A student claims that division always makes things smaller so 5 / (1/2) cannot 10 because 10 is greater than the number 5 she started with. How do you respond?
I don't understand this question at all!!
the smaller the divisor, the larger the quotient.
12/6 = 2
12/4 = 3
12/3 = 4
12/2 = 6
12/1 = 12
12/(1/2) = 24
because there are 24 halves in 12 wholes.
When dividing by a number less than 1, the quotient is larger than the dividend.
Again, if a slice of pie is 1/6 of the pie, then 1/(1/6) = 6. There are 6 slices, each of which is smaller than the whole pie.
In general, when dividing by a fraction, just multiply by the reciprocal.
(2/3)/(4/7) = 2/3 * 7/4 = 14/12
In fact, you use that rule all the time, without thinking about it
3/6 = 3/(6/1) = 3 * (1/6)
Well, it seems like that student could use a little math humor to clear things up! You see, division doesn't always make things smaller, it just depends on the numbers involved. In this case, when you have 5 apples and divide them by 1/2, it's like asking how many times you can fit half an apple into a whole apple. And the answer is... drumroll, please... 10! So, it turns out that division can make things bigger too, just like a growing appetite when faced with delicious apples! 😄🍎
I understand your confusion, allow me to explain. When dividing a number, it is true that the result can be smaller than the original number. However, this is not always the case.
In the example you mentioned, dividing 5 by 1/2 is equivalent to multiplying 5 by the reciprocal of 1/2. The reciprocal of 1/2 is 2/1 or simply 2. So, 5 divided by 1/2 is the same as multiplying 5 by 2, which equals 10.
Therefore, the student's claim that division always makes things smaller is not accurate. Division can indeed result in a smaller number, but it can also yield a larger number, as demonstrated by this example.
I understand that you are facing some confusion regarding the question. Let me explain it to you.
The student's claim that division always makes things smaller is not correct. Division is essentially the process of splitting a quantity into equal parts. It does not necessarily result in a smaller value.
In this particular case, let's consider the division expression 5 / (1/2). To evaluate this expression, we can simplify it by inverting the divisor and multiplying:
5 / (1/2) = 5 x (2/1) = 5 x 2 = 10
So, 5 divided by 1/2 is equal to 10, not smaller than 5 as the student claims.
To objectively respond to the student's claim, you can simply explain that division can result in a smaller, larger, or equal value depending on the numbers being divided. It is important to emphasize that division only makes things smaller when the divisor is greater than 1.