Chapter 1 – Counting Techniques
Note to Students:
Students should be aware that these worksheets are not ‘fill in the blank’ worksheets. We have intentionally given you no space to write answers on the worksheets. You should have a notebook for your work, perhaps even beginning by sketching ideas on a white board or scratch paper until you have sufficient organization to put a coherent explanation in your notebook. Note also that the expectation is to write ‘explanations’ and not ‘answers’. These activities are not designed to teach computational skills, but instead are designed to introduce mathematical concepts. The process used to solve problems is the focus, not the end result.
Activity 1a – Pascal’s Triangle
This first activity on Pascal’s Triangle leads you to some basic patterns of counting the number of ways to perform certain tasks. There are many such techniques, including permutations and combinations, with which you may be familiar. The key here is to discover the recursive pattern; ‘how do we use the third case to help us find the answer to the fourth case?’ The best descriptions are ones that use the previous information to find the answer to the next question in a way that can be repeated.
After completing the individual problems, it is important to find connections between problems. Why are questions 2 and 3 really the same question? How do we translate from coin flips to binomial expansion? Why are the coefficients in the binomial expansion the same as the entries in Pascal’s triangle? Why are the numbers of subsets in question 4 the same as the entries in Pascal’s triangle? How do we translate from subsets in question 4 to coin flips in question 2?
Activity 1b – General Counting Principle, Permutations and Combinations
This activity is a continuation of the first activity on Pascal’s Triangle. It formalizes the connections between the various counting problems and Pascal’s Triangle. Students are encouraged to understand the definitions and notation prior to this activity that can be found in lessons on
- The General Counting Principle
- Binary Counting