Count the Same Situation Two Ways
Lesson 11.5 · 从两个方向统计同一个情境
A total does not change when we regroup it. “Unanimous” means correct for everyone, and “capacity” means the most marks a column can hold under the stated rule. Count by students or by questions, by question totals or by speaker types, and use the equality to uncover hidden information.
One collection. Two ledgers. The totals must agree.
One collection can have two ledgers
Click any square. Then count the check marks by rows and by columns. You are counting the same marks, so the grand totals must match.
Double-counting principle
Every check mark belongs to exactly one row and exactly one column.
Changing the grouping does not create or remove a mark.
Count the worked example test results by students
Four students answer a 15-question test. They get 11, 12, 13, and 14 questions correct.
Complete practice problem
A 15-question test is taken by four students. Their numbers of correct answers are 11, 12, 13, and 14. What is the smallest possible number of questions that all four students answered correctly?
Row ledger
Add the correct-answer totals across the four students:
The corresponding error counts are:
Count the same correct marks by questions
Now treat each question as a column that can hold at most four correct marks.
If no question were correct for all four…
Each question could hold at most three correct marks.
But the row ledger contains 50
The extra marks are:
A column can exceed the three-mark limit by only one mark, and only if all four students are correct. Reaching 50 therefore needs at least five such columns; if other columns have fewer than three marks, even more unanimous columns are needed.
Spread errors to make the overlap as small as possible
Both arrangements keep the row error totals 4, 3, 2, and 1. Only the placement of those ten errors changes.
Build a general guaranteed-overlap rule
Count correct marks by students, then compare with the capacity of questions that are not unanimous. A correct mark is one student’s correct answer to one question. A guarantee of zero means the totals alone do not force any question to be correct for everyone; a particular arrangement may still have shared correct questions.
For these four answers, restore this fixed preset: 5 students, 12 questions, 52 total correct marks.
Count “yes” answers by question totals
The next practice problem looks different, but it uses the same principle: one collection of yes answers can be counted from two directions.
Complete practice problem
An island has 2,014 people. Some always tell the truth and the rest always lie. Every person worships exactly one of three gods: the Sun God, Moon God, or Earth God. Every resident is asked three yes-or-no questions: “Do you worship the Sun God?”, “Do you worship the Moon God?”, and “Do you worship the Earth God?” The numbers of “yes” answers to the three questions are 806, 1,004, and 1,204. How many residents always tell the truth?
Count the same “yes” answers by speaker type
Exactly one god is worshipped. That fact fixes how many “yes” answers each kind of speaker contributes. Let T count truth-tellers and L count liars; T + L is the population, not the number of answers.
Truth-teller
Says “yes” to the one worshipped god and “no” to the other two.
1 yesLiar
Gives the opposite answer to each of the three true yes-or-no answers.
2 yesesTransfer the method to a four-choice survey
Apply the same counting method to four categories instead of three. Track how that changes one opposite-answer student’s contribution.
New transfer problem
There are 120 students. Each belongs to exactly one of four clubs. Every student is asked “Do you belong to this club?” once for each of the four clubs, answering yes or no each time. Some students answer every question truthfully; the rest give the opposite answer to every question. Altogether, the four questions receive 198 “yes” answers. How many truthful students are there?
Contribution per student
A truthful student says “yes” once.
An opposite-answer student says “yes” to the three clubs they do not belong to.
For a student in club A, truthful answers to A, B, C, D are Yes, No, No, No. Reversing every answer gives No, Yes, Yes, Yes. The same counts hold whichever club is chosen.
Two ledgers
Optional connection: compare the two methods
Chapter 11 also includes a 69-person party exercise with consecutive acquaintance counts. That problem was developed fully in Lesson 11.1 because its main structure is a complement-and-extreme-count argument rather than the row/column double-counting model emphasized here.
Independent workshop
Solve at least 6 of 8 fresh problems. Each has a local hint.
Hint
Both directions count the same marks.
Hint
A non-unanimous column holds at most four marks.
Hint
Compare the correct count with all possible marks.
Hint
A guarantee cannot be negative.
Hint
Reverse one yes and three no answers.
Hint
Start with one yes per person, then count each extra pair.
Hint
Subtract the opposite-answer population from ninety.
Hint
Compare the target with the one-per-person baseline.
Independent exit ticket
Solve all five fresh problems. The certificate also requires Missions 1–9.
Hint
Compare with eighteen columns holding at most three marks each.
Hint
There are 4 × 18 possible marks.
Hint
Each opposite-answer person adds three yeses beyond the baseline.
Hint
Check that the two populations add to 150.
Hint
People must be counted in whole numbers.
Optional reflection — not automatically graded
Certificate of mastery
Double-Counting Systems Architect
This certifies that the learner can count one collection in two ways, use row and column capacities, guarantee overlap, and solve weighted response-count problems.
Lesson 11.5 · Grade 5 Mathematical Reasoning
Teaching notes
The small matrix, general overlap calculator, four-club transfer problem, workshop, feedback, and assessment are newly created instructional scaffolds.