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Activity 2-5: What are you implying? PowerPoint PPT Presentation


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www.carom-maths.co.uk. Activity 2-5: What are you implying?. You are given these four cards:. Each of these cards has a letter on one side and a digit on the other. This rule may or may not be true: if a card has a vowel on one side, then it has an even number on the other.

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Activity 2-5: What are you implying?

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Activity 2 5 what are you implying

www.carom-maths.co.uk

Activity 2-5: What are you implying?


You are given these four cards

You are given these four cards:

Each of these cards has a letter on one side

and a digit on the other.

This rule may or may not be true:

if a card has a vowel on one side,

then it has an even number on the other.

Which cards must you turn

to check the rule for these cards?


Activity 2 5 what are you implying

This is a famous question

that has tested people’s grasp of logic for years.

Most people pick Card with A, correctly.

Many pick Card with 2 in addition.

But… if the other side of the Card with 2 is a vowel, that’s fine.

And if the other side of the Card with 2 is a consonant,

that’s fine too!

The right answer is to pick Card with 7 in addition to the A.

If the other side of the 7 is a vowel, there ISa problem.


Activity 2 5 what are you implying

Consider these four assertions:

In the statements below, a, n and m are positive integers

1. a is even

2. a2 is even

3. a can be written as 3n + 1

4. a can be written as 6m + 1


Activity 2 5 what are you implying

If we make a card with one statement on the front and one on the back, there are six possible cards we could make.


Activity 2 5 what are you implying

Define A I B to signify ‘A implies B’,

Define A RO B to signify ‘A rules out B’, and

A NINRO B to mean ‘A neither implies nor rules out B’.

We can see that the statements for Card 1

mean that this is of type (I, I): each side implies the other.

Task: how many different

types of card do we have

with these definitions?


Activity 2 5 what are you implying

We conclude we have four different types of card:

1 is (I, I),

3 and 5 are (RO, RO),

2 and 4 are (NINRO, NINRO),

while6 is (I, NINRO).


Activity 2 5 what are you implying

Are these the only possible types of card?

Yes, since if A RO B, then B RO A,

so (RO,I) and RO, NINRO)

are impossible cards to produce.

Can we be sure that if A RO B, then B RO A?

We can use a logical tautology called MODUS TOLLENS:

if A implies B, then (not B) implies (not A).

Now A RO B means A I (not B).

If A I (not B), then by Modus Tollens,

not (not B) implies (not A).

Thus B implies (not A), that is B RO A.


Activity 2 5 what are you implying

if A implies B, then (not B) implies (not A).

What mistake do people make with the four-card problem?

They assume that A I B can be reversed to B I A.

In fact, A I B CAN be reversed,

but to (not B) I (not A) – Modus Tollens.

So VowelIEven reverses to (Not Even) I (Not Vowel),

which shows we need to pick the 7 card.

Let’s try a variation on our initial four-card problem.


Activity 2 5 what are you implying

This time the obvious reversal works, for ‘Tree RO Panda’ is the same as ‘Panda RO Tree’: you DO need to turn over the two cards named in the question.

You are given four cards below.

Each has a plant on one side and an animal on the other.

Task: you are given the rule:

if one side shows a tree, the other side is not a panda.

Which cards do you need to turn over to check the rule?


Given two circles there are four ways that they can lie in relation to each other

Given two circles, there are four ways that they can lie in relation to each other.

Task:

if you had to assign

(I, I),

(RO, RO),

(NINRO, NINRO),

and (I, NINRO)

to these,

how would

you do it?


Activity 2 5 what are you implying

Let’s invent a new word, DONRO,

standing for DOes Not Rule Out.

Task: is it true that

if A DONRO B and B DONRO C, then A DONRO C?

Pick an example to illustrate your answer.

A: The shape S is a red quadrilateral

B: The shape S is a rectangle

C: The shape S is a blue quadrilateral

A: x = 2

B: x2 = 4

C: x = 2

A: ab is even

B: b is less than a

C: ab is odd


Activity 2 5 what are you implying

With thanks to:

The Open University and my teachers on the

Researching Mathematical Learning course.

Mathematics In School

for publishing my original article on this subject.

Carom is written by Jonny Griffiths, [email protected]


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