Let's work through the questions in the image step by step.

Question 1:

Jules has -3 points in a trivia contest. In the next round, he loses 6 points.

a. Complete the model to find -3 - 6.

• The number line shows starting at -3 and moving 6 units to the left, landing at -9. Therefore: $-3 - 6 = -9$

b. How does the model in problem 2a also show the addition problem -3 + (-6)?

• In subtraction, moving to the left on the number line can also be interpreted as adding a negative number. So, $-3 - 6$ is equivalent to $-3 + (-6)$, both giving the same result: -9.

c. What is -3 - 6? What does that tell you about how many points Jules has after the next round?

• $-3 - 6 = -9$. Jules now has $-9$ points, meaning he lost more points and is further in the negative.

Question 3:

How is the value of -2 the same as the value of 3 - 2? How is it different?

• $3 - 2 = 1$, which is a positive number.
• $-2$ is a negative number.
• While they may involve subtraction, the first results in a negative value, and the second results in a positive difference of 1.

Question 4:

a. On a number line, what is the distance between -9 and -5?

• The distance between $-9$ and $-5$ is 4 units.

b. One way to express the distance between -9 and -5 is $|-5 - (-9)|$. Why do you use absolute value notation to express the distance between two numbers?

• Absolute value measures the distance between two points without considering direction. Distances can't be negative, so we use absolute values to represent positive distances.

c. To find the distance between -9 and -5, does it matter whether you find $-5 - (-9)$ or $-9 - (-5)$? Explain.

• It doesn't matter, because: $|-5 - (-9)| = |-5 + 9| = |4| = 4$ And: $|-9 - (-5)| = |-9 + 5| = |-4| = 4$ Both give the same result of 4 units.

Do you want further details or explanations on any part of this?

Here are 5 questions to expand on this information:

1. What is the mathematical significance of subtracting a negative number?
2. How would the answer change if Jules gained points instead of losing them?
3. What is the geometric interpretation of absolute value on the number line?
4. How can you generalize the method of finding distance between any two integers?
5. What are some real-world applications of negative numbers and subtraction?

Tip: When subtracting integers, visualize the number line to better understand the direction and magnitude of movement.