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EMPOWERR SUMMER 2009 WORKSHOP MATHEMATICS ACTIVITIES FRACTIONS AND ADDITION We recall our definition of fractions in the box below. THE NUMBER LINE DEFINITION OF FRACTIONS We take as our starting object the number line with the whole numbers marked off as dots on it. We also assume the following axiom of the number line to be true. AXIOM OF THE NUMBER LINE: Given any whole number l with l ≠ 0 and given any segment of the number line, it is possible to split that entire segment up into exactly l pieces, each one of which has the same length as each and every one of all the other pieces. Definition. Let k, l be whole numbers with l > 0. Divide each of the line segments [ 0, 1], [ 1, 2], [2, 3], [3, 4], . . . into l segments of equal length (this is possible because of the axiom of the number line given above). These division points together with the whole numbers now form an infinite sequence of equally spaced dots on the number line (in the sense that the lengths of the segments between consecutive dots are equal to each other). The first dot to the right of 0 is labeled 1 . l SO THE DEFINITION OF THE SYMBOL 1 l IS THAT IT IS THAT DOT- it is the first one to the right of 0. The second dot to the right of 0 is by definition EMPOWERR, Summer 2009 2 3 k , the third , etc., and the k-th one is . l l l 1 Fractions in lowest form and GCF We know that any fraction can be expressed in many different ways (in fact, in infinitely many different-looking ways). For example, we have the equalities 12 / 50 = 6 / 25 = 30 / 125 = 54 / 225. In all these different ways of labeling the same dot on the number line, only 6 / 25 has the property that the numerator and denominator have no common factors except one. For example for 12/50 one has GCF (12, 50) = 2 for 6/25 one has GCF(6, 25) = 1 for 30 / 125 one has GCF(30, 125) = 5 for 54 / 225 one has GCF(54, 225) = 9 Definition If a and b are whole numbers with b ≠ 0 then the fraction a/b is said to be in lowest form if GCF ( a , b ) = 1, i.e. a and b are relatively prime (that is, a and b have no common prime factors). In general let x and y be natural numbers and consider the fraction x/y. You can give a formula for the correct expression for x/y in lowest form or simplest form using a combination of x, y, and GCF(x,y) x/y in lowest form is given by [ x / GCF(x,y ) ] / [ y / GCF(x,y ) ] CHALLENGE QUESTION: EMPOWERR, Summer 2009 2 Some students might say that when a and b are relatively prime, the fraction a/b cannot be “reduced any further”, because there are no common factors for a and b except 1 (why are there no other common factors?) so there is no ‘canceling out’ that can occur. Can you explain the idea of reducing a fraction in terms of the number line model of fractions we have adopted? For example, what does it mean in terms of the number line definition of fractions, that 30 / 125 “can be reduced” to 6 / 25, but that it cannot be reduced ‘any further’ ? Can you give a way to think of the lowest form of a fraction in terms of the number line definition of fractions? ADDITION OF FRACTIONS Another advantage of the number line definition of fractions is that we can extend the operations of addition (and subtraction) from whole numbers to fractions very easily. We can define addition and subtraction for whole numbers using the number line. We can then extend addition to fractions using the same ideas. Each fraction is a dot on the number line. Each dot on the number line has associated to it an arrow pointing from 0 to that dot. Whole number addition on the number line is described by arrow concatenation or juxtaposition as we now describe. If a and b are whole numbers and we associate to a and b the corresponding number line arrows, then a + b is the dot on the number line obtained as follows. Take the initial point of the b arrow and place it at the terminal point of the a arrow. This newly placed b arrow points to a new number and we define this new number to be a + b. In the box below we look at the example of 3 + 5 via the number line. EMPOWERR, Summer 2009 3 3 |----------------------------> 5 |----------------------------------------------> +--------+--------+--------+--------+--------+--------+--------+--------+--------+--… 0 1 2 3 4 5 6 7 8 9 3 5 (started at the 3 endpoint) |---------------------------->|---------------------------------------------> 8 |----------------------------------------------------------------------------> +--------+--------+--------+--------+--------+--------+--------+--------+--------+---… 0 1 2 3 4 5 6 7 8 9 We use exactly the same idea to define fraction addition: each fraction is a dot on the number line, so has an arrow associated to it. To add one fraction to another, just juxtapose the arrows in the order like the one for whole numbers. The second addend will point to a new dot on the number line, and that dot is defined to be the sum of the two original fractions. EMPOWERR, Summer 2009 4 Example: 3/5 + 2/3 3/5 |------------------------- |--------+--------+--------+--------+--------|--------+--------+--------+--------+------|--------- 1 2 0 1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 10/5 2/3 |---------------------------- |--------------+--------------+---------------|--------------+--------------+---------------|---------- 1 2 0 1/3 2/3 3/3 4/3 5/3 6/3 3/5 2/3 |-------------------------|---------------------------- = 3/5 + 2/3 |--------+--------+--------+--------+-------|--------+-------+--------+--------+------|--------- 1 2 0 1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 10/5 So 3/5 + 2/3 gives a new dot on the number line-it is the dot between 6/5 and 7/5. But how do you know 3/5 + 2/3 is a fraction according to our definition of fraction? In other words, can you give a reason why that dot is the address or location of a point on the number line, obtained by dividing each unit interval up into a certain whole number of pieces, then counting over a whole number of those pieces from 0? In more mathematical-sounding language, we might ask this: how do we know that the number line definition of fractions is closed under this definition of fraction addition? EMPOWERR, Summer 2009 5