The phrase "Thrice the difference of the integer $N$ and 4" can be translated into a mathematical expression as follows:

1. "The difference of the integer $N$ and 4" refers to the expression $N - 4$.
2. "Thrice" means three times, so you would multiply the expression by 3.

Therefore, the open phrase translates to the mathematical expression:

$3 \times (N - 4)$

Or simply:

$3(N - 4)$

Would you like to go over more details or have any questions?

Here are 5 related questions to further explore this topic:

1. How would you express "Twice the sum of the integer $N$ and 7" in mathematical terms?
2. How do you write the expression for "Four times the square of the integer $N$"?
3. How do you interpret "Half the product of $N$ and 10" mathematically?
4. What is the expression for "The sum of the squares of $N$ and 5"?
5. How would you represent "Five times the difference between $N$ and 6" algebraically?

Tip: When translating word problems into mathematical expressions, identifying keywords like "sum," "difference," "product," and "quotient" can help in correctly forming the corresponding algebraic expressions.