A student claims that division always makes things smaller so 5 ÷ 1/2 cannot be 10 because 10 is greater than the number 5 she start with. How do you respond?
Please help ASAP!!!
dividing by a number less than 1 makes the quotient larger than the dividend.
To respond to the student's claim, you can explain the concept of division and how it doesn't always make things smaller. Here's how you can explain it:
1. Begin by acknowledging the student's understanding that division often results in a smaller value. Division is indeed a process where you split a number into smaller parts. However, it does not always make things smaller.
2. Explain that when you divide by a fraction, it's essential to understand the concept of "dividing by" and "multiplying by its reciprocal." In this case, dividing by 1/2 is equivalent to multiplying by its reciprocal, which is 2.
3. Show the step-by-step calculation:
5 ÷ (1/2) = 5 * (2/1) = (5 * 2)/1 = 10/1 = 10
4. Emphasize that the result of 5 ÷ (1/2) is 10, not because division always makes things smaller, but because dividing by a fraction is equivalent to multiplying by its reciprocal.
5. Highlight that the student's initial assumption is based on dividing by whole numbers, where the result is typically smaller than the original number. However, when dividing by fractions, especially fractions less than 1, the result can be greater than the original number.
6. Encourage the student to explore other examples of dividing by fractions to reinforce this concept. For instance, consider dividing 10 by 1/4: 10 ÷ (1/4) = 10 * (4/1) = 40.
By explaining the concept step by step, the student should gain a clearer understanding of why 5 ÷ 1/2 can be 10 and that division doesn't always make things smaller.