Prove it Mathematically Club at Manchester Muslim Prep School
Teaching Outline
With the exception of week 3, the general structure of the club each week is some form of game along with worksheets which the students complete during the club and a short talk by me supplying the answers to the worksheets and teaching them the new concepts that are necessary for them to complete the proof over the 5 week period. Therefore the outline below is not the complete club content but captures what the children will learn each week as we walk through the proof.
Week 1: I begin by assuming the children do not know any algebra and so I first introduce the idea of replacing an unknown number with a letter and then solving to work out what number that letter equals. We start by solving beginner problems like x+5=10. At this point the children are also introduced to squaring a number and the mathematical nomenclature: a x a = a2. After getting comfortable with solving several algebraic equations (including those involving squares), the children are then introduced to the concept of mathematical functions via a dressing up game. We learn then that y=x2 is a function of x and the concept of the inverse function. By the end of this club the children have learnt that the inverse function of squaring is called “the square root” and has the symbol “√” ie y=√x.
Week 2: This week we begin learning about irrational numbers. Irrational numbers can’t be written down because their fractional part goes on forever. Most mathematically talented primary school children love thinking about infinity. A common question maths teachers get from primary school children is “Sir what does ∞+1 equal?”. Next we learn all fractions, when fully factored, must have an odd number in them. The children discover this fact by exploring what happens when they simplify a set of fractions. Fractions with two even numbers are really like zombie actors in that they are simply fully simplified fractions dressed up. Thus, like zombies, fractions with two even numbers don’t really exist. It is in this week we introduce the children to the concept of a proof by contradiction via them starring in a Murder Mystery Drama where the hapless Watson assumes an old man has died of natural causes only to discover the evidence contradicts his assumption. He is guided by the great detective Holmes to correctly conclude that a young woman has been murdered (because of a dagger sticking out of her head) oops!
Week 3: This is a rest week where we enjoy some fun and games with a mathematical theme in the hall or even outside (weather permitting).
Week 4: We resume our quest. In week 2 we learnt that fractions with an even number in both their numerator and denominator like Zombies don’t really exist. This week we will assume that √2 can be written as a fraction and we let the numerator equal the unknown number a and the denominator equal the unknown number b such that we say √2=a/b. Now if we can do some maths which show that if this assumption is true that both a and b are even numbers then we have come up against a contradiction because we know that no fully simplified fraction can have both an even numerator and denominator (see week 2). Next we explore two functions: one that spits out all even numbers (e=2n) and the other all odd numbers: (o=2n+1). The children then explore what happens when they square numbers generated from the e=2n and o=2n+1 functions. Finally, we prove algebraically that the square of an even number is an even number and by exploration, the square of an odd number appears to always be an odd number. If this is the case, then it means the square root of an even number must always be an even number. Finally from here the children show that if √2 could be written as a fraction, in form a/b, then both a and b must be even numbers. However, since fully factored fractions with two even numbers don’t exist, our assumption has led us to a mathematical contradiction and like Detective Watson in week 2, who after assuming an old man had died of natural causes discovered in fact it was a young woman who had been murdered, so we, after assuming we could write √2 down as a fraction discovered we couldn’t. Therefore √2 must be an irrational number.
Week 5: In week 4 the children proved algebraically that the square of an even number is an even number by squaring the function e=2n. However, while the children also found that squaring odd numbers always gives an odd number, they do not know enough algebra to expand o2=(2n+1)2 which is necessary to prove that the square of an odd number is also an odd number. Therefore in the final week, we will work through a relatively easy to understand geometrical method of expanding this function. By cutting out and arranging shapes, where the area of each shape forms the answer to the expansion of (2n+1)2 the children will learn that this evaluates to 4n2+4n+1 which is always an odd number. The more algebraic way to expand (2n+1)2 introduces BODMAS. The second method is more demanding mathematically so I do not teach it in the final week but I will give all the children a detailed worksheet which you can work through with them at home after the final club if they are interested. This type of algebraic expansion is not strictly necessary to understand the root 2 proof which we effectively completed in week 4 but it does tie up one important loose end because just showing that the squares of all even numbers are even numbers did not eliminate the possibility that the square of an odd number could also be an even number – a possibility which we show this week is impossible.