1. Two of the statements in this box are wrong.

2. There are 604800 seconds in a week.

3. The sum of the first 10 square numbers is 385.

4. A square is also a rectangle.

5. Multiplying a value by a whole number makes it bigger.

6. The numbers from 1 to 20 add up to 210.

• Wikipedia,
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• A paradox is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition.
• Natalie, London
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• I am thankful for providing such wonderful starters. They are of immence help and the students enjoy them very much. These starters have saved my time and have made my lessons enjoyable.
• Rhonda, Arizona
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• The answer states that multiplying by a negative whole number makes the answer negative. However, whole numbers cannot be negative by the definition of what whole numbers are. So that answer is true.
• Wiliam, Lincoln
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• Number 5 is wrong since multiplying a value by 1 which is a whole number gives an answer the same value as before neither smaller or bigger.
• Meilyr Wyn, Ysgol Syr Thomas Jones
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• Excellent Starter - Thank you very much
There has been some debate amongst the department about whether a square is a rectangle. A square is not a rectangle if the definition of a rectangle includes "top and bottom same length as each other, right and left same length as each other but different length to top".
• The Best Maths Class Ever (7cd/m2), King Alfred's Oxfordshire
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• It was a silly starter but it made us all think! Students: We thought that it was not very logical because the statement was true and false at the same time. We found that when it was false it became true.
• Tony Graham, Stevenage, Hertfordshire
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• Sorry, Rhonda, it is possible to have a negative integer.
• Nick Ball, Enoree, South Carolina
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• The definition we use in the USA for whole numbers are numbers 0 and greater. So you can't have a negative whole number. But the value you start with could be negative or a fraction...and one of our social studies teachers says that zero is a concept, not a number. So this was a dumb question.
• Simon, Hampshire
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• Whole number means an integer (from the Latin 'integer'), so whole numbers can be negative. Natural numbers can only be positive - as to whether zero is a natural numbers depends on your view as a mathematician.
A rectangle is defined as a quadrilateral with two pairs of parallel sides at right angles, so a square is a rectangle. Equally, a rectangle and a square are both parallelograms.
• Chas, New York
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• The person who said that there is no such thing as a negative integer is dead wrong!
The answer key's reasoning for statement 5 is wrong, because there is no such thing as a negative whole number.
Statement 5 IS false though, because 0 is a whole number.
• Grace Harrison, West Kirby Grammar
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• I loved this starter it really made me think and involved some good classroom discussions.
Thanks.
• Kiwi, New Zealand
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• Here whole numbers cannot be negative, so multiplying by a negative integer would not be allowed. You are allowed to multiply by one, though, giving an equal but not larger answer so the statement is incorrect.
• RB, UK
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• Multiplying by 1 would also be a counter example for question 5 - so even if you don't want to include 0 and negatives as 'whole numbers' the statement is still false. I hope no one will debate whether or not 1 is a 'whole number'!
• Paula, Gillingham School
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• Unfortunately, the answer sections gives a different statement for number 5 in that the word 'negative' is missing in the question. Will try it on the kids anyway and see if they spot the mistake. Thanks.
• Hannah, South Yorkshire
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• Enjoyable but Made my brain hurt! I loved this and would love to see more of these starters.
• Dartmouth Academy 5/6P, Dartmouth
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• We're confused. If 5 is wrong then 1 is wrong and if 5 is correct then one is correct which makes it wrong.......Or does it????
• Matthew Zhao, Year 7, Brisbane Boys' College, Toowong, Brisbane
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• It was an enjoyable paradox. Good trick!
Keep it up, Transum!
• MrMiss, Essex
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• Not quite a paradox as multiplying by 1 doesn't make things bigger and the first square number is 0 so the first 10 add up to 285.
• Par Radox,
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• I always tell lies.
• St Mark's, Year 5
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• We would like a check button at the end please!

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
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Previous Day | This starter is for 6 May | Next Day

## Answers

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Educational Technology on Amazon

 Teacher, do your students have access to computers?Do they have iPads or Laptops in Lessons? Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Here a concise URL for a version of this page without the comments.

Transum.org/go/?Start=May6

Here is the URL which will take them to a related student activity.

Transum.org/go/?to=paradox

## Visual Paradoxes

If you randomly select one of the possible answers to this multiple choice question what is the probability you are correct?

a) 20%

b) 40%

c) 60%

d) 20%

e)  0%

I ALWAYS
TELL LIES

In 1901, the British philosopher and mathematician Bertrand Russell uncovered a possible paradox that necessitated a modification to set theory. One version of Russell's Paradox involves a town with one male barber who, every day, shaves every man who doesn't shave himself, and no one else. Does the barber shave himself?

From The Math Book published by Sterling

Interesting number paradox

Did you know that all numbers are interesting?

Proof: Assume there exists a set of uninteresting numbers. This set would have a smallest number, which is interesting because it is the smallest uninteresting number. But a number cannot be both interesting and uninteresting, so the assumption that there exists a set of uninteresting numbers must be wrong and hence, all numbers must be interesting.

The image above is inspired by another famous paradox. If you don't know it you could do a little research.
Clue: the man is known as Achilles.

For Students:

For All:

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