pattern.
It takes three slices of bread and two eggs to make two egg sandwiches. There are twenty five slices of bread in a loaf, including crusts. How many eggs will be used if sandwiches are made from five loaves of bread?
In approaching this activity, the students will need to make assumptions. Making different assumptions will mean that their final solutions are not likely to be exactly the same. The students might need to be reassured that this does not mean any solution is wrong, rather that it fits a different interpretation of the question. When discussing the assumptions the students will need to make, there is an opportunity to discuss the term average and to introduce the idea of central tendency and also discuss the likelihood of the mean, median and mode being very similar for a large sample size. This activity may be carried out with step by step guidance, or by allowing the student to follow their own method of solution. The approach should be chosen in sympathy with students' skills and depth of understanding.
The arithmetic approach (show more)
- The student is able to make assumptions and to use a diagram to represent the pattern, in order to solve the problem.
Prompts from the teacher could be:
- How are you going to go about solving this problem? Can you show generations on a diagram?
- Which of the people on your diagram are related to the original ancestor(s)?
- What assumptions do you need to make about the ages of the people involved?
- Which of the people on your diagram are likely to still be alive?
The conceptual approach (show more)
- The student is able to make assumptions, to use diagrams and/or to generalise in order to solve the problem.