A: I am choosing an integer. I then subtract 10 from the integer, take the opposite of the result, add -3, and find the opposite of the new result. My result is -3. What is the original number?
B: Judy wants to do the activity in part (a) with her classmates. Each classmate probably chooses a different number and Judy wants to tell each classmate quickly what number was chosen. Judy figures out that the only thing she needs to do is to add 7 to each answer she gets. Does this always work? Explain why or why not.
C: Come up with your own "trick" similar to the one in part (b) that works for each answer you get from your classmates.
A
n
n-10
10-n
7-n
n-7
n-7 = -3, so n=4
B yes, it always works, as shown above
A: To find the original number, we need to work backwards through the steps given. Let's break it down:
1. Start with the final result: -3
2. Take the opposite of the result: 3
3. Add -3 to the number: 3 + (-3) = 0
4. Take the opposite of the new result: 0
Therefore, the original number chosen is 0.
B: Judy wants to add 7 to each answer she gets because by doing so, she is essentially undoing the subtraction of 10 that was done originally. Let's go through the steps to see why this works:
1. Judy's classmate chooses a number
2. Judy subtracts 10 from the number
3. Judy takes the opposite of the result
4. Judy adds -3 to the new result
5. Judy finds the opposite of the new result
By adding 7 to each answer, Judy effectively cancels out the subtraction of 10 and obtains the original number chosen by her classmates. This works because addition and subtraction are inverse operations.
C: To come up with a similar trick that works for each answer from classmates, we need to identify a consistent operation that reverses the original steps. One such trick could be to multiply each answer by 2.
For example:
1. Classmate chooses a number
2. Multiply the number by 2
3. Add 20 to the new result
4. Divide the new result by 2
5. Subtract 17 from the result
By following these steps, you will obtain the original number chosen by your classmates. This trick works because multiplication and division are inverse operations, and adding and subtracting the same value reverses each other.