Systematic lists and possibility grids · Foundation
도입됨| English | 의미 |
|---|---|
| systematic list/ˌsɪstəˈmætɪk lɪst/ | 체계적 나열 |
Have we listed every choice?
- A cafe offers tea or juice with apple, banana or melon. How can a list prove that no lunch choice was missed?
- This lesson studies systematic list 系统列表: A list made in a fixed order so every allowed outcome appears exactly once.
The method
- Fix one first choice and list every allowed second choice before moving to the next first choice.
- Use a table to check that no outcome is missing or duplicated.
- If combinations are forbidden, remove those entries from the list.
Why is a systematic list better than guessing when counting outcomes?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Fixing one choice and listing every allowed partner before moving on covers the whole space exactly once - guessing has no such guarantee.
A worked case
List tea-apple, tea-banana, tea-melon, juice-apple, juice-banana, juice-melon: six choices. Removing tea-melon leaves five. The codes using 1,2,3 once each are 123,132,213,231,312,321: six. These answers follow from complete lists.
Systematic lists and possibility grids
Fix one first choice and list every allowed second choice before moving to the next first choice
Compare the model with the worked case and explain one change.
How many cafe choices are in the complete list?
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
There are three listed foods under each of two drinks: six listed outcomes.
Where it goes wrong
- State whether order matters and whether repetition is allowed.
- Choosing A then B can be different from B then A for a code, but not for an unordered two-person team.
- A restriction can make a simple product invalid.
Not true: Three available first choices and four second choices give twelve outcomes even if some combinations are forbidden. The product rule assumes every pair is allowed; forbidden combinations must be removed from the 12.
How many remain when tea-melon is removed?
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Remove one of the six outcomes: five remain.
The product rule assumes every pair is allowed.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
True. The product rule assumes every pair is allowed; forbidden combinations must be removed from the 12.
Applying it
- Foundation should justify answers with a complete ordered list or grid.
- Higher may compress a verified structure using the product rule, but must still check restrictions.
How many three-digit codes use 1,2,3 once each?
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
List 123,132,213,231,312,321: six codes.
Match each cafe count to its reason.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Complete lists make the count visible; the product rule is a shortcut only when every combination is allowed.
In your course
- AQA GCSE Mathematics (8300), Foundation tier, specification section 3.1.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Systematic list: A list made in a fixed order so every allowed outcome appears exactly once.
- Fix one first choice and list every allowed second choice before moving to the next first choice.
- Watch for: state whether order matters and whether repetition is allowed.